THEORY OF INSTRUCTION:

PRINCIPLES AND APPLICATIONS

By Siegfried Engelmann and Douglas Carnine

Revised edition

NIFDI PRESS
P.O. Box 11248, Eugene, Oregon, 97440

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 Foreword, 1991 Edition
by Robert Dixon

Many questions regarding Theory of Instruction: Principles and Applications have arisen in the years since the
publication of the first edition in 1982. Is it a textbook? Why wasn't it named, Theory of DIRECT Instruction? Why is it
so difficult to read? How relevant is it to the current Zeitgeist of educational philosophy? And last-and least-is the cover
of the 1982 edition red or orange?

I propose at this publication of the revised edition, that Theory of Instruction is exactly what the title implies, and
further that my proposition is of potentially inestimable significance to the field of education.

Theory

First and foremost, Theory of Instruction is the articulation of a theory-not in the atheoretical sense “theory” is used
in educational jargon, but in the more precise sense well-established among scientists and philosophers of science.

Engelmann and Carnine's theory evolved the same way original natural science theories have evolved, through the
scrupulous application of logical analysis to existing empirical observation. The Engelmann and Carnine theory
possesses the most critical attributes of natural science theories: (1) it is exhaustive in that it covers everything from the
most basic motor skill instruction to the highest of the “higher order” thinking skills, and (2) it does so economically. In
short, it is parsimonious.

Engelmann and Carnine's theory builds logically from just two initial assumptions: that learners perceive qualities,
and that they generalize upon the basis of sameness of qualities. (This is not unlike the way Euclidean geometry derives
logically from a minimum of unproven and unprovable assumptions about points and lines.) If we accept Engelmann
and Carnine's simple assumptions and if we were to employ rigorous logic to any instructional problem, then the
instruction we would derive would fall within the constraints of the Engelmann and Carnine theory. We wouldn't come
up with the same instruction, but rather, with the same or similar instructional principles.

That is highly significant. Engelmann and Carnine don't look at the book when they develop instruction; they
developed most of their instruction before they wrote their book. They haven't memorized various sequences from their
own book, either. They simply apply the logic of their own theory to new content, and essentially recreate
manifestations of their theory. Put another way, one very good indication that Engelmann and Carnine are operating
within the framework of a theory is that they are constrained to adhere to their own theory. One can only religiously
conform to a theory that exists. It strikes me as absolutely fantastic that the published Direct Instruction programs–
before or after the theory book–are consistent in terms of how examples of given types are ordered and sequenced.
(Some variation exists due directly to refinements in the theory.) Absolutely no other published programs of any type
demonstrate such consistency, at such a level of detail. Absolutely no other published programs have an underlying,
consistent rationale for the examples they use and the order they use them in. It's quite likely that few authors of
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published educational materials have ever given the slightest thought to the fact that when we change the examples, we
change the information that is communicated to the learner.

The Engelmann and Carnine theory provides a basis for making predictions that can be tested. In the absence of a
theory, experimentation is driven by random hypotheses based upon “plausible ideas” or intellectual frolicking. If such
hypotheses prove to be false, little is gained, save the rejection of one of an infinite set of plausible (but wrong) ideas. If
such hypotheses prove to be true, very little is still gained: there's an idea that shows promise, but where does it fit?
How does it relate to other ideas that show promise? The current state-of-the-art in educational experimentation is
characterized by this kind of tinkering with plausibility.

If a hypothesis generated by a theory proves false, on the other hand, not only is the hypothesis itself questionable,
but because of the logical interconnectedness of the theory's components, the entire theory becomes questionable. But
if a hypothesis generated by a theory is verified, then the veracity of the entire theory is strengthened. Theory-based
research is worth the time and effort; plausible idea-based theory isn't. When Time charged that the longest running
joke on most university campuses is the Education Department, the black humor tended to obfuscate the reason that so
many non-education academics might feel that way: conducting research in the absence of a theory might be funny,
were it not for the unconscionable waste of money and human resources.

A true theory not only predicts, but explains. For example, if we are interested in why cognitive psychologists have,
after several years of research, concluded that the extent to which learning transfers is dependent upon the relative
salience of surface and structural features of examples, this theory will explain that for us. If we are interested in why a
typical textbook presentation of a new concept must fail to communicate accurately to many learners, this theory will
explain that, too.

Instruction

Theory of Instruction, again as the name implies, is a theory of instruction, not a theory of learning. Learning
theories (if they are really theories at all) are no doubt of value to those interested in how humans learn in the absence
of instruction (which generally is inefficiently).

Theory of (DIRECT) Instruction

There is a relatively simple fact about the Engelmann and Carnine theory that many educators appear to find
disturbing: it is the only theory of instruction and therefore, does not require the “direct” qualifier. Although we hear of
all kinds of “theories” of learning and even a few “theories” of instruction, this book represents the only theory of
instruction that would withstand the rigorous tests of theories required by philosophers of science—disciplined logical
interconnectedness, predictive value, parsimony, etc.

Theory of Instruction does not prescribe any “one best way,” but rather, describes a range of best ways, and
suggests an infinite range of ineffective ways. (If I were to set out to intentionally design the worst instruction possible,
I would still look to Theory of Instruction for guidance on how to precisely lead students far astray.)

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 I have even heard serious educational researchers express the fear that if Theory of Instruction is the first and only
theory of instruction, then there would be nothing left for anyone else to do. That is like arguing that Newton's
Principia and Lavoisier's Chemistry killed physics and chemistry. The emergence of a first theory in a field opens that
field to countless opportunities for inquiry, closing off only those practices akin to changing lead to gold. (Geocentrism
died because it was untrue, not because it was unpopular.)

Reading the Theory

I've never heard of anyone who found Theory of Instruction to be an easy read. In general, any written theory, in
the presence of either no competing theory or no similar theory, will be hard to read. The reader will not possess a
frame of reference necessary for easy comprehension. That's one way of characterizing the purpose of a theory: to
create a new frame of reference. I doubt that Principia was anyone's leisure reading when it first appeared.

Engelmann and Carnine's Theory of Instruction is as clear as it can be for whom I believe to be the principal
intended audience: Engelmann and Carnine. Imagine carrying around in your head a theory so exhaustive, so
economical, on a subject so broad and complicated. My conjecture is that a crucial stage in the development of any true
theory is for that theory to be written down, first and foremost for the benefit of the theorist(s), and then only
secondarily for any of the rest of us who might be interested. The theory seems complex because it is complex.

Summary

Theory of Instruction could easily be the most important educational book ever written, bar none. (Yes, I am aware
of Aristotle and Dewey and Piaget and Skinner, etc.) Instruction is at the heart of education, and Theory of Instruction
is the first true theory that cuts to the heart of instruction. Although I believe the theory of instruction to be
fundamentally correct, even that judgment is secondary to the importance of a theory existing in the field of instruction.
My contentions herein may seem like exorbitant fanaticism. Time and experience will tell.

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 Acknowledgments
The authors would like to acknowledge these original patient readers—particularly Galen Alessi, Robert Dixon,
Joseph Jenkins, Martin Siegel, and Marcy Stein. Final scrutiny came from our editor, Wesley Becker. We are very
grateful to these people. We are also heavily indebted to the graduate students and colleagues who provided us with the
research data on specific instructional or theoretical issues: Abby Adams, Craig Darch, Phil Dommes, Ruth Falco, Bill
Fink, Roger Freschi, Russell Gersten, Patty Hickman, Ed Kameenui, John Kryzanowski, Alex and Robyn Maggs,
Dorothy Ross, Randy Sprick, Bob Taylor, and Paul Williams.

We are grateful to the teacher-trainers and site supervisors we have worked with, because they provided us with
ongoing reminders of what people could do if they had a highly technical understanding of instruction. Each of the
following people has provided incredible demonstrations of teaching the unteachable—Geoffrey Colvin, Karen Davis,
Alex Granzin, Glenda Hewlett, Deborah Lamson, Carol Morimitsu, Linda Olen, and Linda Youngmayr.

Finally, we are grateful to the people who have developed instructional programs with us and have helped us learn
about the amazing difference that good programs can make in the performance of children- Carl Bereiter, Elaine
Bruner, Linda Carnine, Gary Davis, Robert Dixon, Gary Johnson, Jean Osborn, Jerry Silbert, Leslie Zoref, and perhaps
more than anyone else, Susan Hanner.
   

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 Introduction, 2016 Edition
What is Instruction?

The words instruction and teaching do not occur very often in the education literature. In fact, the word instruction
appeared only 18 times in the 230 pages of the Common Core standards. The words teach or teaching appeared only 5
times. Ironically, instruction or teaching is what is supposed to occur in the classroom. Specifically, if the learners do
not have a particular skill or bit of knowledge, the assumption is that the learners will acquire these through some form
of “interaction” or process in the classroom. The interaction or process that is designed to transmit skill or knowledge
is teaching. It may be disguised as a “learning activity” and may be configured so the teacher has no role in directly
transmitting a specific skill or information, but instead does something that is designed to change the learner’s
cognition in specific ways. Practically and pragmatically, whatever the teacher does that is supposed to result in
specific changes in the learner’s repertoire and behavior is “teaching.”

In a rational system, teaching is related to three other processes—standards, curriculum, and testing. The four
processes occur in a fixed order that starts with standards and ends with testing.

Standards curriculum teaching testing

The order is justified on rational grounds. The sequence couldn’t start with teaching without specifying what to
teach and how what is taught is related to other skills and knowledge that are scheduled for students to learn. Logically
the curriculum and standards must be in place before specific teaching occurs. Without these prerequisite processes
there would be no safeguards against first-grade teachers presenting material that is neither appropriate for the subject
being taught nor for the grade level.

1. Standards:

If the curriculum is math level K or 1, a possible appropriate standard would indicate that learners are to “Count
backward from 20 to 0.” The standard, “use information from the text to draw conclusions about where Columbus
would go next” is more advanced (possibly grade 4 or 5) and is not a math standard but a geography, history, or science
standard.

2. Curriculum:

The standards imply specific features of the curriculum. If a skill or informational item is specified in a standard,
there necessarily must be a specific segment of the curriculum that provides the instruction needed to teach the skill or
information. If this provision is not honored, there would be no rational basis for relating the standards to the
curriculum.

A proper curriculum scrupulously details both the order of things that are to be taught and the requirements for
adequate or appropriate teaching.

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 The curriculum is often packaged as an instructional program. A properly developed curriculum would have
detailed “lesson plans” that provide adequate directions for the sequence and content of what is to be presented first,
next, and next in each successive lesson.

The degree to which the teacher’s presentation behavior is specified by a lesson script varies greatly across
programs, but the goal of all instructional programs is the same—to provide students with the skills and information
specified by the standards.

Questions about the adequacy of the teacher presentation are answered empirically, by facts about student
performance. If the teacher presents lesson material the way it is specified, and students learn the skills and content,
whatever training and scripting the program provided are judged to be adequate. Conversely, if students tend to fail, the
presentation the teacher provided is flawed. It may require observations to determine why it failed and what has to
change for the teacher to be successful. Note, however, that it is not possible to observe the presentation in one part of
the program and extrapolate to unobserved portions of the program. A program could have parts that are quite good
with respect to teaching students, and have other parts that are quite bad.

3. Teaching:

Teaching is the process that follows the specifications provided by the curriculum. The relationship is simple: the
teaching must transmit to the students all the new skills and knowledge specified in the curriculum. A test of a valid
curriculum would show that students did not have specific knowledge and skills before the teacher taught them. The
posttest that is presented after instruction shows that students uniformly have the skills. The conclusion is that a process
occurred between the pretest and posttest and caused the specific changes in student performance.

The evaluation of a curriculum that occurs when a high percentage of students fail the posttest is more
complicated. The failure could have been caused by a flawed curriculum, by flawed standards, by a flawed
presentation, or by a combination of flawed curriculum, standards, and presentation. If the grade-one standards have
items that assume skills that are not usually taught until grade 4 or 5, the teacher fails when she tries to teach her first
graders these skills, and the students fail the test items that require these skills.

It is not possible to look at the outcome data alone and infer why the failure of these items occurred. We have to
analyze what knowledge and skills students would need to pass these items, and identify the instructional sequence that
would be needed to teach this information and skill set.

4. Testing:

The final process is testing. Its purpose is to document the extent to which the student performance meets the
standard. Also the testing should be designed to disclose information about each standard. As noted above, if students
fail items on the pretest and pass items of the same type on the posttest, we assume that teaching accounted for the
change in performance.

Ideally the testing would occur shortly after students have completed the teaching. The testing should be fair and

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extensive enough to generate specific information about the standards, the curriculum, and the teaching.

Standards that are unreasonably difficult or inadequately taught are identified by examining test results of the
highest-performing classrooms. Any items that are failed by more than half of the students are possibly poor items or
items that test material that is poorly taught. The most direct way to obtain more specific information about the failed
content is to work with students who failed specific items and observe what they tend to do wrong or what information
they don’t know.

Benefits of Theory of Instruction

Instruction is the essential operation that drives standards, curriculum, and assessment. Instruction provides the
basic evidence of what can be achieved in altering student performance. These facts of achievement, in turn, provide
the basic foundation for standards, curricula, and testing. The problem with current instructional practices is that there
are no widely accepted rules for what instruction is capable of achieving or of the essential details of successful
instruction.

This paucity of information occurs because there are no widely accepted guidelines for using facts about teaching
to formulate standards or assessments. Stated differently, there is no widely recognized theory of instruction that lays
out basic principals of teaching and that provides various empirical tests to facilitate refinement of instructional
practices.

Theory of Instruction fills this gap. It articulates principles of effective instruction in sufficient detail to permit
educational practitioners to develop effective instruction. The effectiveness of the instruction may be measured by
comparing results generated by Theory of Instruction with results of other educational approaches.

A final implication is that if educational institutions have clear information about the extent to which students of all
levels can be accelerated, the institutions are then able to develop and install reasonable standards, effective curricula,
and fair assessments.

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Contents

Foreword
Acknowledgments
Introduction
SECTION I – Overview of Strategies
Chapter 1 – Theoretical Foundations
Chapter 2 – Analysis of Basic Communications
Chapter 3 – Knowledge Systems
SECTION II – Basic Forms
Chapter 4 – Facts and Rules About Communicating Through Examples
Chapter 5 – Non-Comparative Sequences
Chapter 6 – Nouns
Chapter 7 – Comparative Single-Dimension Concepts
SECTION III – Joining Forms
Chapter 8 – Single-Transformation Sequences
Chapter 9 – Correlated-Features Sequence
SECTION IV – Programs
Chapter 10 – Introducing Coordinate Members to a Set
Chapter 11 – Hierarchical Class Programs
Chapter 12 – Programs Derived from Tasks
Chapter 13 – Double Transformation Programs
Chapter 14 – Programs for Teaching Fact Systems
SECTION V – Complete Teaching
Chapter 15 – Expanded Teaching
Chapter 16 – Worksheet Items
SECTION VI – Constructing Cognitive Routines
Chapter 17 – Cognitive Routines
Chapter 18 – Illustrations of Cognitive Routines
Chapter 19 – Scheduling Routines and Their Examples
Chapter 20 – Prompting Examples
Chapter 21 – Covertization
SECTION VII – Response-Locus Analysis
Chapter 22 – New Response Teaching Procedures
Chapter 23 – Strategies for Teaching New Complex Responses
Chapter 24 – Expanded Chains
Chapter 25 – Expanded Programs for Cognitive Skills
SECTION VIII – Diagnosis and Corrections
Chapter 26 – Diagnosis and Remedies
Chapter 27 – Mechanics of Correcting Mistakes
Chapter 28 – Program Revision and Implementation

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 SECTION IX – Research and Philosophical Issues
Chapter 29 – Instructional Research: Communication Variables
Chapter 30 – Instructional Research: Programs
Chapter 31 – Theoretical Issues
References
Subject Index

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 Section I

Overview of Strategies
Section I provides the theoretical foundations for the analysis of cognitive skills and the implications that are
derived therefrom f or how to teach those skills.

The precise analysis of cognitive learning is difficult, if not elusive, because it stands at the juncture of three
separate analyses—the analysis of behavior, the analysis of stimuli used as teaching communications, and the analysis
of knowledge systems or the content to be taught. (See Figure I.1)

1. The analysis of behavior seeks empirically-based principles that tell what is universally true about the
ways in which the environment influences behavior for different classes of learners.

2. The analysis of communications seeks principles for the logical design of communications that effectively
transmit knowledge. These principles allow one to describe the range of generalizations that should
logically occur when the learner receives specific sets of examples. The analysis of communications
focuses on the ways in which examples are the same and how they differ.

3. The analysis of knowledge systems is concerned with logically organizing knowledge so that relatively
efficient communications are possible for related knowledge.

Figure I.1

1. 3.
Analysis of Analysis of
Behavior Knowledge
Systems

ANALYSIS OF
COGNITIVE
LEARNING

2.
Analysis of
Communica-
tions (Stimuli)

The analysis that has received the most attention from psychological theories is the analysis of behavior (Hilgard &
Bower, 1975). Although the other two analyses have received some theoretical attention (e.g. Gagne, 1970; Bloom,
1956; Markle & Tiemann, 1974), there has been little systematic effort to develop precise principles of

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communications used in instruction or to analyze knowledge systems.1 This book frames behavior theory within a
three-way analysis of human cognitive learning.

The three areas of analysis derive directly from the nature of cognitive learning. The first aspect of cognitive
learning is that the learner learns from the environment, which means that the environment is somehow capable of
communicating concepts or skills to the learner. The analysis of communications provides rules for designing these
communications so they are effective transmitters.

Another aspect of cognitive learning is that it always involves some topic or content. When we think, we think
about something, even if that something is a process. This aspect of cognitive knowledge carries basic implications for
designing the communications that we present to the learner. We cannot communicate with the learner without
communicating something. Conversely, if we are to understand how to communicate a particular bit of knowledge
(such as knowledge of the color red, or knowledge about the operation of square root), we must understand the
essential features of the particular concept that we are attempting to convey. Only if we understand what it is and how it
differs from related concepts can we design a communication that effectively conveys the concept to the learner.

The final aspect of cognitive learning has to do with the relationship of a given concept to other concepts. The
word large is related to the word blue because both function as adjectives. The color blue is related to the color red
because both have the properties of color. The relatedness of cognitive knowledge suggests that it is possible to develop
a classification system for various types of knowledge (circle 3 in Figure I.1). If this classification system is to be of
value to the instructional designer, the system should be designed so that the classification of a particular skill carries
information about how to communicate that skill to a learner. Concepts that are structurally the same in some respects
can be processed through communications that are the same in some respects.

Both the analysis of communications and the analysis of knowledge systems are logical analyses that involve
assumptions about the learner. The analysis of behavior, however, investigates the learner and how the learner responds
to specific communications. Chapter 1 presents an overview of the strategy that we will use to unite the three parts of
the analysis; Chapter 2 further develops the analysis of communications; Chapter 3 outlines the organization of
knowledge types that will be used throughout the book.
 

                                                                                                                         
1  Although
analyses such as Gagne’s deal with learning and with the teaching of concepts, principals, etc., the theoretical
development is at best a beginning.  
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 Theoretical Foundations
A theory of instruction begins with the assumption that the environment is the primary variable in accounting for
what the learner learns. The different skills learned by people in different environments suggest that the assumption is
reasonable. People who live in primitive societies learn skills quite different from those learned by people who live in
urban societies. Although the environment is assumed to be the primary cause of what is learned, it is not assumed to
be the total cause. Within any group of people there are individual differences. Also, there are differences that correlate
with the age of the learner. Therefore, the learner is also a variable.

To show the relationship between the role of the environment and the learner, we are faced with the basic problem
of experimental control. We must control one of these variables (the environment or the learner) before we can make
precise observations about the other variable. Ideally, we would rule out or eliminate one of these variables (either the
environment or the learner) and observe the remaining variable in a pure state. This solution is not possible. A possible
solution is to control one of the variables so that it functioned as if it were ruled out. We cannot readily achieve such
control over the learner because we do not know precisely how to do it. However, such control is possible with the
environment. We can design communications that are, ideally, faultless. Faultless communications are designed to
convey only one interpretation. From a logical standpoint, these communications would be capable of teaching any
learner the intended concept or skill. When we present such a communication to the learner, we effectively rule out the
environment as a variable. The communication is not merely standardized; it is analytically or logically capable of
transmitting the concept or skill to any learner who possesses certain minimal attributes discussed later. The learner
either responds to the faultless communication by learning the intended concept, or the learner fails to learn the
intended concept. In either case, the learner’s performance is framed as the dependent variable. The extent to which the
learner’s performance deviates from the performance that would occur if the learner responded perfectly to the
communication provides us with precise information about the learner. The deviations indicate the extent to which the
learner is not a perfect “mirror” of the environment. Furthermore, these deviations are caused by the learner (not the
environment, which has been controlled so that it is faultless).

The strategy of making the communication faultless and then observing the performance of the learner is the basis
for the theory of instructions that we will develop. We will use this strategy in designing instructional sequences and in
deriving principles for communicating with the learner. The following is a summary of the steps in our strategy,
showing where logical analysis is used and where behavior analysis comes into play:

1. Design communications that are faultless using a logical analysis of the stimuli, not a behavioral analysis
of the learner.

2. Predict that the learner will learn the concept conveyed by the faultless presentation. If the communication
is logically flawless and if the learner has the capacity to respond to the logic of the presentation, the
learner will learn the concept conveyed by the communication.

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 3. Present the communication to the learner and observe whether the learner actually learns the intended
concept or whether the learner has trouble. This information (derived from a behavioral analysis) shows
the extent to which the learner does or does not possess the mechanisms necessary to respond to the
faultless presentation of the concept.

4. Design instruction for the unsuccessful learner that will modify the learner’s capacity to respond to the
faultless presentation. This instruction is not based on a logical analysis of the communication, but on a
behavior analysis of the learner.

Note that the behavioral analysis comes into play only after the communication has been designed so that it is
faultless. The faultless presentation rules out the possibility that the learner’s inability to respond appropriately to the
presentation, or to generalize in the predicted way, is caused by a flawed communication rather than by learner
characteristics.

Assumptions About the Learner

The primary problem that we face in pursuing this strategy is that we do not know what constitutes a faultless
communication unless we make some assumptions about the learner. Stated differently, assumptions about the learner
and the communication vary together. The greater the assumed capabilities of the learner, the less the assumed
responsibility of the communication. If we assume that the learner will learn from any exposure to the environment, we
will provide communications that do not control details of the presentation. If we assume that the learner is not capable
of learning from communications that are ambiguous, we will approach the design of communications quite differently.
To provide for control of the maximum number of communication variables, we must postulate a simple learning
mechanism. Also, we must assume that the learner’s behavior is lawful, which means the learner who possesses the
assumed mechanism will learn what the communication demonstrates or teaches.

The learning mechanism that we postulate has two attributes:

1. The capacity to learn any quality that is exemplified through examples (from the quality of redness to the
quality of inconsistency).

2. The capacity to generalize to new examples on the basis of sameness of quality (and only on the basis of
sameness).

These attributes suggest the capacities that we would have to build into a computer that functions the way a human
does. Note that we are not asserting that these are the only attributes that a human possesses, merely that by assuming
the two attributes we can account for nearly all observed cognitive behavior.

1. The Capacity to Learn Any Quality from Examples

This assumption indicates what the mechanism is capable of learning, not how it learns. A quality is any
irreducible feature of the example. The simplest way to identify qualities is to begin with a concrete example. Any
example (such as a pencil) has thousands of qualities, which relate to shape, position, parts, color, texture, etc. All

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differences between a given concrete example and any other concrete example are differences in quality. Also,
anything we do to change the example we start with is a change in quality. We can make the pencil shorter, break the
point, paint it, change its position, and so forth. Each change is related to a quality of the original example.

The assumption that the learner mechanism learns qualities means simply that if an example possesses a quality, no
matter how subtle, the mechanism has the capacity to learn that quality. The only factor that limits the learner
mechanism is the acuity of the sensory mechanism that receives information about qualities. This mechanism, however,
is capable of learning qualities as subtle as the unique tone of a particular violin or qualities that involve the correlation
of events (such as the relationship of events on the sun to weather on the earth).

2. The Capacity to Generalize on the Basis of Sameness of Quality

Attribute 1 above indicates what the learner is capable of learning. Attribute 2 suggests how learning occurs.
According to this attribute, the learning mechanism somehow “makes up a rule” that indicates which qualities are
common to the set of examples presented to teach a concept. By using this rule, the mechanism classifies new examples
as either positive examples of the concept or negative examples. A new example is positive if it has the same
quality(ies) possessed by all the positive examples presented earlier. It is a negative example if it does not have the
same quality(ies).

According to the assumption about the generalization attribute, there is no sharp line between initial learning and
generalization. The rule-construction of the learning mechanism is assumed to begin as soon as examples are presented.
In formulating a rule, the mechanism does nothing more than “note” sameness of quality. Once the mechanism “has
determined” what is the same about the examples of a particular concept, generalization occurs. The only possible basis
for generalization is sameness of quality. If the example to which the learner is to generalize is not the same as the
earlier examples with respect to specific qualities it is impossible for generalization to occur unless the learning
mechanism is empowered with magical properties.

A further implication of attribute 2 is that the generalizations the learning mechanism achieves are completely
explained in terms of the examples presented to the learner and the qualities that are common to these examples.

Table 1.1 illustrates how the learning of conservation of substance is the same as the learning of red.

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 Table 1.1
Learning of a Learning of Red
Cognitive Operation
(e.g., Conservation
of Substance)
Before exposure to exam- Before exposure to exam-
ples, the learner has no ples, the learner has no
knowledge of concept. knowledge of concept.
Only some possible exam- Only some possible exam-
ples are examples of this ples are examples of this
concept. concept.
The learner demonstrates The learner demonstrates
mastery of concept by mastery of concept by
treating selected concrete treating selected concrete
examples of the concept in examples of the concept in
specified ways. specified ways.
The learner generalizes The learner generalizes
to new examples of the to new examples of the
concept. concept.
The appropriate generaliza- The appropriate generaliza-
tions are to examples that tions are to examples that
possess the quality of the possess the quality of the
concept. concept.

Both concepts are learned in the same way—through a communication from the environment that shows the nature
of the concept. The only difference is what is learned. And the “whatness” is the quality that comes from the examples,
not the learner. For the learner to learn these diversely different qualities, the learner must have the ability to detect
both the quality of redness and the quality common to the conservation examples (e.g., the relationship between
changes in appearance and changes in amount).

The Structural Basis for Generalization

The assumptions about the two-attribute learning mechanism imply the type of structure that we must provide to
cause specific generalizations. The two-attribute learning mechanism suggests that the learner operates on qualities and
sameness, and that both the qualities and samenesses come from the concrete examples that have the same quality and
provide information that these concrete examples are the same in a relevant way.

The most general implication of the two-attribute mechanism is the nature of the analysis that we must use for
cognitive learning. If the only primary difference between such disparate cognitive skills as learning the color red and
learning conservation of substance is the quality that is to be learned, and if the quality comes from concrete examples
(and not from the learner), the primary analysis of cognitive learning must be an analysis of qualities of examples and
of the communications that present these qualities to the learner. This analysis focuses on the stimuli that the learner
receives. We refer to this analysis as the stimulus-locus analysis (which is developed further in this and subsequent
chapters).

More specific implications of the two-attribute learning mechanisms suggest the general parameters of a
communication that is capable of inducing a particular generalization. This communication must meet these structural
conditions:

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 1. The set of positive examples presented through the communication must possess one and only one
distinguishing quality. If we assume that the learner learns qualities that are presented through examples,
we must make sure that the set of examples presented demonstrates only one identifiable sameness in
quality—not more than one. If every positive example in the set that is presented to the learner possesses
two distinct qualities, at least two distinct generalizations are implied by the communication. Since one of
these generalizations is inappropriate, the set of examples does not meet the structural conditions
necessary for inducing the intended generalization. For instance, if every example of red presented to the
learner was a circle and every example that was not-red was box-shaped, at least two generalizations are
implied by the same communication. Possibly the learner will generalize according to sameness in shape
(calling any circle “red” regardless of shape). Both generalizations are possible because both are based on
the qualities and samenesses shown by the demonstration examples. Since a given learner is assumed to
have no preknowledge of the concept and must base the generalization solely on the quality and sameness
of demonstrated examples, a given learner may learn an inappropriate generalization from the
demonstration of red circles.
To avoid this problem, we must eliminate the inappropriate quality from the demonstration examples.
Different techniques are possible for achieving this goal; however, the simplest is to modify the set of
examples so that some of the examples identified by the teacher as “not red” are circles. With this
modification, the set does not present circularity as a distinguishing quality of the positive examples.

2. The communication must also provide a signal that accompanies each example that has the quality to be
generalized. This signal is the only means we have for treating examples in the same way. When we
present examples that are physically different (such as two examples of red that are not the same shade)
we must use some form of signal to tell the learner, in effect, that these examples are the same and that the
learner must discover how they are the same. The signal, typically a behavior such as saying “red” for all
examples that are red, also provides the learner with a basis for communicating with us. The learner can
use the same signal, “red,” to let us know which generalization examples have the quality of redness.
The assumption about the signal accompanying the various examples is necessary because our goal is to
induce a particular generalization. However, if we simply present a group of examples that share a
particular quality, we cannot guarantee that: (a) the learner will attend to the common quality; or (b) we
will be able to communicate about this quality, even if the learner does attend to it. For instance, if we
present a group of objects that are red, how do we know that the learner is attending to the sameness in the
quality these examples share? We face other problems if we wish to test the learner to see if the
generalization was induced. How does the learner indicate which generalization examples have the
quality? Unless we use some signal to suggest sameness (such as putting all red objects in one place or
calling them “red” or associating some other unique signal with each example), we cannot demonstrate
sameness; we cannot test sameness; and we cannot correct the learner who responds inappropriately.
For the most basic type of communication, two signals are implied. One is used for examples that have
the quality. Another is used for examples that do not have the quality.

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 3. The communication must present a range of examples that show the physical variation of the examples
that exhibit a common quality. If every example that the communication presents to the learner is exactly
the same shade of red, the communication does not provide adequate information about the range of
variation in the quality that is to be labeled as “red.” Since this demonstration does not imply that other
shades of red share the quality of redness, the communication is incapable of inducing the appropriate
generalization to examples of other shades of red.
To show the quality that is to be generalized, the communication must demonstrate (through examples)
the range of variation that typifies the concept. In other words, the communication must present positive
examples that are physically different, but that share the quality that is to be generalized.
The requirement of showing a range of positive variation derives directly from our assumptions about
the learning mechanism. We assume that the learner is capable of learning any quality exemplified through
examples. For most concepts, the quality is something that is common to variations that are physically
different. We assume that the learner has the capacity to make up a “rule” about this range of variation.
We further assume that if we do not show an appropriate range of variation, the learner is not provided
with the information that is necessary to formulate the appropriate “rule.” Therefore, if the communication
fails to demonstrate the range, the learner cannot be expected to generalize appropriately.

4. A basic communication must present negative examples to show the limits of the variation in quality that
is permissible for a given concept. If we show the learner a range of red examples that differ in shades of
redness, the communication may appropriately induce a generalization to new examples that are red. (The
learner with the two-attribute learning mechanism should appropriately classify any example that falls
within the demonstrated range of variation as “red.”) However, this communication does not show the
boundaries for the generalization, which means that on a test of generalization, the learner may call pink
examples “red.”

To show the learner basic concepts, the communication must demonstrate the boundaries for the range
of permissible generalization. All negatives presented to demonstrate the limits of permissible variation
are the same in that they possess the quality of being “not red.” To signal that these negative examples are
the same, a common behavior is presented with each example. To assure that the learner does not classify
these examples in the same way that the positive examples are classified, the communication presents a
different signal for the negatives (for example, “not red”). The basic communication, therefore, presents
two sets of examples (one for the positives and one for the negatives) and two distinct signals (one to
signal each positive and the other to signal each negative).

5. The communication must provide a test to assure that the learner has received the information provided by
the communication. The test should present positive examples and negative examples that had not been
demonstrated earlier, but that are implied by the range of variation of quality demonstrated for the
positives and the negatives. If the learner has formulated an appropriate “rule” for the quality that had been
demonstrated through the demonstration examples, the learner should be able to respond appropriately to

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 new examples that fall within the range of variation previously demonstrated. A variation of the same
signals that are used to demonstrate positive and negative examples is used when the generalization
examples are tested.

In summary, the two-attribute learning mechanism implies that a communication for basic concepts must meet
these structural requirements.

1. The communication must present a set of examples that are the same with respect to one and only one
distinguishing quality (the quality that is to serve as the basis for generalization).

2. The communication must provide two signals—one for every example that possesses the quality that is to
be generalized, the second to signal every example that does not have this quality.

3. The communication must demonstrate a range of variation for the positive examples (to induce a rule that
is appropriate for classifying new examples on the basis of sameness).

4. The communication must show the limits of permissible variation by presenting negative examples.

5. The communication must provide a test of generalization that involves new examples that fall within the
range of quality variation demonstrated earlier.

Analyzing Whether Communications are Faultless

In addition to serving as guidelines for creating faultless communications, the five points above provide the basis
for analyzing communications to determine whether they are faultless. The primary analysis for the communication
involves no reference to a particular learner. The analysis does not deal with empirical information, but with the
structural basis for generalization that is provided by the communication. A communication is judged faultless if it
meets the five structural requirements outlined above. The set of examples presented to the learner must be
unambiguous about the quality that is to be generalized. The examples must be designed so that only one quality is
unique to all positive examples. The range of positive variation exemplified by the set of demonstration examples must
be sufficient to imply the appropriate generalization. The negatives should be precise in demonstrating the boundaries
of a permissible generalization. The signals presented with the examples must unambiguously provide the basis for
classifying examples as either positives or negatives. The test of generalization that is presented as part of the
communication must assure that the learner appropriately responds to new positive and negative examples that are
clearly implied by the set of demonstration examples. In summary, the communication is judged faultless if it
adequately provides the learner with information about quality and sameness.

The structural requirements that must be met if a communication is to be judged faultless do not refer to specific
techniques that are used to correct an inappropriate communication or to design one efficiently. However, these
techniques (which are discussed in later chapters) follow from the structural requirements. If we understand that a
communication must show that a particular quality is unique to the positive examples, we will investigate possible
techniques that achieve this goal. From the possibilities we will select those that are most efficient and those that show

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the uniqueness most emphatically. Similarly, the design of the test examples can be reduced to some how-to-do-it
formula once we understand what the test examples must do.

The five structural requirements derive directly from our assumptions about the learner. We can appreciate the
implications of the two-attribute learning mechanism by considering how the structural basis for a generalization would
change if we changed our assumptions about the learning mechanism. For instance, if we assumed that the learner
generalized on the basis of similarity, not sameness, we would not be provided with a strict standard about whether the
communication that we design presents examples that are “similar.” The notion of similarity is not precise and it begs
the question of how the “similar” examples are the same. If examples are similar, they must be the same with respect
to some quality, but the notion of similarity does not require us to identify this qualitative sameness. Therefore,
similarity leads to an imprecise standard for evaluating our communication. By assuming that the learner generalizes
only on the basis of sameness, we are required to create examples that are the same in some identifiable way, and the
standard we use is objectively stronger.

If we assumed that the learner’s generalizations are not clearly determined by the common quality of the concrete
examples of a concept, we would not be provided with a standard for judging whether a communication adequately
shows both the quality and the range of variation in the quality across various examples. We might assume that the
generalization would occur simply if the learner received some “exposure” to the concept. But we would not have any
analytical yardsticks for determining whether the “exposure” presented through a particular communication was
adequate.

With the assumed two-attribute learning mechanism, however, we are provided both with general guidelines for
creating structures that will induce specific generalizations, and with more specific implications about what the
communication must do and what it must avoid doing.

Predictions of Generalizations

The procedure for determining flaws in a communication is a logical one, based on observable details of the
communication. The procedure therefore permits us to make predictions about what the learner will learn. These
predictions are independent of the learner. The basic form that these predictions take is that if the communication is
flawless (adequately meets the five structural requirements), the learner will learn the generalization that is conveyed
through the communication. The learner will respond appropriately to the examples that test the generalization and will
respond to additional examples that are implied by the demonstration examples. Conversely, if the communication has
flaws, some learners who receive this communication will learn the inappropriate quality demonstrated by the flawed
aspect of the communication.

Equally important, the development of procedures for determining whether a communication is faultless permits us
to engage in a very precise study of the learner. A faultless communication serves as a standard against which we
compare the learner’s performance. If this communication is analytically faultless (with respect to clarity in
communicating one and only one possible generalization), any learner who possesses the two-attribute learning
mechanism will learn the concept that is presented by the communication. If a learner does not perform in the predicted
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manner, we immediately know three things about that learner:

1. We know that the learner does not have (or is not using) the two-attribute mechanism.

2. We know the precise ways that the learner’s performance deviated from the predicted performance.

3. Because we know that the problem resides with the learner and not with the communication (which is
judged faultless), and because we know precisely how the learner has deviated from the predicted
standard, we know how we must modify the learner so that the learner is capable of performing acceptably
in response to the communication.

We are able to make these strong inferences about the learner because we have ruled out the possibility that the
learner’s poor performance can be accounted for by the presentation. Furthermore (as we observed earlier), we would
not be able to draw precise conclusions about the learner unless we ruled out the possibility that the communication has
flaws and that the learner is responding in a logically reasonable way to the flawed communication. If the learner
generalized to circles following the communication that presented circularity as a quality common to all positive
examples of red, we would be presumptuous if we interpreted this generalization as an indication of a “faulty” learning
mechanism. Only if the communication is faultless can we make strong inferences about the learner.

Stimulus-Locus and Response-Locus Analyses

Although the major goal of this book is to describe procedures for designing effective instructional
communications, not to study the learner’s behavior, the procedure that we use parallels the one that we would use to
study the learner. We use two analyses. The primary analysis is a stimulus-locus analysis, which deals with an analysis
of the stimuli or communications the learner receives. The second analysis is the response-locus analysis, which
focuses on the learner. This analysis comes into play if the learner is unable (for whatever reason) to produce the
responses that are called for by the communication. The response-locus analysis consists of techniques for modifying
the learner’s capacity to produce responses. If the learner does not respond in the predicted manner to a faultless
communication, the assumed “fault” lies not with the communication, but with the learner. Therefore, we must switch
our focus. This switch involves a complete change in orientation, from a concern with the analyses of communicating
quality and sameness in a precise manner, to the laws of behavior. These laws provide us with specific guides about the
amount of practice, the massing and distribution of trials, the schedules of reinforcement, and other variables that cause
the growth or strengthening of the learner’s response to take place. For example, if the learner apparently forgets the
word red and cannot respond to various examples in a faultless presentation that asks the learner, “What color is this?”,
we modify the learner’s capacity to “remember” how to produce the name. When the learner reliably remembers the
words, we return to the original communication. The learner is now assumed to be an adequate receiver, capable of
responding according to the predictions of the stimulus-locus analysis.

The basic difference between the response-locus analysis and the stimulus-locus analysis is that the stimulus-locus
analysis does not involve the learner. It involves the logic of ruling out all the possibilities but the one to be conveyed
through a teaching communication. The response-locus analysis is based on empirical findings on learning.

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 When instruction in skills involves teaching new responses (those the learner has never produced before in
response to any signal), we use the stimulus-locus analysis to design the sequence of skills to minimize possible
conceptual confusion. We also use response-locus techniques to assure that the new responses are induced efficiently.
However, even for the teaching of “motor skills” (such as shoe-tying, ball-throwing, etc.), the stimulus-locus analysis is
the primary one. The reason is that the communications must be clear and must be organized so that the appropriate
generalizations are induced and the appropriate response generalizations are implied. These communications, however,
rely heavily on the application of behavioral principles.

Extending the Stimulus-Locus Analysis to Types of Knowledge

If we follow the stimulus-locus assumptions to their conclusion, we discover that knowledge may be classified
according to the samenesses of communications used to teach various concepts. The samenesses in features of the
communication parallel samenesses in the concepts that are to be taught. Viewed differently, the extent to which
concepts are the same provides a precise measure of the extent to which faultless communications for these concepts
may have the same features or attributes. Let’s say that we design a faultless presentation for a particular concept. The
communication isolates the quality presented to the learner, unambiguously signals the quality through examples, and
provides additional examples for testing the learner’s generalizations. To design a faultless communication for a
concept that is highly similar to the original one, we would create a communication that is highly similar to the original
one. The close logical parallel between the structure of the concepts we wish to teach and the structure of the
communications that convey these concepts faultlessly results because the two concepts are the same with respect to
many qualities. The samenesses in quality of the concepts is reflected in the samenesses in the communications that
convey these qualities. Conversely, if two concepts differ in many ways, the faultless communications that
communicate them will have many differences.

By extending the notion of the parallel between the structure of concepts and the structure of communications that
convey them faultlessly, we are provided with general guidelines for creating classes of cognitive skills. For this
classification, each category consists of concepts or skills that are the same with respect to important structural
features. Since the concepts in each category share samenesses, all concepts within a given category can be processed
through simple variations or transformations of the same basic communication or form. To classify a concept within
this system is to be provided with an algorithm for communicating the concept to a learner who is assumed to possess
the two-attribute learning mechanism.

Summary

The design and analysis of communications are based on assumptions about the kind of information the learner is
capable of extracting from the communication. For analytical purposes, we postulated a learning mechanism that has
these attributes: the capacity to learn any stimulus quality shown through examples, and the capacity to generalize a
sameness of quality to new examples. This assumed mechanism implies that the primary analysis of cognitive learning
must focus on quality and sameness of the examples presented to the learner. Further implications suggest the structural
criteria that must be met by a communication if the communication is to induce a generalization for a basic concept.

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 1. The positive examples of the concept must be distinguished by one and only one quality.

2. An unambiguous signal must accompany each positive example, and a different signal must accompany
each negative example.

3. The examples must demonstrate the range of variation to which the learner will be expected to generalize.

4. Negative examples must clearly show the boundaries of permissible positive variation.

5. Test examples, different from those presented to demonstrate the concept, assure that the generalization
has occurred.

These criteria serve as guidelines for designing faultless communications and for determining whether a particular
communication is faultless. The analysis of communications according to the structural features of the communication
is the stimulus-locus analysis. The stimulus-locus analysis assumes that the learner is a “receiver” capable of attending
to the information presented through a “faultless” communication. However, a particular learner may not learn in the
predicted manner. The difference between the learner’s actual performance and that predicted by the stimulus-locus
analysis suggests the extent to which the learner does not respond to the basic logic of the communication (the logic of
quality and sameness). If the learner is incapable of producing responses that are implied by the stimulus-locus
analysis, our focus shifts from the stimulus-locus analysis to the response-locus analysis. Behavioral principles are used
to induce new responses and to maintain responses.
 

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 Analysis of Basic Communications
This chapter describes basic communications, elaborates the procedures by which sameness and difference are
conveyed to the learner, and outlines a new type of diagnosis of learning failure that is implied by stimulus-locus
analysis.

Basic Communications

Basic communications play a very important role in a theory of instruction. They are the simplest forms that are
used to communicate concepts. More elaborate forms are extensions of basic communications.

A communication occurs when the learner is presented with examples and verbal descriptions that demonstrate
how to respond to particular stimuli. For the stimulus-locus analysis, the most basic communications are those that
deal with generalizations. A non-generalization item, such as the task, “What’s your name?” is perhaps more
adequately viewed through a response-locus analysis. Even this task becomes involved in a generalization when the
learner must discriminate who is to respond when “What is your name?” is presented to different people.

The most basic communication for the stimulus-locus analysis has the following features:

1. A set of examples or instances.

2. Some behavioral signal provided by the teacher with each example.

The set of examples consists of the things or events shown to the learner. All positive examples of the concept
within a set have the quality that is being taught.

The behavior signal demonstrates whether a particular example has the quality. The signal may be complicated or
simple. In the simplest form, the teacher uses two behaviors, the first for signaling, “Yes, this example has the features
you are to attend to.” The other behavior signals, “No, this example does not have the features.” Any two behaviors can
be used to communicate information about the “yes-no” classification of the examples in the set.

Any sameness shared by all examples that are treated in the same way describes a generalization. If the examples
treated in the same way share quality A and if the learner has the capacity to abstract this sameness, the learner will
generalize to examples that have quality A.

If we hand a toddler a crayon and say, “Red,” then present another crayon and say, “Red,” we are asserting that
there is something the same about both examples. If there were no structural sameness in the object, we would have no
basis for treating them in the same way. By treating things the same way, we do not guarantee that we communicate
how they are the same. We merely signal that something about the situations is the same. Perhaps the act of receiving a
crayon is “red;” perhaps the act of coloring is “red;” perhaps anything you can scribble with is “red.” All possibilities
are suggested by the communication above, because both examples that were treated the same way share these

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structural features.

This list of facts in Table 2.1 shows some “samenesses” possessed by both examples. The samenesses are the
possible bases for generalizations.

Table 2.1
If Object 1: And Object 2: Then the
sameness
being demon-
strated is:
Was handed to Was handed to Handing some-
the child the child thing
Was colored red Was colored red Colored red
Was wrapped in Was wrapped in Wrapped in
paper paper paper
Was used for Was used for Used for coloring
coloring coloring

The last column shows that many samenesses are logically implied by the communication. Which will the learner
select? The answer is beyond the stimulus-locus analysis. We do not assume that the child has any preknowledge of the
appropriate sameness. We therefore assume that the possibilities are “equal.” We might modify this conclusion on the
basis of empirical information of how learners respond.

Learning Sets

We might further modify our conclusion on the basis of what the learner has learned in the past. For example, if the
presentation above has immediately followed the successful teaching of green, yellow, and blue, there would be very
little doubt that the presentation would teach red.

The reason, however, is that the teacher is treating various objects in the same way.

• If coordinate objects in group 1 have a particular color (green),

• and coordinate objects in group 2 have a particular color (yellow),

• then coordinate objects in group 3 have a particular color (red).

This is a rather sophisticated extension of the sameness application. The idea is basically simple. The learner is
prompted to attend to a particular set of features because the previous communications have shown that these features
are the basis for the preceding discriminations. The present communication involves examples that have many of the
same visual properties the preceding communications had. The present communication also involves teacher behavior
that has many of the same features as the behavior presented with the earlier communications.

Research on “learning sets” (Harlow, 1959) has shown that learners tend to learn faster or better on subsequent
examples of the same type (e.g., oddity problems). The process can be explained by referring to what is the same about
the various examples and the behavior used to signal them. Puzzles of a given type have solution features that are the
same. By learning to respond successfully to different puzzles, the learner learns how to treat the puzzles in that same

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way. Therefore, subsequent puzzles involve less new learning than initial puzzles. The learner has learned to process
the later puzzles through the same strategy steps that led to solutions with the earlier puzzles.

Tests of Generalization

The stimulus-locus approach treats generalizations as the product of the communication. Samenesses, which form
the structural basis for generalization, are shown through the communication. The learner will select one of the
generalizations communicated. For us to determine whether this selection has occurred, we must test the learner. The
simplest test is to present the learner with examples and see if the learner treats each in the appropriate manner. If we
deal with a very simple communication (one that uses a “yes” signal for some examples and a “no” signal for others),
we would present the learner with new examples that share a particular sameness with the examples identified earlier
as “yes” and new examples that share the sameness with examples identified as “no.” From the learner’s pattern of
responses we can determine whether or not the learner has learned a particular sameness. Note that the new examples
we present may be physically different from the examples presented during the communication with the learner so long
as they clearly possess a sameness shared by all the “yes” examples or by all the “no” examples.

The tests of generalization are limited to certain objective facets of the communication. We do not assume,
however, that the communication conveys only these samenesses. If the receiver of the communication is capable of
learning samenesses that are intentionally shown by the communication, the learner is also capable of learning many
other “associations.” The teacher’s green dress may be the same as someone’s dress in a past situation of some import;
the sound that occurs in the background may be the same as sounds in other learning situations. These “associations”
will certainly occur. For instructional design purposes, however, we do not deal with these generalizations because we
are unable to control them. No matter how carefully we control the ambient aspects of the learning situation, many
uncontrolled generalizations will be possible.

Basic Concepts and Examples

A basic concept is one that cannot be full y described with other words (other than synonyms). A communication
for a basic concept, therefore, is one that requires concrete examples. We cannot explain red to a blind person in a way
that would permit the person to discriminate between red and not-red objects. Similarly, concepts such as smooth,
heavy, over, toward, happy, etc., require concrete examples unless the learner already knows the concept by a
different label (unrough, unlight, above, closer, glad). Examples are needed to teach these concepts because the
words that we use are not containers of the concept. They are merely symbols that stand for particular qualities. Unless
the communication presents the learner with the actual experience of the quality being symbolized, the communication
provides no basis for understanding which quality or property the symbol represents.

A problem with concepts and their analysis is their potential confusion with words. A basic concept is not a word
and does not logically imply words. The qualities (the samenesses that exist in sets of examples) are “real” and
objective. The words used to refer to them are creations. These creations are useful if they serve their primary functions
of signaling the quality. They may have a secondary function of fitting into grammar and possibly possessing features
that are shared by other words that refer to similar events. (The word sixty is related to the word six because both have

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a common part and that part refers to a common feature of reality.) But the word is not the concept and does not imply
the concept to a naive person.

Inducing Patterns of Responses

If we used only one word for each quality we would have no problem with words. The fact that one word has many
possible meanings, however, suggests that if we begin with the analysis of words, we may become so embroiled in
word games that we fail to observe that this approach is irrelevant. We should begin with the idea (basic sensory
concept) that we wish to teach, not the word. Here’s how we could test an idea. We simply test the learner’s response
pattern by requiring the learner to point to different examples that show a concept. For instance, after instruction, we
could tell the learner: “Point to all the objects that are over the table.” The pattern of responses for pointing does not
require the learner to use words. However, the pattern shows the generalization. The same pattern would be observed if
we substituted a verbal response for the pointing response.

If the question presented with each example is, “Is this object over the table?” the learner would say, “Yes” for
every object that had been touched and “No” for every example not touched. If we tell the learner, “Tell me ‘over’ or
‘not-over’,” the learner would say, “Over,” for every object previously touched. If the same examples are responded to
in one way when we require pointing to objects, when we require the learner to say “Yes,” and when we require the
learner to say “Over,” something must be the same about those objects. If the only observable sameness is the quality
of being over, we conclude that: (1) we have induced this quality or idea; and (2) the idea does not depend on any
particular response. (The pattern of responses remains stable over different responses.)

The words that we use are merely signals for the qualities of examples. We make no assumption that teaching the
particular meanings signaled by over will induce the understanding of over when it is used in this way: “The party is
over.” Furthermore, we make no judgment about the relative value of the word over to describe the position, or about
whether the learner should be taught the other meanings that are conventionally labeled with the word over. (“She’s
done that paper over four times.”) The objective of a teaching communication is to convey one meaning, calling the
learner’s attention to a particular sameness. We select a word that is conventionally acceptable for this purpose. Then
we teach the learner about the relevant sameness, using the word that we have selected.

Concrete Examples

We have already alluded to the notion that a set of examples must be shown to induce basic concepts. It is logically
impossible to present a single example of a basic concept that shows only one concept. The reason is that an example
capable of showing only one concept would have to possess only one quality or property. It would have to exist without
any features that are irrelevant to the concept. If we are teaching redness, the example would have to show only
redness—not space, position, duration, shape, or other identifiable non-color features. Such examples do not exist.

To determine the other concepts or qualities that a particular example of red exhibits, we ask, “Could this object be
used as an example of ____?” Let’s say that a concrete example of red is a rectangular piece of red felt placed on a felt
board. The test of the other concepts this example exhibits discloses that the example could be used to demonstrate an

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indefinitely large number of other concepts. The concrete example is an example of cloth, of felt cloth, of felt cloth of
a particular shape, of felt cloth in a particular place, of an object, of a solid object, of an object smaller than a
breadbox, of an object smaller than a dog, of an object smaller than . . ., of the number 1, of the number 4 (four
corners), of on, of above the floor, of . . . The list continues indefinitely. Furthermore, a similar indefinitely long list
can be constructed for any example that is presented to show a basic concept.

Although a concrete presentation is an example of thousands and thousands of concepts, the set of concepts
generated by one concrete example is never the same as the set generated by another example. The reason is that if the
sets were identical, the examples would be the same in every conceivable detail, which means that they would occupy
the same place at the same time. They would therefore be the same example. Given that they are different examples,
there is a difference between them, which means that we can make observations about one of the examples that we
cannot make about the other. By manipulating the differences of examples, we can rule out irrelevant qualities that are
inevitably present in the isolated example.

Analysis of Communicating Sameness

The learning of basic discriminations or concepts is inductive. The teacher treats concept examples one way (uses a
common signal). The learner must identify the qualities that are referred to and learn that the word serves to signal the
qualities.

Figure 2.1 shows five examples, each of which is labeled “glert.” The word glert, therefore, stands for some
particular feature or combination of features. The features of each example are represented by the letters A, B, C and D.
Example 1 is presented first, and example 5 last.

Figure 2.1
Examples 1 2 3 4 5

Features A A A A A
Within B B B B B
C C C Not-C Not-C
Examples
D D Not-D D Not-D

Teacher
Behavior “glert” “glert” “glert” “glert” “glert”

Example 3 shows that D is not necessarily a feature of glert. This communication is achieved through the
following inferences: the examples of glert can be treated the same way only if they are structurally the same. Example
3 is treated the same way as examples 1 and 2. Example 3 does not possess feature D. Therefore, feature D is not a
necessary feature of glert.

The communication provided by the entire set of five examples suggests that A and B are necessary for glert. Both
A and B appear in all examples and they are the only features common to all examples. However, the communication
provides the following possibilities:

Glert refers to A.
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 Glert refers to B.

Glert refers to A and B.

A communication that is effective in showing that A is the only feature relevant to glert follows a juxtaposition
rule for showing sameness. The rule: To show sameness, juxtapose examples that are greatly different; treat each
example in the same way. (To juxtapose examples, present one immediately after the other or position the examples
next to each other.)

Figure 2.2 draws a set of five examples that is consistent with the rule. All examples are positive instances of glert.
All are treated in the same way (by being referred to as “glert”). And the sequence juxtaposes maximally different
examples. The first example has four features (A, B, C, D). The second example has one feature A. Since both
examples are called “glert,” the concept of glert cannot be associated with what is different about the two examples,
but only with what is the same—feature A. Different arrangements of examples would be as effective as the one given;
however, any effective arrangement should juxtapose examples that are greatly different.

Figure 2.2
Examples 1 2 3 4 5

Features A A A A A
Within B Not-B B Not-B B
C Not-C Not-C C Not-C
Examples
D Not-D D Not-D Not-D

Teacher
Behavior “glert” “glert” “glert” “glert” “glert”

Analysis of Communicating Differences

Just as structural sameness is implied if all examples are treated in the same way, a structural difference is implied
if two examples are treated in a different way. The difference assumption is: If two things are treated differently, the
examples must be different with respect to some feature. A poor communication is one that presents negative examples
that are greatly different from the positive examples of the concept. Figure 2.3 shows why large differences are
ineffective.

Figure 2.3
Examples 1 2
A Not-A
B Not-B
Features C Not-C
D Not-D

Teacher
Behavior “glert” “not-glert”

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 These juxtaposed examples differ in A, B, C and D. The behavior of the teacher signals that there is a structural
difference between the examples (calling one example “glert” and the other “not-glert”). The problem is that there are
many differences between example 1 and example 2. The only difference that “causes” example 2 to be treated
differently from example 1 is the absence of A. All the other features are structurally irrelevant. The communication
provided by the juxtaposition of these examples, however, provides many possible options about which features cause
the change in the teacher’s behavior. Stated differently, if a capable learner received the information provided by the
communication above, the learner might not identify the example in Figure 2.4 as “not-glert.”

Figure 2.4
Not-A
B
C
Not-D

The learner may identify it as “glert” or as “not-glert.” Both possibilities exist because the presentation of the two
demonstration examples is not specific about the minimum difference between “glert” and “not-glert.”

Figure 2.5
Examples 1 2
A Not-A
B B
Features C C
D D

Teacher
Behavior “glert” “not-glert”

The rule for articulately communicating differences through examples is: To show difference, juxtapose examples
that are only minimally different and treat them differently. Any structural difference observed in the juxtaposed
examples can function as a possible basis for the different treatment of the examples. There is only one structural
difference between examples 1 and 2 above in Figure 2.5. Therefore, there can be only one structural basis for the
different treatment—the absence of feature A. This prescription for showing differences is contraintuitive, but logically
compelling. If the only difference between the two juxtaposed examples is a small difference and if the examples are
treated differently, the small difference must be solely responsible for the different behavior. Communication of the
very small difference logically implies that differences that are larger than the one shown would also lead to the
examples being identified as “not-glert.”

Operations to Induce Generalizations

Although the procedures for demonstrating sameness and difference will achieve the desired communication goal,
a deeper analysis of operations used to induce generalization provides a clearer understanding of why these operations
work. This analysis identifies three specific operations: interpolation, extrapolation, and stipulation.

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 Interpolation. In its most basic form, the operation of interpolation assumes changes along a single stimulus
dimension, such as color, size or position. Figure 2.6 shows color gradations, from light blue to dark blue. The display
does not suggest that all features of color (hue, saturation, intensity) are taken into account, merely that the examples
are arranged progressively from light to dark.

Figure 2.6
X P X O X
Light blue Dark blue

The three X’s indicate examples that are communicated to the learner as being the same. If they are the same, then
what is being labeled as “blue” must be what they have in common (the range of the color value being labeled “blue”).
According to the operation of interpolation, if the learner receives the communication about the three examples, the
learner will identify any example that is intermediate in blueness as “blue.” For example, the learner should identify
examples O and P as “blue.” (No example shown above would be presented on a continuum. The continuum is used
only to indicate darkness and lightness of our examples.)

There are other forms of interpolation. One form involves the addition or subtraction of parts. Although this form
deals with a generically different type of interpolation, the basic operation is the same. If the generalization example
falls within the range of interpolation described by the demonstration examples, sufficient information for
generalization is implied.

If we deal with very complex examples, we might not be able to precisely express the nature of the various features
that change from one positive example to another. However, if we show examples that are greatly different from each
other, and if we treat these examples in the same way, we imply interpolation of most new examples. The reason is that
the examples we show initially would not be placed close to each other on a continuum. Any new example would
probably fall somewhere between the examples demonstrated. Therefore, interpolation is implied.

The examples that are used to demonstrate blue would not predict that the learner would generalize to a shade of
blue that is quite white and unsaturated. The reason is that this shade does not clearly fall on the continuum implied by
the examples that we had presented. To solve this problem, we must present a larger number of examples. Let’s say
that we present three variations for each shade of blue—each variation presents a different saturation and hue (see
Figure 2.7).

Figure 2.7
Shade 1 X P X O X
Shade 2 X’ P’ X’ O’ X’
Shade 3 X” P” X” O” X”
Light blue Dark blue

The presentation consists of nine examples (the X’s). The test of generalization involves six. The six examples are

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clearly interpolated within the range of variation described by the nine demonstration examples; therefore, the
communication implies that all variations of O and P would be identified as blue.

Extrapolation. According to the principle of extrapolation, if a small change makes a positive example negative,
larger changes will also make the example negative. The differences of the various examples in Figure 2.8 is indicated
by their position on the continuum of change. The learner is shown three examples. Two are labeled “blue” (the X’s)
and one “not-blue” (the Y). The last two examples are quite close to each other on the continuum (to communicate
difference). If the examples are only minimally different and are treated differently (one called “blue,” the other “not-
blue”) the structural basis for the different label is unambiguous, because there is only one apparent difference in the
examples.

Figure 2.8
X X Y O P
blue blue not-blue

Extrapolation is based on this idea: The difference between the second X and the Y is sufficient to make Y a
negative example. Any difference that is greater than this difference implies that the examples will also be negative.
The difference between X and O is greater than the difference between Y and O; Y is negative. Therefore, O must be
negative also. Similarly, the difference between X and P is greater than the difference between X and Y. Therefore, P
must be negative.

Another way to conceive of this sort of extrapolation is to visualize a concrete example, such as a patch of dark
blue. Now consider how much change in the example is needed to convert the example into something that is not-blue.
Changes greater than the amount create examples that are obviously not blue.

Stipulation. Stipulation involves repeatedly demonstrating examples that are highly similar and presenting only
these examples. Each example is treated in the same way. The stipulated set of examples implies that any examples
falling outside the range that had been demonstrated are not to be treated in the same way.

Figure 2.9 shows that the non-naive learner is presented with eight examples that are quite similar to each other in
every respect (X’s). The communication does not present any additional demonstration examples. When tested on
example P or example O, the learner would probably treat it as a negative example. The outcome is problematic
because the learner is not given precise information about the range of variation that is permissible for the X’s. The
communication shows only that when the examples of a particular type are presented, they are treated in the same way.
Examples O and P are not the same type (because their obvious structural differences sets them apart from any
examples that had been demonstrated). Therefore, a reasonable inference is that they are probably not the same as the
X’s, and are therefore negatives.

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 Figure 2.9
XX
XX
XX
P XX O

Stipulation depends on the number of examples that are presented. If we presented the learner with this single

example and labeled it “glup,” we would not be surprised if the learner identified this test example as “glup”: .
However, if we presented the learner with 20 vertical examples and labeled each as “glup,” the probability is greatly
increased that the learner would identify the slanted test example as “not-glup.”

One way to understand what is happening when stipulation occurs is to think of a particular concrete example and
the various features it has (see Figure 2.10).

Figure 2.10
A
B
C
D

If we repeat this example again and again, we imply that the essential sameness possessed by the positive examples
includes all features, A, B, C, D. If we later present an example that is different—one that has only A and B—the
learner will probably reject it because features that have been implied to be essential are missing.

Examples of Complex Concepts

When we deal with complex learning, the communications involve a combination of stipulation, interpolation, and
extrapolation. Consider the situation in which we present an operation for figuring out how to add numbers. The
operation involves the same counting steps when it is applied to different examples. The operation therefore stipulates
some behavior for all examples encountered. The examples presented show some range of variation. Some examples
involve one-digit numbers, some involve two. Some problems present the largest number first. These examples
communicate sameness by demonstrating that the same addition operation holds for a range of counting numbers and
arrangements. Also, if the examples suggest that a change from one digit to two digits does not affect the operation and
that a change from smaller numbers to larger numbers does not affect it, the learner could reasonably be expected to
“extrapolate” and conclude that the operation holds for any number.

In summary, we communicate with the learner through examples. The game is something like trying to view a vista
through a very small peephole, a glimpse at a time. The learner has the capacity to make a consistent “whole” or gestalt
from what we show. The glimpses we provide are examples. The idea is to provide the learner with enough information
to make a consistent whole from the examples. We do not want to induce distortion or misrules by showing poorly-

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selected glimpses.

One and Only One Generalization

Our objective, as noted earlier, is to design communications that lead to only a single generalization or
interpretation. In other words, a communication should be faultless. Only if it is faultless do we receive unambiguous
information about the learner. If there is more than one sameness for the various examples that are treated in the same
way, the communication has faults. The learner may respond to an inappropriate sameness. Each sameness describes a
generalization. Therefore, the inappropriate samenesses imply inappropriate generalizations. Conversely, if there is
only one sameness that is possessed by all positive examples, there is only one possible generalization.

A communication that is analytically faultless is faultless for any learner. Viewed differently, any naive learner
needs the same information about the nature of the concept—about which changes create negative examples, and about
which changes do not. Although, by chance, a learner may pick up the appropriate information from a presentation that
has faults, the learner is as likely to pick up inappropriate generalizations. This situation is possible whether the learner
is “very bright,” or “slow.” The variable is not the learner, but the communication.

Faultless communication does not imply that there is only one possible communication that will work for “all”
learners. Since there are countless examples of a given concept, there are potentially countless variations of faultless
communications. Each variation would use different examples. However, all variations would be the same in many
respects. All are based on the quality or concept being taught.

Let’s say we wish to teach the naive learner the grade of a hill, angle of the hill, or slant of the hill. Our goal is to
teach one of these labels, and to show the learner what it means. Let’s say that we choose to refer to the steepness of
the hill.

The most efficient way to communicate with the learner is through the continuous conversion of examples.
Continuous conversion occurs when we change one example into the next example without interruption of any sort.

Hold up your hand in an angle about like this: . Give a behavioral indication for this example: “See this.” Now

move your hand to about this angle: . Say: “It got steeper.” You have created an example through continuous
conversion. You change the first display into the example (“It got steeper”). Continuous conversion of examples
logically provides the most precise communication with the learner. The reasons are:

1. Many aspects of the display appear in all examples and are therefore shown to be irrelevant. If you
presented a series of continuous conversion examples with your hand, your hand would be in every
example. The hand, therefore, could not account for the fact that some examples are “positive” (steeper)
and some are “negative” (not-steeper). The differences in the slant of the various examples would be the
only basis for difference in the labels of the examples.

2. It is possible to show only those changes in the example that lead to a change in label. Let’s say that you
start with your hand in a particular position. You then create the simplest change that permits the next

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 example to be labeled “it became more slanted.” You must change the aspects of the display that have to
do with slantedness (you must rotate your hand). No other change will achieve the objective of creating an
example more slanted. It is possible to make that change and no other change when examples are
presented through continuous conversion.

3. It is possible to show the type of changes in the relevant dimensions that do not lead to a change in label.
The relevant dimension is the one that is used to create a position from a negative example or vice-versa
(rotation of the hand for the concept slanted). If we show the learner that certain changes in this dimension
lead to changes in the label, we should also show a very small movement of the hand that is labeled, “It
got steeper,” and a very large movement that is labeled, “It got steeper.” Both involve rotation, but both
are obviously different from each other. The amount of change from positive example to positive example,
therefore, is not relevant to “It got steeper.”

For many concepts, continuous conversion is not possible. However, the communication through continuous
conversion provides us with the guide or model for creating non-continuous-conversion sequences. The controls that
are automatic in a continuous conversion sequence must be carefully constructed if we are required to present static
examples to communicate the same information so well provided through continuous conversion.

Individualized instruction as it occurs in the home (when the mother instructs the child) often involves continuous
conversion. The mother tells the child to do something: “Put a fork here.” The child makes mistakes, putting a spoon in
the designated place. The mother converts one example to the next: “No, honey, a fork, not a spoon. Here’s a fork.”
The spoon is replaced with the fork and the learner receives specific information on the correlation between the
differences in label and difference in example.

A Faultless Sequence

A faultless communication consists of two parts. The first shows what controls how the example is treated. The
second shows more about the context in which the concept or quality may occur. Figure 2.11 shows a faultless
presentation for the concept getting steeper.

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 Figure 2.11
Example Teacher Wording

Watch my hand, I’ll tell you if it gets steeper.

It didn’t get steeper.

1

It didn’t get steeper.

2

3 It got steeper.

4 It got steeper.

5 It got steeper.

6 Did it get steeper?

7 Did it get steeper?

8 Did it get steeper?

9 Did it get steeper?

10 Did it get steeper?

11 Did it get steeper?

12 Touch the line that is steeper.

13 Hold up a pencil so that it is steeper than this pencil.

14 A B Which hill is steeper? Hill A or Hill B?

The first 11 examples show what controls getting steeper. The first five are “modeled” by the teacher, which
means that the teacher tells the answers. The next six examples are tested. Following example 11 are examples that
show the learner something about the range of contextual variation in which steeper occurs. The examples present
objects not shown in the initial part of the communication (lines, pencils, hills) and different types of tasks (touching,
naming a hill, etc.).

Different examples in the first part of the sequence have different functions. Some show differences (examples 1-3
and 5-6). Minimally different examples are juxtaposed and are treated differently. Sameness is shown by examples 3-4-
5. Greatly different juxtaposed examples are treated in the same way. Examples 6 through 11 have the functions of
testing the learner and of presenting examples whose values are implied by interpolation or extrapolation created with
the first five examples.

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 The communication should be nearly faultless because it does an adequate job of demonstrating what controls the
label and of not stipulating that the label is used only in connection with the presentation of hands. Examples 12
through 14 are different enough from hands to prompt the extrapolation of the concept to a wide range of examples not
shown in the teaching. The presentation for getting steeper is faultless enough to permit rigorous study of the learner.
If the learner does not generalize to new examples that are presented with the hands or does not extrapolate to new
examples that do not involve the hands, the outcome cannot be explained with reference to the communication or the
“stimulus.” An explanation must be sought by reference to the learner.

Related Sequences for Related Concepts

The presentation above is adequate to communicate the structure of the concept getting steeper. With
modifications, it is adequate for communicating any closely-related concept. Any changes in the structure of the
concept, however, imply changes in the communication. If we present the concept greater grade instead of getting
steeper, we could use the same examples and change only the label that we use. The reason is that the only difference
is the label. If we change the concept so that both the label and the structure change, additional changes are needed. For
instance, if we present getting faster, the label must change and the nature of the examples must change. We cannot
demonstrate “faster” by showing “steeper.” However, since both concepts (getting steeper and getting faster) are
comparatives, they have structural details that are the same. These structural details imply samenesses that should be
retained in the communication.

To teach getting faster, we can start with something that is moving. We can then model the changes that occur in
the example after the type of changes that occur in the sequence for getting steeper. If there is a big change in the
positive direction to create an example in the getting-steeper sequence, we will introduce a big change in the positive
direction for the corresponding example of getting faster. If there is a small positive change in the getting-steeper
sequence, we will make the corresponding example for the getting-faster sequence by introducing a small change in
the positive direction. Figure 2.12 shows the first five examples of a positive sequence for getting faster. For all
examples, the object moves in a circle. The size of the circle and the direction of the object remain the same for all
examples.

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 Figure 2.12
Examples Wording
Object moving 40 Watch the dot. I’ll tell
revolutions per minute. you if it goes faster.

1. Object moving 30 It’s not going faster.

revolutions per minute.
2. Object continues moving 30 It’s not going faster.
revolutions per minute.
3. Object begins moving 40 It’s going faster.
revolutions per minute.
4. Object begins moving 70 It’s going faster.
revolutions per minute.
5. Object begins moving 90 It’s going faster.
revolutions per minute.

These examples precisely parallel the first five examples of the sequence for getting steeper. The structural
aspects of the two that are the same appear in both sequences—particularly the size of the change from example to
example and the direction of the change. The prediction would be that if the sequence for getting steeper is faultless
and if the concept of getting faster has the structural samenesses that are reflected in the samenesses of the two
sequences, the sequence for getting faster should also be faultless.

Stimulus-Locus Diagnosis of Learning Failure

An implication of the stimulus-locus analysis is that the primary diagnosis of learning failure should be a diagnosis
of the instruction the learner receives. If the learner fails to generalize, the problem may lie with the learner or with the
communication the learner receives. We can rule out one of these possibilities by assuming that the communication is
responsible for the observed problem. The remedy is to identify faults in the communication the learner is receiving
and correct them so that the communication is faultless. If the faultless instruction fails, we know that the problem is
with the learner and not with the communication. If the faultless instruction succeeds, we know that the initial problem
was indeed with the communication.

This diagnostic procedure is the opposite of the traditional diagnostic procedure, which assumes that the learner is
at fault for any learning inadequacies. The fact that the traditional diagnostic procedures hold the learner responsible
can be ascertained by referring to the percentage of case histories in which the learner’s deficiency in reading, for
instance, is judged to be caused by poor teaching. The percentage consistently hovers around zero, implying that the
traditional diagnostic paradigm assumes that the learner has the deficiency. Not only is this position improbable, it is
also illogical. Any diagnosis begins with an observation of behavior. This behavior may be influenced both by
deficiencies in instruction and deficiencies in the learner. To assert that the behavior is “caused” either by learner
inadequacy or by teaching inadequacies is to go far beyond the data.

The value of the initial hypothesis of the problem (that the teaching is the sole cause of the learner’s problem) is

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that it requires us to rule out the possibility that instructional variables could account for learner failure. The
hypothesis requires us to identify flaws in the instruction the learner has received and to provide faultless instruction.
The prediction is that when we provide faultless instruction, the learner’s problems will be solved.

Regardless of the outcome of this test, we will receive very precise information about the actual status of the
learner’s problem—the extent to which the problem is caused by learner inadequacy and the extent to which it is
caused by instructional deficiencies. If the learner responds to the faultless instruction by learning the samenesses or
generalization conveyed by the faultless instruction, we conclude that the learner’s initial problem was caused primarily
(or solely) by instruction. If the learner remains virtually unchanged after the introduction of the faultless
communication, we conclude that the learner’s original problem was inadequacies in the learner, not inadequacies in
the instruction. An intermediate outcome provides us with an intermediate conclusion: part of the learner’s problems
are caused by the learner, and part by the faulty communication.

If we accept the traditional approach to learning failure, we receive no diagnostic information that translates readily
into instruction. If the learner’s deficiencies are caused by learner inadequacies, how do we perform an instructional
remedy that will reduce the learner inadequacies? The instruction is not suspect, so there is no reason to change it.

Illustration

Let’s say that we observe a naive learner who has been taught speech behavior by one teacher in the same setting
and the learner does not practice the newly-learned behaviors with other people in other settings. The instruction has a
serious fault. Specifically, it is guilty of stipulation. All examples of learning language or speech skills have a large set
of common features-the teacher, the details of the setting, and nature of the tasks. A prediction based on this fault is that
the learner will not perform with other people. The skills will not “transfer.”

If we attempt to diagnose the learner instead of the instruction, we will probably conclude that the learner is poor at
generalizing, implying that the problem is caused by learner inadequacy. This diagnosis does not suggest how we can
make the learner adequate or how we can rule out the possibility that the learner has responded in a perfectly
reasonable way to the communication.

The stimulus-locus diagnosis assumes that although the learner clearly failed to “transfer,” this failure is not caused
by learner inadequacies, but is consistent with the instruction received. The solution is to modify the instruction.

Summary

Basic communications are the most important units for a technology of instruction. They present concepts or
qualities that cannot be fully described to the learner in words because the learner lacks knowledge of which quality is
being labeled. The communication consists of examples and behavioral signals presented with the examples.

Any samenesses shared by all examples treated the same way describes a generalization.

The test of this sameness may be performed with any stable response from the learner. For the simplest form of the
test, the learner is presented with a series of test examples (some of which may be different from those presented during

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the earlier demonstration) and the learner indicates whether each is an example of the concept. If we test the learner
first by requiring a pointing response, then by requiring some other response to the same examples, we discover that the
pattern maintains, although the response changes. Since all that remains is the pattern, the teacher must have
introduced this pattern.

The examples that are used in a basic-communication sequence have an indefinitely large number of qualities, each
of which could theoretically serve as the basis for possible generalizations. One example, therefore, cannot possibly
teach. Subsequent examples must be used to rule out some possible generalizations and to confirm others. In the end,
the teaching of the concept requires a set of examples. The learning process is inductive. The learner is simply shown
which examples are the same. The learner must identify the sameness that binds them.

In manipulating the examples to rule out particular interpretations, we follow juxtaposition principles.

To show sameness, we juxtapose examples that are greatly different and we treat each example in the same way.

To show difference, we juxtapose examples that are only minimally different and we treat the examples differently.

The three operations that describe our primary manipulations with the set of examples are: interpolation,
extrapolation, and stipulation. Interpolation is based on a display that treats obviously different examples in the same
way. If the range of variation shown by these examples does not cause the label to change, an intermediate value or
change should also be treated in the same way.

Extrapolation is efficient for ruling out a range of negative examples. If the change from a given positive example
to a minimally different negative causes the negative to be labeled differently, a greater change in the same direction
from the positive will also result in a negative example.

Stipulation is the repeated presentation of examples that have a great many samenesses. The presentation implies
that all features of these examples are necessary to the label. The result is that if the learner is presented with variations
in any features, the learner will not treat the example in the same way as the original examples.

The most efficient way to show relevant changes in the examples and to label these unambiguously is through the
process of continuous conversion. The process involves presenting an example and then converting it into the next
example, providing the learner with a demonstration of only the difference between the examples. The difference is
then labeled. The two examples are either treated in the same way (implying that the change was relevant to sameness)
or in a different way (implying that the change caused a change in the label).

By combining the principles of showing sameness, showing difference, and testing on generalizations, we can
create a basic communication—a set of examples accompanied with wording that tells and wording that tests. If a given
presentation proves to be faultless, we are provided with a possible model for creating faultless sequences for
discriminations that share many structural features with the concept taught in the original sequence. We simply change
the parts of the sequence that process structural details that are different.

Finally, we may use the analysis of communication as the basis for diagnosis of learning failure. Instead of

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diagnosing the learner, we begin by diagnosing the instruction. We identify flaws in the instruction and correct them,
with the assumption that the learner’s problems were caused by flaws in the instruction. This hypothesis requires us to
control the instruction or communications and then test the learner. Regardless of the outcome, we will receive very
precise information about the learner.
 

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 Knowledge Systems
Chapter 2 demonstrated that concepts with the same basic structure (getting steeper and getting faster) could be
processed through faultless sequences that had the same basic structure. By extending the relationship of sameness in
concept structure and sameness in communication, we are provided with a basis for categorizing types of knowledge.
In this categorization, concepts that have the same structure are placed in the same class. Concepts within a class share
structural similarities. Because these structural similarities parallel similarities in the structure of the communication
used to convey the concepts to naive learners, concepts within the same class may be processed through variations of
the same communication.

The two objectives of organizing different types of knowledge are:

1. To provide an exhaustive system that permits classification of any cognitive operation, from simple
discriminations to complex operations.

2. To link the classification system with instructional procedures, so that all concepts within a particular class
or category may be processed through variations of the same communication form.

This chapter describes the classification system. Subsequent chapters articulate the instructional procedures
implied by the various categories of the classification system. We will first examine the stimulus-locus classification
for cognitive skills (based on analysis of the cognitive operations). We will then outline the response-locus
classification (based on analysis of learner’s characteristics when learning new responses).

Classifications for Cognitive Knowledge

Basic Forms (sensory-feature concepts)

• Non-comparatives

• Comparatives

• Nouns

Joining Forms (relationships between sensory-feature concepts)

• Transformations

• Correlated-features relationships

Complex Forms (chains of joining forms)

• Cognitive problem-solving routines

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 • Communications about events (fact systems)

The three major categories within this system are presented in order of ascending complexity. Concepts within the
first category (basic forms) are the simplest concepts. The communications that are appropriate for communicating
them present single-step tasks or questions with the same, single question used for all examples in the sequence.
Getting steeper is a basic form concept. Each test item for this concept asks a single question: “Did it get steeper?” All
positive examples are responded to with the same response, “Yes.” The goal of each of these communications is to
teach a single concept.

The communications for the second category, joining forms, are more complicated. For some of these sequences,
more than one question or task is presented with each example. For others, a single question is presented with each
task; however, the appropriate response changes from positive example to positive example. The goal of each
communication within this category is not to teach a single concept, but to teach a single relationship.

The communications for the third category, complex forms, are more complicated than the joining-form
communications. For each example presented in a complex form communication, the learner must perform a series of
steps. These communications are not designed to teach a single concept or a single relationship. Rather, each is
designed to teach a set of relationships that are appropriate either for solving problems of particular types or for
learning about the set of features that distinguishes one event from another. In either case, the communication presents
a series of familiar concepts or relationships that are combined in a unique manner.

The classification indicates the most precise communication that is logically possible for each type of concept. The
most precise communication possible for basic forms is basic forms. The most precise communication possible for
joining forms is joining forms; however, it is possible to treat any joining form concept as a basic-form concept.
Similarly, it is possible to treat any complex form as a joining form or as a basic form. The matrix on page 20 shows
the relationship between the type of concept and the classification options available for that type. The cells in which the
X is circled show how each type of cognitive operation is classified in the system. According to the matrix, any concept
may be treated as a basic concept. If we were to treat the concept of carrying out a long-division operation as a basic
form, we would present it through a communication that follows the same pattern used for all basic forms. We would
present examples of working long division problems. For the examples that test the learner, we would ask the learner,
“Is that the right way?” For some examples, the answer would be “yes” and for others, “no.” This is not the most
precise communication that is possible for the concept of the long division operation (or of a particular algorithm for
working long division problems). A more precise way would be to show the learner the steps involved in working the
problem. The concept of a long-division operation is a complex form—one that involves many steps, many concepts,
and discriminations linked together.

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 Type of Classified As
Concept

Basic Joining Complex

Basic X

Joining X X

Complex X X X

Basic Forms

Basic-form concepts are the simplest form because they cannot be reduced to simpler forms, and they cannot be
clarified through verbal explanations. Basic-form concepts refer to specific meanings of words like red, under, back,
truck, door, sit, horizontal, and girl. To communicate any basic form, we must present examples, some of which
show what the concept is (positive examples), and some of which show what the concept is not (negative examples).
There are several ways to test a concept to determine whether it is a basic form concept. The simplest way is to start
with any sentence that conveys a relatively specific meaning, such as:

“Look at the red block under the table.”

Earlier we noted that we are not dealing with words when we analyze concepts, merely meanings. The use of a
sentence as a starting point does not contradict the earlier statement. In fact, the use of the sentence illustrates the
difference between analyzing words and analyzing concepts. In the sentence, each word has a single, clear meaning.
However, each word in the sentence has many possible meanings when the word is considered apart from the meaning
conveyed in the sentence. Look could be a noun or a verb. It could refer to the appearance of something (“The new
look”). Similarly, the word red apart from the sentence could refer to color or anger (“I saw red”). The sentence
conveys the various meanings that we are to deal with. It also presents conventional labels for each meaning. The
meanings that we will consider are those conveyed by the sentence. The words that we will use to signal each meaning
are those contained in the sentence.

The smallest conceptual units signaled in the sentence are basic-form concepts. The concept look is a basic-form
concept. So are the concepts red, block, under, and table. We cannot reduce these concepts or break them into
components. If the learner does not know the concept under, we gain nothing by telling the learner, “It’s under
because it is below,” or “When it’s beneath, we say it’s under.” We would still need to teach the meaning of the new
word that we introduce in the explanation—beneath or below. Since these words have the same communication
function in the sentence as under, we would create an unnecessary step by introducing the synonym. The most direct
and precise communication would be one that teaches the meaning of under that is used in the sentence. To convey
this meaning, we would present different examples that show an object in different positions with respect to another
object. We would label some examples “under” and some “not-under.” Then, we would require the learner to respond
to additional examples, answering this question for each example: “Is it under?”

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 Variations of the same procedure are used for all basic-form concepts. The basic form concept under is a non-
comparative, which means that a given static example that shows something under is always an example of under. The
concepts of getting steeper and getting faster are different. They are comparatives. An example of a particular grade
is not always an example of steeper. It is steeper only when compared to an example that is less steep.

The final type of basic-form concept is a noun concept. Nouns are the meanings referred to by words like shoe,
building, magazine, chalk, and other noun words. Nouns are basic forms that are structurally different from both
comparatives and non-comparatives with respect to the number of dimensions that can be operated on to change a
positive example of the concept into a negative. For non-comparatives (like under), only a single change in a positive
example will make it a negative example. (For under, this change involves the spatial relationship between the two
objects in the example.) Similarly, for comparatives (like getting steeper), changes in only one dimension will convert
positive examples into negatives (for the concept of getting steeper, the change is the slope of the object). Both
comparatives and non-comparatives are single-dimension concepts. Nouns are multiple-dimension concepts, which
means that it is possible to change a positive example of a noun into a negative example by manipulating many
dimensions of the object. (We can change a jacket into a non-jacket by changing the material, length, shape, by
removing parts, and by adding parts.) The structural uniqueness of nouns suggests some unique features of
communications that convey nouns faultlessly. However, nouns, like comparatives and non-comparatives, are simple,
irreducible concepts that are based on the sensory features of objects or examples.

Joining Forms

Non-comparatives, comparatives, and nouns are signaled by a single label or by a group of words that functions as
a single label. Joining forms, in contrast, involve relationships between basic-form concepts. Therefore, joining forms
involve two independent labels that are related in some way. The joining forms are the simplest ways that logically
unrelated concepts such as under and table may be combined. We can illustrate the relationship between basic forms
and joining forms by referring to the concept under. After the learner has mastered the discrimination of under—not-
under (basic form discrimination), we may combine under with other concepts that are logically unrelated to under.

One way to create a link between under and another concept is through a transformation. A transformation is
systematic ordering of examples and a parallel ordering of symbols used to describe the examples. If the learner knows
the positional meaning of under and understands the meaning of basic noun labels, such as table, shoe, etc., and we
could combine these concepts with under using a transformation sequence. We would present different examples of an
object in different positions. The learner, however, would not simply indicate whether the object is “under,” or “not-
under.” Instead, the learner would produce a unique verbal response for each test example by telling where the object
is. The learner’s responses to different examples would be: “Under the table . . . under the shoe . . . under the bed . . .
under the shelf . . . under the book . . .” If the sequence used to communicate the joining that occurs when under is
linked with these objects is faultless, the learner should be able to generalize to a new example, one that had not been
presented during instruction. For instance, if the learner is able to identify “pencil,” the learner should be able to
produce the new, appropriate response, “Under the pencil,” when presented with an example that shows the target
object under the pencil. Note that this response is one the learner had never produced before. The fact that the learner
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produces it implies that the learner has learned the transformation procedure for how to express the basic way that
under is joined with other basic-form concepts.

Another way that basic-form concepts are joined together is the correlated-feature joining. This joining is not
based on a system of ordered responses that change according to a transformation rule, but rather on empirical
associations. If two things happen together, they are correlated. However, the two things involved in this correlation
are logically unrelated, which means that the learner could exhibit complete basic-form understanding of both things
involved in the correlation and yet not know the correlation. The following sentence expresses a correlated-feature
joining involving under: “If it is under, it is below.” The learner could be proficient at identifying all possible
examples that require the learner to label the example as “under” or “not-under” without understanding that each
example has another label—below. The relationship between the words is based on an empirical fact—by convention,
examples of under have another label, below.

To teach the relationship between under and below, we would use the same sequence of examples used to
communicate under. The reason is that under and below vary together. If something is labeled under, it is always
labeled below. If the object is non-under, it is not-below. Therefore, the same set of examples and sequence of
examples used to communicate under would effectively show what controls the label of below. To assure that the
relationship between under and below is made explicit, however, we would not ask the same question that we use in
the sequence for under (“Is it under?”). Instead, we would present a pair of different questions to show the relationship.
The first requires the use of the new label (below). The second requires the learner to express the correlation between
under and below.

Figure 3.1 shows the first three examples from a possible faultless sequence.

Since all correlated-feature relationships are the same (based on an empirical relationship between the basic-form
concept that is known to the learner and the concept that is correlated with it), we can use the same communication

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procedures to faultlessly convey any correlated-feature concept. Below is the first part of a correlated feature concept
that involves steeper grade. Note that the sequence of examples is the same as that used to communicate steeper
grade. However, the questions are changed to assure that the learner learns the relationship between the grade and the
speed of the object.

Figure 3.2
Example Teacher Wording

Watch my hand. It shows how steep the grade of the stream is. I’ll tell you if the
water moves faster down the grade.

Did it move faster? No. How do I know? Because the grade did not get steeper.
1

Did it move faster? No. How do I know? Because the grade did not get steeper.
2

Did it move faster? Yes. How do I know? Because the grade got steeper.
3

Whether the joining form involves a transformation or correlated features, the concept is not a basic form concept.
It is a relationship between basic-form concepts, and the relationship cannot be reduced to simpler concepts. The
relationship is irreducible because knowledge of any basic form concept does not imply its relationship to other
concepts. Knowledge of under does not imply the system of ordered responses used to refer to “under the table . . .
under the chair . . .” etc. Knowledge of under does not imply knowledge of the word below. Knowledge of steeper
grade does not imply knowledge of moving faster. To connect these logically unrelated concepts, we use
communication forms that show how the basic concept that is familiar to the learner is related either to the system of
ordered responses or to some empirically associated concept.

Complex Forms

There are two types of complex forms—cognitive problem-solving routines and communications about events (fact
systems). Both cognitive problem-solving routines and communications about events require the learner to attend to
various details present in the example. The various details require multiple responses from the learner—each dealing
with different aspects of the problem or event. Therefore, complex forms are distinguished from simpler forms by the
multiple responses involved in processing each example.

If complex forms are to be unambiguously introduced to the learner, they should process concepts through a series
of verbal instructions or directions. The directions tell the learner what to do, what to attend to, or how to label some
features of the examples. Since any series of verbal instructions is composed entirely of basic-form concepts or joining-
form concepts, the complex forms are of a higher order than either joining forms or basic forms.

Cognitive Problem-Solving Routines

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 The juxtaposition pattern for the complex forms is different from that of basic or joining forms. A sequence for a
basic form presents juxtaposed examples that involve the same response dimension. The sequence strongly prompts the
learner to attend to this dimension, because attention is never drawn from the single response dimension. With slight
exceptions, the joining forms work in the same way. The juxtaposed examples deal with the same relationship and there
is very little interruption between presentations of examples involving the response dimension being taught. (An
exception is the correlated-feature sequence, which presents two questions with each example. However, both
questions deal with different facts of the same relationship: “Would it move faster? . . “How do you know?” . . .)

With complex forms, the same response dimension is not referred to in juxtaposed tasks or questions. Figure 3.3
shows a cognitive problem-solving routine for working problems of the form: 5–3=☐.

Figure 3.3
Teacher Learner
5–3=

1. Read it. Five minus three equals

how many?
2. (Touches under 5.)
What does this tell us? Start with five.
3. Do it. (Learner makes 5 lines:
5–3= )

4. (Points to –3.)
What does this tell us? Minus three.
5. Do it. (Learner crosses out 3 lines:
5–3= )

6. (Points to equal sign.)

What does this tell us? We must have the same
number on both sides.

7. What number is that? Two.

8. Make the sides equal. (Learner makes lines and

writes answer:)
5–3=2

9. Read the problem and

the answer. Five minus three equals
two.

Notice that the entire routine deals with one subtraction problem (one example). Obviously, variations of the
routine could be used for any problem within the same class as 5–3=☐; however, each example would be processed
through the same nine steps. And no two steps deal with the same relationship. In step 2, the learner responds that the 5
tells us to “start with five.” If the number were different, the response would be different. Therefore, the first step

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would have been pretaught as a transformation, with the teacher presenting different beginning numbers and asking
about each, “What does this tell us?” (The responses are ordered according to ordered changes in the examples;
therefore, a transformation is involved.)

This transformation is dealt with in only one step of the routine, however. Step 3 tells the learner to start with 5 and
the learner makes five lines. A different transformation is involved in this step (one that would also be pretaught). Step
4 involves another transformation, in which the learner’s responses vary as the symbols in the problem vary. Note that
each step deals with a new feature or step in the solution to the problem. Note also that each step is composed entirely
of basic concepts or joining-form concepts.

Cognitive problem-solving routines are appropriate for any task that may be treated as a series of steps that lead to
a solution. The judgment of whether a routine is possible depends on whether the learner is logically required to
process a series of concepts, details, or discriminations to arrive at the appropriate solution. An assumption is that if the
logical analysis of the operation under consideration discloses that the learner must attend to a variety of
discriminations, a cognitive routine is more appropriate than any other form for communicating the structure.

For example, simple word decoding for the learner who is assumed to be naive, logically implies attention to the
different letters in the word and to their order. If the learner does not attend to the m in mat, the learner logically may
confuse mat with hat, cat or at. If the learner does not attend to the a, the learner may confuse the word with met. If
the learner does not attend to the t, the learner may confuse mat with mad or map. This analysis suggests that we
should design a routine that deals with all the various discriminations or concepts. This routine should permit the
learner to produce strong behavioral signals that leave little doubt about whether the learner is appropriately processing
the various discriminations.

Figure 3.4 shows a possible routine for teaching initial word-reading.

Figure 3.4
Teacher Learner

mat
(Touches ball of arrow.)
When I touch under each
sound, say that sound.
Keep saying it until I
touch the next sound.
Touches under m, a, t. Says “mmmmaaat”
without pausing between
sounds.
Say it fast. “Mat.”

The routine assures that the sounds are produced, processed in order, and then transformed (through “say it fast”)
into the word spoken at a normal speaking rate.

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 We observed earlier that any complex cognitive operation may be treated either as a joining form or a basic-form
concept, we would simply present examples of words and give the learner instructions to say the words that we show
(“look and say”). The cognitive routine is logically superior to procedures that leave the steps of the operation covert
because the routine reduces the possibilities for misgeneralization. When many steps are involved in a solution, and the
steps are not explicit, the learner may learn spurious strategies that work in the initial-teaching situation but that will
not work later. If the initial set of words to be read contains only one three-letter word that begins with m (mat), the
learner who is taught to decode mat as if it were a basic-form concept may process the example by attending to the
length of the word and the beginning letter. Although this strategy will permit the learner to discriminate between mat
and the other words introduced early in the sequence, the learner will encounter serious problems when mad and man
are introduced, because the learner’s strategy will lead to these words being identified as “mat.”

The highly overt procedure provided by the cognitive routine is superior to the procedure that leaves the steps
covert because of the nature of all cognitive operations. These operations are quite different from physical operations.
Yet many tacitly stated analogies about cognitive operations proposed by some cognitive psychologists are based on
assumed parallels between physical operations and cognitive ones. The two primary differences between physical and
cognitive operations are:

1. The physical environment provides continuous and usually unambiguous feedback to the learner who is
trying to learn physical operations, but does not respond to the learning attempts for cognitive operations.

2. There is no necessary overt behavior associated with any cognitive operation.

The physical environment provides feedback to the learner for all applications of physical operations. Physical
operations include fitting jigsaw puzzles together, throwing a ball, “nesting” cups together, swimming, buttoning a
coat. When the learner performs any physical operation, the physical environment provides feedback. This feedback
takes the form of contingencies that occur if the operation is not being performed correctly. The physical environment,
when viewed as an active agent, either prevents the learner from continuing or provides some unpleasant consequences
for the inappropriate action. If the learner is not performing the operation of buttoning a coat properly, the physical
environment “prevents” the coat from being buttoned. If the learner is not nesting a series of bowls correctly, the
physical environment “prevents” the nesting from occurring. If the learner does not carry out the operation of hopping
correctly, the physical environment “interferes” when the operation is not being performed correctly. The “responses”
from the physical environment (the negative consequences or the prevention of the operation from continuing) have a
precise communication value. They indicate that some behavior must change if the task is to be completed.

For any physical operation, we can state the behaviors that account for the completion of the operation. Also, we
can completely account for any outcome by referring to the overt behaviors produced by the learner. The learner cannot
open the door without producing certain overt behaviors. Furthermore, if the door has been opened by the learner, we
can account for every aspect of the outcome by referring to the different overt, observable things the learner did.

For any cognitive operation, there are no necessary overt behaviors to account for the outcome that is achieved.
The practiced learner does not have to “write out” formulas to solve complex problems. The learner may solve them

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covertly. Cognitive operations do not exist in the sense that physical operations exist. We cannot account for the “silent
reading” of a practiced learner by referring only to the overt behaviors that the learner produces. Clearly, if we are to
observe the behaviors that lead to the outcome of a cognitive operation, we must design the steps so they are overt and
so the outcome is accounted for by these overt steps.

The physical environment does not provide feedback when the learner is engaged in cognitive operations. If the
learner misreads a word, the physical environment does nothing. It does not prevent the learner from saying the wrong
word. It does not produce an unpleasant consequence. The learner could look at the word form and call it “Yesterday”
without receiving any response from the physical environment.

The basic properties of cognitive operations—from long division to inferential reading—suggest both that the
naive learner cannot consistently benefit from unguided practice or from unguided discovery of cognitive operations.
Unless the learner is provided with some logical basis for figuring out possible inconsistencies (which is usually not
available to the naive learner), practicing the skill without human feedback is likely to promote mistakes.

To build adequate communications, we design operations or routines that do what the physical operations do. The
test of a routine’s adequacy is this: Can any observed outcome be totally explained in terms of the overt behaviors the
learner produces? If the answer is “Yes,” the cognitive routine is designed so that adequate feedback is possible. To
design the routine in this way, however, we must convert thinking into doing.

Although cognitive routines are composed entirely of basic-form and joining-form concepts, the routine is not a
good vehicle for teaching these concepts. The reason has to do with the pattern of juxtapositions that occurs in a
routine. Of ten only one step of the routine presents a particular discrimination. If the routine has five steps, a great deal
of interference occurs before the learner receives a second example of the concept in a particular step. The learner does
not receive massed practice on the critical concept (because the routine does not present juxtaposed examples that deal
with this concept). Therefore, the learner’s memory requirements are increased enormously and the total amount of
time needed to teach the concept is increased. Components, concepts, and skills should be pretaught before bringing
them together in the routine.

A final point about cognitive problem-solving routines: they should be designed so they apply to the widest
possible range of examples. Cognitive routines are inventions. Two opposing considerations are involved in the
inventing process. The first is the need to make the learner’s processing steps overt. The opposing consideration is the
generalizability of the routine. If each routine applies to only a very small class of problems, many different routines
would be required to teach the learner how to process the entire range of problems encountered. This situation is
ineffective because: (1) each routine involves preteaching (and with many routines, a great deal of preteaching is
involved); and (2) if the learner must select from a variety of routines, additional discrimination training is required.

Figure 3.5 shows an example of a generalizable routine. The teacher wording is the same wording presented in the
routine for processing 5–3=☐. The example, however, is a negative-number problem.

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 Figure 3.5
Teacher Learner
3–5=

1. Read it. Three minus five equals

how many?
2. (Touches under 3.)
What does this tell us? Start with three.
3. Do it. (Learner makes 3 lines:
3–5= )

4. (Points to –5.)
What does this tell us? Minus five.

5. Do it. (Learner crosses out 5

lines: 3–5= )

6. (Points to equal sign.)

What does this tell us? We must have the same
number on both sides.
7. What number is that? Minus two.
8. Make the sides equal. (Learner makes lines and
writes answer:)
3–5= –2

9. Read the problem and

the answer. Three minus five equals
minus two.

The routine exhibits the basic properties of a well-designed routine.

1. Details that appear in one problem are treated in the same way when they appear in another problem (such
as making lines for each positive counter and slashes for each negative counter).

2. The approach is maximally overtized, which means that the routine requires the learner to “show” exactly
which steps the learner takes in solving the problem. Also, the learner must respond to the various details
that are logically necessary to the solution.

3. Because the learner responds overtly to every detail that is logically necessary to solve the problem, the
operation has the same feedback potential as a physical operation. The outcome is totally explained in
terms of the overt behaviors. We know what the learner is “thinking” and we can respond to the overt
steps that are functionally necessary for performing a physical operation. This feedback must be provided
by humans or machines.

Fading. Successful teaching of cognitive problem-solving routines involves “fading out” the overt steps and
“covertizing” the operation so that the learner performs independently. This feature further distinguishes cognitive

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routines from physical operations. Fading of steps is never necessary for physical operations because such steps are
always overt.

The judgments about how quickly the routine should be covertized is made by considering two competing facts:
(1) the learner’s proficiency, and (2) the problem of stipulation. We should not covertize the routine until the learner is
reasonably proficient with overtized routines. If our communication is to be consistent with a single interpretation, we
must make sure that the learner is performing the appropriate steps. We will have this assurance only if we know that
the learner performs appropriately when the steps of a routine are overt. However, the longer we work on the problem
overtly, a step at a time, the more reliant we make the learner on the teacher direction. The covertizing process must
proceed as quickly as it reasonably can (to reduce the stipulation problem). However, the process must not begin until
the learner is proficient with the overt routine.

Communications About Events

These complex communications are like cognitive problem-solving routines in several ways. The communications
are composed of steps that guide the learner. Also, juxtaposed steps deal with different response dimensions or features
of the example. The primary difference between cognitive problem-solving routines and communications about events
is that communications about events deal with learning about a new “whole” by learning about unique relationships of
the different parts that make up this whole. The whole may be an object, such as a particular refrigerator. However, the
goal is not to use that refrigerator as an example of some concept that is common to many other refrigerators. Instead,
the goal is to attend to the features that make the particular refrigerator unique. To appreciate the uniqueness, the
learner must attend to the sum of details that distinguish it from other refrigerators. Each distinguishing feature is
expressed as a fact. “It has a scratch on this side . . . it is yellow . . . it has two handles . . .” etc.

There are many applications of communications about events. The primary one is the “expansion” of basic and
joining form concepts that have been taught. Once a concept such as yellow has been taught through a basic-form
sequence, the initial teaching has been accomplished. However, the use of the concept has not been established. To
demonstrate the use, the concept now becomes a step in various communications about events. Let’s say that the
learner has recently been taught numerals, their relationship to counting, colors and the comparative concept of bigger.
For an expansion activity, the teacher writes 2, 7, and 5 on the chalkboard. The numerals 2 and 7 are white; 5 is yellow.
5 is also written bigger than the other two numerals.

Teacher points to 5. “What numeral is this? . . . What number does it tell you to count to? . . . Let’s hear you count
to five . . . .What color is this numeral? . . . Yes, it’s yellow. Tell me if it’s bigger than the other numbers or not bigger .
. . .Yes, it’s bigger.”

Unlike the steps in a cognitive routine, the steps that the teacher presents for communication about an event do not
occur in a particular order, because the objective is not to convey a procedure for solving a problem (which requires a
particular ordering of steps), but to deal with the features that make the 5 unique. These features may be presented in
any order. In the case of the expansion activity above, the communication also plays an important role in demonstrating
that the various concepts that have been taught are useful components in formulating a precise understanding of what

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makes the five unique. Knowledge of each component concept adds to the learner’s ability to express details of the 5’s
uniqueness.

One of the more sophisticated types of communications about events involves a symbolic event that is created to
teach a system of facts. Figure 3.6 shows a display that shows how factories work. The display functions as the event.
The features that distinguish this display or event from others have to do with the unique spatial arrangement of details
and the specific words that appear on the display. The display functions like a super outline that shows higher-order and
lower-order relationships (the relationship between raw material is changed into product and the more specific
instances that derive from this rule: “Cotton is changed into cloth,” etc.)

Figure 3.6

If the learner memorizes the words that appear in different parts of the display, the learner will be provided with an
outline of how factories operate. Therefore, the goal of the communication is to teach the learner to memorize the
wording in the various cells.

To achieve this goal, the communication first rehearses the learner on the wording in the various cells, then tests
the learner with a display that is the same as the original, except that the cells are empty. The learner has to indicate the
words that go in each cell.

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 Table 3.1 shows the communication for the teaching of the display followed by testing with the empty-cell display
(Figure 3.7).

Table 3.1
Teacher Teacher Student Teacher Teacher Student
Does Says and Does Says Does Says and Does Says
1. Present This chart shows what a “Factory.” 9. Touch F. (Follow arrow and touch each “Crude oil
Figure 3.6 factory does. Factory. Say it. space as you say) Crude oil is is changed
Touch A. (Signal.) A factory takes (touch changed to gasoline. to gaso-
B) raw material and changes it Say it. (Signal.) (Follow arrow line.”
into (touch C) a product. and touch each space as children
Remember: (follow arrow from respond.)
B to A.) Raw material goes Some factories take crude oil
into a factory. (Follow arrow and boil it and put things in
from A to C.) A product it. When they are done, they
comes out of the factory. have changed the crude oil to
gasoline. Crude oil also makes
2. Touch B. First I’ll tell you about some “Raw plastic and oil. But the big
raw materials. Raw material. material.” product is gasoline.
Say it. (Signal.) Remember: crude oil is
Touch iron, Remember: Raw material changed to gasoline.
cotton, crude goes into a factory.
oil. 10. Touch A. A factory needs things if it is
to change raw material into
3. Touch iron. Iron. Say it. (Signal.) “Iron.” products. Here are the things
Iron is a raw material. Iron it needs.
comes from iron mines. Touch G. A factory needs transpor-
4. Touch cotton. Cotton. Say it. (Signal.) “Cotton.” Touch H. tation.
Cotton is a raw material. It Touch I. It needs power.
grows on plants. Touch J. It needs labor, and
It needs markets.
5. Touch crude Crude oil. Say it. (Signal.) “Crude
oil. 11. Touch G. A factory needs transporta- “Needs
Crude oil is a raw material. oil.”
tion. Needs transportation. Say transporta-
It comes from under the
it. (Signal.) tion.”
ground.
Transportation is what we do
6. A factory takes a raw material “Product.” when we move something
and changes it into a product. from one place to another. We
Product. Say it. (Signal.) need transportation to bring
Touch steel, These are products. the raw material to the factory.
cloth, Remember: A product comes We need transportation to
gasoline. out of a factory. move the product from the
7. Touch D. (Follow arrow and touch each “Iron is factory to some place where
space as you say) Iron is changed to we can see it.
changed to steel. Say it. steel.” Touch K. Here are the big types of “Waterway,
(Signal.) (Follow arrow and transportation used by facto- railway,
touch each space as children ries: waterway, railway, high- highway.”
respond.) way. Waterway, railway,
Steel is used to make many highway. Say it. (Signal.)
things—cars, lawnmowers, (Touch boat.) A waterway is
cans, and hundreds of other what boats use. A waterway is
products. Remember: iron is a river or a lake or an ocean.
changed to steel. (Touch train.) A railway is what
trains use.
8. Touch E. (Follow arrow and touch each “Cotton is (Touch truck.) A highway is
space as you say) Cotton is changed to what trucks use. Remember:
changed to cloth. Say it. cloth.” waterway for boats, railway
(Signal.) (Follow arrow and for trains, highway for trucks.
touch each space as children
respond.) 12. Touch H. A factory needs power. Needs “Needs
Some factories take the raw power. Say it. (Signal.) power.”
material cotton and change it
into cotton cloth. Some shirts
and blue jeans and sheets are
made of cotton cloth.
Remember: cotton is
changed to cloth.

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 Table 3.1
Teacher Teacher Student Teacher Teacher Student
Does Says and Does Says Does Says and Does Says
Some factories use steam 15. Touch G. Let’s go over the needs of a “Needs
power. They heat water with factory one more time. Say it. transporta-
coal until the water turns into (Signal.) tion.”
steam. Then the steam runs
the machinery in the factory. Touch K. What kinds of transportation “Waterway,
Some factories use electric are there? (Signal.) railway,
power. The electricity runs the highway.”
machinery. Touch H. Say it. (Signal.) “Needs
Remember: (touch A) a factory power.”
(touch G) needs transportation
and (touch H) needs power. Touch I. Say it. (Signal.) “Needs
labor.”
13. Touch I. A factory needs labor. Needs “Needs
Touch J. Say it. (Signal.) “Needs
labor. Say it. (Signal.) labor.”
markets.”
Labor is another name for
the workers who operate the 16. Get ready to tell me all the
machines in a factory. A lot of facts about a factory.
work is done by the machines,
but some products need a lot 17. Touch the (For each space, say) Say it. (Children
of labor. Remember: (touch A) spaces in this (Signal.) respond.)
a factory (touch G) needs order: (A,B, (Repeat until firm.)
transportation and (touch H) C,D,E,F,G-K,
needs power (touch I) and H,I,J).
needs labor.
18. Remove
14. Touch J. The last thing a factory needs is “Needs
Figure 3.6.
a market. Needs market. Say market.”
it. (Signal.) 19. Present The spaces are empty on this
A market is a place where the Figure 3.7. chart. Get ready to tell me the
product is sold. The big mar- exact words that go in each
kets are the big cities. The big space.
cities are big markets because
there are a lot of people in the 20.Touch the (For each space, say) Say it. (Children
big cities. Remember: needs spaces in this (Signal.) respond.)
markets. order: (A,B,
C,D,E,F,G-K,
H,I,J).

Although the classification system of cognitive skills is exhaustive and is capable of generating specific
information about how to communicate any concept within a given category, the specificity of the teaching information
is less for the complex forms than it is for either the basic forms or the joining forms. There are two reasons for this
reduced specificity.

1. The procedure or event may be approached in different ways (each way leading to a different arrangement
of steps or facts).

2. Different wording is possible for each step or fact and for each response the learner is to produce.

Even with the less specific guidelines for complex forms, the classification system is capable of providing
important information about the structure of any concept, and therefore carries important implications about how to
teach the concept.

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 Figure 3.7

Response-Locus Classification

The primary analysis for all cognitive operations is the stimulus-locus analysis. The development of sequences and
routines does not place heavy emphasis on teaching the learner to produce new responses. The primary domain for new
responses is the physical world. However, once a routine is designed, it functions a great deal like a physical operation.
The learner is presented with specific stimuli that call for specific, overt responses. The learner may not be able to
produce the responses acceptably. For instance, the learner may be completely incapable of saying the word “four” so
that it is impossible to distinguish the response from the learner’s response for “five.” At this point, we enter the
domain of response-locus analysis. We must modify the learner so that the learner is capable of producing the new
response. Similar problems occur if the learner is not able to make lines for numerals, or even if the learner tends to
forget the label for the numeral 5.

The response-locus analysis is also appropriate for teaching any simple physical response or chain of responses
used for a complex physical operation. The learner may not be able to turn a door knob. Although the physical
environment will assist in the teaching of this operation (by providing the learner with feedback on every trial), we can
simplify the operation, streamline it, and provide the learner with prompts about how to approach the task.

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 Classifications for Response-Locus Communications

Simple Responses

• Response-context shaping

• Response-form shaping

Complex physical problem-solving operations

Simple Responses

Context and form. There are two primary types of response-locus instruction.

The first is context instruction. This type of communication is appropriate if the learner can produce the desired
response in some contexts, but not in the context of the operation being taught. For instance, the learner can assume a
“tuck” position while lying on the gym mat, but not in the response context of doing a back somersault from the diving
board. The context of the response must therefore be shaped or changed so the learner learns to perform in the desired
context. A different type of context-shaping problem is one in which the learner can say a word, such as stegosaurus,
within the context of the task, “Say stegosaurus,” but not within the context of the situation in which the learner is
asked, “What’s the name of this dinosaur?” For whatever reason, the learner is unable to remember the name.

The second type of response-locus instruction has to do with form. The learner cannot produce the response in any
context. The learner is told to “Say stegosaurus,” and the learner responds with something like, “Dedgustus.” When we
try other contexts, we can find no behavioral context in which the learner produces the desired response.

Shaping. The analysis of the learner discloses that practice is the primary variable for learning new contexts for
responses or learning responses of a new form. The technique used to provide effective practice is shaping. There are
two generically different ways a task may be made easier for the learner. The context of the task can be made relatively
easier and then progressively modified to shape behavior in the new context. (The context for the task is easier if the
task immediately follows a successful trial on the same task.) Also, the criterion for an acceptable response can also be
made progressively more demanding. Instead of requiring the learner to produce a response of a certain configuration,
we initially permit the learner to produce an approximation—any response that falls within a broader range of variation.

In summary, if the learner can produce the response called for by the task but not within the specified task context,
the shaping focuses on context changes. If the learner can produce only an approximation, the shaping focuses on the
response form.

Response-locus communications are unlike stimulus-locus communications in that they do not involve a particular
number of trials. The reason for this difference becomes apparent if we consider what these two analyses deal with. The
stimulus-locus analysis deals with concepts and determines the type of information that is needed to communicate how
the concept works—which changes affect it and which do not. The response-locus analysis is an analysis of the learner
and how the learner learns. Without possessing a great deal of information about the particular learner for whom the

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analysis is being used, we don’t know how many trials are required to bring the learner to an acceptable criterion.
Instead of introducing a specific sequence, therefore, the response-locus communication involves general rules about
how to change the example or tasks when the learner performs unacceptably on initial tasks.

Context shaping. The context-shaping procedure sequences contexts of varying difficulty. The easiest context is
one in which an example of the task is juxtaposed to another example of that task. The most difficult context is one in
which the task is temporarily removed from another example of that task. For the sake of simplicity, context-shaping
may be conceived of as involving three levels of context difficulty:

Level 1. A A

Level 2. A B A

Level 3. A B C D. . . A

The circled A in each context is the target task. Level 1 shows the task immediately preceded by a successful
presentation of the same task. Level 2 shows the task preceded by the task and by some interference. B is a familiar
task that is not similar to A. Level 3 shows more interference in the form of familiar tasks B, C, D, and possibly other
familiar tasks not highly similar to A. The memory requirements for A on level 3 are far more difficult than those
required for levels 1 and 2.

The teacher stays on a particular level until the learner is able to perform correctly on perhaps four consecutive
trials. The teacher then moves to the next level. The goal is consistent performance on level 3. Throughout the training
the learner will most probably receive reinforcement on at least 70 percent of the trials. Corrections, whether they occur
within the current task training program or later after the skill is supposed to have been learned, involve returning to
step 1 and quickly going through the various levels before repeating the task in the con text in which the mistake
occurred.

Form shaping. The procedures for shifting the response criterion in form shaping involve transitions similar to
those indicated for context shaping. By observing the learner on a number of trials, we can determine the learner’s
baseline of performance. We establish a criterion that permits the learner to receive reinforcement on at least 70 percent
of the trials if the learner performs no better than on baseline. When the learner’s performance improves to perhaps 85
percent, the criterion is changed. The cycle is then repeated. An effective variation is to have standards for single
reinforcement and for double reinforcement. The requirement for “doubles” is higher than for “singles.” The higher
requirement for double reinforcement gives the learner more immediate information about the direction in which the
shaping will proceed. (The difference between responses that receive singles and those that receive doubles suggests
the desired direction in which the responses are to be modified.)

Complex Physical Operations

Complex physical operations, such as swimming, hitting a baseball, throwing a baseball, soldering, and dialing a
number on the telephone, have some of the same features as complex routines designed to teach cognitive problem-
solving behavior. Both the physical operation and the routine involve discrete steps. Both involve creating a desired

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outcome by performing in a certain way. Both are composed of components that can be removed from the context of
the complex behavior.

The samenesses suggest that we can use some of the same techniques designed for communicating cognitive
operations when we teach physical operations; however, the approach must be modified to accommodate shaping, our
primary technique for inducing the desired behavior,

Features of complex operations. All complex physical operations have component behaviors. A component
behavior is one that retains the same form it has in the complex operation when it occurs in other contexts. (In other
words, the same component can be identified in other tasks.)

There are two types of components—essential behaviors and enablers (or non-essential components). Essential
behaviors are those that account for the outcome of the task. Enablers are behaviors that must be performed if the
essential behaviors are to be performed. The essential behavior for the operation of brushing teeth is moving the brush
while it is in contact with the teeth. One enabling behavior is holding the toothbrush. (If the learner does not hold the
toothbrush, the direct behavior of bringing it into contact with the teeth cannot be achieved.) Note that the essential
behavior occurs at the same time as the enabler. This situation is common to physical operations. The learning of
physical operations, therefore, implies teaching not only the component behaviors, but also the coordination of these.
Conversely, the operation would be analytically easier if it required less coordination because it would require less
learning.

There are three basic strategies for beginning the instruction in a way that requires less coordination. They are:

1. The essential-response-feature approach.

2. The non-essential-response-feature approach.

3. The removed-component approach.

The essential-response-feature approach begins with an operation that has been simplified by eliminating some of
the enabler-response components. The learner produces the essential-response components and thereby produces the
behaviors responsible for achieving the outcome of the operation. (In the toothbrushing example, the learner would do
the brushing; however, the toothbrush is rigidly attached to a glove, which means that the learner does not have to grasp
it. The enabling component—grasping—is eliminated so that the learner performs on a simplified version of the
operations. Note that the learner does the actual brushing.)

For the first step in the non-essential-response-feature approach, the learner produces the enabling or non-essential
features of the operation, but is not responsible for the essential response feature. (The learner would put the toothpaste
on the toothbrush, and possibly hold the toothbrush; however. the act of brushing would be performed by somebody
else.) The learner would perform only the components that accompany the essential behavior.

The third approach, the removed-component approach, begins with a particular component removed from the
context of the operation in which it is to occur. The instruction may begin by requiring the learner to hold an object like

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a pencil horizontally and then move it up and down while maintaining the horizontal orientation.

In practicing this behavior with the wrist turned in different positions, the learner practices a component of the
brushing operation. The context is simplified because it does not require maintaining contact with teeth or turning the
brush so the bristles are oriented properly, etc.

The three different strategies dictate different first steps of instruction. Shaping is used to achieve the desired
performance. After the first step, the learner is given increasing responsibility for the total response until the learner is
performing the operation with no prompts.

Each approach has the basic problem of distortion. Distortion is the counterpart of stipulation. It comes about
because the learner is permitted to perform in a way that will not transfer to new examples or applications of the
operation. The first step of the essential-response-feature program provides the learner with a very easy example. If the
learner works too long on this presentation, the learner may learn to perform the response successfully, but not exactly
the way it should be performed when the non-essential components are included in the operation. Although it is
possible for the learner to perform on this example in a manner that is perfectly continuous with all examples of the
operation (including the very difficult ones), the learner will probably perform in a way that is not perfectly continuous.
The more response latitude the easy example permits, the greater the probability that serious distortion will be evident
when new more difficult applications are presented.

Consider the problem of teaching the learner to button. If we begin with a very large button, we are providing the
learner with a great deal of response latitude. The probability is therefore great that the learner will learn behaviors that
achieve the goal of the operation, but that will have to be modified when more difficult examples are introduced. (The
learner may grab the flat sides of the button between the thumb and index finger, a behavior that cannot readily be
performed with smaller buttons.) The same sort of distortion is implied when the learner performs the non-essential
response features of the operation. Distortion is observed when the components are integrated or when more difficult
examples are introduced. Some distortion is inevitable because the simplified operation is just that—simplified.
Therefore, component responses that are produced in this context are not under the constraints they will be under when
they are integrated with other component responses.

To reduce the amount of distortion resulting from the initial practice, we may alternate from “easy” to “hard”
examples. For instance, part of the time the learner works on a removed component; part of the time on the entire
operation with no prompting. When the learner works on the entire operation, heavy use of reinforcement is used for
shaping specific features of the operation. However, the primary value of this work would be to provide the learner
with some sort of “advance-organizer” information about the direction the learning is to take and about the relationship
between the removed component and the entire operation. A variation of this strategy can be used with either the
essential-response-feature approach or the non-essential-response-feature approach. For instance, the learner might use
something like a walker when learning to walk (non-essential-feature approach); however, instead of permitting the
walker to support 100 percent of the learner’s weight, we would use a spring-loaded walker so that the amount of
support it provided varied. During one practice session, the learner might do some walking when 90 percent of the

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body weight is supported by the walker, some when 60 percent of the weight is supported, and some when 45 percent is
supported. The percentages would change as the learner becomes more proficient at each of the initial percentages.

Effective strategies for teaching physical operations provide a great deal of practice. Children do not become
proficient at cursive writing, at typing, or at dribbling a basketball without a great deal of practice. Within the
framework of providing practice, however, the approach must be an effective communication. The complex operation
should not be presented as something that is atomized into a number of pieces that can be put together through a
backward chain. The best programs are the simplest, with the learner performing actual instances of the operation as
quickly as possible.

Communicating physical operations to the learner is different from communicating simple discriminations only in
the sense that if the learner does not respond quickly or well to the initial instruction, the learner needs practice.
However, the basic stimulus-locus principles of communication still hold. If the learner works too long on a particular
set of examples, serious stipulation or distortion will probably occur. If the examples that are practiced show a range of
difference and show that the same basic response can be used for all applications, the communication implies a
generalization. (It shows sameness of response across a range of examples.) To show differences in responses, we
would use basically the same technique that we would use to show differences in examples of a concept. When dealing
with physical operations, however, we must overlay the stimulus-locus procedures with the facts about practice. Even
though we know that we are working on a particular example or application too long, we may be faced with a double-
bind problem. We cannot go on to more difficult examples until the learner performs. However, the learner will not
perform without distortion unless we proceed to other examples. The solution is a compromise. We must make the
tasks easy enough to assure that the learner will succeed. At the same time, we must try to minimize the potential
misgeneralization that occurs if the learner works too long on the easy examples. We reduce misgeneralization by
interspersing some difficult examples early in the program. In the end, the communication that we provide will violate
some principles, either in creating the initial examples or in providing the amount of practice the learner needs. The
principles are violated only when necessary.

Summary

The classifications for cognitive knowledge forms and for response forms are as follows:

Cognitive Knowledge Forms (stimulus-locus analysis)

Basic Forms

Non-comparatives (single-dimension concepts)

Comparatives (single-dimension concepts)

Nouns (multiple-dimension concepts)

Joining Forms

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 Transformations

Correlated feature relationships

Complex Forms

Cognitive problem-solving routines

Communications about events (fact systems)

Physical Operation Forms (response-locus analysis)

Simple Responses

Context shaping

Form shaping

Complex Physical Problem-Solving Operations

The basic forms are the simplest concepts. By considering the various labels that occur in a sentence, we are
provided with a single meaning of each label. The Basic-form communications consist of a series of examples that
focus on the specific meaning that has been determined for a label. The sequence shows which features of the examples
lead to changes in the label and which features do not affect the label. The sequence begins with a series of examples
that are paired with statements about the label, followed by a series of examples that test the learner. Juxtaposed
examples in the sequence deal with the same concept.

Joining forms do not deal with single labels. They involve the relationship between two logically unrelated basic-
form concepts. The two types of simple relationships are a transformation and a correlated-feature relationship. The
transformation sequence shows the learner how changes in the examples lead to ordered changes in the responses,
thereby inducing a generalization that permits the learner to produce new responses to examples that have not been
presented earlier. The goal of the transformation communication is to teach the relationship between the features of the
examples and systematic changes in the responses. The correlated feature sequence shows empirical relationships
between two things that occur together. This relationship is based on empirical fact. If it is an empirical fact that a
steeper grade is correlated with faster movement of an object down the grade, a correlated-feature sequence is implied.
The communication for this relationship presents the same set of examples that would be used to teach steeper grade;
however, the questions are different. The first asks about “faster.” (Did the object go faster?) The second links this
conclusion with the evidence presented in the example. (How do you know?)

Complex forms are characterized by the logical requirements that the learner must attend to various dimensions or
features of the example to understand the concept. The communication makes the learner’s attention to these
dimensions or features explicit or covert. Because the communication deals with sets of relationships rather than a
single relationship (which is what the joining forms communicate), the juxtaposed questions or tasks in the complex
communication do not deal with the same details or features and do not call for responses that deal with only one

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dimension. Therefore, these communications are not well designed to teach basic-form concepts or joining-form
concepts.

The two types of complex communications are cognitive problem-solving routines and communications about
events. The problem-solving routine is a cognitive counterpart of a physical operation—a creation that provides the
learner with the various steps that are logically required to solve any problems of a specific type. (In other words, the
same series of steps would be presented for all problems of a given type.) The routine is designed so that the learner
produces overt responses for the various discriminations or details that logically must be processed if the learner is to
solve the problem. The overt character of the processing assures that the teacher is able to observe the relevant details
of the learner’s processing and therefore is able to provide feedback (in the same way that the physical environment
provides feedback on the learner’s attempts to perform physical operations). The cognitive problem-solving routine is
an initial-teaching communication. After the learner has mastered the overt routine, the steps are “faded” or made
covert so that the learner processes these steps independently.

Communications about events consist of a series of tasks and instructions that are designed to articulate the unique
character of events by pointing out the unique character of the individual features and the relationships that exist among
the features. Unlike cognitive routines, the communications about events do not involve a particular order for
processing the information. Communications about events, therefore, require the learner to learn the parts and
relationships when these are referred to in any order. The visual-spatial display represents a sophisticated event that
functions as a super outline showing the key relationships between various facets of a system of facts.

The response-locus-analysis is an analysis of the learner and therefore involves a classification system quite
different from that for cognitive operations. Since the learner learns new responses and overcomes response
deficiencies slowly, it is not possible for these communications to specify a set number and type of example to teach a
particular skill. Instead, the classification is based on procedures for inducing simple responses and more complex
ones.

The two response-induction procedures for simple responses involve shaping. Context shaping is used if the learner
is capable of producing the desired response in some contexts but not in others. The context is systematically modified,
from the one that the learner can perform into the targeted context. If the learner can answer a question such as,
“What’s your name?” only immediately following the answer to the question, this juxtaposition context is the starting
point—the simplest context. Interruptions between presentations of “What’s your name?” are systematically presented
until the learner is able to perform when a great deal of interference and time is presented between presentations of the
task, “What’s your name?”

Form shaping is different. This technique is used if the learner is unable to produce the desired response in any
context. The learner is able to produce only an approximation of the response. The shaping procedure involves
establishing a criterion for reinforcing the learner so that the learner will be able to receive reinforcement for producing
approximations that are as good as or better than those the learner typically produces. The criterion for reinforcing the
learner shifts as the learner improves until the learner is reinforced for producing responses that are deemed

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appropriate.

The final response-locus category involves communications for complex physical operations (opening a door, tying
a shoe, working a jigsaw puzzle, etc.). These operations involve parts that are to be chained together (in much the same
way that a cognitive problem-solving routine chains parts or steps together to form a solution to the problem). The
communication for inducing these operations is a program.

The key difference between these programs has to do with how they start. For the essential-response-features
program, the first step may require the learner to perform the operation when the operation has been simplified. The
first step for the enabler-response program requires the learner to produce the behaviors that are not essential to the
outcome of the operation. The removed-component program initially presents practice in a context that is different
from that of the operation.
 

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 Section II

Basic Forms
Section I presented the rationale for the stimulus-locus analysis, its relationship to the response-locus analysis, and
an overview of the classification system that is based on common structural features of knowledge and skills. Section I
provided a perspective of the major topics that will be dealt with in the following sections of this book.

Section II and the sections that immediately follow it shift from an emphasis on broad descriptions to a focus on
specific how-to-do-it issues. The goal of these sections is to provide the degree of specificity needed for a person
interested in creating faultless sequences to classify the concept being taught and to generate an acceptable sequence.

Chapter 4 provides a restatement of some of the facts and principles presented in Chapters 1 and 2. These
principles are important because they apply to virtually all the communications that we will discuss. They indicate how
to show samenesses, how to show differences, and how to test to assure that the generalization has been transmitted to
the learner.

Each of the remaining chapters in Section 2 deals with a specific type of basic-form communication. Chapter 5
presents procedures for creating non-comparative sequences; Chapter 6 deals with nouns; and Chapter 7 deals with
comparatives.

The communications that are developed for the basic-form concepts in Chapters 5, 6, and 7 are initial teaching
sequences, designed to show the learner what controls the concept being taught and how the examples of the concept
are to be labeled. The communications do not indicate how to expand the concept and incorporate it in contexts other
than the initial-teaching sequence.

These initial-teaching communications are quite close to being faultless. They are probably more faultless than
many instructional situations require. However, if you understand how to construct these communications, you will
have little trouble creating approximations that are not as intricate or carefully controlled. The value in studying the
more faultless forms is that when the communication must be very precise, you will be able to respond to the situation.
Also, fewer problems are created if the communication errs in the direction of providing too much information rather
than too little.

The forms that we will work with serve as models of faultless presentations. They also serve as a basis for
evaluating communications that are less than faultless.

However, there are alternative ways to present various concepts that may actually be preferable to the forms that
we will develop. For example, if we wished to teach getting wider, our first choice would be a communication based
on the sequence presented in Chapter 7. However, the teacher would have to follow it precisely. The potential of the
communication would be achieved only if the teacher did what the sequence specifies to do. If the teacher did not
appropriately time the presentation of each example and use the wording presented for the example, the communication

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might fail. If the teacher talked and moved the hand at the same time, the learner might not understand the relationship
between the talking and the changes. The teacher must show the change; stop; and then produce the specified wording.
Since these and other behaviors are highly relevant to the transmission of the communication, we must control them.
We can do this by giving the teacher an elaborate list of do’s and don’t’s, by having a master teacher present the
examples or by using the printed page. When we use the printed page, however, our examples must become static
because we cannot readily use continuous conversion. Those details that are controlled so gracefully through
continuous conversion must now be approximated through contrivances that exaggerate samenesses or differences and
that provide for “relatively easy” comparison of examples. In the process, we make many compromises. Necessity
usually dictates that we reduce the number of examples and deviate from the ideal in other ways. Often these deviations
work reasonably well; however, they must be seen as “good solutions,” rather than the ideal.

Although single dimension non-comparatives, nouns, and single dimension comparatives are different in structure,
they comprise the sensory-based concepts, generalizations or discriminations. We will use the terms concept,
discrimination, and generalization interchangeably. A concept is a generalization to the appropriate range of examples.
A generalization is not possible unless a discrimination is involved. (You cannot generalize something unless what you
generalize is specific and different from other possible things that might be generalized.) The basic nature of a concept
is a qualitative irreducible feature that makes the particular concept different from all others. If such qualitative
structure cannot be identified, we can reduce the concept to another, simpler concept (or concepts). We can avoid
possibly confusing discussions about the nature of concepts by referring to sets of examples. If a set of examples has an
observable sameness, that sameness is a concept, the basis for a discrimination or a generalization.

As noted in Chapter 3, communications for sensory-based discriminations share these features:

1. The communication involves presenting concrete examples and labeling each.

2. All positive examples receive the same response.

3. Negative examples receive responses that are different. (The set of negatives may receive a single
response, “No,” or a variety of responses, each different from the response used for positive examples.)

Examples of non-comparative single-dimension concepts include: horizontal, between, over, three, more than
one, gradual, convex, curved.

Examples of nouns include: dog, factory, car, shoe, quotation mark, sentence, adverb, animal.

Examples of comparative single dimension concepts include: steeper, louder, hotter, faster, getting cloudier,
becoming more intense.

The structure of the different concepts is summarized in Table II.1.

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 Table II.1
Single Dimension Nouns Single Dimension
Non-Comparatives Comparatives
Changes in single dimension Changes in many dimensions Changes in single dimension
create positives and negatives. create positives and negatives. create positives and negatives.
Positive and negatives have an Positives and negatives have an ab- Positives and negatives have a
absolute value and a fairly solute value and an imprecise relative value and a precise
precise boundary. boundary. boundary.

Both non-comparatives and comparatives are the same with respect to how positives and negatives are created. For
both types, changes in a single dimension create the examples. For instance, positive examples of the non-comparative
concept over are created by manipulating the position of the target object. By manipulating the position of the object,
negative examples are created. For the comparative concept faster, changes in the speed of the object create positive
examples, changes in the speed also create negatives. Nouns are “conjunctive” concepts, which means that an example
is positive only if a number of features are present. For an object to be a shoe, it must have different parts, dimensions,
relationships. We can change the object into a non-shoe by manipulating any of these dimensions or features.

The precision of the boundaries between positives and negatives varies considerably for the three types of sensory-
based concepts. For single dimension non-comparatives, the boundaries are fairly precise; however, there may be some
ambiguous examples. For instance, when is a stationary object over another object? Clearly, if the target object is in a
“shadow” projected upward from the other object, the target is over. But what if the target is on the edge of this
projection? Precisely when is it called over? The area of uncertainty for the non-comparatives is typically quite limited.

The boundary for comparatives is usually quite precise. Either something is louder, or not louder than another
example of an audible signal. If we present the concept through sensory examples, we must make sure that the
differences are perceptible through sensory observation. However, even if the differences are very small, the example is
not ambiguous because the dividing line between positives and negatives is usually very precise for comparatives.

For nouns, the dividing line is very vague. When a shoe becomes a not-shoe is not known by knowledgeable
adults because a precise dividing line does not exist and must become a matter of personal interpretation.

The three types of sensory-based concepts also differ with respect to the absolute value of the examples. The
question that determines whether the concept is absolute is: Is a positive example of the concept always a positive
example of that concept? For non-comparatives, the answer is, “Yes.” An example of over the table, is always an
example of over the table. Also, an example of a noun, such as dog is always an example of dog. Comparatives,
however, are different. “Is this line longer?”

The answer depends on the line to which the line is compared. The same line can be used as an example of longer
and of not longer.

The structural samenesses and differences of these sensory-based concepts suggest structural samenesses and

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differences of the communications designed to teach them faultlessly. The sequences for teaching nouns must address
the structural problem that nouns do not have a precise boundary between positive and negative examples. The
sequences for teaching comparatives must somehow show that a given outcome (such as the line above) may be
positive or negative.
 

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 Facts and Rules About Communicating Through Examples
When we teach a concept, we must communicate a message to the learner. Our communication must be
unambiguous—consistent with only one interpretation—and complete enough to permit the learner to apply what has
been taught to different situations. To achieve the teaching we present a series of examples that:

1. Show the difference between positives and negatives.

2. Show the range of positive examples.

3. Provide a fairly thorough test of the learner’s understanding of the concept.

Facts About Presenting Examples

Basic cognitive teaching involves presenting the learner with some examples that will induce a generalization to
other examples. There are logical facts about these communications with the learner and there are rules or principles for
achieving different communication goals, such as showing how things are the same or how they are different.

The facts and principles will be illustrated by referring to sensory-based concepts that are shown by presenting
concrete examples and labeling each example.

Fact 1. It is impossible to teach a concept through the presentation of one example.

The teacher presents a positive example of a concept by showing the object below and labeling it: “This is glerm.”

Glerm could refer to any feature of this example or any combination of features. Since the example has an
enormous number of features, glerm could mean any one of an enormous number of things. The learner might select
the right interpretation; however, the learner’s chances are not very good. Glerm could mean “horizontal,” “straight,”
“pointed,” “made of wood,” “writing instrument,” “object with eraser,” “something that floats,” etc.

Fact 2. It is impossible to present a group of positive examples that communicates only one interpretation.

We can limit the number of possible interpretations by presenting positive examples only. This presentation may
strongly imply the desired interpretation; however, a set of positive examples is always capable of generating more than
one possible interpretation. Therefore the sequence of examples must contain negatives as well as positives.

Fact 3. Any sameness shared by both positive and negative examples rules out a possible interpretation.

The behavior that the teacher uses to signal positive examples is different from that used for negatives. Any

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sameness that is observed in both the positive and the negative examples, therefore, cannot account for treating the
negative examples differently.

If we present example A as a positive example of flot and example B as a negative example of flot, any
samenesses in the examples rules out possible interpretations.

A B
“This is a flot.” “This is not-flot.”

The block is the same in both positive and negative example, therefore, flot cannot mean block. The horizontal
orientation is the same in both examples, therefore, flot cannot refer to the horizontal orientation. Since any feature
that is the same in both examples cannot be the basis for referring to one example as flot and the other as not-flot, the
difference in behavior must be explained by reference to the features that are different from one example to another.

This fact about how negative examples rule out possible interpretations has great implications for teaching. To rule
out a particular interpretation, we simply show the same feature in the positive example and a corresponding negative
example.

Fact 4. A negative example rules out the maximum number of interpretations when the negative example is
least different from some positive example.

If negative examples rule out possible interpretations, it follows that, the more samenesses shared by positives and
negatives, the more interpretations the negatives rule out. The reason is that these negatives show that a greater number
of features do not play a role in determining whether the object is positive or negative. Figure 4.1 shows negatives that
are increasingly more like the positive:

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 Figure 4.1

A
“This is gloof.” “This is not gloof.”

B
“This is gloof.” “This is not gloof.”

C
“This is gloof.” “This is not gloof.”

D
“This is gloof.” “This is not gloof.”

Set A generates a number of possible interpretations, including: gloof means “being held in the hand”; gloof means
“something with corners”; gloof means “higher than”; gloof means “dark.”

By changing the negative example (as in set B) so that it has more of the same features observed in the positive
example, we can reduce the number of possible interpretations.

Some of the interpretations that were generated by set A are ruled out by the negative in set B. “Gloof” cannot
mean block, because the block is present in both the positive and the negative. “Gloof” cannot mean dark, because
darkness is a feature of both the positive and the negative examples. “Gloof” could mean higher than, or in the hand,
or horizontally-oriented.

Set C rules out the interpretation that “gloof” means horizontally-oriented because the horizontal orientation is
now in both the positive example and the negative example, and therefore cannot be the basis for determining what
gloof is: “Gloof” could still mean being in the hand.

Set D rules out this interpretation by making being in the hand a feature of both positive examples and negative

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examples.

Set D generates only a few possible interpretations of “gloof.” Perhaps it means suspended. To clarify the precise
meaning of gloof, we would need more examples—both positive and negative.

The fact about samenesses shared by positive and negative examples is important in designing communications. If
we design a negative example so that it is highly similar to a positive example, we rule out the greatest number of
possible interpretations. Only the difference between the positive and the negative can account for one example being
positive and the other negative. Since there is only a minimum difference, that difference must be the basis for treating
the examples differently.

In summary, the four facts indicate the basic ways that we can control the possible interpretations communicated
through the examples by controlling the details of the examples. If we use a single positive example, we communicate
an enormous range of possible interpretations. Additional positive examples rule out possible interpretations. For a
feature to be the basis for a possible interpretation, that feature must be present in all positives. If it is present in only
some, the feature cannot be a basis for classifying the examples as a positive. By presenting a wide variety of positives,
we show which features are relevant to the concept.

Negative examples rule out possible interpretations when samenesses occur in both positives and negatives. For a
feature to be the basis for classifying the example as positive, that feature must be present in all positives and in no
negatives. Therefore, any feature that is present in a positive and in a negative cannot be the basis for a possible
interpretation. The negatives that rule out the greatest number of possible interpretations are those that are least
different from some positive. In this situation, the differences are few, so the range of possible interpretation is limited.

Juxtaposition Principles

The fact that a set of examples must be presented to convey a concept to the learner introduces a new variable—
juxtaposition of the items. The examples that precede an example make that example relatively difficult or easy. There
are five principles of juxtaposition. These are expressed as how-to-do-it rules.

1. The wording principle: To make the sequence of examples as clear as possible, use the same wording on
juxtaposed examples (or wording that is as similar as possible).

By using the same wording with all examples, we assure that the learner focuses on the details of the example and
is not misled by variations in the wording. If the wording presented with each example is the same, the learner is shown
that each example is processed in the same way.

We can make an example more difficult by changing the wording on juxtaposed items. The sequence below makes
this task relatively easy: “Say the last letter in the word hog.”

1. Say the last letter in the word man.

2. Say the last letter in the word dog.

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 3. Say the last letter in the word hog.

To make the task relatively difficult, we precede it with a series of tasks that require the learner to perform a
different operation. Here is such a sequence:

1. Say the first letter in the word man.

2. Say the first letter in the word dog.

3. Say the first letter in the word frog.

4. Say the last letter in the word hog.

This procedure is similar to that used in the game “Simon Says.” The leader presents a series of tasks in which he
does the same thing he says. He then presents a task in which he does one thing and says another.

Easy juxtapositions are those in which the learner does the same thing with different examples, because these
juxtapositions do not require the learner to process as much information. The learner does the same thing with all
examples.

2. The setup principle: To minimize the number of examples needed to demonstrate a concept, juxtapose
examples that share the greatest possible number of features.

This juxtaposition principle deals with the examples in the same way the first principle deals with wording. The
greater the number of variables shown in the juxtaposed examples, the greater the number of total examples needed to
demonstrate a concept. If we teach under by using a dog or a cat that is under a table or chair, we need quite a few
examples. We must show:

1. The cat under the chair, the cat not-under the chair (on it, next to it, etc.),

2. The cat under the table, the cat not-under the table,

3. The dog under the table, the dog not-under the table,

4. The dog under the chair, the dog not-under the chair.

To show each of these positions, more than one example would be needed.

If we increase the number of features shared by juxtaposed examples, we decrease the number of examples needed
to demonstrate the concept. If we eliminate the cat and use only the dog, we are required to show only:

1. The dog under the table, the dog not-under the table.

2. The dog under the chair, the dog not-under the chair.

If we eliminate the chair, we are required to show only:

1. The dog under the table, the dog not-under the table.

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 All juxtaposed examples contain the same dog and the same table. The only variable from example to example will
be the relative position of the dog, which is the variable relevant to under.

The setup principle provides an easy formula for constructing the shortest, most efficient sequence: construct all
examples so they share the greatest possible number of features. Not only will this formula permit us to demonstrate
how the concept works with fewer examples, it will facilitate showing the learner which features are critical to the
concept. If examples share the maximum number of features, they differ in the minimum number of ways. These are
the ways that are relevant to the concept. The probability is therefore great that the learner will attend to the features
and changes that are relevant to the concept.

3. The difference principle: To show differences between examples, juxtapose examples that are minimally
different and treat the examples differently.

Positive and negative examples provide the maximum information when they differ only slightly from each other;
however, positive and negative examples must be juxtaposed to guarantee that this information will be

Figure 4.2
Set A
Example Teacher Wording
1 This line is not horizontal.

2 This line is horizontal.

Set B
1 This line is horizontal.

2 This line is not horizontal.

3 This line is not horizontal.

4 This line is not horizontal.

transmitted to the learner. Consider example set A in Figure 4.2. There are two minimally different examples created
through continuous conversion.

When these examples are juxtaposed (one immediately following the other in time), the difference is conveyed
relatively easily because the only change occurring from one example to the next is a slight change in orientation.
When other examples are interpolated, however, the learner is not provided with a demonstration that shows this
difference. Consider examples 1 and 4 in set B. They are the same examples that appear in set A. Although examples 1
and 4 are slightly different from each other and are labeled differently, the difference is not obvious because they are
not juxtaposed.
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 When we juxtapose differently labeled examples that show the minimum difference, we call the learner’s attention
to the fact that the small difference is the only basis for the change in labels.

4. The sameness principle: To show samenesses across examples, juxtapose examples that are greatly
different and indicate that the examples have the same label.

If we show under using only a table top and eraser, we can create many different positive examples of under.
Some would be close to the table; others farther away; some would be near the middle of the table; others would be
near the ends. The juxtaposed examples in Figure 4.3 show sameness. Note that the examples follow the wording
principle (same wording for all examples), the setup principle (same objects appearing in juxtaposed examples), and the
sameness principle (great difference from example to example with the same label from example to example).

Figure 4.3
Example Teacher Wording

1 The eraser is under.

2 The eraser is under.

3 The eraser is under.

4 The eraser is under.

Since juxtaposed examples are treated in the same way (“The eraser is under”), the changes from example to
example are shown not to affect the label. The learner is told in effect, “These examples are the same, so whatever
difference you observe in them is a difference that is irrelevant to under.”

5. The testing principle: To test the learner, juxtapose examples that bear no predictable relationship to each
other.

After the learner has been shown sameness and difference, the learner is tested on the generalizations implied by
the communication. If the demonstration is adequate, the learner should be able to handle some new examples of the
concept that are presented within the constraints of the setup principle. Note that the testing principle refers to the test
that is provided immediately after the demonstration of sameness and difference. This test involves the same setup
features as the sameness and difference demonstration. Following successful performance on this immediate test, the
learner will be given expansion tests that involve new setups and new wording. The most immediate communication
question, however, is: Did the learner receive the information needed to generalize to new examples presented within
the original setup? The test that immediately follows the demonstration examples answers this question.

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 According to the testing principle, no pattern of responses should be evident from one test example to the next.
(For example, no alternating sequence from positive to negative examples.) Examples presented in the test may be
similar or dissimilar. The response called for by these examples may be the same or may be different. The test segment
should repeat some examples that had been used to demonstrate difference and sameness. The segment should also
contain new examples.

Applying the Five Principles

To provide a clear initial-teaching communication to the learner, we must design a sequence that takes into account
all the facts and all five juxtaposition principles. The sequence must show relevant samenesses and relevant differences
in the examples. The wording associated with the various examples must be precise. The setup must be designed to
permit the smallest number of examples. And the learner must be provided with an immediate test, one that requires the
learner to respond to examples created within the setup.

The sequence in Figure 4.4 is consistent with the five principles. As shown in Figure 4.4, different examples are
designed to meet the requirements of different juxtaposition principles. The sequence is a comparative sequence
designed for the initial teaching of the concept, getting wider. Each example involves a space (diagrammed as a line
with a marker at either end). To present the sequence above, hold your hands about a foot apart and say: “Watch the
space.” Then hold your hands stationary and say the wording for example 1: “It didn’t get wider.” Move one hand in
slightly, stop, and say the wording for example 2: “It didn’t get wider.” Move the same hand out slightly. Then say the
wording for example 3: “It got wider.” Continue in this manner for the remaining examples.

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 Figure 4.4
Symbol Principle Examples
A1 A2 The wording principle 1-13
B The set-up principle 1-13
C The difference principle 2-3, 5-6
D The sameness principle 3-5
E The testing principle 6-13

Example Model Teacher Wording

Watch the space.

1 It didn’t get wider.

2 It didn’t get wider.

C
3 It got wider.
A1

4
D It got wider.

5 It got wider.
C
6 B Did it get wider?

7 Did it get wider?

8 Did it get wider?

9 E Did it get wider?

A2
10 Did it get wider?

11 Did it get wider?

12 Did it get wider?

13 Did it get wider?

Bracket A shows that the “same wording” is used for all examples. The wording could have been made even more
uniform if the teacher demonstrated on examples 1 to 5 by asking the same question presented to the learner and then
answering it: “Did it get wider? No....Did it get wider? No....” etc. The wording used in the illustration above is fairly
uniform, however. All demonstration examples and all test examples refer to the same response dimension: getting
wider.

Bracket B shows that all examples involve the maximum number of common setup features. Note that only one
hand is involved in changing an example to the next example. (The same hand moves in all examples that involve
movement.) The hands appear in all examples. The orientation of the hands, the background, and other features of the
situation remain the same. Therefore, the setup principle is satisfied. The set of examples contains the maximum
number of common features.

Bracket C shows that the difference is shown in two parts of the sequence. Both involve small differences in the
juxtaposed examples that lead to different wording.

Bracket D shows sameness, created by juxtaposing examples that differ as greatly as possible within the constraints
of the setup. Note that the size of the change from example to example is controlled for this concept because the
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communication must convey the idea that the size of the change has nothing to do with getting wider. The change
from 2 to 3 is a very small change. The change from 3 to 4 is quite large. The change from 5 to 6 is intermediate. All
are labeled in the same way: “It got wider.”

Bracket E shows the test juxtaposition, an unpredictable sequence that contains some minimum-difference
examples and some greater difference examples. The test sequence attempts to avoid any pattern such as an alternating
pattern of positive, negative, positive, etc.

Continuous Conversion

As noted earlier, the procedure by which you present the examples when you change one example into the next
(e.g., holding your hands up and then moving one of the hands to create the different examples) is called continuous
conversion. The use of continuous conversion forces you to create relevant changes in the examples. When you create
examples through continuous conversion, you do not create total examples, only changes. The learner is therefore not
required to attend to all details of the examples, merely those involved in the change. The fact that you must change
examples in a way that changes positives to negatives means that you will create changes that are relevant to the
concept.

To appreciate the difference in communication potential between continuous conversion and non-continuous
conversion, present the sequence of examples for getting wider non-continuously. To do this, use exactly the same set
of examples you used before. However, between each example, put your hands at your sides. Then put your hands up
again to create the next example. Start with your hands about a foot apart. Say, “Watch my hands.” Put them at your
sides. Raise them to where they had been and say, “They didn’t get wider” (example 1). Again put your hands down.
Return them, this time slightly closer together and say, “They didn’t get wider.” Continue through all examples in this
manner.

The advantages of the continuous conversion sequence become immediately apparent when examples are
presented non-continuously. It would be difficult for the learner to see that the space for example 2 is slightly less than
that for example 1. The reason is that non-continuous conversion requires creation of all details for each example.
First, all details of example 1 are created. Then all details of example 2 are created. To compare two examples, the
learner is required to compare the two sets of details and observe possible differences. This task is logically more
difficult than one in which the first example is changed into the next.

Continuous conversion makes small changes perceptible because one example is changed just enough to create the
next example. If the two examples are highly similar, only a small change occurs; however, it is easy to detect because
it is the only thing that happens.

Analyzing Concepts By Using Continuous Conversion

Continuous conversion of examples can be used to show us which dimensions are relevant to the concept. If we are
in doubt about what causes an example to be positive or negative, we use continuous conversion to provide the answer.
First we start with a negative example and, through continuous conversion, change it into a positive example. The

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dimension that we changed is a relevant dimension. (If we changed position, position is relevant; if we changed
intensity, intensity is relevant, etc.) Next, we see how much change along the relevant dimension we can apply to a
positive example without making it negative. We must operate within the constraints of the setup. For example, after
observing that moving the hands in a certain direction causes an example to change from negative to positive, we
conclude that movement of the hands is a dimension that is relevant to the concept. Next we see the extent to which we
can move our hands without changing a positive example into a negative. This variation of the relevant dimension
shows the range of positive variation that is possible within the constraints of the setup.

By applying the two-step conversion described above, we discover that many concepts are single-dimension
concepts, which means that if we start with a negative example we can change it into a positive example only if we
manipulate one dimension. We discover that other concepts are multiple-dimension concepts, which means that it is
possible to change a positive into a negative by manipulating various dimensions.

Not all concepts can be presented through continuous conversion. If the example is the verbally presented
statement, “The crow flew over the tree,” we cannot continuously convert that example into another verbal example.
However, if we pretend that examples can be created continuously, we more readily see which changes are relevant to
the concept, how the various examples are the same, and how they differ.

Perspective. The rules and principles presented in this section apply in some form to any instructional design
problem, because all these problems start with some “need.” You want to show how things are the same or how they
are different. You want to reduce the variables and make the relevant changes as obvious as possible. You want to test
the learner on non-trivial generalizations of what you have taught. The principles apply because they are logical
principles about examples.

Learn the principles of juxtaposition, and try to think of them as logical rules that apply to any situation that
involves communicating through examples.

Summary

The most basic communications for teaching discriminations or concepts are ones that present examples and that
permit the learner to perform on examples not shown in the initial presentation. Basic communication facts and
principles are described in the context of communicating through examples. The facts deal with basic logical properties
of communications that involve examples. The principles tell how to achieve particular communication goals, such as
showing sameness, or showing difference.

The first fact is that the presentation of only one example cannot logically show one concept. The reason is that any
concrete example is an example of thousands of concepts because it has thousands of features. The labeling of the
concept does not indicate which features are being referred to.

The addition of different positive examples rules out possible concepts if the examples differ in various ways, but
are labeled in the same way. The common label must relate to how the examples are the same, not how they are
different. All observed differences among the examples are ruled out.
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 By presenting only positives examples, it is possible to strongly imply a concept, but logically impossible to show
it precisely. The addition of negative examples rules out possible interpretations by calling attention to differences in
the examples that are correlated with differences in the labels. The smallest differences provide the greatest
information. If a very small change occurs and that change is associated with a change in the label, that change must be
the cause of the different label. Furthermore, since it was a very small change, the identified cause is very precise. The
change marks the boundary between the positives examples and the negatives.

A final fact is that samenesses and differences of examples are more obvious when the examples are juxtaposed.
This fact implies that the continuous conversion of examples provides the clearest presentation of samenesses and,
differences because it creates the changes that occur from one example to the next.

The principles of juxtaposition are based on the notion that the examples must be presented in sequence to the
learner. The order of the examples is important. The general procedure in presenting sequences is to demonstrate with
some examples and test on others.

Five principles provide specific direction for designing the wording, the setup, demonstrations of sameness,
demonstrations of difference, and the test. These principles of juxtaposition are:

1. The wording principle. To make the sequence of examples as clear as possible, use the same wording on
juxtaposed examples.

2. The setup principle. To minimize the number of examples needed to demonstrate a concept, juxtapose
examples that share the greatest possible number of features.

3. The difference principle. To show differences between examples, juxtapose examples that are minimally
different and indicate that the examples have different labels.

4. The sameness principle. To show sameness across examples, juxtapose examples that are greatly different
and treat the examples in the same way.

5. The testing principle. To test the learner, juxtapose examples that bear no predictable relationship to each
other.

For a sequence to communicate unambiguously, all principles must come into play. The wording must be uniform,
and the setup efficiently designed. The demonstrations of sameness and difference must precede the test examples. The
test involves the same setup as the demonstration, and provides the most fundamental information about the sequence:
Does the learner generalize within the setup? If so, the sequence is immediately followed by one that uses variant
setups and variant wordings. However, our first concern is the performance within the setup.

The principles of juxtaposition apply to any set of examples. Become familiar with these principles before
proceeding to the communications for sensory-based concepts.
 

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 Non-Comparative Sequences
The non-comparative sequences that we will develop in this chapter are appropriate for teaching positional
relationships (under, near, in); some actions (running, hopping); and simple adjective features (pointed, bent,
horizontal, red). Variations of the basic non-comparative sequence are possibly appropriate for more advanced
applications (adjectives such as gradual and despondent, actions such as evade and inflect).

There are two basic types of sequences for these concepts—positive-first sequences and negative-first sequences.
As the labels imply, the positive-first sequences begin with demonstrations of positive examples. The negative-first
sequences begin with demonstrations of negatives. Both sequences provide demonstrations of positives and negatives,
followed by a test.

Figure 5.1 shows the basic formula or format for a negative-first sequence. By following this formula for any
single dimension non-comparative concept, we can create an initial-teaching communication that is faultless. The
concept taught through the sequence below is slanted, meaning not vertical and not horizontal. Note that the wording is
not provided (merely the letters N and P to indicate whether each example is positive or negative) so that the focus on
the positive and negative examples is more obvious. Present this sequence through continuous conversion, using a
pencil. After creating the first example, rotate the pencil counter-clockwise. Then stop when the pencil is horizontal to
produce example 2. Continue through the remaining examples by quickly rotating the pencil, then holding it stationary
for the next example.

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 Figure 5.1
Example Example Type

1 N

2 N

C
3 P

4 P
D

5 P

C
6 N

B
7 P

8 N

9 N

E
10 P

11 N

12 P

13 P

Juxtapositions

The sequence begins with two negatives. Both are minimally different from some positive examples, but the
negatives are not highly similar to each other.

Following the negatives are three positives, juxtaposed to show sameness (examples 3, 4, 5).

Following the third positive is a negative (example 6).

The remainder of the sequence consists of examples in an unpredictable order with respect to whether they are
positive or negative and with respect to the difference between juxtaposed examples.
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 The brackets next to the series of examples indicate that the sequence is consistent with juxtaposition principles:

B. The setup involves your hand and the same pencil in all examples.

C. Difference is created by juxtaposing two minimally different examples, each consisting of a positive and a
negative (examples 2-3 and 5-6).

D. Sameness is created by juxtaposing maximally different examples 3, 4 and 5. The juxtaposed examples
vary according to the direction of the point, the amount of rotation from the preceding example, and the
absolute difference from the preceding examples.

E. Testing is provided by a series of examples that bear no predictable relationship to each other.

Constructing Non-Comparative Sequences

Negative-First Sequences

Since all non-comparatives have the same structure, all can be effectively conveyed through a sequence that is of
the same form as the sequence above. To construct such a sequence for any single dimension non-comparative:

1. Begin with two negative examples that differ minimally from some positive example. For the sequence in

Figure 5.1, we could have begun with any of the following pairs of examples: or with
the two used in the model sequence.

2. Make sure that the second negative (example 2) is minimally different from example 3. In the model
sequence the second example is horizontal with the point facing right. Example 3 is slightly slanted with
the point still facing right.

3. Make juxtaposed positive examples 3, 4 and 5 differ as much as possible from each other within the
constraints of the setup. In the model sequence, example 3 points slightly downward, 4 points up and to
the left, and 5 is pointed down and almost vertical. This order shows that the label of slanted does not have
to do with the direction of the point or with the amount of slant. Each example differs in direction of point
and in slant.

4. Make juxtaposed examples 5 and 6 minimally different, but do not use the same minimum difference used
in examples 2 and 3. In the sequence above, examples 2 and 3 showed a minimum difference involving a
horizontal example (2). Examples 5 and 6 show a minimum difference involving a vertical example.

5. Following example 6, present a series of 6 to 8 test examples that show no predictable order. The test
segment should repeat some earlier examples; it should also contain new positives and new negatives. It
should contain a sufficient number of positives to provide a good test of the learner’s understanding, but it
should not show a predictable order or pattern of positives (P) and negatives (N). The pattern for examples
6 through 13 in the sequence above is: NPNNPNPP. This pattern is acceptable. The following patterns are

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 not acceptable: NNPPNNPP or NPNPNPNP.

Illustration. The negative-first form (used to communicate slanted) is used in Figure 5.2 to communicate a
different non-comparative—suspended. The sequence follows the same five rules:

1. The sequence begins with two negative examples, each of which is different from some positive example.

2. Example 3 is minimally different from example 2. (The only difference is that example 3 is held against
the bottom surface of the table.)

3. Examples 3, 4 and 5 present sameness juxtapositions. Note the height and position of the block changes in
all three of these examples.

4. A new minimum difference occurs between examples 5 and 6.

5. Beginning with example 6 is a series of 7 non-patterned test examples. (The sequence for suspended
contains a total of 12 examples, compared to 13 for slanted. This difference is somewhat arbitrary. For
some concepts more test items are needed to test the concept adequately.)

Figure 5.2
Example Example Type Teacher Wording

1 N The block is not suspended.

2 N The block is not suspended.

C
3 P The block is suspended.

4 D P The block is suspended.

5 P The block is suspended.

C
6 N Is the block suspended?
B

7 P Is the block suspended?

8 P Is the block suspended?

9 E N Is the block suspended?

10 P Is the block suspended?

11 N Is the block suspended?

12 P Is the block suspended?

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 Present this sequence. Use any object and a table surface that has space beneath the surface. Follow the sequence
of examples. First position an example. Then say the specified wording. Quickly position the next example and say the
wording. As you present the examples, observe how the different possible interpretations are ruled out by the
minimum-difference negative examples and the positive demonstration examples (3, 4, 5). Make sure that you hold the
object in the same way for all examples.

Teacher wording. The teacher identifies the first five examples, saying either: “The block is suspended,” or, “The
block is not suspended.” The teacher tests on the examples 6 through 12 by asking, “Is the block suspended?” The
teacher wording could say, “Tell me suspended or not-suspended.” This wording would be used for all test examples.
The positive response for the question, “Is it suspended?” is “Yes.” The positive response for “Tell me suspended or
not-suspended” is “Suspended.”

Variation Within the Setup

According to the setup principle, all examples should share as many features as possible. In the sequence above, all
are held. All involve a horizontally-oriented object (not a vertical or slanted object). To show the range of positive
variation (examples 3, 4 and 5), you might be tempted to present a slanted example. Such an example would certainly
not make the sequence unacceptable, merely more difficult to construct. Here’s why: When you violate the setup
principle, you show a “variation” in some dimension that is incapable of changing an example from positive to
negative. However, the learner does not know that this is the case. To show the learner that the variation is a change
along some dimension other than the critical one you must therefore add more examples to the sequence. If you show a
slanted positive example, you must show a slanted negative example to indicate that the slant of the object is
unimportant and can appear in both positives and negatives. An easier way to show that it is unimportant is to remove it
from the sequence. If no positive examples are slanted, the slant of the object cannot be relevant to whether the object
is suspended. Similarly, if no objects have stripes, having stripes cannot be relevant to whether the object is suspended.

Determining Which Features Should Appear in All Examples

As noted earlier, the easiest test to determine which features should not be varied from example to example is to
begin with a positive example and observe all possible changes that will convert it into a negative example. Will
rotating the eraser convert a positive to a negative? No. If a change does not convert a positive example to a negative,
that change deals with an irrelevant feature.

There are two basic ways to show that a feature of a positive example is irrelevant:

1. Present another example that does not have the feature, but is identical to the original in every other
respect.

2. Present a negative example that has the feature, but is minimally different from the positive.

We could show that the vertical orientation of an object is irrelevant by:

1. Presenting another positive example that is not vertically oriented, but that is suspended over the same

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 place of the table.

2. Presenting a negative example that is vertically oriented and that is in nearly the same place.

Figure 5.3 shows a 12-example sequence that parallels the model sequence of suspended. The only difference is
that the orientation of the object shifts between being vertical and being horizontal. The addition of this variable creates
a communication that does not rule out all possible misinterpretations. The sequence is consistent with the
interpretation that a positive example in the middle of the table must be vertically oriented. Also a positive on the right
end of the surface must be horizontal.

Figure 5.3
Example Example Type Teacher Wording

1 N The block is not suspended.

2 N The block is not suspended.

C
3 P The block is suspended.

4 D P The block is suspended.

5 P The block is suspended.

C
6 N Is the block suspended?
E

7 P Is the block suspended?

8 P Is the block suspended?

9 B N Is the block suspended?

10 P Is the block suspended?

11 N Is the block suspended?

12 P Is the block suspended?

We could eliminate these misrules by adding 3 or 4 examples to the sequence. In the end, however, we would
convey the same basic message that we convey through the 12 examples of the original sequence.

Stipulation

Stipulation occurs when the learner is repeatedly shown a limited range of positive variation. If the presentation

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shows suspended only with respect to an eraser and a table, the learner may conclude that the concept suspended is
limited to the eraser and the table.

Stipulation is an “undergeneralization.” The learner does not know the range to which this concept applies.
Misrules are different from stipulation. Misrules occur when more than one cause for positive-negative changes is
implied. It is a misgeneralization. The sequence provides a false idea of what makes an example positive or negative.

The initial teaching sequences for non-comparatives are guilty of stipulation. They clearly show the minimum
necessary difference between positive and negative examples. They also show the range of variation that can be
achieved within the constraints of the setup; however, each sequence is limited to one setup. All examples of slanted
are shown with a pencil. All examples of suspended are shown with a block and a table.

The stipulation implied by these initial-teaching sequences is that the concept may be limited to a particular
context. The learner, for example, may pick up the stipulation that the concept suspended applies only to blocks and
tables. The sequence does not provide any examples to discredit this possible implication. To counteract stipulation,
additional examples must follow the initial-teaching sequence. Following the learner’s successful performance with the
sequence that teaches slanted, for instance, the learner would be shown that slanted applies to hills, streets, floors,
walls, and other objects. Following the presentation of suspended, the learner would be systematically exposed to a
variety of suspended objects.

Stipulation of the type that occurs in initial-teaching sequences is not serious if additional examples are presented
immediately after the initial teaching sequence has been presented. The longer the learner deals only with the examples
shown in the original setup, the greater the probability that the learner will learn the stipulation. If the learner is
presented with the new examples immediately following the initial teaching sequence, no significant new learning
should be necessary. The learner is simply shown that the same dimension involved in the judgment that a block is
suspended is involved in judgments involving other things and other tasks. (Remember, the initial- teaching sequence
requires only about one minute to present.)

To avoid stipulation during the initial teaching demonstration, we must violate two principles—the wording
principle and the setup principle. The consequences of these violations are communications that are unfortunately
elaborate and possibly unclear. To compensate for violations of the setup principle we must add examples. To
compensate for violations of the wording principle, we must add examples. If both changes in setup and wording occur
at the same time, a large number of examples must be added to make the sequence consistent with the desired
interpretation. Even with these additional examples, however, the sequence will be “crude.” The idea is not merely to
make the sequence consistent with a single generalization, but to suggest that generalization as early in the sequence as
possible. If the single generalization is apparent only after ten examples have been presented, the sequence is far
inferior to one that reveals it after four. The simplest solution to the problem of stipulation is to follow the setup and
wording principles. Then immediately follow the initial-teaching sequence with examples and tasks designed to
counteract any possible stipulation. Follow-up sequences are discussed in Chapter 15.

Positive-First Sequences

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 Theoretically, negative-first sequences reveal the desired interpretation faster than positive-first sequences.
However, for this interpretation to occur, the learner must be reasonably sophisticated and must realize that the
negatives shown first are related to the positives that will follow. The first two examples show the sophisticated learner,
“This is not it yet.” The change involved in creating example 3 reveals to the learner, “Now you see what it is.” At this
point (example 3) the learner should have a very good idea of the concept. Some additional questions about how it
works may still have to be answered through subsequent examples; however, the basic structure has been shown,
through only three examples. Theoretically, positive-first sequences are not as capable as negative-first sequences in
showing the concept as early. The reason is that the positive-first sequence starts with a positive example. Like all
positive examples that are labeled, this positive is consistent with many possible interpretations. Some interpretations
are ruled out by the positives that follow; however, possible misrules last until the negatives are introduced.

A possible communication problem with negative-first sequences occurs if the learner misunderstands the intent of
identifying the first examples as “This is not slanted.” The learner may conclude that the example has nothing to do
with slanted and may ignore it. The result is that the learner does not benefit from the negatives.

In any case, the learner should be presented with some concepts through positive-first sequences and some through
negative-first sequences. If only negative-first sequences were presented, the learner might develop the strategy of
“memorizing” the pattern, rather than that of attending to the features of the example.

Constructing Positive-First Sequences

Here are the procedures for constructing positive-first sequences for any single dimension non-comparative.

1. Begin with three positives that are juxtaposed to show sameness (examples 1, 2, 3).

2. Follow with two negatives (examples 4, 5). The first is minimally different from example 3. The next is
minimally different from example 6.

3. Follow with a positive (example 6) that is minimally different from example 5.

4. Follow with a series of 6 to 8 examples that follow the test-juxtaposition order.

5. Model the first five examples; test on the remaining examples.

Figure 5.4 illustrates a positive-first variation for suspended.

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 Figure 5.4
Example Example Type Teacher Wording

1 P The block is suspended.

D P The block is suspended.

2

3 P The block is suspended.

4 N The block is not suspended.

C
5 N The block is not suspended.

6 P Is the block suspended?

B

7 N Is the block suspended?

8 P Is the block suspended?

9 E P Is the block suspended?

10 N Is the block suspended?

11 N Is the block suspended?

12 P Is the block suspended?

The sequence is created by rearranging the same set of examples presented in the negative-first sequence. The
order of appearance for the examples in Figure 5.4 occur in the negative-first sequence (Figure 5.2) as: 7, 4, 5, 6, 2, 3,
1, 8, 10, 9, 11, 12.

One advantage of the positive-first sequence is that it permits us to show a wider range of positive variation. The
negative-first sequence constrains the range of positives somewhat because there must be a minimum difference
negative on either side of the positives. Since the positive-first sequence begins with positive examples, greater latitude
is possible. The implication is that if you find it difficult to arrange the first three positives when working with a
negative-first sequence, try a positive-first sequence.

Narrow-Range Concepts

Some concepts have a very limited range of positive variation. (They can only be demonstrated through a setup
that provides for a very narrow range of variation.) For instance, we could teach the concept gradual turn (meaning
not abrupt) in a way that requires a great many variables; however, if we follow the setup principle and try to keep as

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many features as possible common to all examples, we will control the size of the turn, the object that is turning, and
the direction of the turn. The boundary between gradual and not-gradual is not clear. (We can make it clear by
teaching the concept more gradual and not-more gradual; however, this concept is a comparative.) If we keep the
original concept—gradual—we must make some arbitrary decisions about what is gradual and what is not. The safest
convention is this: Show all examples of gradual turn at one rate and all examples of not-gradual turn at a fast rate
that is quite discriminable from that labeled gradual.

Figure 5.5 illustrates the resulting sequence. It begins with one positive and one negative.

Figure 5.5
Example Example Type Teacher Wording
Watch the turns that I
trace with my finger:

1 (gradual) P It turned gradually.

2 (abrupt) N It didn’t turn gradually.

3 (gradual) P Did it turn gradually?

4 (gradual) P Did it turn gradually?

5 (abrupt) N Did it turn gradually?

To present this sequence, start with your finger pointing up. Turn it gradually clockwise 90 degrees. Stop and say
the wording for example 1. Turn another 90 degrees abruptly in the same direction. Stop and say the wording for
example 2. Note that you turn your hand 90 degrees for each example. You move it either abruptly or gradually. Try to
present all abrupt examples at the same speed and all gradual ones at the same speed—a speed quite different from that
shown for the abrupt examples.

The sequence might be improved if it contained more modeled examples at the beginning (possibly two positives
and one negative). However, because the difference between positives and negatives is obvious and because there is no
range of variation for the positives, one demonstration should be enough.

For all limited-range variations of non-comparatives, begin with either a positive or a negative, demonstrate on 1 to
3 examples and test on 3 to 5.

Conversely, for concepts that have an unusually wide range of variation, present more than three examples to show
sameness. For concepts that have many types of negatives that are minimally different from positives, include more
than two minimally different negatives in the demonstration part of the sequence.

Non-Continuous Conversion Sequence

Continuous conversions are not always possible or practical, particularly when we teach symbolic concepts.

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Assume that we wanted to teach the learner whether spoken words end in voiced sounds or not-voiced sounds. We
cannot actually say words as a continuous vocalization. We say a word, then another word. This vocalization is not
continuous. One word is presented. It disappears. Then another is presented. (The procedure is quite different from the
presentation of slanted, which consists of examples that are retained until a change creates the next example.)

The rule to follow in dealing with such cases is to design the sequence in basically the same way you would design
it if continuous conversions were possible. Figure 5.6 shows a negative-first sequence for teaching whether spoken
words end in a voiced sound or in a non-voiced sound. The brackets show that the same rules of juxtaposition used for
the other sequences apply to this one.

Figure 5.6
Examples Example Teacher Wording
Type
I’ll say words. Then I’ll
tell you if each word
ends in a voiced sound.
1. Mack N It doesn’t end in a voiced
sound.
2. Mass N It doesn’t end in a voiced
C sound.
3. Maz P It ends in a voiced
sound.
4. Mag D P It ends in a voiced
sound.
5. Mad P It ends in a voiced
A sound.
C
6. Mat B N Does it end in a voiced
sound?
7. Mass N Does it end in a voiced
sound?
8. Mab P Does it end in a voiced
sound?
9. Maff E N Does it end in a voiced
sound?
10. Mal P Does it end in a voiced
sound?
11. Mat N Does it end in a voiced
sound?
12. Mav P Does it end in a voiced
sound?

To present the sequence, say the example, then say the specified wording for that example. Note that the teacher
begins with an explanation of what the teaching will show. The A bracket shows that the wording principle is followed.
The B bracket indicates that one part of each word (ma) is common to all examples. Each C bracket shows a pair of
minimum-difference examples. The D bracket shows the three examples juxtaposed to convey sameness. The E bracket
indicates the test examples.

Because these examples are presented verbally, they cannot be converted continuously. However, if we presented

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written words to the learner, we could present the same examples through continuous conversion.

The teacher would begin by writing the first word (mack) on the board and saying, “It doesn’t end in a voiced
sound.” The teacher then erases only part of the word to create the next example. (The teacher erases ck and replaces it
with ss. This change converts the first example into the second example.) Each following example would be created in
the same way. The teacher would erase the ending of a word and replace it with another ending.

Remember, written examples can be continuously converted. Oral examples of words cannot. If we used written
examples, however, we might choose to modify the teacher wording. The reason is that our communication depends on
the learner correctly identifying the ending of each word. If the teacher simply says, “This word doesn’t end with a
voiced sound,” there is a question about whether the learner was actually registering the correct pronunciation or
whether the learner silently misread it. To reduce the ambiguity, the teacher would say the word, then tell about the
ending or direct the learner to read each word aloud. (See Figure 5.7)

Figure 5.7
Example Teacher Wording
1. mat What word?
Mat doesn’t end in a voiced sound.
2. map What word?
Map doesn’t end in a voiced sound.
3. mab What word?
Mab ends in a voiced sound.
Etc.

Illustrations

Ordinarily pictorial illustrations do not permit continuous conversion because they are static and there is no easy
way to change one picture into another. However, there are three ways that continuous conversion can be created with
illustrations:

1. By covering or uncovering parts of the picture.

2. By adding or removing cutout parts to make a “background.”

3. By pointing to parts of the picture.

The simplest procedure is to point to different parts. For some illustrations, however, this procedure is not
practical.

We can illustrate the use of pictures with a sequence designed to teach the concept they. The teacher presents a
group of illustrated characters, two girls, two boys, two men, two women, two dogs, and two cars. The teacher then
creates continuous conversion by pointing to different characters or combinations of characters.

Figure 5.8 shows the first part of the sequence. The examples indicate what the teacher points to.

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 Figure 5.8
Example Teacher Wording
I’ll tell you if I show they.
Watch what I point to.
1. A boy It doesn’t show they.
2. A girl It doesn’t show they.
3. A girl and boy It shows they.
4. Two dogs, a car, and
a man It shows they.
5. A dog and a woman It shows they.
6. A dog Does it show they?
Etc.

By placing appropriate cutouts in a scene and by removing them, the teacher could create the same series of
examples through continuous conversion. The teacher would first place the cutout of the boy in the scene. For item 2, a
boy would be exchanged for a girl. For item 3, the girl would remain and a boy would be added to create a minimum-
difference positive example. Example 4 would be created by removing the boy and girl and replacing them with two
dogs, a car, and a man. This example is greatly different from example 3. The teacher treats it the same way example 3
was treated, thereby demonstrating sameness.

Note that the series of examples parallels the other negative-first series we have created, with respect to sameness
and difference.

Two-Choice Tasks

These tasks present two choices, both of which are usually named in the task. “Tell me if it’s pink or red.” “Is this
line vertical or horizontal?” “Is this fraction more than one or less than one?”

Two-choice tasks should be avoided in initial teaching sequences that are presented to relatively naive learners.
The reason is that the tasks require the learner to use a more complicated formula for responding. The learner must
remember which features of the examples go with each choice. Since there are two choices, the task may induce
reversals. The tasks that are capable of inducing the most serious reversals are those that involve a transformation, with
part of the response the same for both choices. The task, “Is this fraction more than one or less than one?” is such a
task. The responses that the learner must produce (more than one, less than one) are the same except for a single word.
Another example of choices with common parts is presented in this task: “Tell me if it is on the table or over the table.”

Two-choice tasks that have common parts should be used with only the most facile learners or those who are
already somewhat familiar with the responses that are being taught. Two-choice tasks that do not have common parts
are more appropriate, even for facile learners. Examples of these tasks are: “Tell me if it is tilted or flat.” Since the
responses “Tilted” and “Flat” have no common parts and are highly dissimilar from each other, the probability of
reversals is reduced.

For the more naive learner, single-choice tasks should be presented. “Tell me if it is tilted,” “Is this fraction more

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than one?” “Is this on the table?” Sometimes the task may call for a response that involves the word not. If the learner
is facile with this word, these tasks are not difficult; however, they may be quite difficult for the very naive learner who
is not familiar with the meaning of not. For this learner, the task may function as a two-choice task that involves
common response parts and therefore calls for a transformation, e.g., “Tell me if it is under or not-under.” The learner
may have mechanical problems in combining the word not with the appropriate features of the example.

Determining Positives in Two-Choice Tasks

For many teaching situations, we are required to teach facile learners related concepts, such as tilted and flat.
Since no transformation is involved in the responses and the learners are facile, we could teach the concept through a
two-choice task. The obvious advantage of this approach is that it is fast. With one demonstration, we could teach both
words.

If we decide to pursue this approach, we are faced with the problem of deciding which we should make the positive
examples—tilted or flat. This problem is unique to two-choice tasks. Here is a rule to follow: Make the examples with
the greatest range of positive variation the positive examples and the examples with the narrower range the negatives.

For tilted-flat, the concept tilted has the greatest range of variation, so tilted becomes the positive and flat the
negative. Figure 5.9 shows a possible beginning of a sequence. It is a positive-first sequence beginning with three
examples of tilted. Note that there is only one modeled example of flat because, within the context of the setup, there is
no range of variation for flat.

Figure 5.9
Example Example Type Teacher Wording
I’ll tell you if it’s tilted
or flat.

1 P It’s tilted.

2 P It’s tilted.

3 P It’s tilted.

4 N Now it’s flat.

5 P Tell me: tilted or flat.

6 N Tell me: tilted or flat.

Etc.

The only difference between this sequence and the earlier positive-first sequence comes after example 4. The
reason is that there is only one negative in the sequence above. The teacher presents the same task or question for each

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test example; however, note that the learner produces different responses.

Non-Comparative Concepts as Components of Complex Tasks

In later sections we will deal with more complex forms of communication. These forms are composed of more
basic forms. The following are two illustrations of how the basic non-comparative sequence is incorporated into more
complex communications.

In our first example, we wish to teach the learner various spelling relationships that involve the discrimination of
short vowel and not-short vowel. One relationship is:

• If the /ch/ sound immediately follows a short a, e, i, or o, it is spelled t-c-h.

• If the /ch/ sound does not immediately follow one of these vowels, it is spelled c-h.

We could teach at least part of the relationship through a sequence that begins like the one in Figure 5.10. This
sequence is a non-comparative that follows the same pattern as the simpler non-comparatives.

Figure 5.10
Example Teacher Wording
Listen: Batch. Does a short vowel sound come just
before the ch sound? Yes.

Listen: Hitch. Does a short vowel sound come just

before the ch sound? Yes.

Listen: Itch. Does a short vowel come just before

the ch sound? Yes.

Listen: Inch. Does a short vowel come just before

the ch sound? No.

Listen: Pinch. Does a short vowel come just before

the ch sound? No.

Listen: Pitch. Does a short vowel come just before

the ch sound? Yes.

The sequence is a positive-first communication that presents a range of variation for the three positives at the
beginning of the sequence, followed by two pairs of minimum-difference examples.

Note that the sequence above would not provide the entire teaching for the rule that the ending is spelled tch only
if the ending is immediately preceded by a short a, e, i or o vowel. After the discrimination is taught, it would become a
component in a cognitive routine that makes the entire operation of figuring out the appropriate spelling overt. Here is a
possible routine:

1. Listen: Hatch. Say it. “Hatch.”

2. Does a short vowel come just before the ch sound? “Yes.”

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 3. So how is the ch ending spelled? “t-c-h.”

4. Spell the word hatch. “H-a-t-c-h.”

Note that step 2 in the routine is the same task that would have been pretaught through the non-comparative
sequence above. The point is that non-comparatives are not merely simple discriminations; they may be quite complex
and may require a long verbal explanation.

Before we leave the short-vowel illustration, we should note that instruction would not begin with the non-
comparative sequence presented above. This sequence assumes that the learner is able to discriminate short-vowels
from other vowels. Short vowels are interesting because they are not a single discrimination, but a group of different
sounds that are arbitrarily called “short.” They do not share a compelling set of sameness that would permit any sort of
generalization from some of the short vowel sounds to others. (If we brought the learner to criterion on short a, short o,
short i, there is no logical way for the learner to generalize to short u or short e. Therefore, we must teach each short
sound as a separate discrimination and then combine them under the family name, “Short _____.”)

To teach each of the short sounds, we could use a non-comparative sequence (or sequences). The biggest problem
in designing these is to follow the setup principle. Although there are different solutions to the problem, perhaps the
easiest is to keep the first part of each word constant. Figure 5.11 shows a negative-first sequence for teaching the
short-a discrimination.

Figure 5.11
Example Example Type Wording
par N Is the a short? No.
pay N Is the a short? No.
pan P Is the a short? Yes.
pass P Is the a short? Yes.
patter P Is the a short? Yes.
pater N Is the a short?
pap P Is the a short?
pall N Is the a short?
pail N Is the a short?
pass P Is the a short?
pain N Is the a short?
pant P Is the a short?

The decision about the range of positive variation is arbitrary. Perhaps the sequence should deal with only one-
syllable words. Clearly, the sequence is not capable of teaching short a in all its contexts. It must be followed by
parallel sequences. In the end, however, the sequence is a single dimension non-comparative.

In our second example, we wish to teach the learner to discriminate between story problems that involve

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multiplication and those that involve addition. The rule: If a problem deals with the same number again and again, it is
a multiplication problem; if it does not deal with the same number again and again, it is not a multiplication problem.
We could convey this idea to the learner through rules or through elaborate teacher-directed routines. The
underpinnings of the rules or the routines, however, would be the non-comparative discrimination. In its simplest
communication form, the concept of dealing with the same number again and again is presented through a series of
stories that are similar in as many details as possible. Some problems make reference to using the same number again
and again. Others do not.

Here are the first four examples from a possible negative-first sequence:

“I’m going to say parts of the problems. Some parts are from multiplication problems. Other parts are not from
multiplication problems.”

1. Listen: The man went to the store and bought five apples and two oranges. Is that a multiplication
problem? No.

2. Listen: The man went to the store and bought five apples. Is that a multiplication problem? No.

3. Listen: Every time the man went to the store, he bought five apples. Is that a multiplication problem? Yes.

4. Listen: The man went to the store seven times. Is that a multiplication problem? Yes. Etc.

This sequence, like the one that taught the discrimination of the short-vowel sound immediately before ch involves
a complex verbal example; however, it is clearly a non-comparative. The non-comparative sequence may not always be
the best choice for processing these examples; however, if the decision is made to treat the discrimination or concept as
a “yes-no” concept, and if the concept is absolute (rather than relative), the same form used for simple non-
comparatives can be used to communicate the concept to the learner. The rules that apply to communicating the simpler
non-comparatives apply to the more complex communications. Three juxtaposed examples show sameness; two pairs
of minimum-difference examples show difference between positives and negatives; and a series of test examples follow
the demonstrations of sameness and difference.

Summary

To construct non-comparative sequences, follow one of the basic forms shown in this section. After constructing it,
modify it if a problem of communication exists. First decide whether the concept is a non-comparative single-
dimension concept. The easiest way is to begin with a concrete example of the concept. If the example you select
would always be a positive example of the concept, the concept is absolute, and you may be dealing with a non-
comparative concept. Now change the positive example into a negative. If you can achieve this change only by
manipulating a single dimension of the positive example, you are dealing with a non-comparative single-dimension
concept.

To get more information about how to construct a sequence for teaching the concept, refer again to the concrete
example. Figure out how to convert that example into other positive examples that are different from it. This exercise

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will reveal the sameness the examples share.

Now figure out the minimum-difference negatives. To do that, take a positive example and change it the very least
possible amount to create a negative. Make sure that the conversion involves a perceptible manipulation. Our goal is
not to obscure, but to amplify. The change may be small, but it must be quite perceptible.

Now follow the form, perhaps the negative-first form. Begin with two negatives, follow with a minimum
difference and three positives, follow with a minimum difference, and end with a series of test examples.

Add a variation of the same wording for all examples. Model the first three to five examples and test on the others.

Test the sequence to make sure that it is consistent with only one interpretation. If it is not, change it. Perhaps more
examples are needed (for concepts with a wide range of variation). Perhaps you can show the range better if you use a
positive-first sequence rather than a negative-first.

Make sure that the single interpretation emerges as early as possible. It must occur by example 7; however, it
should be strongly implied by example 5.

Check the minimum-difference negatives to make sure that they are different from each other.

Check the positives to make sure that they follow the juxtaposition principle for showing sameness.

Check the test examples to make sure that there are enough to provide a reasonable test, and that they are presented
in an unpredictable order.

Make sure that the sequence is not needlessly laborious. If there is a very narrow range of positives or negatives,
you can probably shorten it without jeopardizing the communication.

Use two-choice tasks as an economy measure if the learner is facile or already has some understanding of the
concept. Try to avoid two-choice tasks that involve pairs of responses that have common words or part.
 

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 Nouns
Nouns are labels for object classes: bottles, things, solid objects, clouds, eyes, shoes, dogs, animals. Also,

symbols such as the letter R are nouns. The braille symbol .: is a noun, and so is the numeral 142.

Some nouns have subtypes with names (dogs include terriers, spaniels, and other named subtypes; vehicles
include boats, cars, and other named subtypes). These nouns are called higher-order nouns and will be covered later in
this chapter.

Procedures for communicating the others—nouns such as bottle, car, shoe—are considered first.

Higher-order and “regular” nouns share a set of features. They are multiple-dimension concepts; differences
between positives and negatives are not precise, but are absolute.

Concepts processed through a non-comparative sequence are single dimension concepts. We can make objects hot
only by manipulating a single variable—the temperature of the object. In contrast, nouns are multiple-dimension
concepts. In traditional terms, the noun is a “conjunctive” concept requiring the presence of more than one feature or
attribute. When something is labeled car or tree or bush, the label stands for a number of features, each of which is
necessary in some form. We could change a shoe into a non-shoe by adding parts (such as an upward extension, which
would change a shoe to a boot); by subtracting parts (such as the entire upper, which would make the shoe a sandal);
changing the material (such as the material of the sole, which would make the shoe a slipper); or by amplifying some of
the features (filling the entire inside of the shoe with a solid-leather core, so that the shoe cannot function as a shoe).
Some of these negatives are nameless, such as a shoe that is sewn to a pair of dress slacks.

Because non-comparatives are single-dimension concepts, it is usually practical to show relatively small difference
between positives and negatives. Because nouns are multiple-dimension concepts, however, it is not only impractical to
show minimum differences; it is virtually impossible, because even knowledgeable adults do not agree on the boundary
line between positives and negatives of common nouns. To demonstrate this fact, present a group of adults with 20
examples of footwear and ask them to identify whether each object is shoe or not-shoe. Include tennis shoes,
moccasins, flats, sandals, “ankle-high” shoes, and other “marginal” shoes. Although there will be perfect agreement on
some items, there will be disagreement on tennis shoes, flats, and all other “marginal” shoes. In contrast, if we
presented examples of getting wider to the same group, we would find virtually no disagreement.

The point of this demonstration is that with multiple-feature concepts there is no precise minimum difference
between positives and negatives. Even if we tried to specify a definition or classification criterion for shoe that seemed
totally precise, we would discover that we could create examples that are ambiguous because there are examples at the
boundary line that cannot be classified. The interplay of the various features involved in the positive examples of nouns
suggests that a precise formula for the positive examples is all but impossible. So it is with all nouns. The implication
for instruction is that we should not try to create a precise boundary line (minimum differences) when very
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knowledgeable adults do not agree on one. Fortunately, teaching minimum-differences is not necessary.

Continuous conversions are not necessary and not practical for the initial teaching of most nouns. The purpose of
continuous conversion of examples is to call the learner’s attention to the precise details or differences between one
example and the next. When we deal with nouns, there are not precise boundary lines between positives and negatives;
therefore, the basic rationale for using continuous conversion is not present.

Since there are many differences between positives and negatives of most nouns, the probability is increased that
the learner will attend to some feature that permits reliable discrimination of positives and negatives.

When dealing with the small difference between something that is on a surface and something that is not, the
learner must attend to a specific difference. The situation with nouns is quite different. There may be 20 observable
differences between a positive and the most highly similar negative of a noun. Because any one of these differences
will serve the learner to discriminate between positives and negatives, we should not have to call attention to any
particular difference, because any one of the differences will serve the learner in discriminating between positives and
negatives.

Finally, there are serious mechanical problems in trying to use continuous conversions with nouns. Converting
examples of car into some existing negative—truck, bus, train, etc.—could be achieved through animated motion
pictures; however, for the resources that are generally available, continuous conversion sequences for nouns are
impractical.

Constructing Initial-Teaching Sequences for Nouns

The structure of nouns suggests differences between the sequences for teaching non-comparatives and sequences
for teaching nouns. The major features of the noun sequences are these:

1. Since it is impossible to determine precise boundaries between positives and negatives, the sequence does
not show the smallest minimum differences that are possible. Instead, the sequence limits the negatives to
nouns already known by the learner. They are the least-different examples in the learner’s repertoire from
the noun being taught. In other words, the learner should be able to already identify all negatives used in a
noun sequence. When we refer to minimum negatives in a noun sequence, we do not mean the smallest
differences that are possible, merely the smallest differences that are included in the sequence.

The noun sequence changes as the knowledge of the learner changes; therefore, the minimum
differences are relative to what the learner knows. For one learner, the negatives in a sequence that teaches
train may be trucks. For another learner, the negatives may be handcars and streetcars.

2. Since the names of negatives have already been taught, the learner labels both negatives and positives in
the noun sequence. Instead of saying, “Not-pen” to indicate a negative of pen, the learner names the
negative: “Crayon.”

3. Because the learner names both positives and negatives, a new type of minimum difference may be present

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 in the noun sequences—the similarity of names.

The symbols b and v are not similar in shape; however, their names are minimally different. If the
learner already knows b when v is being taught, b would be a minimum difference negative based on
similarity of name. Note, therefore, that the learner might confuse positives and negatives of nouns either
because of similarity in name or because of similarity in features of the examples.

4. Because very small differences in examples are usually not shown in a noun sequence, the sequence is
usually fairly short, consisting of enough positive examples to show the range of variation and those
negatives that might be confused with the positives. The number of negatives and positives depends on the
range of positive variation and on the negatives known to the learner.

The Table 6.1 summarizes the major differences between non-comparative sequences and noun sequences.

Table 6.1
Non-Comparatives Nouns
Present minimum-differ- Do not present minimum-
ence negatives. difference negatives, but
the most highly similar
negatives already known
to the learner.
Usually require the learner Require the learner to label
to respond to negatives or name the negatives.
with “No” or “Not _____.”
Present examples through Present static examples (not
continuous conversion. created through continuous
conversion).
Call for responses that do Call for responses that may
not involve minimum-dif- involve minimum-differ-
ferences in names. ences in names between pos-
itive and negative examples.
Call for the same response Call for different responses
for all negatives. for different negatives
(because each negative is
labeled).

The Sequence

Figure 6.1 shows a sequence designed to teach truck to a four-year old. The learner is able to identify train, bus,
and car, which are the negatives in the sequence.

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 Figure 6.1
Example Teacher Wording

1 This is a truck.

2 This is a truck.

D
3 This is a truck.

4 What is this?
E1
5 What is this?
C
What is this? A
6

7 What is this?
C
8 E2 What is this?
C
9 What is this?
C
10 What is this?

11 What is this?

Juxtapositions

The A bracket shows that the wording principle is followed. Variation of the same wording is used for all
examples.

The B bracket is missing, because the setup principle is not followed. All objects are presented at the same time
and in the same place; however, there is no further attempt to increase the sameness shared by positives and negatives.

The D bracket shows that the sequence begins by showing sameness shared by positives. A noun sequence always
begins with positives. The reason is that information about negatives does not imply anything about the positives. Many
features change when we go from negatives to positives; therefore, the sequence does not begin by showing what the
noun is not. It begins with positives. Greatly different examples are juxtaposed and labeled in the same way. This
juxtaposition provides an idea about how examples are the same by showing some of the features they share.

The C brackets (showing difference) occur only in the test segment, after positives have been demonstrated. All
negatives are minimum-difference negatives (with respect to the learner’s knowledge). Therefore, each time a negative
is juxtaposed with a positive, a minimum-difference juxtaposition occurs.

The E bracket is divided into E1 and E2. The E1 bracket is a test of wording. It is a test of two of the examples that
had just been modeled in segment D. It assures that the learner can produce the labeling response for the positive
examples. Following E1 is the E2 test, which requires the learner to discriminate between objects labeled truck and

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similar objects with other labels (bus, car).

Minimum-Difference Examples

The reason for including only minimally different familiar examples in the sequence is this: If the learner can
discriminate between examples that differ in the smallest practical way, the learner should automatically discriminate
between examples that differ in larger ways. If we make sure that the learner can discriminate between truck and those
things or names known to the learner that are most highly similar to truck, we assure that the learner will not confuse
truck with any other familiar discrimination or label. Knowledge of the difference between a truck and a train
guarantees that the learner will never confuse a truck with a butterfly. Similarly, if the learner can discriminate between
a chair and a couch, the learner will never confuse a chair with a dog.

Test Examples

No new examples of truck are included in the test segment. The sequence would not suffer from a greater variety
of test examples; however, no generalization items are necessary in the test segment (if the range is adequately shown
in the D segment). If you are in doubt, add items to the sequence. The test sequence should be long enough to assure
that the learner can respond to the examples of the new noun when they occur within the context of familiar nouns.

Within the test segment, there should be more positive examples than negatives.

The Naive Learner

If the learner who is to be taught the noun truck is quite naive, we would use a different sequence from the one
above. If the learner did not know train, bus, and car, these would not appear as negatives. If the learner could identify
no objects that have reasonable similarity to truck (in name or in features of the example), we would simply select
some objects that are familiar to the learner and include them in the E2 segment of the series. These objects serve two
functions. They provide a more difficult juxtaposition for responding to truck in the test sequence. They also assure
that the learner is attending to details that discriminate truck from the other objects. Figure 6.2 shows a possible
sequence.

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 Figure 6.2
Example Teacher Wording

1 This is a truck.

2 This is a truck.
D
3 This is a truck.

4 What is this?
E1
5 What is this?
C
What is this?
A
6

7 What is this?
C E2
8 What is this?
C
9 What is this?
C

10 What is this?

11 What is this?

Remember, minimum differences are relative to the learner’s knowledge. Do not create them where none exist for
the learner. Although many objects are more highly similar to truck than hats and blankets, for this learner, hats and
blankets are highly similar for the learner being taught the discrimination.

If no close minimum-difference negatives are apparent, or if you are in doubt about the learner’s knowledge:

1. Present positives.

2. Follow with response test (E1).

3. Follow with a test segment that contains possibly two objects that are familiar to the learner. Create a
pattern of juxtaposition appropriate for a test segment (E2).

4. Keep the sequence as brief as you practically can.

Narrow Range Sequence

The number of examples in a noun sequence depends on the range of positive variation and the differences
between positives and negatives.

1. The range of positive variation the learner is expected to deal with. The range of positive variation for the
letter b is very narrow. We can write the symbol b larger or smaller, in bolder print, and with other minor

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 variations; however, the range of b is far less than the range of truck, cup, or shoe.

2. The degree of difference between examples of familiar concepts and the new concept. As a rule, add one or
two examples of familiar concepts to the E2 segment for every highly similar discrimination presented in
the sequence.

Figure 6.3 illustrates the procedure with the teaching of b. If the learner has been taught no letters that are highly
similar in name or in shape to b, but has been taught the letters r and s, we could use this sequence to teach the symbol
b:

Figure 6.3
Example Teacher Wording
1. b This is b.
2. b D This is b.
3. b What is this?
4. b
E1 What is this?
C A
5. s What is this?
6. r E2 What is this?
C What is this?
7. b
8. b What is this?

Two examples of b appear in E1 and two appear in E2. The sequence is short because the concept has a very
narrow range and the learner had not been taught any names or shapes that are minimally different from b. (The
sequence is a slightly shortened version of the noun sequence specified for truck).

If a learner had been taught the letters h and q, the sequence would be longer because symbols for h and q would
appear in the minimum difference test segment (E2). See Figure 6.4.

Figure 6.4
Example Teacher Wording
1. b This is b.
2. b D This is b.
3. b What is this?
E1
4. b What is this?
C What is this?
5. h
6. q C What is this?
7. b E2 What is this?
8. q
C What is this?
9. h What is this?
C What is this?
10. b
11. b What is this?

The D segment remains unchanged because the range of positives has not changed. The E2 segment of the
sequence is longer, however, because highly similar negatives were included. Note that h and q replaced r and s
because, for the learner being taught, h and q are the known symbols most highly similar in shape to b. The test of h

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and q is more elaborate than that of r and s because there is more reason to believe that the learner might reasonably
confuse h or q with b.

Differences in Features and Responses

The rules for sequencing different types of negatives in the E2 segment are:

1. Juxtapose minimum differences in features first.

2. Juxtapose minimum differences in name next.

3. Juxtapose minimum differences in name and features last.

The sequence involving b, h and q contains negatives that are different from b in shape only. If minimum
differences in name were included, they would be tested after the shape differences in the E2 segment.

By first dealing with shape difference in E2, we assure that the learner is attending to the features of the new
concept. Next, we introduce examples that may create name confusion. At this point, we know that any mistake the
learner makes is a name mistake, not a mistake of understanding the features.

Illustration. If the concept being taught is f and the negatives include j, t, and s, we would first present minimum
difference juxtaposition of f-j-t. These examples are minimally different in shape, but their names are not similar. Next,
we would present juxtapositions of f and s. These examples are not similar in shape, but in name. Since the set of t-j-s
does not include a member that is highly similar to f in both name and shape, rule 3 does not apply to the sequence.

Illustration. The concept being taught is d.

The learner has been taught:

a (Minimally different in shape only.)

t (Minimally different in name only.)

p (Minimally different in shape and name.)

Figure 6.5 shows the sequence. Note the juxtaposition of E2.

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 Figure 6.5
Example Teacher Wording
1. d This is d.
2. d D This is d.
3. d What is this?
4. a E1 What is this?
C What is this?
5. d
6. d What is this?
C
7. a What is this?
8. t E2 What is this?
C What is this?
9. d
10. p C What is this?
11. d C What is this?
12. p
C What is this?

The E2 segment first juxtaposes d and a, then d and t, and near the end of the sequence, the sequence begins to
alternate between examples of the new concept and negatives. Although this alternation is not desirable, it is far less
dangerous in a noun sequence than it is in the non-comparative. The reason is that the learner must produce a different
response for each example. Instead of saying “Yes” and “No” or saying “Slanted” and “Not-slanted,” the learner must
identify everything that is shown. Because the response requirement is stronger for the noun sequence (less probably a
function of guessing), the presence of the pattern is not a serious problem.

Dealing With Discriminations That Involve Small Minimum Differences

Because of the multiple-feature structure of nouns, they are usually easy to teach. The reason is that examples of
the concept possess many features that are not shared by any discriminations in the learner’s repertoire. If the learner
does not attend to a particular difference between the new discrimination and a familiar one, no problem is created
because there are usually many other differences available. All will serve to discriminate the positive (the new noun)
from the negatives (familiar nouns).

If differences between concepts familiar to the learner and the new concept are very small, the learner may have
serious problems learning the discrimination. As a rule of thumb, serious problems occur when highly similar names
are involved. Problems are also possible, however, when sets of examples highly similar in features are presented.
However, the most serious problems are those that involve examples similar in both name and features.

There are two ways to adjust the noun sequence for the learner who will probably have problems. The first
involves presenting the noun sequence a part at a time rather than all at once. The second involves adding some “easy”
items to the sequence, as well as presenting the sequence a part at a time.

Presenting the sequence a part at a time. Response problems are of ten predictable if the new discrimination calls
for a response highly similar to that for a familiar discrimination. If response problems seem probable, present the part
of the sequence that precedes the similar-name juxtapositions in E2. At a later time, when the learner is firm on the first
part of the sequence, present the entire sequence. For instance, instead of presenting the entire sequence that involves
the negatives d, a, and t, present examples 1 through 7 first. The learner is presented with d, tested on producing the

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response for d, and tested on discriminating between d and a. The learner is not yet required to deal with t (which is
highly similar to d in name) or p (which is highly similar in shape and name).

During the next session with the learner, repeat the first part of the sequence. If the learner performs reasonably
well on it, continue with examples 8 through 12 of the sequence. The point to remember is that the entire sequence does
not have to be presented during the first session that introduces the new discrimination.

Adding easy examples to the sequence. If the type of learning required by the material is dissimilar from any kind
the learner has achieved, anticipate more severe problems. If the learner has never learned from an adult in a teacher-
learner situation, has never learned to discriminate between highly similar objects (such as letters), and has never been
required to remember the various names for the objects, anticipate severe problems. The learner will probably exhibit
discrimination problems as well as response problems. The learner may make mistakes on examples that require
discrimination of a and d, which are similar in features, but not in label.

1. Include only those discriminations that are not similar to the new discrimination (d) in either shape or
label.

2. Insert these easy discriminations immediately after the E1 test.

3. End the sequence with the test of the new member (d) and the non-similar members.

Figure 6.6 illustrates these adjustments.

Figure 6.6
Example Teacher Wording

1. d This is d.

2. d This is d.

3. d What is this?

4. d What is this?

5. What is this?

6. d What is this?

7. What is this?

8. What is this?

9. d What is this?

10. What is this?

11. d What is this?

Ultimately, we want the learner to respond to d in the context of t and p. The sequence in Figure 6.6 presents d in

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the context of highly discriminable negatives—boat and scissors. Confusion is minimized by this context. Although
the context does not do much to facilitate the discrimination of d and p, it provides practice in producing the response
for d.

This easy discrimination would be repeated until the learner performs well on the letter d. At other times during the
day (or the lesson), the teacher continues to review p and t, but p and t do not occur in the same sequences as d. (More
information about the nature of these reviews is provided in later chapters. The point here is that the familiar
discriminations are kept firm while the new one is introduced within an easy context.)

The context in which d occurs becomes increasingly more difficult following success with the easiest context. The
learner is introduced to the shape differences (d and a). Next, the name differences are added (d, a and t). Finally, the
most difficult context is introduced: d, a, t and p.

We do not know the extent to which such a carefully staged introduction and modification of the context is needed:
that remains an empirical question. However, it is possible to adjust the context level of difficulty from a very easy
context to the terminal context. The context changes in the following ways:

• First, the new discrimination and non-similar negatives.

• Next, the new discrimination and negatives similar in shape only.

• Next, the new discriminations and negatives similar in shape, followed by negatives similar in name.

• Finally, the new discrimination, followed by negatives similar in shape, negatives similar in name, and
negatives similar in shape and name.

Throughout the introduction all negatives should be reviewed in a context that does not involve the new
discrimination.

Wide Range Nouns

The range of examples that we must show for a noun depends on what the learner knows and on the range of
variation for the noun. The concept dog involves examples that vary far more than examples of b. The easiest way to
show this wide range of variation is to include more positive examples in the sequence. The sequence illustrated in
Figure 6.7 begins with 3 positives, followed by a response-production test (E1) on 2 of them. New positives are
introduced later in the sequence as test examples.

The sequence in Figure 6.7 assumes that the learner has been taught cat, dog, and mouse.

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 Figure 6.7
Example Teacher Wording

1 This is a dog.

2 This is a dog.

D
3 This is a dog.

4 What is this?
E1

5 What is this?

C
6 What is this?

7 What is this?
C

8 What is this?
A
9 What is this?
C
10 What is this?
E2

11 What is this?

12 What is this?
C

13 What is this?
C
14 What is this?

C
15 What is this?

C
16 What is this?

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 The particular negatives in the C segments should not be inordinately similar to dogs. The learner should be
expected to discriminate between most small dogs and cats that are roughly the same size; however, we should not
search for cats that look unusually doglike.

The E juxtapositions first test examples that are minimally different in shape (cat and mouse versus dog) and later
in the sequence, examples that are minimally different in name: dog and frog. Note that the E segment is quite long (13
examples) because the range of variation of dog is large. Note also that there are more examples of dog than of not-dog
in the test summary.

The addition of new dogs to the E2 segment is somewhat arbitrary. The sequence would not be a communication
failure if none were added. Possibly the sequence could be modified so that it began with four models, instead of three,
and no additional new examples were added in E2. However, as a general guide, when a noun has a large range of
positive variation, add examples later in the sequence, if possible or practical.

Advanced Applications

The more the learner knows, the greater the amount of teacher wording that can be added to the sequence and the
fewer the examples needed to assure learning.

Suppose we wish to teach the discrimination between a mattock and other tools familiar to the learner, such as a
hoedad and a pick. Figure 6.8 shows a possible presentation.

Figure 6.8
Example Teacher Wording
The mattock has a
blade like a hoedad
on one side and like
a pick on the other.
This is a mattock.
What is it?
1 (mattock)

What’s this?
2 (pick)

What’s this?
3 (mattock)

What’s this?
4 (hoedad)

Suppose we wanted to teach the learner the difference between black oak leaves and white oak leaves. We could
begin with a rule that expresses the difference and follow with a series of examples that shows the difference.

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 “If the lobes have points, it’s a black oak leaf. If the lobes are rounded, it’s a white oak leaf.” (See Figure 6.9.)

Figure 6.9
Example Teacher Student
Wording Response

1 This is a
black oak.

2 This is a
white oak.

3 What’s this? White oak.

4 What’s this? Black oak.

5 What’s this? Black oak.

6 What’s this? White oak.

Suppose the learner is required to discriminate between black oaks (the new discrimination) and sycamores,
maples, and white oaks (the familiar discriminations). The rule for identifying black oaks in this context is: “Black oak
leaves have the same vein pattern as the white oaks, but not the same lobe features. Black oaks have the same pointed
lobe features as the sycamores and maples, but not the same vein pattern.” The rule points out the discrimination
problems confronting the learner. The learner must discriminate between some examples on the basis of lobes (white
versus black oaks) and discriminate between other examples on the basis of configuration or vein pattern (sycamores
and maples, versus black oaks). The sequence must provide a sufficient number of examples to show these differences
and to show the range of variation of black oaks. The sequence, therefore, must contain more examples than the
sequence for showing black oaks and white oaks.

As shown in Figure 6.10, for each minimum difference negative, at least one example is added to E2.

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 Figure 6.10
Example Teacher Wording Student Response

1 This is a black oak.

2 This is a black oak.

3 D This is a black oak.

4 What’s this? Black oak.

5 What’s this? Black oak.

C
6 What’s this? Sycamore

7 What’s this? Maple.

C
8 What’s this? Black oak.

9 What’s this? Black oak.

C A
10 What’s this? Sycamore.

C
11 What’s this? Black oak.

C
12 What’s this? Maple.

C
13 What’s this? Black oak.

14 What’s this? Black oak.

C
15 What’s this? White oak.

C
16 What’s this? Black oak.

C
17 What’s this? White oak.

C
18 What’s this? Black oak.

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 This sequence adequately shows the difference between black oak leaves and the others. Possibly, we could reduce
the number of examples if we preceded the sequences with a verbal rule, such as: “It is a black oak leaf if the veins do
not meet in one place and the lobes are pointed.” (We will discuss verbal rules later.) Note that the example with
minimum difference name (white oak) is presented near the end of the sequence.

The black oak sequence illustrates a critical design problem that comes about when several similar members are
taught. Nouns are multiple-dimension concepts. Each dimension of the new member is a possible basis for similarity.
The new member may be the same as familiar member A with respect to one dimension. It may be the same as familiar
member B with respect to another dimension and the same as familiar member C with respect to another dimension.

For each minimum difference, add one or more test examples. If the sequence becomes too laborious, divide it into
parts. Possibly, use verbal rules to call attention to the relevant details of the examples; however, do not use the verbal
rule as a substitute for the examples. The sequence must provide a sufficient number of examples in E2 to test the
learner’s application of the rule.

Higher-Order Nouns

A higher-order noun is one that has various subtypes, all of which have names. Vehicles is a higher-order noun that
includes various named subtypes—cars, trucks, etc. Games is a higher-order type that includes a diverse group of
named activities (from solitaire and chess to handball and baseball).

To teach a higher-order noun, use wording that preserves the critical nature of the concept. For the nouns that we
have dealt with, the negatives are coordinate with the positives. The situation is different with higher-order nouns. If
we present positive examples of vehicle and negatives that are labeled chair, swing and motor, we tend to imply that
these negatives are coordinate with vehicle. To make them truly coordinate, however, we must classify them according
to their “vehicleness.” To achieve this goal, we refer to positive examples of higher-order nouns as “vehicles” and the
negatives as “not vehicles.” The higher-order noun sequence, therefore, becomes a variation of the non-comparative
sequence. All positives are responded to in the same way. A different response (“No,” or “Not vehicle”) is used for all
negatives.

The learner may be familiar with lower-order class names for the positives that we present (boats, trucks, cars)
and for the negative examples (chair, swing, motor). The practice that the learner has received with these names may
have induced stipulation. The learner has learned one name, and the learner may resist calling the examples by another
name. The sequence for higher-order noun concept counteracts this problem by requiring the learner to classify a series
of juxtaposed examples as “vehicles” or “not vehicles.” If the learner responds that a positive example is “a car,” the
teacher simply repeats the question, “But is it a vehicle?”

Figure 6.11 presents a sequence that teaches the higher-order concept vehicle to a learner who may be familiar
with lower-order names for all the positives and negatives presented in the sequence.

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 Figure 6.11
Example Teacher Wording

1 This is a vehicle.

2 This is a vehicle.

3 This is a vehicle.

4 This is a vehicle.

5 This is not a vehicle.

6 This is not a vehicle.

7 This is a vehicle.

8 This is a vehicle.

9 This is a vehicle.

10 This is a vehicle.

11 This is a vehicle.

The sequence is a positive-first variation with six examples modeled. Negatives are selected to rule out possible
misinterpretations. The swing rules out the notion that sitting and moving is sufficient for something to be a vehicle.
The lawn mower rules out the possibility that moving and being motor-powered is adequate.

The same wording that is used for non-comparatives is used in the sequence. The teacher asks, “Is this a vehicle?”
for each test example. Also, the same juxtaposition pattern used for non-comparatives is followed. Following four
positive examples are the minimum-difference negatives, swing and lawn mower. Note that some new positives are
introduced in the test segment. The examples in the sequence are created non-continuously (through illustration).

Selecting Negatives for Higher-Order Noun Sequences

The higher-order noun sequence has the same basic form as a non-comparative sequence. Although the sequence is
the same for these two types of concepts, the concepts are different. Non-comparatives are usually single-dimension
concepts. Higher-order noun sequences process nouns, and nouns are multiple-dimension concepts. Because they are
multiple-dimension concepts, a positive example may be changed into a negative by manipulating various dimensions
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of the example. This fact carries implications for the negatives that we select for the higher-order sequence. We must
select negatives that show the multiple-dimension nature of the concept. To achieve this goal, we use different
negatives to rule out the possibility that a single feature is sufficient for the object to be a positive example. The last
black oak sequence presented some negative examples to rule out the possibility that pointed lobes was sufficient for
an example to be positive. Other negatives ruled out the possibility that a particular vein pattern was sufficient to make
an example a positive. The vehicle sequence followed the same formula for using negatives. Some examples ruled out
the possibility that movement is sufficient for an example to be a vehicle. Other negatives rules out the possibility that
having a motor is sufficient, or that accommodating one in a sit- ting position is sufficient.

As a group, the negatives must show that a set of features must be present before an example is a positive. The
strategy for showing the role of the various features derives from the difference principle: To show differences,
juxtapose examples that are minimally different and treat them differently.

Suppose the essential features of a noun are A, B and C. If we juxtapose an example of the noun with a negative
example having these features: A, B, S, we show that C is an essential feature. (The absence of C makes the example
negative.) If we juxtapose A, B, and C with negative B, C, S, we show that A is necessary also. If we juxtapose a third
positive with the negative (A, C, S), we show that B is necessary. Through this series of juxtapositions, we show that
the positive condition is brought about only through the presence of A and B and C. The series of juxtapositions shows
that all features are necessary.

The sequence for vehicles includes the examples boat, train, truck and car. If we present this set of positives to the
naive learner with no negatives, misgeneralizations are implied. All the examples move, therefore a possible misrule is
that anything that moves is a vehicle. All examples are associated with somebody sitting; therefore, another possible
misgeneralization is implied.

The negatives must counteract these possible misinterpretations by showing the learner that: (1) the examples do
move, but moving is not a sufficient feature; and (2) they may be associated with somebody sitting, but somebody
sitting is not a sufficient feature.

If we present negatives of someone sitting in a seat swinging on a swing, we rule out movement and sitting. Note
that the idea is not necessarily to find a single negative example that rules out all possible misrules. Identify different
negatives, each designed to rule out a misgeneralization based on one of the features (or possibly two). Also make sure
that your negatives are not questionable. For instance, are examples of someone on roller skates or someone on a pogo
stick negative examples of vehicles?

The vehicles that we present as positives have a power unit. To rule out the possibility that anything with a power
unit is a vehicle, we present a lawnmower (which also rules out movement).

Illustration. The concept games is difficult because games are diverse. All involve following a set of rules, a
criterion for completion, and some sort of contingencies or unpredicted events that affect the completion. For some
games, the contingency takes the form of chance, as in card games. For some, direct action of a competitor affects
performance of the opponent (as in basketball, where shots can be blocked). For others, the contingency is independent
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performance of a competitor (as in bowling). In this case, you may lose even though you perform very well.

If we were to teach the higher-order noun games to a learner, we would create a group of positives and negatives.
The positives would include a card game, a running game such as basketball, a game such as chess, and a contact game
like football. The negatives must show that the presence of rules is not sufficient for something being labeled game,
that a contingency or unpredicted outcome is not a game, or that the component motion (activity) is not sufficient for
something to be a game.

The sequence in Figure 6.12 is of interest only as a model. We selected games for this example because it has been
the subject of some philosophical discussions. In real life, any learner who knew a dozen games would probably
already know how to classify any example as a game. The teaching of the discrimination game would therefore be
completely unnecessary. The illustration below simply shows that very diverse subclasses of objects can be processed
through a sequence that shows the common features of the positives.

Figure 6.12
Teacher Wording
1. Is following traffic rules a game? No.
2. Is running a game? No.
3. Is running a relay race a game? Yes.
4. Is football a game? Yes.
5. Is playing cards a game? Yes.
6. Is shuffling cards a game?
7. Is following instructions a game?
8. Is basketball a game?
9. Is checkers a game?
10. Is counting stars a game?
11. Is fighting a game?
12. Is bowling a game?
Etc.

The sequence begins with two negatives, followed by three positives. All these examples are modeled. Following
is a series of seven test examples. The negatives rule out the notion that following rules is sufficient (examples 1 and
7), that component actions are sufficient (examples 2 and 6), and that scorekeeping or counting behaviors are sufficient
(example 10).

Regular noun sequences always begin with positives. Higher-order sequences are modeled after non-comparatives
and therefore may be either positive-first sequences or negative-first sequences.

Certainly examples could be added to the sequence and possibly better negatives could be created. However, the
sequence above meets the objectives of communicating the higher-order concept. It implies what all games have in
common, and it suggests that a set of common features is needed for an examples to be a game.

Below is a summary of the steps to follow when designing a higher-order noun sequence.

1. First determine that it is a higher-order noun sequence. If it is, it is composed of subtypes that have names.

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 2. Select a set of positives that show the various subtypes. Juxtapose 3 to 6 examples to imply the sameness
they share.

3. Determine the individual features that are shared by all the positives to be presented. Each feature suggests
possible interpretation or generalization. (If all positives move, then movement is a possible basis for
generalizing to new examples. If all positives involve action, the presence of action may become the basis
for a generalization.)

4. Construct a negative that rules out each individual feature. That negative will possess the feature to be
ruled out, but will be a negative. (An example that moves, but that is labeled as a negative, rules out a
generalization based only on movement. An example that involves action, but that is negative, rules out a
generalization based only on action.)

5. Design a sequence that begins with either positives or negatives. Juxtapose positives to show sameness.
Juxtapose positives and negatives to show difference. (Minimum-difference juxtaposition consists of a
positive and a negative that rules out a particular generalization.)

Model and test both positives and negatives.

Use non-comparative wording, requiring the learner to classify each example as a positive or negative
example of the higher-order category (e.g., “Is this a vehicle?”).

Noun labels for conglomerates. The higher-order sequence is based on the assumption that all members of the
higher-order set possess the same quality or set of features. It is possible to design “nouns” that are simply
conglomerates, nouns that cover a diverse group of entities that do not share a single quality or set of features. An
example is short vowels. Another example is sentence. The type of utterances designated as sentences bears no
compelling sameness. We could designate the letters a, f, g, m, u, and w as frombers. All other letters are not-
frombers. The higher-order sequences would not be particularly appropriate for teaching frombers because no
generalization is involved. A more appropriate procedure would be to present the group through a communication that
treats each letter as an essential feature of a particular system. This communication would present a visual-spatial
display. (See Chapter 14.)

Summary

The structure of nouns has intrigued philosophers, particularly the fact that diverse things can be classified
according to the same noun label. Clearly, the learning of nouns cannot be the simple process of “associating” name
with objects. Learning nouns involves learning the critical common features, with different features serving to
distinguish the examples of a noun from various negatives.

From the standpoint of communicating nouns, the important facts are:

1. Nouns are multiple-feature concepts.

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 2. The boundary line between positive and negative examples of nouns is imprecise.

To teach nouns, first decide whether you are teaching it as a higher-order noun. If it is not higher-order, follow
these steps:

• Present non-continuous examples.

• Use negatives that are known to the learner.

• Design the sequence with the test question, “What is this?”

• Always begin with positives.

• Juxtapose the positives to show sameness (great difference between juxtaposed examples). For wide-range
nouns, show at least three examples. For narrow-range nouns, show one or two examples.

• Follow positives with two types of tests—response test followed by discrimination test. The response test
(E1) presents examples that have been shown and requires the learner to label the examples in response to
“What is this?” The discrimination test (E2) introduces negatives.

• Include only minimum-difference negatives for the sophisticated learner in E2.

First test on negatives that are similar in shape.

Then test on negatives that are similar in name.

Then test on negatives that are similar in name and shape.

• For non-sophisticated learners, modify the sequence. Possibly divide it into parts and present the first part
through the E1 test, until the learner is facile; then add the remaining parts. Or, change the sequence by
adding negatives that are not similar to the positive. When the learner is firm on this difference, add the
minimum-difference negatives.

Add new negatives to E2 if the noun is a wide-range noun.

Add one or two positives to E2 for every minimum-difference negative.

• If you do not know whether the learner knows a particular negative, or if the sequence is to be presented to
a group, either teach the negatives first or play it safe by eliminating all negatives that would be
questionable. (The sequence will fail if the teacher must correct on negatives that are supposed to be
known.)

• If the noun is higher-order, follow the same general procedures used for non-comparatives that are
presented non-continuously. Use a question form that requires classification. (“Is this a ___?” or, “Tell me
___ or ___.”)

• Begin the sequence with either positives or negatives.

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 • Design the negatives to rule out possible misrules.

• Select positives that sample from among the various subtypes that have names.

• Present an adequate number of test examples.

To show that a positive example must have multiple features, juxtapose negatives and positives that have a
particular feature. This juxtaposition rules out the possibility that the single feature is adequate for an example to be
positive.

Above all, remember that nouns have uncertain boundaries. Do not become embroiled over those examples that are
questionable. If an example is not clearly negative or not clearly positive, do not use it. The communication with the
learner is designed to convey what convention has already established, not to create classification conventions or
precise boundaries where none exist.
 

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 Comparative Single-Dimension Concepts
Comparatives include getting wider, getting hotter, getting colder, getting heavier, and similar concepts. The
concept full is a non-comparative: Getting fuller is a comparative. Green becomes a comparative when we change the
concept to getting greener. Similar conversions can be made for any non-comparative: happy, worried, shiny, rough,
troublesome, abrupt, and slanted. In Chapter 5 the concept abrupt was presented as a non-comparative. By changing
the concept to more abrupt, it becomes a comparative.

Comparatives are single-dimension concepts; therefore, they can frequently be processed through continuous
conversion.

Comparatives have precise boundary lines between positives and negatives. Therefore, the communication should
show minimum differences.

Comparatives are relative. The communication should demonstrate this relative nature.

The structure of comparatives indicates that they are quite similar to single-dimension non-comparatives. This
similarity is deceptive. For non-comparatives, we label a particular event as the example in a continuous conversion
sequence. For comparatives, we label a change from one event to the next. An event presented in a non-comparative
sequence as a positive example is always a positive example. An event presented in a comparative sequence as a
positive example, such as an object that is slanted 45°, is not always a positive example. Positive examples of a
comparative always assume a reference point. A “small” star is not “large” or “larger” unless it is compared with other
stars. An image that is “more brilliant” assumes some other images as the reference point.

The unique structure of comparatives suggests differences between the sequences designed to communicate
comparatives and those for teaching non-comparatives and nouns. The following statements describe comparative
sequences that use continuous conversions:

1. The comparative sequence always begins with a starting point—an example that is neither positive nor
negative.

2. Every positive or negative example in the sequence is compared to the preceding example (with example 1
being compared to the starting point).

3. The change that occurs from example to example is the basis for determining whether the example is
positive or negative.

4. When no change occurs, the example created is negative.

As point 4 above indicates, the no-change negative is unique to the comparative sequence. When comparatives are
presented through a series of continuous-conversion examples, a change must occur to create a positive example. If we

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say, “It got wider,” a change in width from the preceding example had to occur. If we say, “It got heavier,” there was a
change in weight from the preceding example. However, if we say, “It didn’t get heavier,” it is possible that the weight
did not change at all. This possibility is the basis for the no-change negative. Two types of minimum-difference
negatives are therefore used in the comparative sequence: the minimum-difference negative change and the no-change
negative.

Negative-First Comparative Sequence

Figure 7.1 shows a model for the negative-first comparative sequence. The concept being taught is getting
heavier. The learner’s hand is placed, palm up, on a table surface. The teacher’s finger presses downward against the
learner’s palm to create the sensation of the finger getting heavier and non-heavier. (The reason for placing the
learner’s hand on the table surface is to prevent it from moving down. We do not want the presentation to be consistent
with the misrule that “getting heavier” means that it moves down further.)

Figure 7.1
Example Teacher Wording Example Type
Starting point: Pressure 3 “Feel this.”
1. Pressure 2 “It didn’t get heavier.” –
2. Pressure 2 “It didn’t get heavier.” –
Pressure 3
C “It got heavier.”
3. +
4. Pressure 7 D “It got heavier.” +
5. Pressure 9 “Did it get heavier?” +
C
6. Pressure 9 B “Did it get heavier?” – A
7. Pressure 5 “Did it get heavier?” –
8. Pressure 6 “Did it get heavier?” +
E
9. Pressure 3 “Did it get heavier?” –
10. Pressure 4 “Did it get heavier?” +
11. Pressure 2 “Did it get heavier?” –
12. Pressure 5 “Did it get heavier?” +
13. Pressure 10 “Did it get heavier?” +

The amount of weight or pressure is indicated in the middle column in Figure 7.1. The weight is indicated on a
scale of 1-10. These units do not refer to actual weight. They are arbitrary assignments designed to show how hard the
teacher is pushing down with each example and the change in downward pressure from example to example. The least
pressure that will be presented is 1. The greatest pressure is 10. A change from 4 to 5 is a slight change. A change from
4 to 8 is a fairly large change. Note that the teacher does not mention these numbers. The teacher says what appears in
the third column.

The teacher begins by applying a pressure of 3 and saying “Feel this.” This starting point is not a positive example
or a negative example. It is a “starting point,” a basis for showing the change that is to follow. The teacher then changes
the pressure slightly (to 2), creating example 1. “It didn’t get heavier.”

Present the sequence to yourself, pressing in your own palm. You are not expected to maintain precise values for
each of the pressures specified in the sequence; however, try to approximate each value. (Try to keep the value of 5

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almost in the middle of the range of positive variation in your examples.)

The brackets A, B, C, D, and E are quite similar to the brackets for a negative-first non-comparative.

Wording. The A bracket shows that the wording for all examples refers to “heavier.” Note, however, that the first
pressure presented is not referred to as “heavier.” The wording merely calls attention to the “feeling.” The change from
this example to the next is the basis for labeling example 1.

Setup. The B bracket indicates that all examples are created through the same setup—the finger creating pressure
on the learner’s palm.

Difference. All negatives modeled to show difference in a comparative sequence (through example 6) involve
either a small change or no change from the preceding example. Example 2 is negative. It is created through a small
change from example 1; example 1 is created through a small change from the “starting point” stimulus. This
convention is unique to comparatives. Note that when we construct a negative-first non-comparative sequence, we
begin with two negatives, each of which is minimally different from some positive, but not necessarily minimally
different from each other. When we construct negative-first comparatives, however, both negatives at the beginning of
the sequence are related to each other—created through a small change or through no change. A small change is used
from negative example 2 to positive 3 and no-change from example 5 to example 6 (C brackets).

Sameness. The D brackets indicate that comparatives follow the same basic pattern used in the non-comparative
sequence. Three positive examples are sandwiched be- tween minimum-difference negatives. Examples 3, 4 and 5
show the range of positive variation within the constraints of the setup. The amount of change from example to
example is not the same and is not progressive. If the same increase is shown from positive example to the next
positive, a misrule is implied. Examples 3, 4, 5 below are unacceptable:

Example Pressure Change

2 3 –
3 5 +2
4 7 +2
5 9 +2

Each positive example is two units heavier than the preceding example. The misrule implied is that the increase
must be of a certain amount for the example to be positive. Also, the sameness shared by positives is not clearly shown
if the changes are ordered this way:

a change of 1

a change of 2

a change of 3

This ordering is progressive, implying that a given change must be progressively greater than the preceding change if
the example is to be positive.

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 In the model sequence, the changes between examples 2 through 5 are 1, 4, 2—a small change, a quite-large
change, and an intermediate change. Follow this pattern for all comparatives. It implies that any change in the positive
direction creates a positive example.

Test items. Examples 5 through the end of the sequence are test items. For each, the teacher asks, “Did it get
heavier?” The test segment could easily be reduced to seven examples. Note that it contains various positives, large
negative changes, and minimum-difference negatives.

Summary. The steps to follow to construct a negative-first sequence for comparatives are:

1. Establish the “starting point” of your sequence slightly below the middle of the range you wish to
demonstrate. If your range of variation is to be from 1 through 10, start with 3 or 4. If you are going to
extend your arms all the way for the extreme example of getting wider, start with your arms slightly less
than half-extended. If the extreme example of getting fuller is a glass that is entirely full, begin with an
example that is slightly less than half-full. If your starting point is too high, you cannot show a great range
of variation in the first positive examples that are designed to show sameness—examples 3, 4, and 5. If
your starting point is 1, you cannot show a minimum-negative change.

2. For the negative-first sequence, open with two negative examples—a no-change negative and a minimum-
negative change. It does not matter which comes first.

3. Follow the second negative with three positives. Each should differ from the preceding examples in: (a)
absolute value, and (b) the amount of change required to create the example. If you were demonstrating
getting wider, you would show an example that got wider through a very small increase in width, an
example that got wider through a very large increase in width, and one that got wider through an
intermediate-sized increase in width.

4. Make example 6 a negative that is minimally different from the preceding positive. This negative may be a
no-change or a minimum-negative change. A minimum-negative change from a value of 7 is a value of 6;
a minimum-negative change from something that is 50 centimeters long is something that is 48
centimeters long. To create a no-change negative, retain the same value as that of the preceding example.
Example 6 in the getting heavier sequence is a no-change negative (example 5 has a value of 9, example
6 has the same value).

5. Follow with a series of unpredictably sequenced test examples.

a. These should show some large differences, some small differences, and some no-changes.

b. These should sample the range. (In the sequence for heavier, the test segment spanned a range from a
pressure of 2 to a pressure of 10.)

c. At least one value should be repeated as a positive and as a negative example. (In the test segment for
heavier, the pressure of 5 is a positive example and, later, a negative example. Also, the pressure 9 is

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 presented both as a positive example and as a negative.)

6. Limit the range of values shown to 1-10. Do not present examples of no-pressure, for instance, or no-
width, or no-fullness, or no-speed. The basic question, “Did it get wider?” or “Did it get fuller?” does not
make much sense with these examples. It did not get wider, but it has no width: therefore the question is
not entirely fair for the introductory sequence. Save it for later amplifications of the concept.

Sensory Qualities and Perceptible Differences

The two most common mistakes that designers make when dealing with comparatives are: (1) to present indirect
sequences that do not teach the basic sensory quality, and (2) to become preoccupied with “measuring” the absolute
size of the “minimum differences.”

Indirect sequences. When teaching the naive learner, do not use numbers that refer to measurement units. To teach
wider, do not give the learner information about numbers such as, “The stream was five feet wide. Then it was six feet
wide.” These examples do not convey information about width to the naive learner. They convey information about
numbers. If the learner knows how something looks or feels when it gets wider, then of course the learner can work
from the numbers. But remember this about numbers: The same numbers may refer to any measurable relationship.
“The stream was 4 glums. Then it was 5 glums.” This information does not clarify the meaning of glum. (Does glum
refer to a particular type of eddie on the surface, the speed, the width, the amount of pollution, the presence of sludge
worms, or the temperature of the stream?) Numbers tell nothing unless you understand the meaning of the units. To
teach these units, you must show changes in relevant dimensions. The basic sensory quality of heavier is not the
appearance of a scale moving down. It is the feeling of downward pressure. The basic sensation of hotter is not
something that is observed visually. The teaching of hotter, therefore, would involve examples that are felt. Steeper is
something that can be felt or seen. Therefore, it can be presented either visually or tactually. Greener is something that
is perceived visually and must be taught that way. Remember, a blind person can deal with measurements of green and
deductions associated with green. (If it starts out with a greenness value of 4 and goes up to a greenness value of 6, it
got greener.) However, after receiving information about green, a blind person knows no more about the basic quality
we label green than before.

Perceptible differences. To construct minimum differences in the comparative sequence make sure that the
differences are small, but quite perceptible. In the sequence for teaching heavier, the difference between a downward
pressure of 3 and one of 4 is small but is assumed to be perceptible. Also, the difference between a downward pressure
of 8 and one of 9 is small, but perceptible.

If we were teaching the concept getting hotter, for example, and were using a faucet that mixes hot and cold
water, we would try out the sequence and determine small differences that are clearly perceptible. These small
differences are not a simple product of so many physical units. The fact that we turn the faucet a certain amount (such
as 1/12 of a turn) does not count for anything if the results of this physical manipulation is an imperceptible difference.
Furthermore, we do not make assumptions about the “equality” of units. To make a perceptible change in the
temperature range of cold water does not necessarily require the same physical manipulation as that required to make a

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perceptible change with hotter water. The question is empirical. The goal is to identify differences that are perceptible
and relatively small. Do not try to identify the absolutely smallest difference that you can perceive. This difference
leaves no margin of error. Try a difference that is slightly larger.

Wording

The teacher models the first four or five examples and then tests on the remaining examples. (In the sequence for
heavier, the teacher modeled only four.) For modeling positive examples, the teacher says, “It got ___,” or answers a
question, “Did it get ___? Yes.”

An addition to the teacher wording is required by the starting point. The teacher presents the starting point example
and says, “Watch,” or “Look at this,” or “Feel this.” The purpose of this instruction is simply to call the learner’s
attention to the example and to the subsequent change.

The test wording may be designed to call for “yes-no” responses. “Did it get ___?” A stronger task is one that
requires the learner to use the new word being taught. “Tell me hotter or not-hotter.”

After presenting example 5, the teacher may give general instructions that apply to all test examples: “Your turn.
Tell me if it gets heavier or not-heavier.” For each test item, the teacher would then ask: “What happened?” or would
say, “Tell me.” The learner would respond, “It got heavier,” or, “It didn’t get heavier.”

Illustration. A sequence for the concept getting hotter is shown in Figure 7.2. The setup is a stream of water from
a faucet. The learner’s hand is in the stream. The faucet is obscured from the learner’s view so that the learner cannot
see the teacher’s manipulation of the hot and cold water handles. The units (1-10) referred to in the sequence cover a
range of temperatures from water that is very cold to water that is on the verge of being uncomfortably hot (10).

Figure 7.2
Example Teacher Wording Type
Starting point: Heat Level 4 “Feel this water.”
1. Heat Level 4 “It didn’t get hotter.” –
2. Heat Level 3 C “It didn’t get hotter.” –
3. Heat Level 4 “It got hotter.” +
4. Heat Level 7 D “It got hotter.” +
5. Heat Level 9 “It got hotter.” +
General instructions for test:
C “Your turn. Tell me whether the
water gets hotter or not hotter.”
6. Heat Level 8 B “What happened?” – A
7. Heat Level 10 “What happened?” +
8. Heat Level 4 “What happened?” –
9. Heat Level 1 “What happened?” –
10. Heat Level 4 E “What happened?” +
11. Heat Level 2 “What happened?” –
12. Heat Level 4 “What happened?” +
13. Heat Level 5 “What happened?” +
14. Heat Level 8 “What happened?” +

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Variations in the Negative-First Sequence

Although the sequence in Figure 7.2 is a negative-first comparative (like the sequence for getting heavier), and
although it follows the same principles of juxtaposition, it differs from the earlier one in five ways:

1. The starting point of this sequence is a higher value (4) than the starting point of the sequence for getting
heavier; furthermore, the values throughout the sequence are not identical.

2. The pattern of no-change and minimum-change negatives is different. In the getting-heavier sequence, the
first negative (example 1) is a minimum-negative. In the getting-hotter sequence, the first negative is a
no-change. Also, example 6 in the getting-heavier sequence is a no-change, while example 6 in the
getting-hotter sequence is a minimum negative.

3. The number of examples modeled differs, with four items modeled for getting-heavier and 5 for getting-
hotter.

4. The teacher wording is different, with the getting-heavier sequence calling for “yes-no” responses (“Did
it get heavier?”) and the getting-hotter sequence requiring the learner to use the name for the concept
(“What happened?”). Associated with the use of this strong-response question for the getting-hotter
sequence, general instructions are needed to show the learner which responses are expected. Without
these, the learner might attempt to talk about “cooler” or “the same hotterness.” The desired responses are
“It got hotter,” or “It didn’t get hotter.”

5. The number of examples in the sequences differ—13 in the getting-heavier sequence and 14 in the
getting-hotter sequence.

Despite these differences, both sequences follow the same pattern of juxtapositions. The differences between them
demonstrates the latitude that is available to the instructional designer, even within the constraints of a fairly “tight”
formula.

Positive-First Comparatives Sequences

If the learner is to be taught six comparative concepts, approximately half should be presented through negative-
first sequences and half through positive-first variations. Like the non-comparative positive-first sequences, the
comparative positive-first initially shows how positives are the same, and then presents minimum differences, and
finally tests. Figure 7.3 shows a positive-first presentation of the concept closer together. The setup is the teacher’s
hands, which are shown closer or farther apart. Note the general instructions after example 5 in the sequence.

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 Figure 7.3
Example Teacher Wording Type
Width of Hands

Starting point: “Watch my hands.”

1. “They got closer together.” +

2.
D “They got closer together.” +

3. “They got closer together.” +

C
4. “They didn’t get closer together.” –
B
5. “They didn’t get closer together.” –
“Tell me whether or not my hands
C get closer together.”

6. “What happened?” +

7. “What happened?” –

8. “What happened?” +

9. E “What happened?” +

10. “What happened?” –

11. “What happened?” +

12. “What happened?” –

13. “What happened?” +

Setup Features. For this sequence, the teacher moves only one hand. The changes could have been created so that
each change involves two hands. However, to be consistent with the setup principle, the same sort of change used for
one example should be used for all. If one hand moves to create one example, one hand must move to create all
examples. If a two-hand change creates one example, that change must create all examples that involve a change.

Sameness. To show sameness, the same pattern used for the other sequences is presented in examples 1 to 3: a
small change (example 1), a large change (example 2), and an intermediate change (example 3).

Difference. Two minimum differences are modeled, one that involves a minimum-negative change and one that
involves no change. Note that all changes occurring in examples 3-6 of the positive-first comparative sequence are
small.

Test items. The test items present a variety of types.

Advanced Applications

Advanced applications possibly involve two-choice wording, fewer examples, and a verbal explanation.

Two-choice wording. Many comparatives involve the words more and less. Something becomes more intricate or

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less intricate. It becomes more strongly accented or less strongly accented. If the learner has demonstrated ability to
perform on some concepts that involve changes labeled more or less, two-choice variations in the test question can be
presented for comparatives expressed with more ____ or ____ less. If the learner is not facile with other concepts that
refer to more or less, the two-choice variation should not be used.

Verbal explanation. Any explanation provided by the teacher should not be a highly-detailed description or a
substitute for the examples. If the particular concept can be explained in a sentence (or two), the verbal explanation
may precede the sequence. “I’ll show things that get more involved. When something becomes more involved, it gets
more parts.” The explanation is not an attempt to exhaust the possible meanings of getting more involved, nor to
specify every aspect of a particular meaning. The examples are expected to provide the details.

Number of examples. With sophisticated learners, 6 to 8 examples are usually enough. Two or 3 are modeled,
followed by some test examples.

Illustration. Figure 7.4 shows a possible sequence designed to teach an advanced learner the meaning of more
permeable. It incorporates verbal explanation, reduced number of examples, and a two-choice task.

Note these points:

1. Only six examples are used in the sequence.

2. The teacher models one positive and one negative.

3. Test items include minimum negatives and a fair range of positives.

Figure 7.4

Teacher: “When something is more permeable, liquid moves through it faster.”

Teacher then presents an adjustable filter on the end of a hose. Initially, the water flows at the rate of 4.

Example Water Flow Teacher Wording Student Response

Starting point: 4 Watch the water flowing through this
filter. I’ll tell you what happens.
1 5 The filter became more permeable.
2 4 The filter became more permeable.
Your turn: Tell me if the filter becomes
more permeable or less permeable.
3 5 “What happened?” It became more permeable.
4 4 “What happened?” It became less permeable.
5 8 “What happened?” It became more permeable.
6 10 “What happened?” It became more permeable.

Two-Response Comparative Sequences for More Naive Learners

When dealing with more permeable, less permeable, the sophisticated learner actually deals with only one new
word that is in a familiar context. When providing responses for bigger-smaller, however, the learner is required to use

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two words, both of which are new. The learner must also remember which new word goes with which observed change
in the examples.

For teaching pairs such as, faster-slower, bigger-smaller, older-younger, or happier-sadder to naive learners,
first introduce one of the labels (slower, not-slower). After the learner has applied this label in a range of activities,
introduce the two-response sequence. The label taught earlier (slower) would be the negative for the new sequence; the
new term (faster) would be the positive. The sequence would contain a fairly large number of examples to assure that
the learner receives practice in applying both words. The general instructions would appear after positives and
negatives are modeled: “Tell me if it moves faster or slower . . . What happened?”

Indirect Sequences

A direct sequence is one in which we present the learner with actual examples of the concept being taught. For
steeper grade, we present examples that show the changes in the grade; for more abrupt, we present examples that
show the abruptness the learner is to label. An indirect sequence does not present the actual sensory qualities for each
example. Instead, a description of the quality is presented. The description often involves numbers. Indirect sequences
should not be presented to the naive learner. If the learner knows the sensation that is associated with getting hotter,
and understands that the temperature refers to how hot something is, we can use measurement numbers instead of
sensory examples to teach concepts.

The following sequence teaches the concept temperature increasing to a sophisticated learner:

Teacher Wording
Listen. The temperature starts out at 45 degrees. Then
it goes to 46 degrees. My turn: What happened to the
temperature? It increased.
Then it goes from 46 degrees to 73 degrees. My turn:
What happened to the temperature? It increased.
Then it goes from 73 degrees to 73 degrees. My turn:
What happened to the temperature? It didn’t increase.
Your turn to tell me if the temperature increases or
doesn’t increase. It goes from 73 to 72 degrees. What
happened to the temperature?
It goes from 72 to 91 degrees. What happened to the
temperature?
It goes from 91 to 91 degrees. What happened to the
temperature?
It goes from 91 degrees to 134 degrees. What happened
to the temperature?

The sequence is a positive-first sequence that is modeled after the presentations for communicating basic sensory
qualities. The differences are that:

1. Fewer examples are modeled and tested.

2. The examples are described verbally by the teacher.

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 3. Numbers are involved in the description of what happened.

Note, however, that the same pattern for showing sameness and difference used in earlier sequences applies to this
sequence.

We could use a variation of this presentation for teaching greater height, more permeable, longer distance,
faster, and other concepts for which conventional measurement units are available. The sequence can be designed as a
negative-first or a positive-first sequence. Numbers, however, should not be used unless the learner is familiar with the
sensory quality that is labeled by height, permeable, distance, faster, etc.

Other indirect sequences may be designed to illustrate the nature of the concept. For instance, if we were to teach
the concept of momentum, we might begin with a sequence that teaches how momentum works. To achieve this goal,
we might alter the concept so that it deals with more momentum and less momentum. The setup involves something
moving and a means for measuring how resistant it is to stopping.

Teacher Wording
I’ll tell you about a car that drives into a brick wall.
Listen. The first time the car hits the wall, it goes through
four feet of brick before it stops.
The next time it doesn’t go through four feet of brick, it
goes through three feet of brick before it stops. It had less
momentum this time.
The next time, it doesn’t go through three feet of brick. It
goes through four feet of brick. It had more momentum.
The next time, it doesn’t go through four feet of brick.
It goes through seven feet of brick. Tell me about the
momentum.
The next time, it doesn’t go through seven feet of brick, it
goes through five feet of brick. Tell me about the momen-
tum.
The next time, it doesn’t go through five feet of brick, it
goes through nine feet of brick. Tell me about the mo-
mentum.

This sequence is basically the same as the others that we have developed. The wording is different, to assure that
the learner will remember the two numbers that are involved in the comparison called for by each example.

A simpler variation of this sequence could be designed so the teacher draws a line into a representation of a brick
wall to indicate how far into the wall the car moves on each trial.

Here’s how far the car goes

into the wall the first time:
Here’s how far the car goes
into the wall the next time:
The car had more momentum.
Etc.

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Note that for the second example, the teacher simply adds to the line or subtracts parts of it. The teacher does not draw
a new line. The choice of which sequence to use is somewhat arbitrary. The verbal presentation has the advantage of
forcing the learner to operate from number descriptions. If these descriptions are to be used later, the verbal description
is probably preferable. The sensory presentation (using the line into the wall) provides a more compelling presentation.
If there is some doubt about the learner’s facility with the number descriptions, this sequence is preferable.

Identifying and Analyzing Comparatives

Some comparative concepts are not immediately apparent. The concept of increasing is a good example.
Increasing means getting more of something (increasing speed, increasing duration, increasing other measurable
qualities). The simplest way to analyze a concept to determine whether it is comparative is to see if it has the attributes
of a comparative. To apply this strategy to increasing, start with an example of something increasing in some
dimension. After you present the example, look at the display that you have created. (After showing increasing in
width, look at the distance between your hands after you have shown the increase.) Ask: “Could that same outcome
also be used as a negative example of increasing? It could. Therefore, the concept is a comparative. When we apply
the same strategy to the concept of a fraction that is more than one, we see that although the label uses the word more,
the concept is non-comparative. A particular concrete example of a fraction that is more than one is always an example
of more than one—never of less than one.

Remember, start with a positive example of the concept; ask if the event that is presented in the example could
possibly be used as a negative example of the concept. If the event could be positive in some cases and negative in
others, the concept is comparative regardless of the wording used to describe the concept.

When to Make a Concept a Comparative

It is possible to change many non-comparatives into related comparative concepts. Running is a non-comparative
discrimination, but running faster is a comparative. Near is a non-comparative; nearer is a comparative. Properties
that are expressed through adjectives are particularly amenable to being converted into comparatives.

Happy is a non-comparative. Happier is a comparative. Angry is a non-comparative. Angrier is a comparative.

High-pitched is a non-comparative. Higher-pitched is a comparative.

When we convert a non-comparative into a comparative, we change the concept. A non-comparative requires the
learner to classify the example as a positive example of the concept or as a negative. The difference that the learner
must attend to is the difference between those qualities of the example that make the example positive and those that
make it negative.

When we change the concept to a comparative, we deal with an entirely different difference. The difference now
has to do with the kind of change that occurred. The simplest way to see how the concept changes when we convert a
non-comparative concept into a comparative is to look at the negatives. The negatives for a non-comparative sequence
are non-examples of the concept. A negative of the concept moving is not-moving. It has no movement. The negative
of moving faster has movement, however. It moves, but not as fast as an example of moving faster. So it is with all
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comparatives. Below is a comparison of three other concepts:

Non-comparative

Positives Negatives
turning not-turning
running not-running
typing not-typing

Comparative

Positives Negatives
turning sharper turning, but not sharper
running farther running, but not as far
typing faster typing, but not as fast

The best test of whether the most appropriate sequence is a comparative or a non-comparative has to do with the
difference that we want the learner to understand. If we want the learner to learn the difference between running and
other actions that are not running, a non-comparative sequence is implied because it shows this difference. If our goal is
to show the different speeds of running or different types of running, the difference implies a comparative (running
faster, or running with larger steps, or running with more arm movement, etc.).

Comparative sequences are appropriate for showing changes within a type of action, quality, or event.
Non-comparatives are appropriate for classifying a type of action, quality or event. Discriminations such as big-little
(or big, not-big) may be taught with non-comparatives (with the big examples all the same size and all the little
examples a different size). If the discrimination is taught as a comparative, the concept changes to bigger. The
examples would show the basis for something becoming bigger or not-bigger. This sequence would actually show the
basis for the concept better than the non-comparative sequence. As a rule, all polars (wet-dry, fast-slow, fat-skinny,
loud-soft, etc.) are best presented as comparatives, introducing only one label in the initial-teaching sequence.

Identifying Concepts as Comparatives, Nouns, or Non-Comparatives

As noted in the earlier chapters, if we start with a sentence we can usually determine the type of concepts that are
named in the sentence. Once we classify a concept, we know the basic procedures involved in communicating that
concept faultlessly to the learner.

Below is a sentence that might be used to describe a procedure for reaching the top of a hill in a motor-driven
vehicle.

If you don’t make it to the top of the hill, back the vehicle all the way down and go faster before you reach the
steeper grade.

We can quickly identify some words or phrases.

The top of something is a non-comparative.

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 Hill is a noun.

Back or back down is a non-comparative.

Vehicle is a noun.

Go faster is a comparative.

Before something happens is a non-comparative.

Reach something is a non-comparative.

The steeper grade is a comparative.

Each of these concepts that we identify would be taught so that the meaning conveyed in the sentence is
transmitted through the initial-teaching sequence. For hill, we would present a noun sequence. The negatives might
include valley and mountain (depending on the assumed knowledge of the learner).

For back down, we would present a non-comparative sequence. The negatives should include the vehicle backing
up an incline and the vehicle going forward down an incline. The sequence, however, is a regular non-comparative
sequence with the test question: “Did it back down?” presented with each test item.

Go faster is a comparative. The sequence used to teach this concept would show some object moving. The speed
of the object would change from example to example. For the test examples, the learner would be required to label each
example as either moving faster or not-moving faster (or answering the question: “Did it move faster?”).

Reach something is a non-comparative. To communicate the concept, we could present a setup that shows an
object moving toward something, like a truck moving toward a lake. The truck either reaches the lake or does not. The
learner responds to “Did it reach the lake?” for each test example. (Some positive examples would show the truck just
reaching the water line. Others would show the truck continuing farther into the water. Note that these examples would
be presented through a diagram or by tracing the route of the truck on a picture that shows a road leading to a lake.)

Not all the concepts named in the sentence are best presented as non-comparatives, comparatives, or nouns. The
concept before something happens may be presented as a non-comparative; however, it is best presented as a
transformation (discussed in the following chapters).

In addition to identifying the single concepts named in the sentence, we can also identify the groups of concepts,
described by the phrases in the sentence. For instance, instead of teaching top and hill as separate concepts, we could
simply teach top of the hill. The concept is a non-comparative. It could be presented with a picture of the hill. For each
example, the teacher touches part of the hill. For each test example, the teacher asks, “Did I touch the top of the hill?”
Note that this concept is communicated through the basic sequence used for other non-comparatives. The fact that the
label involves more than one word is irrelevant. The concept is a non-comparative with minimum negatives and a
possible range of positive variations. The range of positive variation depends on the shape of the hill. If there is some
clear boundary for the top (a flat place that is not so extensive that the hill becomes a plateau), a range of positive

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variation is possible.

Also, we could teach the concept of make it to the top of the hill. This concept is a non-comparative. The
sequence is similar to that for reach something. The difference is that a hill is involved.

Back all the way down is another non-comparative. The negative examples would show backing but not all the
way down, going down all the way but not backing, and backing all the way up the hill.

Finally, we could teach Go faster before reaching the steeper grade as a comparative concept. The negatives
would show going the same speed (as the previous trial) before reaching the steeper grade, going slower before
reaching the steeper grade, and possibly starting out fast but going slower before reaching the steeper grade.

If we understand the meanings conveyed by words and phrases of a sentence—any sentence—we can classify the
concepts that are conveyed. We can immediately classify many of the individual words as comparatives, non-
comparatives, and nouns. Furthermore, we can frequently group the words that occur in phrases (or possibly clauses)
and classify each group of words as a comparative or as a non-comparative. Note that groups of words would rarely be
classified as nouns. The phrase, “the little boy with the red cap” names several nouns, but is a non-comparative
concept. The negatives are a little boy with a cap that is not red, a not-little boy with a red cap, a little girl with a red
cap. By applying the test for determining whether the concept is a comparative or non-comparative (Is a positive
example always a positive example?) we can readily determine how to classify the concept. To classify the word,
phrase, or clause is to understand how to convey the meaning of that expression in a faultless manner. The power of the
basic-form sequences becomes apparent when we recognize that any sentence is composed of basic-form concepts. In
later chapters we will consider possibly more appropriate ways of presenting some concepts; however, the fact remains
that any concept may be treated as a basic-form concept.

Practice with sentences. A useful exercise is to practice identifying how words or groups of words are classified.
Do not become concerned over whether you exhaust all possible ways of grouping the words. A large number of
groupings is possible when a sentence presents elaborate wording. You do not have to deal with all possible groupings.
But you should become facile at classifying words, or groups of words, according to their most basic classification—as
comparatives, non-comparatives, or nouns.

Summary

Comparatives are different from non-comparatives in the following ways:

• Comparatives assume a comparison of two things.

• Whether an example in a comparative sequence is positive or negative does not depend on the absolute
value of the stimulus, but on the difference between that stimulus and the one with which it is compared.

Comparatives differ from nouns in the following ways:

• Comparatives involve precise minimum differences.

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 • Negatives for any particular comparative have the same label.

The procedures for creating continuous-conversion comparative sequences are:

1. Follow the same general wording rules and setup rules specified for non-comparative sequences. If
possible, use continuous conversion of examples.

2. Begin the sentence with a starting point—a stimulus that is neither positive nor negative.

3. Make sure that the starting point is somewhere near the middle of the range that you will show. (This
convention assures that you can “maneuver” toward either extreme.)

4. Follow the starting point with either positives or negatives.

• For sequencing positives, show a small change, a large change, and a change of intermediate sizes, in
that order.

• For all modeled negatives, create a minimum difference from the preceding example.

• Use two types of minimum differences—a very small negative change and a no-change negative.
Model both positives and negatives. Four models are usually sufficient.

• Construct the series of test examples according to the same rules used for non-comparatives.

5. Use shorter sequences for sophisticated learners.

6. Use two-choice tasks for comparatives if the same word appears in both the positive and negative changes
(such as higher pitch and lower pitch, or more pressure and less pressure, or more parallel and less
parallel). As a rule, do not use two-choice tasks for naive learners if different words are used to describe
the positive change and the negative change (bigger-smaller, faster-slower, happier-sadder).

Convert polars such as big-little into comparatives (initially teaching getting bigger). This modification makes the
basis of the concept precise.

The test of whether discriminations should be presented as comparatives has to do with the difference we wish to
show. For most non-comparative concepts, such as leaning, listening, lurching, or losing, we wish to show how they
differ from other conditions (not-leaning, not-listening, not-lurching, not-losing). To show this difference, we would
use a non-comparative sequence. To show increases in an action—leaning more, listening more carefully, lurching
farther, or losing more—we use comparatives.
 

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 Section III

Joining Forms
Anything that is immediately obvious in examples—anything that we can point to, sense, or show through the
presentation of a concrete example—can be effectively communicated through one of the basic forms presented in
Section II. Section III deals with the simplest forms that relate one concept to another. These are joining forms. As the
name implies, they join basic form concepts. There are two types of joining forms—transformations and correlated-
feature relationships.

A transformation is possible only when there is a systematic parallel between changes in examples and changes in
responses for the examples. When a small change from one example to the next occurs, there is a corresponding small
change in the response. The changes in the responses follow rules. They are not random changes.

The diagram below shows the nature of the relationship between examples and responses used for the examples.

Features of Examples Responses

X Y
XX YY
XXX YYY
XXXX YYYY
XXXXX YYYYY
Etc.

The X column simply represents the way the examples change. The Y column shows that there is a parallel
between the features of the example and the features of the response.

If the learner were provided with some examples and shown the response for these examples, the learner should be
able to generalize to any example of the transformation.

There are two types of transformation relationships between example and response. For the first type, a part of the
response changes as part of the example changes. (The example with the X’s and Y’s above demonstrate this type.) The
other type of transformation is based on logical systems, such as the number system. A small change in the example
may change the response totally; however, the response is logically a small change. The nature of this logic is evident
only if the learner knows the system of responses. We can illustrate both types of transformations with the following
comparison.

Type 1:
Example Teacher Wording
000 00 Tell the numbers being added.
(Three plus two)
0000 00 Tell the numbers being added.
(Four plus two)

Type 2:
Example Teacher Wording
000 00 Tell me how many. (Five)
0000 00 Tell me how many. (Six)

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 The type 1 shows that when one group of circles in an example changes, a corresponding part of the response
changes. The group that remains the same is signaled by the part of the response that remains the same across both
examples (plus two). This type of transformation is very important for many verbal skills.

Type 2 is different because no physical part of the response remains from example to example. One response is
five, the other is six. No physical details of the response “five” are present when the response changes to “six.” If we
used this response for five: “one-one-one-one-one,” and this response for six: “one-one-one-one-one-one,” the
transformation would be a type 1, with part of the response remaining constant. However, adding the one unit is not
made explicit by the responses “five” and “six.” The relationship between the change in the response and the change in
the example is logical if one knows the nature of the counting numbers. A small change occurred in the second
example (the addition of one counter), and the response reflects this change, a change in the total of the counting
numbers. Obviously, the learner must know the names of the various counting numbers to use this transformation.

Whether the transformation is a type 1 or a type 2, it is a system for relating changes in the example to changes in
their responses. It permits generalizations to new examples because all examples follow the same transformation rule.

The second type of joining form, the correlated feature, is completely different. It joins a given sensory
discrimination to another discrimination that happens to occur with the first. The red line on a thermometer rises and
the air gets hotter. These sensory events happen together. Observation of one therefore predicts the other. The sensory
discrimination of the red line moving higher (a comparative concept) leads to the prediction that the sensory
discrimination of “hotter air” would be experienced. Because these events are reliably correlated, the observer could
predict the hotter air even if it is not experienced.

We can see the difference between the correlated event and the transformation by referring to a situation such as,
“He fell down.” A transformation would be a different situation expressed as, “She fell down,” or a third situation
expressed as, “He stood up.” Parts of the original response are substituted to show changes in the situations. We treat
the fact, “He fell down,” quite differently when we deal with correlated features. If we have factual information that
“He fell down,” we know about events that occur together. He is correlated with falling down. If we are shown the
person who is identified as he, we draw the conclusion that the person so identified fell down. Even if we do not
observe the act of falling down, we can draw this conclusion. Every sentence that expresses a fact conveys the
correlation of features expressed by the subject of the sentence (in this case, he) with the features expressed by the
predicate (in this case, fell down).

All correlated feature joinings take the same form: a factual relationship exists between two sensory
discriminations. When this relationship is conveyed to the learner, it serves as a “premise” for drawing a conclusion
that is based on the observations of what is mentioned first in the fact. For the fact, hotter things expand, the
observation involves hotter things and the conclusion drawn is about whether the object expands.

1. Fact: Hotter things expand.

2. Observation: Thing getting hotter.

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 3. Conclusion: Thing expands.

Note that the observer does not experience the expansion of the object, merely that it becomes hotter. As the diagram
shows, the observation joins the discrimination observed directly (getting hotter) with the fact about an event that is
correlated with getting hotter.

There are only two types of joining forms because there are only two ways that we can relate a given response for a
specific sensory discrimination to other responses for other sensory discriminations. We can make the response to a
sensory discrimination a part of an orderly, logical system of responses, or we can relate the sensory discrimination
empirically to some other sensory discrimination. If we relate stimulus events and responses in a logically orderly way,
we create a transformation. If we relate stimulus events empirically, we create a correlated-feature joining.
 

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 Single-Transformation Sequences
The sequences that we have dealt with so far teach single labels (names for concepts). All positive examples in a
basic-form sequence have the same label. A transformation sequence is different because the concepts that we teach
through transformation sequences are different. The objective is not to teach a single label, but to teach a system of
responses that applies not only to the examples that we present, but to a wide range of other examples. The only
practical way to communicate the system of responses is to present various pairings of example and response.

A series of examples is used to demonstrate how each example is correlated with a particular response. After
performing on a series of examples that sample a fair range of variation, the learner is provided with a basis for
understanding what is the same across the various pairings of examples and responses. This sameness is a rule, an
unstated relationship that indicates how responses are related to examples. With an understanding of this rule, the
learner can produce unique responses to examples that had not been presented in the initial-teaching sequence. (Unique
responses are possible because the new examples are the same as all other examples of the transformation with respect
to the rule that governs the response.)

To assure that the learner receives clear information about the relationship between example and response, we will
use a sequence that is different from basic form sequences in two primary ways:

1. The transformation sequence has no negatives.

2. The learner produces different responses for different positive examples.

Both of these features derive from the structure of transformation concepts.

There are no negatives in the transformation sequence because the sequence is designed to show pairings of
different examples with different responses. Therefore, the task of wording that is used with each example requires the
learner to produce a unique response. The two segments from sequences which follow show the basic differences
between the concept of rhyming with at when it is treated as a transformation (the first sequence) and when it is
treated as a non-comparative (the second sequence).

The teacher presents the same wording with each example in the transformation sequence. However, the wording
for the transformation sequence requires the learner to produce different responses for different positive examples.
When the learner starts with the letter m and rhymes with at, the learner produces the response “mat.” When the learner
starts with the letter s and rhymes with at, the learner produces the response “sat.” No negatives are presented in the
sequence because the learner rhymes with at for each example presented.

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 Segment of Transformation Sequence
Example Wording
(visual)
m Start with this letter and rhyme with
at. (mat)
s Start with this letter and rhyme with
at. (sat)
f Start with this letter and rhyme with
at. (fat)
b Start with this letter and rhyme with
at. (bat)
Etc.

Segment of Non-Comparative Sequence

Example Wording
(visual)
mat Does this word rhyme with at?
sat Does this word rhyme with at?
sam Does this word rhyme with at?
Etc.

The non-comparative is different. All positive examples are responded to with the same response: “Yes.” As we
noted earlier, any concept can be treated as a basic-form concept. Rhyming with at, therefore, can be treated as a non-
comparative. However, it is handled more effectively as a transformation, if our concern is with the learner actually
producing the rhyming words rather than classifying or identifying them.

Identifying Transformation Concepts

The simplest way to identify transformations is to look at the task that is to be presented to the learner and the type
of response the learner produces.

The answers to three questions determine whether a transformation is implied. If all are answered “Yes,” a
transformation is implied. The questions are:

1. Can the same task be used with a variety of examples of the same type?

2. Would the learner produce different symbolic responses for different examples?

3. Is there a sameness shared by all responses?

If we apply the three-criteria test to the task “Is this a cup?,” we see that the same task can be used for a variety of
examples. We could present various positive examples of cups, present them to the learner, and use the same task, “Is
this a cup?”

When we test the task with the second criterion, however, we discover that the task does not imply a
transformation. The reason is that for every positive example, the learner produces the same response: “Yes.” The

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learner does not produce different symbolic responses for different examples.

If we test the task “Where is the dog?,” we discover that it too can be used with a variety of examples of the same
type (the dog under the bed, under the chair, under the table, etc.). For each different example, however, the learner
produces a different response. Furthermore, there is an obvious sameness shared by all responses—the word under.
Therefore, “Where is the dog?” implies a transformation. The task “Rhyme with this word” could be presented with a
variety of examples of the same type (the words mat, eat, fish, brother, etc.). For each example that is different, the
learner produces a different response. And there is an obvious sameness across all examples. The last part of each
response contains the vowel sound and the ending of the word presented in the example. This pairing is the same for all
words: mat—sat, eat—meat, fish—dish, brother—mother, etc.).

When we apply the three-criteria test to the task “When I stop counting, tell me the next number,” we see that the
same task can be applied to a variety of examples of the same type (counting to 5, to 7, to 34, to 3, etc.) We note that
for examples that are different the learner produces responses that are different. They always present the number that
occurs next in the counting order. (For 1, 2, 3, the response is 4. For 1, 2, 3, 4, 5, 6, the response is 7. This relationship
of example to response is the same across all examples.)

The line between the transformation and basic forms depends on our criterion for judging a variety of examples to
be “of the same type” (criterion 1). If we presented the task “What is this?” with examples of cups, the learner would
not produce different responses for different examples. Instead, the learner would say “a cup” for every example.
Therefore, within this context of “examples of the same type” the task does not imply a transformation. (It implies a
noun classification.) If we judge any common object to be “of the same type,” a transformation is implied, however,
because the learner will produce different responses for different examples presented with “What is this?” (The learner
would respond, “a cup,” “a ball,” “a pillow,” “a dog,” etc.) Furthermore, there is a sameness across all responses. All
object names are preceded by a. (However, the learner would not be able to respond to an example unless the learner
knew the object name.)

If the example presented to the learner is symbolic (a word, a sentence, a number, a numerical expression), and if
the learner does anything but label the object as a word, a sentence, or a number, a transformation is clearly implied. If
the learner reads the word, adds a past-tense ending to the word, spells the word, etc., the task that directs the learner
implies a transformation.

If the example presented with a task is not symbolic, but the task requires the learner to respond symbolically, a
transformation is probably implied. If the learner is directed to ask a question about the example, tell where the
example is, count the parts of the example, say words that have the same initial sound as the example, etc.,
transformations are implied.

Transformation Sequences

Basic Structure

The three-criteria test suggests the basic structure of the transformation sequence:
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 1. The sequence presents the same wording with all examples.

2. The sequence requires different responses for different examples.

3. The communication clearly shows what is the same about all examples and how changes in the examples
are correlated with changes in the responses.

The communication is designed to demonstrate the unstated rule that governs the transformation; therefore, the
communication is designed to induce a generalization to new examples. Note, however, that the transformation
sequence assumes that the learner understands the names or symbols that are related by the transformation.
Transformations involving the counting numbers assume that the learner already knows the counting numbers. A
communication that teaches the learner to identify the predicate of sentences assumes that the learner has some
knowledge of sentences (at least the understanding of what they say).

The Single-Transformation Sequence

We will use a standard format for designing all single-transformation sequences. We will begin the sequence by
juxtaposing examples to show differences. We will then juxtapose for sameness. By beginning with differences, we
show how a small change in the example controls a corresponding small change in the response. After we show this
relationship with a series of examples, we present sameness juxtapositions. These require the learner to apply the
transformation to examples that do not carefully “prompt” the relationship between the example and response. The
sameness juxtapositions provide the basis for generalizing to new examples.

Figure 8.1 shows a single-transformation sequence designed to communicate making sides of an equation equal.
The sequence requires the learner to specify the missing number in different equations.

Figure 8.1

Example Teacher Wording

1. 15 = My turn. What makes the other side equal? Fifteen.

2. 16 = My turn. What makes the other side equal? Sixteen.
3. 61 = Your turn. What makes the other side equal?
C
4. 6= What makes the other side equal?
5. 5= What makes the other side equal?
6. 5J = What makes the other side equal?
7. 27M = B What makes the other side equal? A
8. 9= E What makes the other side equal?
9. K= What makes the other side equal?
10. 23 = What makes the other side equal?
D
11. 0= What makes the other side equal?
12. R= What makes the other side equal?
13. C+1= What makes the other side equal?
14. 12 = What makes the other side equal?

The sequence is created through continuous conversion. The teacher begins with the first problem written on a
chalkboard, erases the 5, and replaces it with 6. The teacher models the first two examples by saying: “My turn,”

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presenting the task to herself (“What makes the other side equal?”), and saying the answer. Beginning with example 3,
the teacher asks, saying: “Your turn. What makes the other side equal?”

The sequence is continuous because part of the set up remains for all examples. That part is the equal sign. For all
minimum-difference examples, only part of the preceding example is changed. The teacher can create the change from
16 to 61 by erasing only one digit. The 1 in 16 is erased and replaced by 1 after the 6.

Juxtapositions. The B bracket indicates that the examples (created through continuous conversion) share the same
“set up” features, the same equal sign. The sequence moves from showing differences (C) to showing sameness (D).
The test (E) involves new examples. According to the sameness principle, we show sameness if we treat juxtaposed
examples that are quite different in the same way. Note that the D segment achieves such juxtaposition. The same
question applies to 27M and 0. The learner carries out the same operation with both examples. Therefore, the
juxtaposition implies that 27M and 0 are the same with respect to “equality.” When the learner completes the sequence,
the learner should be able to respond to a variety of new examples, such as: 3xr + 1, or 340.

Progressive minimum differences. The pattern for creating minimum differences in transformation sequences is
progressive. Only one change occurs from one example to the next. The progression includes the various types of
minimum difference changes that show the range of variation. For the sequence in Figure 8.1, example 1 is 15 and
example 2 is 16—a minimum difference involving counting. Example 3 involves a different type of change from the
preceding example. The same digits are used (1 and 6); however, they are reversed from 16 to 61. For the next
example, one of the digits in 61 is removed, creating 6. Example 5 is a minimum change in counting numbers 6 to 5,
and in example 6, a letter is added to the 5, forming 5J.

Obviously, other minimum differences could be introduced, such as those that involve fractions, decimals, and
various operational signals, such as the times sign. Later, we will provide guidelines for identifying the variety of
“types” that are to be included in the sequence. However, the basic rules apply for showing differences: only one
change occurs from example to example. That change is a small change. The change from 5J to 15J is a relatively
minimal change. The change from 5 to 15J is not. It involves both the addition of a digit and a letter. A change from 5J
to 500J is not relatively small, because it involves the addition of two 0’s. The difference between 5J and 50J is a
relatively small difference. Remember, to show difference, create one difference from example to example.

The analysis of minimum differences cannot determine which type of minimum difference is the “smallest,” nor
are we particularly interested in questions such as: “Does dropping a digit create a ‘smaller’ change or difference than
reversing the digits?” The question is an empirical one. For us, it is irrelevant because we should show both types.

Constructing Single-Transformation Sequences

1. Begin the sequence with a mid-range example. If you are constructing a sequence that is designed to teach the
subject of sentences, do not begin with an example that has a very long subject. (“The former president of the
organization”) or one that has a very short subject (“He”). Do not begin with one that is extreme in any way. Begin
with a subject (“Five little ducks” or “That broken table”) that can be converted into a shorter subject (“Five ducks,”

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“That table”) and that can be expanded (“Five fat little ducks,” “That broken table in the hall”). Think of the starting
example in the same terms as you would for the starting point of a comparative. If you start somewhere in the middle,
you have latitude for showing a full range of changes.

2. Follow the first examples with 3 to 6 minimum-difference examples arranged progressively. Create each by
making only one small change from the preceding example. Do not repeatedly change the preceding example in the
same way.

A sequence for the beginning of a sequence for subject of statements could open with these examples:

a. Five little ducks went walking.

b. Five ducks went walking.

c. Five rabbits went walking.

d. Five rabbits and a boy went walking.

Each example is created by changing the preceding example in a relatively small way; however, each type of
change is different. Example b is created by subtracting the word little. Example c is created by substituting rabbits for
ducks. Example d adds the words and a boy to the subject.

3. Through the minimum-difference examples, try to show the learner the range of variation that will be covered in
the test examples. If the test examples for the equality relationship are to include examples of this type: 4/3 = , include
this type in the minimum-difference part of the sequence (perhaps preceding the example by: 4 = ). If you cannot show
all types of minimum differences in six progressively arranged examples, you should probably use more than one
sequence.

4. Follow the minimum differences with a series of juxtaposed examples that are as different as possible within the
constraints of the setup (and the types shown in segment C.) If the transformation has a fair range of variation, use as
many as twelve examples in the C segment. For concepts that have a narrow range, use only 4 to 8 examples. The
entire sequence should not exceed 18 items.

5. Model the first 2 to 5 examples. Test on the remaining examples. The simplest modeling procedure is to say,
“My turn,” and then present the same test wording that will be used with examples that are not modeled.

The test should always include at least two minimum-difference examples. If four examples are modeled at the
beginning of the sequence, the sequence should have six minimum-difference examples. If only two examples are
modeled, as few as four minimum-difference examples could appear at the beginning of the sequence.

The first test examples may be repetitions of examples that had been modeled. This procedure is appropriate if
there is doubt about the learner’s ability to produce the response. If there is no doubt, the first test example should be a
new minimum difference.

Illustration. Figure 8.2 shows a single-transformation sequence for changing questions into positive statements.

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Each example is a question phrased in the past tense and beginning with the word did. The response that is called for is
a past-tense statement.

Figure 8.2

Example Teacher Wording

1. Did she smile? My turn to say the positive statement: She smiled.
2. Did she laugh? My turn to say the positive statement: She laughed.
3. Did she laugh at the clown? My turn to say the positive statement: She laughed at the clown.
C
4. Did she sit with the clown? My turn to say the positive statement: She sat with the clown.
5. Did they sit with the clown? Your turn to say the positive statement.
6. Did they run with the clowns? Say the positive statement. A
B
7. Did Herman eat breakfast? Say the positive statement.
8. Did those ducks fly high? Say the positive statement.
9. Did his sister buy a car? E Say the positive statement.
10. Did Mr. Jones close his store? D Say the positive statement.
11. Did the other girls come with you? Say the positive statement.
12. Did the words seem to be clear? Say the positive statement.

The C bracket indicates that there are six examples progressively ordered. The changes show the range of variation
that will occur in segment D. Not all possible minimum differences are included in the first six examples. Example
three could have been: Did she laugh loudly? This example would certainly be acceptable. It includes all remaining
words presented in the question, e.g., “Did the other girls come with you?” is changed by starting with the word after
did: “The other girls came with you.” However, the verb is changed to a past-tense verb. The minimum-difference
examples show that the creation of the past-tense verb is not a mere mechanical operation. For the question, “Did those
ducks fly high?” the learner does not respond by saying, “Those ducks flied high.” Since these irregular verbs (Those
ducks flew high.) are included in the test, they are demonstrated in the minimum-difference examples. Examples 4, 5,
and 6 involve irregular verbs.

Note that this sequence nicely shows that what the learner must learn is a rule and that the rule may be quite
complicated if we express it verbally. (Drop the word did. Start with the word that appears after did. Say all words as
presented, but change the verb from present to past.)

Irregulars and Transformations

If responses are irregular, they are a subtype of examples that require knowledge beyond the knowledge implied
for regular responses. If the learner does not know how to form the past-tense of words such as eat, these words should
not be included in the sequence because they are irregular. If the learner is naive, but capable of handling regular past-
tense transformations, we would expect the learner to produce responses such as: “She sitted with the clown,” “They
runned with the clowns,” “Herman eated breakfast,” “Those ducks flied high,” “His sister buyed a car.” Conversely, if
we expect the “appropriate responses,” we must make sure that the learner has the knowledge of how to produce them.

Subtypes

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 A subtype is a group of examples that have a set of common features not observed in any other examples. When we
deal with transformations, we consider subtypes. The analysis of subtypes provides a better “guess” about difficulty of
the sequence.

As the amount of sameness possessed by all members of a subtype increases, the number of examples in that
subtype decreases. Consider extreme examples. An elephant and the word if are the same in some ways and can
therefore be included in the class or type of entity. (They are the same because they are events observed on the planet
Earth.) The number of examples in this class is very large because the samenesses are very few compared to
samenesses possessed by a group that consists of the word if and other two-sound words that begin with vowel sounds
(if, in, on, up, at, etc.). The membership of the two-sound-word class is smaller because things may be included in this
class only if they have many samenesses. In addition to being events on Earth, they must also be verbal events; they
must be words; they must be words of only two sounds; they must be words that begin with a vowel.

If we take the subtype classification to its extreme, we would create a subset for each word. The word if would be
in a class composed only of two-sound words that begin with the sound /i/ and that end with /f/. The examples in this
class have more samenesses than the examples in the class of two-sound words that begin with a vowel. The
membership of the new class, however, would be smaller, because an example must have more sameness to be
included in the class.

We may group examples according to many shared samenesses, or to relatively few shared samenesses. But the
basis that we use for identifying subtypes is always the features of the examples. Conversely, if examples are of a given
subtype, they will have common features not possessed by examples that are not of that subtype.

Subtypes that are irregular or difficult. Some subtypes should be difficult for the learner. These subtypes require:
(1) additional behavior that is not required by the simpler examples, and (2) behavior that is contrary to that required
by the larger set of examples.

All irregular subtypes require the learner to replace the behavior for producing a regular response with another
behavior. Some irregular subtypes, however, require behavior that is contrary to that required by the regular subtypes.

Producing the response for, “Did she fly over the barn?” (“She flew over the barn”) represents additional behavior.
In addition to expressing the verb as a past-tense verb, the learner must produce the word that is used conventionally to
denote the past tense of fly.

An example of contrary behavior would be identifying the number of tens for teen numerals. For most two-digit
numerals, the number of tens is mentioned first in the name—forty-six, eighty-six, etc. For teens, however, it is
mentioned last: sixteen, eighteen. The formation is contrary to that of the larger group; therefore, we would predict
problems with this subtype.

An easy way to find difficult subtypes is to make statements about the features of the examples or the responses,
then see which examples have the features. The feature must be possessed by all members of a subset. Figure 8.3
presents a simple analysis of two-digit numerals. All X’s in the same row describe the membership for a given feature.

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All X’s in a column describe the features for a given subtype. The feature of names of tens distorted is possessed by
the 20’s, 30’s, 50’s, and by the teens. (We do not say “three-tee-two,” we say, “thirty-two.” We do not say “one-tee-
four,” we say “fourteen.”)

Figure 8.3

Full Set Irregular Subtypes

13, 15,
ends in 0: 20’s (21, 22, . . .) all 20’s,
(10, 20, 30’s (31, 32, . . .) 30’s, and
30 . . .) 50’s (51, 52, . . .) 50’s
10-99 13-19 11, 13, 15

1. Numeral consists of 2 digits. X X X X X X

2. Name for each numeral X X X X X X

3. Name refers to 10’s digit only X

4. Name refers to 10’s digit first,

ones next X

5. Name refers to 10 digit last X

6. Name of ones digit distorted X X X X

7. Name to ten distorted X

If an X appears in a cell, all members listed at the top of the column have the feature indicated. For instance, all the
listed numerals in column 5 have feature 6 (name of the ten distorted). We do not say “two-tee-three,” for 23, but we do
say six-tee-three for 63.

As Figure 8.3 shows, the samenesses shared by all the numerals 10 through 99 are fewer than the samenesses
shared by members of any subset. Figure 8.3 also shows that membership in one subset does not exclude membership
in another subset. The numerals 13 and 15 are in the subset of numerals with distorted tens names (we do not say
thirten), and distorted one’s names (we do not say threeteen). Thirteen is also in the group with the tens digit named
last. To make this name regular, we would have to reverse the order of the parts of the name and straighten out the
distortion of both the name of the tens and the name of the ones (calling the numeral one-tee-three).

Note that not all possible irregular subtypes are specified in Figure 8.3.

The various irregular features suggest where inappropriate generalizations will occur. Conversely, the irregulars
must be taught separately to assure that the learner recognizes them as irregulars and does not generalize
inappropriately.

Figure 8.3 shows the product of grouping subtypes according to samenesses. This categorization is possible only
after you have identified what is the same about all the subtypes. The simplest procedure for determining subtypes is
that used in sequencing examples to show sameness. You are trying to discover unique samenesses; therefore, the
sameness principles provide you with the most efficient guideline for demonstrating sameness.
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 1. Make a list of juxtaposed examples that differ greatly from each other. For the numerals, you might start with 52
and then follow with an example that is quite different from 52, such as 17. Then follow with an example that is greatly
different from 17, such as 90. Continue until you have exhausted the range of variation of two-digit numerals.

2. Specify all the differences between the first three or four juxtaposed examples. How is 17 different from 52 with
respect to the relationship of numerals and names? How is 90 different from 17? etc.

3. For each difference that you describe, list all the numerals that share this difference. If you note that the name
for 17 starts with “seven,” list all the other numerals that have the feature of starting with the name for the ones number
and not the tens number.

4. Cross out all examples farther down your list that differ from the preceding example in the manner that you have
already described.

This procedure requires some trial and error. Expect some initial difficulty in expressing the differences from one
example to the next. However, by following the procedure, you will probably identify all subtypes that are important
and you will have a precise understanding of the features shared by all the examples within each subtype.

Setup for subtype variations. If a subtype involves additional behavior to form a response, or if a subtype calls for
contrary behavior, the subtype should not be presented in the initial sequence.

For the setup features of the initial sequence, limit the examples to a large subtype that does not involve additional
responses or contrary responses. If we deal with two-digit numbers, the initial sequence might contain only examples of
the subtype that consists of regularly-formed numerals in the 40’s, 60’s, 70’s, 80’s and 90’s. The group is large and
generalizable. If we were teaching rhyming with single-sound beginnings, we might limit the examples in the initial
sequence to those that involve continuous sounds that are voiced. This subtype is relatively large in membership and
calls for responses that are relatively easy to produce. (Additional behavior is required by stop-sound beginnings b, t, p,
g because the speaker must create a transition sound that is based on the sound that follows the stop-sound.)

Remember, for the introductory sequence, identify a relatively large subtype that involves producing the simplest
class of responses.

The sequence presented in Figure 8.4 illustrates how the subtype analysis is translated into an initial sequence. This
sequence teaches numerical expansion. The learner is presented with numerals, such as 46, and says the addition fact
for that numeral that is based on the two digits (“Forty-six equals forty plus six”). The initial sequence avoids all the
irregular types (the teen numbers, 20’s, 30’s, and 50’s).

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 Figure 8.4

Example Teacher Wording

1. 79 My turn to say the addition facts for this numeral:

Seventy-nine equals seventy plus nine.
2. 78 My turn again. Seventy-eight equals seventy plus eight.
3. 87 My turn again. Eighty-seven equals eighty plus seven.
4. 87 C Your turn. Say the addition fact.
5. 97 Say the addition fact.
6. 96 B Say the addition fact.
7. 69 Say the addition fact.
A
8. 42 E Say the addition fact.
9. 88 Say the addition fact.
10. 71 D Say the addition fact.
11. 45 Say the addition fact.
12. 93 Say the addition fact.
13. 64 Say the addition fact.

The sequence follows the same pattern as that for other transformations. The minimum differences continue
through example 7. Note, however, that the same example is repeated as example 3 and example 4. The teacher models
only three examples, then tests on the last modeled example. Following example 7 greatly different examples are
juxtaposed to communicate sameness.

Following the learner’s successful performance on this sequence, the learner would be able to generalize to all
regularly-formed numerals. Obviously, however, the learner would not be able to generalize to the irregulars,
particularly the teen numerals (because the teen names are formed in the reverse way the regulars are formed). The
dilemma that faces us is that if we work too long on the regularly-formed numerals, we will stipulate this type and the
learner will have great difficulty when the teens are introduced. On the other hand, if we introduce the teens before the
learner is reasonably firm on the regulars, introduction of teens could lead to serious reversal problems, with the learner
confusing the formulas for the regular numerals and the teens (calling 51 ”ten plus five,” for example).

Introducing the other subtypes would be relatively easy. The learner might generalize to the 20’s, 30’s, and 50’s
following the initial format. Also, the numerals that end in zero (40, 60, etc.) should present little problem, because the
numeral shows that the addition fact will refer to zero. (“Forty equals 40 plus zero”). Following the introduction of the
initial sequence above, we could design a second sequence that introduces both the 20’s, 30’s, 50’s and the two-digit
numerals that end in zero.

Application of the Subtype Analysis

The illustration with numerals shows the basic function of the subtype analysis, which is to identify the range of
examples for which generalization will be possible, and to identify subtypes that will interfere with the generalization.
The analysis suggests a strategy, which is to start with a relatively large, but easy subtype, then introduce the other
subtypes systematically and integrate them with all subtypes that have been taught. The mechanics of this integration
process is presented in Section IV, Programs. The issue that we are primarily concerned with here is that of identifying

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a good initial sequence, one that is efficient with respect to its generalization potential and yet one that is easy enough
for the learner to master.

To show how the subtype analysis permits us both to identify problems and to solve them efficiently, we will take
a relatively simple skill, oral blending, and apply the analysis to designing an initial sequence, creating a second
sequence for one of the more difficult subtypes, and using the subtype analysis to diagnose problems the learner may
have with a sequence. Although we will be dealing with a relatively simple skill, the same procedures would apply to
any transformation concept because these concepts deal with symbolic manipulation (identifying the predicate of a
sentence, expressing numerals as log functions, rhyming, converting commands into statements, etc.). One of the most
prominent features of concepts that deal with symbolic manipulations is that they generate an incredibly large range of
variation, which means that there are many possible minimum differences. These minimum differences may be
categorized into subtypes, some of which should be more difficult than others.

Even if some of these subtypes are perfectly regular, they may involve additional behavior and should therefore be
more difficult than subtypes that require less behavior. The skill of oral blending illustrates this situation. Although
none of the subtypes that we identify when analyzing oral blending is irregular in the sense that the name for 15 is
irregular, some subtypes are more difficult than others.

The same analysis that applies to the identification of irregulars applies to the identification of difficult subtypes.
This subtype analysis will suggest where we should begin when teaching the skill to a naive learner.

For all examples of oral blending, the teacher says a word slowly. The learner then says the same word at a normal
speaking rate. The subtype analysis of words said slowly reveals three large subtypes.

1. Words in which the parts are words familiar to the learner and there is a pause between the parts, e.g.,
lawn (pause) mower.

2. Words in which the individual continuous sounds of the word are held for a longer period of time than
they are when the word is spoken at the normal speaking rate (mmmmaaaannnn). (Note that if a stop-
sound occurs in the word, that sound would not be held: mmmaaat.)

3. Words in which the individual sounds are produced slowly and there are pauses between each sound, e.g.
mm (pause) aaa (pause) nnn.

We can identify subtypes for these three main types. For instance, we could introduce words that have one part that
is a word spoken at a normal speaking rate followed by sounds that are held for a longer than normal duration
(motorzzzzz). Many other combinations are possible, such as sounds spoken at a normal rate with pauses between the
sounds.

The subtype analysis shows us that the main groups are based on pauses (subtype 1) or exaggerating sounds
(subtype 2). Subtype 3 is created by combining pauses and exaggerated sounds.

If we are to teach the skill of oral blending, we should teach all types. The most direct route would be to begin with

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subtype 3. This subtype would teach both the transformation created by pausing and that created by exaggerating
sounds. Because this subtype presents both variations of the transformation, however, it is the most difficult subtype.
We would arrive at the same conclusion if we simply asked which subtypes share the greatest amount of sameness with
words spoken at a normal speaking rate. Subtype 3 obviously presents examples that share the least amount of
sameness with the familiar word (the counterparts that are spoken at a normal speaking rate). Therefore, we would not
introduce this subtype in the initial sequence if our goal is to begin with a sequence that is relatively easy.

Instead, we would consider either subtype 1 or subtype 2 for the initial transformation. Subtype 1 is the easiest
subtype because the transformation involves only one alteration—eliminating a single pause. Subtype 2 involves
transforming each sound to a sound of less duration. Since each sound must be modified, a more extensive
transformation is implied.

Figure 8.5 presents a possible introductory sequence.

Figure 8.5

My turn to say words fast. Listen:

Motor (pause) cycle. Say it fast.
Motorcycle
My turn again. Motor (pause) oil. Say it
fast. Motor oil.
Your turn. Motor (pause) cycle. Say it
fast. C
Motor (pause) oil. Say it fast.
A
Oil (pause) can. Say it fast.
Paint (pause) can. Say it fast.
Paint (pause) brushes. Say it fast. E
Fence (pause) post. Say it fast.
Birth (pause) day. Say it fast.
Lawn (pause) mower. Say it fast.
D
Racing (pause) car. Say it fast.

All examples in the series are of the same subtype. The first five words in the series are minimally different,
arranged so that the change from one word to the next involves a minimum change. The changes are created by: (1)
substituting a new last part while retaining the first part; (2) substituting the first part while retaining the last part of the
preceding word. The D segment of the sequence juxtaposes examples that are greatly different.

This subtype stipulates that the only way that unblended words are transformed into blended words (spoken at a
normal speaking rate) is by eliminating pauses; therefore, we would introduce the subtype 2 as soon as the learner
demonstrated a generalizable skill with the subtype 1 examples. This generalization should occur with the presentation
of the first sequence or certainly with the presentation of a second sequence that introduces different subtype 1
examples.

Minimum Differences Within a Subtype

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 When we design a sequence for subtype 2, we discover that we are confronted with a new problem—a wide
possible range of minimum-difference examples. According to our procedure for creating transformation sequences, we
begin with a series of examples that are progressively minimally different (one change occurring from example to
example). When we consider the range of variation that is possible within subtype 2, however, we discover that we may
create examples that are so different that they may be difficult for the learner to discriminate. These minimum
differences would be self-defeating because they would not show the learner how the transformation works.

The simplest way to determine whether the sequence attempts to include too wide a range of variation is to arrange
the minimum-difference examples progressively (one change from example to example). If it is not possible to cover
the range of variation for a particular subtype through six minimum-difference examples, the subtype is too broad and
will probably prove to be too difficult for the naive learner. The remedy is to reduce the range of variation presented by
the sequence or make the minimum-difference larger and therefore easier.

The sequence in Figure 8.6 illustrates the problem. The sequence attempts to show minimum differences for the
full range of variations that would occur in a sequence that teaches the subtype 2 say-it-fast skill.

Figure 8.6

Example Teacher Wording

Listen: aaammm Say it fast.

Listen: aaannn Say it fast.
Listen: iiinnn Say it fast.
Listen: llliiit Say it fast.
Listen: fffiiit Say it fast.
Listen: fffllliiit Say it fast.
Listen: fffllliiick Say it fast.
Listen: fffllliiip Say it fast.
Listen: sssllliiip Say it fast.
Listen: sssiiip Say it fast.
Listen: sssiiilll Say it fast.
Listen: diiilll Say it fast.
Listen: drrriiilll Say it fast.
Listen: drrriiip Say it fast.
Listen: trrriiip Say it fast.

The seventeen examples in the sequence have not covered the range of variation for sounds in short words;
therefore, presenting the smallest minimum differences for the subtype is probably unreasonable. The sequence would
probably be too difficult for the naive learner. To correct the sequence, we would: (1) try to identify a range of
variation that could be covered through no more than six progressively arranged minimum-difference examples; and
(2) try not to create the smallest minimum differences available to us. (The examples that are particularly difficult in
the sequence above are words like flip, drill, drip, and trip.)

The first six examples of a more appropriate sequence for introducing the subtype 2 say-it-fast words are presented
in Figure 8.7.

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 Figure 8.7

Example Teacher Wording

Listen: mmmeee Say it fast.

Listen: mmmaaannn Say it fast.
Listen: aaammm Say it fast.
Listen: aaaat Say it fast.
Listen: fffaaat Say it fast.
Listen: fffeeet Say it fast.

The sequence introduces two-sound words and three-sound words. It introduces words that begin with vowels,
words that begin with a voiced consonant (mmm) and words that begin with an unvoiced consonant (fff). Only two
vowel sounds are used in the words (eee and aaa). The minimum differences are not as fine as they could be
(particularly with the first three examples). However, the sequence shows that changes in the vowel or changes in the
consonants lead to changes in the word.

Feedback About Difficult Subtypes

The rule about limiting minimum-difference examples to six is a handy way to avoid possible difficulties. In the
end, however, the subtype analysis will not provide us with precise information about how difficult the discriminations
are for the learner. To determine difficulty, we must refer to the learner’s performance. If we create a sequence that
contains a subtype that is too “difficult,” the learner’s performance will provide clear evidence about the subtype. The
learner will tend to miss items of that subtype and will not tend to miss items of other subtypes. Conversely, to analyze
whether the learner has subtype problems, we simply classify the errors the learner makes. If all (or nearly all) items of
a particular subtype are missed by the learner, the subtype is difficult. Note that the learner may make mistakes on
items other than those of a particular subtype; however, if the learner misses nearly all items of a particular subtype, the
sequence presents a subtype that is too difficult. To classify the items the learner misses, we determine if they are the
same in some way. If we can identify features possessed only by these items (and not by other items), we identify the
subtype.

Illustration. Let’s say that the learner has trouble with the items marked with an X in the following say-it-fast
sequence:

My turn to say it fast. Listen: mmm-et. Say it fast: met.

My turn again: vvv-et. Say it fast: vet.

Your turn: fff-et. Say it fast. X

j-et. Say it fast.

b-et. Say it fast.

p-et. Say it fast. X

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 nnn-et. Say it fast. X

sss-et. Say it fast. X

mmm-et. Say it fast.

The learner misses four items: fet, net, pet, and set. We first try to determine if this set of items possesses a
sameness not possessed by other items in the sequence. There does not seem to be any such sameness. Next, we see if
we can identify a “trend.” To do this, we try to find a sameness possessed by most of the mistaken items. Three of the
missed items begin with an unvoiced sound (fet, pet, and set).

If the learner has trouble with this subtype, the learner should tend to miss items of the subtype whenever they
appear. To determine the extent to which this happens, we examine the sequence to find unvoiced beginnings that were
not missed. The conclusion is that although .the learner missed the voiced-beginning word net, this miss was probably
just a lapse of some sort. The major mistake tendency involves unvoiced beginnings.

To summarize the procedure for identifying subtype errors:

1. First determine if all items missed are of the same subtype.

2. If they are, examine the items not missed and determine the extent to which the learner consistently missed
items of this subtype. A strong trend is indicated if the learner misses all items of a subtype.

3. If the items missed do not fall into an obvious subtype, try to divide them into groups based on common
features. Identify the largest possible groups.

4. For each group identified, test the trend by examining the extent to which the learner consistently missed
items of each subtype when they occurred in the sequence.

Remedies for Difficult Subtypes

If the learner’s error pattern suggests a problem with a regular subtype (or subtypes), follow this procedure:

1. Create a sequence that is like the original sequence, but that does not contain the difficult subtype(s).

2. Create a second sequence that consists only of the difficult items. Present this sequence after the learner
has mastered the first remedial sequence.

3. Present an integration sequence that consists of the subtypes that were presented in the original sequence
and of the difficult-subtype items.

Note. This procedure would not necessarily be performed if the subtype was an irregular or contradictory subtype.
The teaching and integration of such subtypes would be delayed until the learner is well-practiced in the regular
subtypes.

Illustration. The mistakes on the earlier say-it-fast sequences suggest remedial sequences. First, a sequence is

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presented that is a variation of the original sequence with the difficult subtype removed (words that begin with
unvoiced sounds).

My turn to say it fast. Listen: mmm-et. Say it fast. Met.

My turn again: vvv-et. Say it fast. Vet.

Your turn. lll-et. Say it fast.

j-et. Say it fast.

w-et. Say it fast.

b-et. Say it fast.

nnn-et. Say it fast.

g-et. Say it fast.

yyy-et. Say it fast.

A second sequence would be presented next to teach the difficult subtype (unvoiced sounds).

My turn to say it fast. Listen: sss-et. Say it fast. Set.

p-et. Say it fast. Pet.

Your turn: p-et. Say it fast.

fff-et. Say it fast.

t-et. Say it fast.

sss-et. Say it fast.

ch-et. Say it fast.

After the learner had successfully mastered both the remedial sequences, an integration sequence would be
introduced. No examples would be modeled. The sequence would begin with minimum differences and would present
both the difficult subtype and other subtypes. The integration sequence begins with the most recently taught subtype or
the most difficult subtype. The reason is that performance on this subtype will tell whether the learner is “ready” for the
remainder of the sequence. If not, we can add items to review the type before introducing the part of the sequence that
presents the other types.

Advanced Applications

For sophisticated learners we may modify the transformation sequences in the same way we modify other
sequences:

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 1. We do not model as many examples.

2. We do not model all subtypes.

3. We introduce new subtypes in the D segment of the sequence.

4. We shorten the entire sequence.

The use of transformation sequences is appropriate for teaching virtually any skill in which the learner is required
to manipulate symbols. For the learner who is learning a foreign language, the sequence can be used to show how
changes in the sentences signal changes in meaning. For the learner who is learning arithmetic, variations of the
sequence may be designed to teach how to add ten to any number, how to multiply by a fraction to change any number
into one, or how to express number values using exponents. The sequences are appropriate for teaching the learner
what nouns, adjectives, and prepositions are. For each of these applications, the subtype analysis becomes important
because of the wide range of minimally different types.

The sequence presented in Figure 8.8 shows the learner how to construct fractions that equal one.

Figure 8.8

1. My turn: If the top is twelve D, what’s

the fraction that equals one?
Twelve D over Twelve D.
2. My turn: If the top is twelve, what’s the
fraction that equals one? C
Twelve over twelve.
3. If the top is two, what’s the fraction that
equals one?
4. If the top is seventeen, what’s the fraction
that equals one?
5. If the top is three plus R, what’s the
fraction that equals one?
6. If the top is 100 JD, what’s the fraction
that equals one?
D E
7. If the top is nine over three halves, what’s
the fraction that equals one?
8. If the top is Z minus 5, what’s the fraction
that equals one?
9. If the top is two thirds, what’s the fraction
that equals one?
10. If the top is one, what’s the fraction that
equals one?

The C segment consists of three examples. Only two are modeled.

Examples 4 through 10 are juxtaposed to show sameness. Included in D are types that involve addition (3 plus R),
subtraction (Z minus 5), fractions (2/3), and regular counting numbers (17).
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 Item 7 involves a fraction over a fraction (nine over three halves over nine over three halves). This subtype could
be expected to present response-production problems for some learners (who may become confused about how many
times to say the word over). The remedy would be: (1) remove the type from the sequence; (2) create a new sequence
that involves only fractions over fractions; and (3) integrate the two types after the learner performs adequately on the
fraction-over-fraction sequence.

The first part of a sequence to teach fractions over fractions follows:

1. My turn: If the top is two thirds, what’s the fraction that equals one? Two thirds over two thirds.

2. My turn: If the top is twelve thirds, what’s the fraction that equals one? Twelve thirds over twelve thirds.

3. Your turn: If the top is twelve thirds, what’s the fraction that equals one?

4. If the top is three twelfths, what’s the fraction that equals one?

5. If the top of the fraction is 12 over R, what’s the fraction that equals one?

6. If the top of the fraction is A over 5, what’s the fraction that equals one?

Etc.

Note that one of the modeled examples is presented as a test.

Each task in the sequences above begins with the same setup feature: “If the top is . . .” For the advanced learner
there would be no particular problem with introducing a variation in the setup such as, “If the bottom number is . . .” To
do this, we would add one more modeled example at the beginning of the sequence:

1. My turn: If the top number is twelve, what’s the fraction that equals one? Twelve over twelve.

2. My turn: If the bottom number is twelve, what’s the fraction that equals one? Twelve over twelve.

3. My turn: If the bottom number is twelve D, what’s the fraction that equals one? Twelve D over twelve D.

4. Your turn: If the top number is 21 D, what’s the fraction that equals one?

5. If the top number is two, what’s the fraction that equals one?

6. If the bottom number is seventeen, what’s the fraction that equals one?

7. If the bottom number is three plus R, what’s the fraction that equals one?

8. If the top number is 100 JD, what’s the bottom number?

Etc.

The items in the sequence after item 8 would refer to either the top number or the bottom number. Possibly, one or
two more test items could be added to the sequence; however, the advanced learner probably would not need them.

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Same-Response Minimum Difference

A unique minimum-difference is possible for many single-transformation concepts. This is the same-response
minimum difference. This minimum difference involves a change in the example that results in no change in the
response. Example 2 in the sequence just presented is a same-response minimum difference. Below is another example.

23 = What’s the answer? Twenty-three.

= 23 What’s the answer? Twenty-three.

The answer is the same even though a change has occurred in the example.

To decide whether same-response minimum differences should be introduced in the sequence, answer this
question: Do other minimum differences involve the dimension or detail manipulated to create same-response
minimum difference? If the answer is “Yes,” same-response differences should be presented.

The example: =23 is created by moving the numeral across the equal sign. If we did not use this dimension of
change (moving something across the equal sign) to create other minimum differences, the no-change minimum
difference should not be introduced in the sequence.

Let’s say that the equality sequence contained the following minimum-difference examples:

12 = (12)
12+1 = (13)

We showed that 12 is changed into 13 by adding a plus sign and a numeral. Since we introduced this dimension to
show a change in value, we should introduce the same-response minimum differences involving the plus sign:

12 = (12)
12+0 = (12)
12+1 = (13)

The response to the first two examples is the same.

Consider another example, the beginning of a possible sequence for teaching the concept of subject of a
statement:

Example Teacher Wording

A fast runner went to My turn. What’s the subject?

the park. A fast runner.

A runner went to the My turn. What’s the subject?

park. A runner.

A little girl went to the What’s the subject?

park.

The dimension manipulated is words in the sentence. We can manipulate this dimension and not change the
response from that of a particular example. We should therefore include same-response changes in the sequence as

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illustrated in Figure 8.9. The same-response minimum differences are in italics.

Figure 8.9
Example Teacher Wording
1. A fast runner went in the My turn. What’s the
park. subject? A fast runner.
2. A runner went in the My turn. What’s the
park. subject? A runner.
3. A runner went up the hill. My turn. What’s the
subject? A runner.
4. Five girls went up the hill. Your turn. What’s the
subject?
5. That girl went up the hill. What’s the subject?
6. That girl sat on the hill. What’s the subject?
7. Five men sat on the hill. What’s the subject?
8. A dog and five men sat What’s the subject?
on the hill.
9. My cat ate beans. What’s the subject?
10. An old alligator was tired. What’s the subject?
11. He sang songs. What’s the subject?
12. My mother likes to cook. What’s the subject?
13. The yellow pencil and the What’s the subject?
while pencil are on the
desk.
14. She slept through What’s the subject?
breakfast.
15. Phone books are very What’s the subject?
useful.
16. A slow turtle sat on a log. What’s the subject?
17. They went in the park. What’s the subject?
18. Hank and Mary had an What’s the subject?
argument.
19. That dream is interesting. What’s the subject?

This sequence may contain too many subtypes (suggested by the number of minimum-difference examples). The
learner may have trouble with the single word subjects (he, they, she) and with the verbs other than action words (is,
has, are). Note also that the sequence does not include subjects that contain ing words (running is fun), subjects that
begin with the word to (to work is fun), or subjects that have verb words (the best vacation we took was in Colorado).

Figure 8.10 gives a sequence that presents a smaller range of sentences. The setup for this sequence is designed so
that part of every predicate is the same. The same-response minimum differences are in italics.

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 Figure 8.10
Example Teacher Wording
1. Fast runners went to the My turn. What’s the
park. subject? Fast runners.
2. A runner went to the My turn. What’s the
park. subject? A runner.
3. Five runners went to the Your turn. What’s the
park. subject?
4. Five runners played in the What’s the subject?
park.
5. That runner sat in the What’s the subject?
park.
6. That runner liked the park. What’s the subject?
7. Henry’s dog ran in the What’s the subject?
park.
8. I went to the park. What’s the subject?
9. Those pigeons flew over What’s the subject?
the park.
10. My brother hates the What’s the subject?
park.
11. He wanted to go to the What’s the subject?
park.
12. Her dog loves the park. What’s the subject?
13. Old people walk in the What’s the subject?
park.

The two sequences for presenting subject of a sentence represent extreme positions. The first sequence processes
a wide range of subtypes and the second sequence involves a single relatively tightly circumscribed subtype. A very
reasonable question is: Which sequence is appropriate for learner X? Stimulus-locus analysis is mute on this issue. The
question is answered through a field tryout.

Recognize the limitations of the analysis. The analysis can tell us only how to achieve certain goals. If we want to
show how two things are different from each other, the analysis provides us with specific procedures for doing so. The
analysis suggests that if the sequence is guilty of stipulation, it must be followed by other sequences specifically
designed to counteract this stipulation. If we decide that the learner should be able to handle a wide range of subtypes,
the analysis suggests the most efficient procedures for communicating the sameness—juxtaposing examples that are
greatly different (examples from various subtypes). If we discover that one of the subtypes included the original
sequence is difficult for the learner, the analysis suggests how we can make that subtype easier (by juxtaposing
examples of the subtype with other examples of the same subtype so the learner does the same thing with juxtaposed
examples). The fastest way to secure information about these “ifs” is to design a sequence that we judge to be slightly
too difficult for the learner. This sequence could contain too many subtypes, too much variation, or subtypes that are
too difficult. If our judgment is correct, we will observe a trend of mistakes.

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Examples that Differ Only in Subtype Details

A final application of the subtype analysis is limited to some symbolic concepts. These concepts permit us to create
a set of examples that is the same except for one part. For instance, if we were teaching the concept of verb of a
sentence, we could create a series of examples that differ only in the verb. The other words in the sentence would
remain constant. This type of sequence is, in one sense, analogous to the non-comparative and comparative sequences.
The setup features are the same for every example and only a single dimension varies.

Figure 8.11 presents a sequence that employs this strategy. This sequence teaches the generalization of the different
types of verbs. It shows that there are one-word verbs, two-word verbs, three-word verbs, and four-word verbs.

Figure 8.11
Example Teacher Wording
1. She runs in the park. My turn. The verb is
runs.
2. She ran in the park. My turn. The verb is ran.
3. She will run in the park. My turn. The verb is will
run. Your turn. What’s
the verb?
4. She would run in the What’s the verb?
park.
5. She was running in the What’s the verb?
park.
6. She has run in the park. What’s the verb?
7. She ran in the park. What’s the verb?
8. She has been running in What’s the verb?
the park.
9. She may have been What’s the verb?
running in the park.
10. She should run in the What’s the verb?
park.
11. She had been running in What’s the verb?
the park.
12. She could have been What’s the verb?
running in the park.
13. She is running in the What’s the verb?
park.
14. She runs in the park. What’s the verb?
15. She might have run in the What’s the verb?
park.

Note, however, that it does not teach different types of verbs. It stipulates that the verb may be limited to action
words (or to the verb run). Subsequent sequences would be needed to introduce the different subtypes of verbs. The
value of the sequence in Figure 8.11, however, is that it shows the general structure of all verbs.

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 Note that the sequence is based on subtype variations. The only difference from example to example is that the
verb is of a different subtype.

To create subtype-variation sequences, follow these steps:

1. First determine whether it is possible to make examples that are identical except for the variation in the
subtype.

2. Identify the various subtypes.

3. Follow the pattern (as closely as possible) for introducing minimum differences at the beginning of the
sequence, followed by juxtaposed examples that differ as greatly as possible within the constraints of the
setup.

Consider this type of sequence when the subtype analysis discloses that there is a staggering range of possible
subtypes (as in the case of verbs). Instead of trying to design an initial teaching sequence that presents a single subtype
(such as one-word verbs), consider an initial sequence that teaches a large group of subtypes. Usually, less stipulation
will result from this strategy than results from the introduction of a single subtype. The reason is that the subtype
sequence (with the only difference between examples being a subtype difference) shows the learner more about the
concept than a series dealing with a single subtype can. If we initially introduce one-word verbs, we may induce serious
stipulation about the structure of all verbs. We may have to introduce three or four subsequent sequences based on a
single subtype before we effectively counteract this stipulation. The subtype sequence shows the basic structure of all
verbs through a single sequence. The sequences that follow the first sequence simply show more about the range of
variation (e.g., that the subjects can change without affecting the verb; that different verbs function the same way as
running when they are substituted for running; that verbs function as verbs only when they follow the subject in a
regular-order sentence; and that some words, such as usually, may split the verb). In the end, we are required to teach
all these relationships. The subtype sequence provides us with the most efficient initial sequence.

Perspective

The subtype analysis raises an issue that we will encounter again—the easy-to-hard sequence. The traditional
orientation to designing a program that proceeds smoothly is based on the idea that it is possible to identify examples of
progressive difficulty. This orientation then draws the false conclusion that the most efficient program orders the
introduction of the various subtypes identified in the easy-to-hard order, so the learner first masters the easiest subtype,
then the second-easiest subtype, etc. There are two serious problems with this orientation. The most obvious is that it
does not take into account the teaching of sameness. The easy-to-hard order is well designed not to show sameness
across the subtypes, but instead to induce great stipulation. If we used this orientation to teach fractions, we would first
introduce those that involve one piece of a pie (1/2 1
/3 1
/4). Working on this subtype first would induce serious
stipulation. If we followed the example with another relatively easy type (whole numbers expressed as fractions over 1
( 5/ 1 7
/1 4
/1), we would induce both reversals and further stipulation (that a fraction must have the number 1).

Certainly, the initial sequence should be easy. But ideally, it must also show the learner what is the same about all

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possible examples that the learner will deal with. Unless it conveys information about this sameness, it is a poor initial
sequence.

A second problem created by the traditional easy-to-hard orientation is that the program typically begs the question
of efficiency by using mastery criteria for each subtype. Although the sequence of subtypes from easy to difficult is
poorly designed, the learner will most probably achieve mastery if enough trials are provided. However, the fact that
the learner masters each subtype does not imply that the program is efficient. In fact, the mastery requirement may
actually promote greater stipulation by requiring the learner to work on a relatively ungeneralizable subtype for a
longer period of time than would be required if the mastery criterion were ignored.

We will return to the issue of easy-to-hard when we deal with more complex communications. The problem that
we will deal with is the same one that emerged from the subtype analysis. We must show all types as quickly as
possible. Only when we show the learner what is the same about all examples do we teach the concept. The longer we
delay dealing with the full range, the more problematic the sequence of subtypes is.

Summary

We can present any concept as a choice-response discrimination. Some concepts, however, are transformations.
Although there are many variations of transformation sequences, their structure is basically the same.

• Design the sequence so that the same wording is used for each test example and so this wording leads to
different responses for examples that are different.

• Begin the sequence with a series of 4 to 6 minimum-difference examples. These should be arranged
progressively, with a change in one detail occurring from example to example.

• Use no more than 6 minimum-difference examples to show the range of subtypes. If more than 6 are
needed, the sequence probably covers too many subtypes. Adjust by removing some of the subtypes.

• If the total sequence involves more than eighteen examples, it is probably too long.

• After the minimum differences in the sequence, juxtapose examples that differ greatly (a change in more
than one detail occurring from example to example).

• At least two examples should be modeled. Test on the rest. Limit the examples that are tested to those that
are presented in the minimum-juxtapositions of the sequence. (For sophisticated learners, this requirement
may be waived.)

• If the original sequence does not present the entire range of variation for the concept, it will be guilty of
stipulation. Counteract the stipulation by following the original sequence with a sequence that counteracts
the stipulation (negates it).

Two approaches are efficient for introducing a transformation that involves many subtypes:

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 1. Introduce a relatively large regular subtype and follow it with other regular or irregular subtypes.

2. Introduce a sequence in which the examples differ only in subtype differences; follow the sequence with
others that expand the range of application for the various subtypes.

The second approach avoids possible stipulation more readily than the first; however, the first is analytically more
predictable. We do not know how difficult an example of the second type will be. We can design the total teaching
more easily if we use the first option, however. The reason is that this solution permits us to introduce one new subtype
at a time.

Because transformations deal with concepts that have many possible minimum differences and a wide range of
possible variation, the most efficient analysis of the concept involves identifying subtypes based on regularity (or
generalizability) and difficulty (based on the number of behaviors called for to process the examples in a particular
subtype). To perform the subtype analysis:

• Select a range of examples.

• Sequence them to show sameness.

• List the features that are the same for all members of a subtype.

• Apply the subtype analysis to determine possible initial sequences (a generalizable type that does not
involve complicated behaviors).

• Apply the analysis to determine a strategy for introducing the other subtypes.

• Apply the analysis to determine whether the learner has difficulty with a particular subtype.

• Also use the analysis to deal with transformation skills that involve great variation across many
dimensions.

• If the concept permits, design a transformation sequence in which the only difference from example to
example is a subtype difference.

Transformations are very important concepts. The latitude that we have in introducing these concepts is much
greater than that for basic-form concepts simply because transformations are joining-form concepts. Because they
combine basic-form concepts, they generate a very large range of possibilities. If we combined the subjects (he, she
and it) with the verbs (runs, sits, and eats) we have nine examples. If we include double and triple combinations (he
and she, sit and eat), there are forty possibilities.

This great range of variation for transformation concepts suggests that there are many different strategies for
approaching the teaching of a given transformation concept. A further implication is that we will not be able to teach
the transformation through a single sequence. Although there are many possible efficient ways to approach a
transformation, we must remember that the initial sequence is the most important. It should be very carefully designed
to show as much as possible about the structure of the concept and the sameness across different examples. If the initial
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sequence is poorly designed (dealing with a small subset of examples), the teaching will become relatively inefficient.
If in doubt about whether to start with a very easy (but possibly limited) subset, or to show same- ness by including a
wider range of variation in the initial sequence, show the sameness. The performance of the learner will indicate
whether the subtype you select is too broad or difficult.
 

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 Correlated-Features Sequence
The second-joining form, the correlated-features form, creates a joining quite different from the transformation
joining. The transformation is a rule for replacing parts of a response with coordinate substitutions. The correlated-
features joining involves linking things that happen to occur together. The goal of the correlated-features sequence is to
teach the learner that the presence of specific stimulus features indicate that other unobserved features or events also
occur. The features that are joined by correlated-features communications have names or labels. The joining involves
two discriminations. One is shown; the other is not. The learner is asked to predict the status of the unobserved
discrimination from the observed discrimination.

The communication of correlated-features is the transition between the immediately observable features of
examples and the complete symbolized representation of events presented in facts (verbal statements). The correlated-
features communication is most appropriate for teaching science rules, spelling rules, unfamiliar words that have
familiar meanings, and similar applications.

Correlated-Features and Statements of Fact

The correlated-feature joining is the same type that occurs in statements of fact. By dealing with statements of fact,
we observe the basic properties of the correlated-feature joining.

All facts may be regarded as having two parts and each part specifies a discrimination. The first part names an
entity and the second part indicates the link or relationship between the entity named and some other features.

Here’s a fact: Ducks fly north in the spring. The entity named is ducks. The discrimination correlated with ducks
is flying north in the spring. If we accept this fact and saw a duck, we would know that the entity flies north in the
spring. We would not have to observe this flying if we relied on the fact.

Another fact: China cups are breakable. The first-named entity is China cups. This entity is linked to being
breakable.

Another fact: Only small glerms with big ears hunt at night. Even though we would not be able to identify
concrete examples of “small glerms with big ears,” this entity is named in the fact. The feature that it possesses is
hunting at night. Therefore, if we are provided with information that a concrete entity is a small glerm with big ears,
we would know that the entity hunts at night.

First-Mentioned Discriminations

All facts name two discriminations. Correlated-features joining is based on the discriminations that are joined in a
particular fact.

The first-mentioned discrimination tells which examples are presented in a correlated-feature sequence. If we wish

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to teach the link that is expressed by this statement of fact: hotter objects expand, we would present hotter objects.

If the fact is: ducks fly south in the spring, we would present examples of ducks. If the fact is: When the sound
/oy/ occurs at the end of a word, the sound is probably spelled o-y, we would present words with the sound /oy/ at
the end.

How the Examples are Sequenced

The second part of the fact tells how the examples are arranged in the sequence. The second part implies either: (1)
a transformation ordering of the examples; or (2) a simple single-dimension discrimination ordering.

Note that transformations are possible only if the first-mentioned entity is symbolic. The following fact implies a
transformation: Words rhyme with their ending. (Rhyming with the ending can be demonstrated with a variety of
words, each leading to a different rhyming response.)

The following fact does not imply a transformation: Words are the basic unit of language communication. (If
something is a word, we draw one conclusion about it. It is the basic unit of language communication. The only
response that is possible for any example of a word is that it is the basic unit of language communication or that it is not
the basic unit of language communication.)

This fact does not imply a transformation: Two digit numbers are bigger than nine. (For any positive two-digit
number, only one conclusion is possible: It is bigger than nine.)

This fact does imply a transformation: The first digit of two-digit numbers tells the number of tens. Every two-
digit number has the property of telling the number of tens. (How many tens for this numeral?) Different answers are
possible for different first digits.

If the second-named discrimination in a fact describes a transformation, arrange the examples as you would in a
transformation sequence.

If the second-named discrimination in a fact does not describe a transformation, arrange the examples as you
would for a basic sensory discrimination (either negative-first or positive-first.)

Wording

The wording used for each example in a correlated-feature sequence consists of two questions.

The first question requires a conclusion. It asks about what is not shown.

The second question requires the learner to link the conclusion with the correlated-features of the examples. This
question is: How do you know?

If the fact being taught is: Hotter air rises, we would show examples of hotter air and not-hotter air. We would
first ask about whether the air rises (which outcome is now shown) and then, “How do you know?” The questions
could be: “Will this air rise? . . . How do you know?” The instructions could be: “Tell me rises or not-rises . . . How do

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you know?” In responding to the two tasks, the learner responds to the relationship that is being taught.

Example Wording Response

Air gets hotter. What will happen It will rise.
to this air?
How do you know? Because it
got hotter.
Air does not What will happen It will not rise.
get hotter. to this air?
How do you Because it
know? didn’t get
hotter.

For positive examples of this relationship, the learner is saying “The air will rise...because it gets hotter,” and for
negatives “The air will not rise. . .because it didn’t get hotter.” The learner is predicting the outcome, by applying the
rule to actual changes in the concrete examples. And the learner’s pair of responses provide clear evidence that the
learner is responding to the appropriate features of the examples and is linking these to the appropriate relationship.

Constructing Single-Dimension Sequences

1. To construct a single-dimension correlated feature sequence, we begin by constructing the fact that we
wish to teach. That fact will provide us with the wording that we will use in the sequence. It will also
imply the type of juxtapositions that we will use and the form of the primary task that we will present with
each example.

If we wish to teach the notion that when free air becomes heated it rises, here is a possible expression of
that idea: Hotter air rises.

2. We analyze the fact to determine whether the second part of the fact (rises) implies a transformation. Since
the concept rises, not-rises is most appropriately treated as a single-dimension comparative, the second
part of the fact does not imply a transformation.

3. We select the juxtaposition pattern that is appropriate for the second part of the fact. Since the second part
of the fact hotter air rises does not name a transformation, we will use the juxtaposition pattern that is
appropriate for single-dimension concepts.

4. We now create a sequence that would show the concept named in the first part of the fact (hotter air). We
use the same juxtaposition pattern and the same examples that would be used to communicate hotter air.

5. We construct a pair of questions that is to be presented with each example, the first asking about rising or
not-rising, the second asking “How do you know?”

Figure 9.1 presents a correlated-feature sequence for teaching the relationship between hotter air and rising. Note
that the examples presented in this sequence are sensory examples. (The learner feels the differences from example to
example. The numbers simply represent discriminable differences, with 4 and 5 being minimally different, but
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discriminably different.)

Figure 9.1

Heat Level of Example Teacher Wording

4 I’m going to tell you whether this air would rise. Feel it now.
4 It won’t rise. How do I know? Because it didn’t get hotter.
3 It won’t rise. How do I know? Because it didn’t get hotter.
4
C
It will rise. How do I know? Because it got hotter.
7 D Your turn: Will the air rise? How do you know?
9 Will the air rise? How do you know?
8 C B Will the air rise? How do you know? A’
1 Will the air rise? How do you know?
3 E Will the air rise? How do you know?
2 Will the air rise? How do you know?
5 Will the air rise? How do you know?
5 Will the air rise? How do you know?
7 Will the air rise? How do you know?

Juxtapositions

The examples are the same as those used to teach hotter versus not-hotter. The juxtapositions are also the same as
those used to teach hotter versus not-hotter. The sequence begins with a starting point, then two negatives, followed
by three positives and a series of unpredictably ordered examples.

The A’ bracket suggests a variation in wording conventions. Although the learner does the same thing with
juxtaposed examples, the learner does more than one thing with each example. Therefore, an interruption occurs
between answering different presentations of the question, “Will the air rise?” (Before another instance of this question
is presented, the learner is presented with the question, “How do you know?”)

Sequence Length

The sequence above is as long as it would be if we were designing a safe presentation for teaching hotter to a
naive learner. The learner who is being taught the relationship hotter air rises, however, is not naive. The learner has
already learned the component concepts, hotter and rises. Therefore, the sequence above is needlessly long.
Remember, the goal is not to teach the component discriminations, but to establish the fact that rising depends solely on
the relative temperature of the air (so far as the present relationship is concerned). Therefore, we can shorten the
sequence. Figure 9.2 presents a shortened version that begins with one negative and two positives. Only the first two
examples are modeled:

The sequence consists of six examples, which is probably enough to establish the correlation.

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 Figure 9.2

Heat Level Wording

of Example

4 Feel the air. I’m going to tell

you if it would rise.
4 It won’t rise. How do I know?
Because it didn’t get hotter.
5 It will rise. How do I know?
Because it got hotter.
7 Your turn: Will it rise? How do
you know?
7 Will it rise? How do you know?
2 Will it rise? How do you know?
5 Will it rise? How do you know?

Representations Rather than Real-Life Objects

We do not have to present sensory examples if there is no communication problem associated with representations
of the sensory qualities. Let’s assume that the learner understands the relationship between heat increases and
increases in number of degrees. We could now teach the relationship hotter air rises by presenting references to heat
units rather than direct sensory presentations. (See Figure 9.3.)

Figure 9.3

Example Wording

40 I’ll tell you if the air rises. It starts

out at 40 degrees.
40 Then it stays at 40 degrees. Did it
rise? No. How do I know?
Because it didn’t get hotter.
50 Then the air goes up to 50 degrees.
Did it rise? Yes. How do I know?
Because it got hotter.
70 Then the air goes to 70 degrees.
Did it rise? How do you know?
70 The air stays at 70 degrees. Did it
rise? How do you know?
20 The air goes to 20 degrees. Did it
rise? How do you know?
50 The air goes to 50 degrees. Did it
rise? How do you know?

Note that the various temperature examples were created simply by adding a zero to each level shown in the
preceding sequence.

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Unambiguous Responses

The primary goal is to create communications that are consistent with a single interpretation. The sequence above
prompts a possible misrule, which is that the rising refers simply to an increase in temperature (the temperature rises)
not physical movement in the air. To correct this possible misrule, we could modify the primary response required by
the learner. Instead of giving the learner the choice of labeling rising and not-rising, we could give the learner the
choice of showing whether the air will rise or not rise. (See Figure 9.4.)

Figure 9.4

Wording

I’ll point to show you which way the air moved as it got hotter and colder. It starts out at 40 degrees. Then it goes to 50
degrees. Which way did it move? How do I know it went up? Because it got hotter.
Then the air goes to 40 degrees. Which way did it move? How do I know it didn’t go up? Because it didn’t get hotter.
Then it goes to 20 degrees. Show me which way it moved. . . . How do you know it didn’t go up?
Then it goes to 70 degrees. Show me which way it moved. . . . How do you know it did go up?
Then it goes to 90 degrees. Show me which way it moved. . . . How do you know it did go up?
Then it goes to 80 degrees. Show me which way it moved. . . . How do you know it didn’t go up?

The wording is changed so that it refers to going up rather than rising. The reason is simply that the past tense of
rising is rose and the question, “How do you know it rose?” does not parallel the question, “How do you know it didn’t
rise?”

The selection of examples is modified for practical reasons. There are no no-change negatives. The reason is that
the fewer pointing conventions we introduce, the better. The sequence therefore presents a simple relationship. If the
temperature goes up, you point up. If the temperature goes down, you point down.

Summary of Variations for Single-Dimension Sequences

We start with a fact that expresses a sensory-based link. If the second part of the fact does not suggest a
transformation, we use the single-dimension sequence format. We show the first-mentioned discrimination in the fact.
We present two questions. The first requires a conclusion about the feature that is not observed. The second question,
“How do you know?” requires the learner to make an observation about the sensory evidence that led to the conclusion.

Because the learner already knows the component discriminations, we can shorten the sequence. We would retain
at least one minimum difference juxtaposition, but we can probably shorten the sequence to 6 examples.

Because the learner is familiar with the component discriminations, we may be able to describe each example
rather than presenting it as a sensory example.

Because the response the learner produces may be ambiguous, we can sometimes reduce the ambiguity by
requiring the learner to point or to produce some other physical response.

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 Illustration. If the learner does not know the conventions for expressing heat in degrees, we can design a
correlated-feature sequence that demonstrates the relationship. We start with the fact: When the number of degrees
gets bigger, the object gets hotter. This fact may seem backward. Why isn’t it expressed this way: When the object
gets hotter, the number of degrees gets bigger? The latter fact would require us to present examples of hotter and
not hotter. The learner would have to respond by telling us if the number gets bigger or not bigger. If we present the
numbers and require the learner to tell whether the object gets hotter or not hotter (or hotter or colder), however, we
are provided with an unambiguous response. Figure 9.5 presents a possible sequence.

Figure 9.5

Example Wording

40 This number tells how hot the

object is. Watch the number. I’ll
tell you if the object gets hotter.
50 Did the object get hotter? Yes.
How do I know? Because the
number got bigger.
40 Did the object get hotter? No.
How do I know? Because the
number did not get bigger.
70 Your turn: Did the object get
hotter? How do you know?
90 Did the object get hotter? How do
you know?
80 Did the object get hotter? How do
you know?
10 Did the object get hotter? How do
you know?
50 Did the object get hotter? How do
you know?

A variation of this procedure can be used to teach any measurement relationship. Note that the sequence assumes
that the learner is attending to the numbers and does not misidentify them. The only way we would know whether this
is the case, however, would be to require the learner to produce some sort of overt response. In other words, we would
add an identification task to each example. Two test examples are shown in Figure 9.6.

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 Figure 9.6

Example Wording

70 Did the object get hotter? Yes.

90 How many degrees now? Did
the object get hotter? How do
you know?
80 How many degrees now? Did
the object get hotter? How do
you know?

This procedure has both an advantage and a disadvantage. According to the wording-juxtaposition principle, the
same wording on juxtaposed tasks is the easiest juxtaposition. The example that involves three tasks violates this
principle, which means that the difficulty level of the sequence increases. The advantages are that: (1) the three-task
examples permit us to introduce the label degree, and they make all the steps overt, thereby reducing ambiguity about
what the learner is doing to process the example.

Correcting Repeated Mistakes

If the learner has repeated problems with any correlated-feature presentation, the best procedure is to improve the
wording juxtapositions. Simply remove the question, “How do you know?” from each example (and possibly increase
the number of examples). If three-task examples are involved, eliminate two of them, retaining only the primary
question. Figure 9.7 shows a correction sequence for mistakes on either of the temperature sequences above.

Figure 9.7

Example Wording

40° Watch the number. It tells how

hot the air is. I’ll tell you if
the air gets hotter.
40° The air didn’t get hotter.
30° The air didn’t get hotter.
40° The air got hotter.
70° The air got hotter.
90° Tell me about the air.
90° Tell me about the air.
10° Tell me about the air.
30° Tell me about the air.
80° Tell me about the air.

This sequence presents juxtaposed tasks that involve the same response dimension. The learner simply concludes
whether the air got hotter. The sequence presents a full range of examples. The same basic procedures would be used if
the learner had trouble with the sequence for other relationships. Only one task would be presented with each example,

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the task requiring the learner to draw conclusions. Figure 9.8 below shows the first five examples of a possible
simplified sequence for hotter air rises.

Figure 9.8

Example Wording

40° The number tells about the temper-

ature of the air. I’ll tell you whether
that air goes up or doesn’t go up.
30° The air didn’t go up.
30° The air didn’t go up.
40° The air went up.
70° Your turn: Tell me about the air.
90° Tell me about the air.

The examples in these correction sequences are virtually identical to those used to teach sensory-based
discriminations, but instead of asking about what is shown, the question asks about a correlated feature. It would be
possible to present all correlated-feature sequences with only a single question. The problem is that this sequence
suggests, for instance, that going up is simply another name for getting hotter. The question pairing that requires a
conclusion and then an answer to, “How do you know?” prevents this possible misinterpretation. Following the
presentation of the single-task sequence, we would again repeat the original sequence that requires answers to two
questions for each example.

Non-Comparative Correlated-Feature Sequence

The preceding examples have involved comparatives—the more one thing changes, the more another changes.
Figure 9.9 shows a non-comparative correlated feature sequence.

Figure 9.9

Example Wording
If a leaf that I show you has a
pinnate pattern, it is an oak leaf.

My turn: Is this an oak? No. How

do I know? It doesn’t have a
pinnate pattern.

My turn: Is this an oak? Yes. How do

I know? It has a pinnate pattern.

Your turn. Is this an oak? How do

you know?

Is this an oak? How do you know?

Is this an oak? How do you know?

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 This sequence shows minimum-difference changes. Two examples are modeled (a positive and a negative)
followed by three test examples. The sequence begins with a statement about the examples that are to be presented. The
opening statement qualifies or limits the relationship to the set of examples that is to be shown. The teacher does not
tell a “lie,” which is what would happen if the teacher suggested that the presence of a pinnate pattern always predicts
an oak leaf. The statement at the beginning of the sequence could be designed to make this point even clearer than it is
in the sequence above: “I’m going to show some leaves that are oak leaves and some leaves that are not. The ones that
are oak leaves have a pinnate pattern.” A variation of this wording could be used in any case involving a relationship
that is not limited to the concept shown. (“A rabbit has legs;” “Windows are pieces of glass,” etc.)

Negative Wording

We might be tempted to avoid the problem of the qualified examples by using a negative rule, such as: “If a leaf
has a palmate vein pattern, the leaf is not an oak leaf.” Certainly this solves the problem from a logical standpoint. The
problem comes when we create examples. Responses to all examples will contain negative wording, implying a tricky
relationship between the vein pattern and the name. This wording creates a spurious transformation. (It’s not an oak
because it’s palmate; It’s an oak because it’s not palmate.”)

Figure 9.10 shows the first examples from a possible sequence based on negative wording:

If possible avoid patterns in which the presence of something leads to a negative response. Try to make the rule of
the form: the presence of something signals the presence of something else.

Figure 9.10

Example Wording

I’ll tell you if these leaves are oak

leaves. Is this an oak? No. How do
I know? Because the vein pattern is
palmate.

Is this an oak? Yes. How do I know?

Because the vein pattern is not
palmate.

Correlations Involving and-or

Some relationships that we wish to teach involve the presence of more than one feature or the presence of either
one feature or another feature. The structure of these relationships carries implications for the design of the sequence.

and. When dealing with a correlation that involves the conjunction of two properties (an and relationship), begin
with positive examples. For example, this rule: “If it has stripes and it looks like a horse, it’s a zebra,” expresses an and
relationship. For an example to be positive it must: (1) have stripes, and (2) look like a horse. If we begin with a
positive example, the first example will specify the criteria for classifying members as zebra or not-zebra.

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 “My turn. Is this a zebra? Yes. How do I know? Because it has stripes and it looks like a horse.”

If we begin with a sequence with a negative example (see Figure 9.11), however, both criteria are not mentioned
with each example.

Figure 9.11

Example Wording
My turn: Is this a zebra? No.
How do I know? Because it
doesn’t have stripes.

My turn again. Is this a zebra?

No. How do I know? Because
it doesn’t look like a horse.

The presentation of two examples has not yet provided the information that both features must be present. The
learner might believe that the criteria mentioned in examples 1 and 2 are two of many possibilities.

The problem is compounded if more than two criteria are used. “If the object grows, reproduces, and dies, the
object is alive.” This relationship would require a minimum of three negatives to present the three criteria. One positive
would do the same job more clearly (e.g., “My turn. Is this object alive? Yes. How do I know? Because it grows,
reproduces, and dies.”).

or. Just as it is efficient to begin an and relationship with positives, it is efficient to begin an or sequence
(disjunctive relationship) with negatives. By beginning with a negative for or relationships, we express both the criteria
involved in the classification. Let’s say the relationship is: “If it’s a reptile, an amphibian, or a fish, it is cold-blooded.”
Figure 9.12 shows the information the learner receives through a negative example.

Figure 9.12

Example Wording
Horse My turn. Is this animal coldblooded?
No.
How do I know? It’s not a reptile or
an amphibian or a fish.

If we begin the sequence with positives, we do not efficiently provide the learner with information about the
criteria. Figure 9.13 shows a sequence that begins by presenting an alligator, a perch, and a frog.

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 Figure 9.13
Example Wording

Alligator My turn. Is this animal cold-blooded?

Yes. How do I know? Because it’s
a reptile.
Perch My turn. Is this animal cold-blooded?
Yes. How do I know? Because it’s
a fish.
Frog My turn. Is this animal cold-blooded?
Yes. How do I know? Because it’s
an amphibian.

At this point, the learner might assume that all vertebrates (including mammals and birds) are cold-blooded. After
all, every type of vertebrate that is mentioned is cold-blooded.

In summary, if a correlated-feature relationship involves and-criteria (the conjunction of two or more criteria),
begin the sequence with positives.

If the correlated-feature relationship involves or-criteria (the disjunction of two or more criteria), begin the
sentence with negatives.

Equivalent Meanings

An equivalent meaning is the substitution of one word or phrase for another that serves as an equivalent for the
original. There are many different ways to teach equivalent meanings. Some are relatively faster and more efficient
than others. We can teach a “definition” that indicates the relationship such as: “When something expands, it gets
bigger.” Then we can approach the definition problem as a transformation problem:

I’ll say things one way, you say them another way, using the word expanded.

Listen: The cloud got bigger. Say it another way.

Listen: The balloon got bigger. Say it another way.

Listen: The steel ball got bigger. Say it another way.

A second way is to require the learner to correlate the meaning with actual examples. This procedure would be
most appropriate for learners who had not demonstrated great proficiency in verbal transformations or who tended to
play “word games” without actually attending to concrete examples. We would express the relationship as a fact:
“When things get bigger, they expand.” We would then present examples of things getting bigger or not-getting bigger.
We would require the learner to indicate whether each example expands. Then we would require the learner to relate
this conclusion to the features of the example. (Note that this treatment supposes that the learner already knows how to
label examples as “bigger” and “not bigger.”) This approach provides the most precise information about whether the
learner is forming the appropriate relationship between sensory features, the familiar word (bigger), and the new word

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(expand). Figure 9.14 shows a possible sequence.

Figure 9.14

Example Teacher Wording

Shows space between hands.

Watch this space. I’ll tell you if it expands or doesn’t expand.

1. My turn. It didn’t expand. How do I know? Because it didn’t get bigger.

2. My turn. It expanded. How do I know? Because it got bigger.

3. Your turn. What happened? How do you know?

4. What happened? How do you know?

5. What happened? How do you know?

6. What happened? How do you know?

7. What happened? How do you know?

8. What happened? How do you know?

The primary question is designed so that the learner produces the word that is being taught. Instead of answering
“yes” or “no” to a question such as, “Did the space expand?” the learner responds to, “What happened?” by using the
word expand.

Remember, for equivalent meaning, ask first about the new word, then about the old. (The primary instruction
could change to: “Tell me expand or not-expand.” The second question would be: “How do you know it expanded?”

Verifying Conclusions

Developmentalists and those concerned with learning by doing, often exhibit an impractical approach to empirical
relationships. The learner conducts an experiment and observes the outcome. From this conclusion, the learner is
supposed to learn the correlation between the presence of some features and the presence of others. However, the
communication is ambiguous. Let’s say that an experiment is designed to show how low pressure affects the movement
of air. One experiment involves turning on a shower and observing the direction in which the shower curtain moves.
The learner concludes that the shower curtain moved toward the stream of water.

The learner might indeed learn about the correlation between the pressure of the air and the movement of the
shower curtain from this presentation; however, the presentation is not designed to teach this relationship because it
shows the outcome before the learner predicts the outcome. (The presentation is also poorly designed in a number of
other ways. It does not provide for quick juxtaposition of examples that change along particular dimensions. It does not
provide for wording that would assure precise communication of juxtaposed examples. It does not provide any sort of

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reasonable test of the generalization.) To communicate the desired relationship we would redo the communication to
teach the fact: Air moves toward a place of lower pressure. (See Figure 9.15.)

Figure 9.15

Example Wording
My finger will point to the way the air moves. I’ll tell you which is the place of lower pressure.

1. J is the place of lower pressure. How do I know? Because the air moved toward J.

2. Your turn: Which is the place of lower pressure? How do you know?

3. Which is the place of lower pressure? How do you know?

4. Which is the place of lower pressure? How do you know?

After the learner has demonstrated an understanding of the basic correlation, the learner could then apply the
correlation to the actual-life situation (or an illustrated situation). This activity is perfectly reasonable as an application.
However, it is not a reasonable presentation for teaching very much about the relationship. The learner who is first
taught the basic relationship through a sequence such as the one in Figure 9.15 would be able to frame the application
as a simple extension of the correlation. The learner is shown a diagram of a shower as in Figure 9.16.

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 Figure 9.16

Teacher: “The shower curtain doesn’t move when the shower is not turned on. So circles R, J, P and T are not
places of lower pressure.” After the shower is turned on, the curtain moves toward R. (We can point to show the
direction it moved.) “So which is the place of lower pressure?” (Point to R.) “How do we know? Because the air moved
toward point R.”

The application is now continuous with the general rule for air pressure. Similarly, every example of air movement
can be diagrammed in exactly the same way as the shower problem.

When dealing with scientific facts, provide the learner with verifications or demonstration that the “rule” does
actually predict what will happen in real-life situations. Provide this demonstration or verification, however, after the
learner has demonstrated an understanding of the correlation of features implied by the fact. The initial instruction
should require the learner to predict or draw a conclusion.

Correlated Features Involving Transformations

The sequences that we have dealt with are single-dimension sequences. The learner does not produce different
responses for different examples. The learner, instead, classifies different examples—indicating whether each is a
positive or a negative.

Correlated-features sequences that involve transformations are different. They do not require the learner to choose
the class for various items, but to produce different responses.

The tasks require the learner to DO IT, not simply to CLASSIFY IT. This production requirement is a major
difference between a transformation and a sensory discrimination. If the learner creates different verbal responses for
different related examples, the correlated features is presented as a transformation.

If a transformation is involved, the fact we start with must meet two criteria:

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 1. It must begin by naming a class of things that are symbolic—any word, fractions that are more than 1,
prime numbers, adjectives, contractions, vowels, inverted word orders, etc.

2. The second part of the fact must suggest a transformation relationship. If the second part describes a
property that is possessed by a large number of examples, a transformation is implied. Any word that
rhymes with its ending. Fractions that are more than one have numerators larger than denominators.

If a transformation is involved, it is possible to make up a number of examples, and each example will involve a
different response.

We can demonstrate how a correlated-feature concept involving a transformation works by starting with this fact:
If you make the top and bottom of the fraction the same, you make a fraction that equals one whole. The first part
of the fact names a symbolic class—tops and bottoms of fractions. The second part of the fact suggests that a
transformation is possible. If we try naming the fraction that equals one whole, we discover that there are many
possible fractions that could be named. Therefore, we can design a task that requires the learner to name some of those
fractions. The example could be fractions with either the numerator or denominator missing. The test tasks could be:
“Make this fraction equal to one whole . . . How do you know it’s equal to one whole now? . . .”

The transformation is not a necessity because all discriminations may be presented as choice discriminations (those
that require the learner to classify examples). Only some, however, may be presented as transformations. Those are
discriminations that deal with symbolic matter (words, numbers, etc.) organized in a manner that presents a parallel
between changes in specific features of the examples and corresponding changes in specific parts of the responses.

Figure 9.17 and 9.18 show two sequences that derive from the fact: If you make the top and bottom of a fraction
the same, you make a fraction that equals one whole. The first sequence treats the relationship as a single-dimension
discrimination and uses a choice-response test. The second sequence treats the relationship as a transformation and
requires the learner to produce the various transformation responses.

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 Figure 9.17
A Choice-Response Sequence

Example Wording
1. 5 My turn. Does this fraction equal one?
4 No. How do you know? Because the
top and bottom are not the same.
2. 4 My turn, again. Does this fraction equal
4 one? Yes. How do I know? Because
the top and bottom number are the
same.
3. 98 Your turn. Does this fraction equal one?
98 How do you know?
4. 7R Does this fraction equal one? How do
7R you know?
5. 7 Does this fraction equal one? How do
7R you know?
6. 14 Does this fraction equal one? How do
8 you know?
7. 12 Does this fraction equal one? How do
12 you know?
8. 81 Does this fraction equal one? How do
5 you know?
9. 241P Does this fraction equal one? How do
241P you know?

The sequence begins with a negative and three positives, followed by a range of test examples. The learner is
required to classify each example.

The correlated feature transformation sequence in Figure 9.18 presents the same basic examples that appeared in
Chapter 8 as a single-transformation sequence dealing with fractions equal to one.

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 Figure 9.18
A Transformation Sequence

Example Wording

I’m going to show fractions that equal one, but part of each fraction is missing.
1. 12D =1 My turn to say the fraction that equals one: 12D over 12D. How do I know 12D over 12D equals
one? Because the top and bottom are the same.
2. 12 =1 My turn again: What fraction equals one? 12 over 12. How do I know it equals one? Because the
top and bottom are the same.
3. 2 =1 Your turn: Say the fraction that equals one. How do you know?

4. =1 Say the fraction that equals one. How do you know?

2
5. =1 Say the fraction that equals one. How do you know?
17
6. =1 Say the fraction that equals one. How do you know?
3+R
7. 100R =1 Say the fraction that equals one. How do you know?

8. =1 Say the fraction that equals one. How do you know?

2/3
9. 5R/7 =1 Say the fraction that equals one. How do you know?

The transformation sequence begins with progressive minimum differences, examples in which either the top part
or the bottom part is missing. The fact that both bottom numbers and top numbers are replaced in the sequence may
seem to be in violation of the setup principle. Actually, however, example four shows that the variation in setup is
needed because it serves as a “same-response difference” (i.e., the response for example 4 is the same as that for 3).

The learner is required to say the fractions that equal one. The learner is then required to indicate why the fraction
equals one.

To create a correlated-features transformation sequence, follow these procedures:

1. Create the same order of examples that would be used for a single-transformation, starting with a series of
progressive minimum differences and then proceeding to greater differences.

2. Use the same primary question that would be used for a single transformation. (This question is the same
one presented with each example.)

3. Follow with the same second question used for choice-response correlated features sequence—How do
you know?

Creating a correlated-features transformation sequence is not difficult if you think of it as a single transformation
sequence (perhaps somewhat abbreviated) with two questions for each example instead of one. The question that is
added is: “How do you know?”
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Which Sequence?

We have introduced four different sequences that could be used to teach concepts such as fractions that equal
one—the single transformation, the correlated-feature choice response, the correlated-features transformation, and a
choice-response single dimension discrimination sequence. Which sequence is most appropriate? In most cases, either
the single transformation or one of the correlated-features sequences would be preferable. The reason is that these give
us more information about what the learner is attending to. These sequences make the responses that are relevant to
processing the various examples more overt. The more overt the procedure, the less ambiguity there is about what the
learner actually does when making a mistake, and what sort of correction is implied.

When dealing with symbolic concepts, remember that it is possible to present either single-dimension
discrimination sequences or transformation sequences. If it is possible to express a fact about correlated features (such
as, when the top and bottom of fractions are the same the fraction equals one), a correlated-features sequence may
be used. For the strongest responses from the learner, use a transformation sequence or a correlated-features sequence
that is based on the transformation sequence. The Figure 9.19 diagrams the choices:

Two important points about this decision flow chart are:

1. The question: “Can a fact be designed to express the relationship of correlated features?” can always be
answered, “yes.” If we know what the features are and how they work, we can always make up a fact that
expresses the relationship. The test of whether we should use a particular fact depends on the amount of
teaching required to prepare the learner. A fact names various discriminations. If it names many
discriminations the learner probably does not know, a great deal of teaching must be provided before we
can teach the new concept. Is it worth teaching these component skills so that we can teach the fact? If not,
the option is to treat the discrimination as a single-transformation concept and not require the learner to
express the relationship between correlated features.

2. The ease of producing the responses should be seriously considered in deciding whether to use a choice-
response or a transformation. If the fact that we make up is very long, even though it involves no new
concept, the juxtapositions are weakened. Think of the fact as interference that occurs between examples
in which the learner does the same thing. The longer the fact, the greater the interference. The longer fact
also increases the probability that the learner will have response problems. A similar problem occurs when
the fact is shorter, but involves expressions that are difficult to produce. If we have reason to believe that
the fact is “difficult,” we should not introduce a correlated-features transformation sequence.

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 Figure 9.19
Decision Flow Chart for Choice of Programs

Is the Use a noun

concept Yes sequence
a noun?

No

Is the
Only a choice concept
response sequence No symbolic?
is possible (involving numbers,
words, or other
symbols)

Yes

Can a fact
Use a single be designed
transformation Not to express the
sequence readily relationship of
correlated
features?

Yes

Are the
responses
Use a correlated- for a correlated- Use correlated-features
features choice No features transformation Yes transformation
sequence(s) or a single reasonably sequence
transformation easy to
produce?

Applications

Many rules for arithmetic, spelling, decoding, and language comprehension can be processed through correlated-
features transformation sequences.

Figure 9.20 shows a sequence in which a side of an equation is either “revalued” or “rewritten.” It is revalued if the
amount on the side is changed. The implication is that the other side must be changed by the same amount. The types of
revaluing and rewriting are limited to operations and notations the learner already understands. Note: The teacher
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points to the left side when presenting all tasks.

Figure 9.20

Example Wording

1. 5 = 5 Here’s the equation we start with. I’ll make changes. I’ll tell you if each
change means that we have to change the other side of the equation.
2. 4+1 = 5 What do we have to do with the other side? Nothing. Why? Because we
didn’t revalue this side.
3. 4+1+1 = 5 My turn again: What do we have to do with the other side? Add one.
Why? Because we added one to this side.
4. 4+1+1-1 = 5 My turn again: What do we have to do with the other side? Nothing.
Why? Because we didn’t revalue this side.
5. 4+1+1-1-1 = 5 Your turn: What do we have to do with the other side? Why?
6. 6-1 = 5 What do we have to do with the other side? Why?
7. 6-2 = 5 What do we have to do with the other side? Why?
8. 5 3 = 5 What do we have to do with the other side? Why?
9. 5 2 = 5 What do we have to do with the other side? Why?
10. 5 1 = 5 What do we have to do with the other side? Why?

This sequence may have too many minimum-difference examples for a transformation; however, the value on the
left side for all examples is near 5; therefore, the minimum-difference does not apply in a strict fashion. There is a large
difference between example 5 and example 6 although the amount on the right side is the same.

The presentation may be ambiguous because we do not know whether the learner is actually figuring out the
amount on each side. To correct this problem, we could add a question for each example (e.g., How many are on this
side now? What do we have to do with the other side? Why?).

The advantage of this solution is that it makes the operation overt. The problem with this solution is that it weakens
the wording juxtaposition.

Another possible solution would be to use simpler examples, or to first go through the sequence asking the single
question, “How many are on this side?” After the learner has responded correctly to the items in the sequence, we can
present the two-question sequence with reasonable certainty that the learner is correctly figuring out how many are on
the left side.

The wording of the second question for the sequence above is, “Why?” This wording is sometimes possible and
may be more appropriate than, “How do you know?” However, the wording, “How do you know?” can be used in
nearly every situation, including the sequence above. If in doubt, use the wording, “How do you know?”

The correlated-features transformation sequence may also be used for reading and spelling rules. Since these rules
are not universal, the sequence must begin with some sort of limiting statement (see Figure 9.21).

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 Figure 9.21

Example Wording

For some words below the vowel is /a/ . For the others it is /a/ .
1. ate My turn. What is the vowel in this word? /a/ . How do I know? Because the
word ends with v c and e.
2. at My turn. What is the vowel in this word? /a/ . How do I know? Because the
word does not end with v c and e.
3. rat Your turn: What’s the vowel in this word? How do you know?
4. rate What’s the vowel in this word? How do you know?
5. rafte What’s the vowel in this word? How do you know?
6. rafe What’s the vowel in this word? How do you know?
7. safe What’s the vowel in this word? How do you know?
8. stand What’s the vowel in this word? How do you know?
9. ale What’s the vowel in this word? How do you know?
10. Stane What’s the vowel in this word? How do you know?
11. malte What’s the vowel in this word? How do you know?

All words are presented in written form. Not all are real words. The instruction at the beginning of the sequence
indicates that the vowel is either /ā/ or /ă/. A variation of the sequence would have to be repeated for other vowels (o,
and u particularly).

Divergent Responses

A very difficult communication problem is associated with divergent responses, situations in which responses are
acceptable and the learner produces a set of them. The teacher must have a solid understanding of responses that are not
acceptable. Figure 9.22 shows the first part of a divergent-response sequence.

Figure 9.22

Example Wording

and My turn: Band rhymes with and.

How do I know? Because band
ends in and.
and Your turn: Make up another word
that ends in and.
How do you know that
rhymes with and?
Etc.

Often a modification is possible by restricting the learner’s behaviors. For example, we might present a list of word
beginnings: s, st, l, b, bl, br . . . We could then instruct the learner to use any one of these beginnings to make up a
word that rhymes with and. For subsequent tasks, we could tell the learner: “Use another beginning to make up a word
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that rhymes with and.” This procedure provides the learner with more options about the word that is created and about
the order or sequence of examples. At the same time, it assures that the range of acceptable variation will be covered.
There is nothing wrong with divergent-responses. Ultimately, we want the learner to produce examples of what we
have taught. The problem is simply one of manageability. The communication must be effective. If we work one-to-one
with the learner, such communications are easily created. Working with more than one learner implies much more
careful treatment of divergent-responses, at least for the initial instruction.

Summary

The correlated-features relationship can be expressed as a fact. The first part of the fact tells about the particular
entity or concept that is correlated with some other features. The second part of the fact describes these features. There
are two basic types of correlated-features sequences—those that require choice-responses for each item and those that
require production-responses. The choice-response sequences are science-type rules, non-symbolic relationships, or an
expression that means the same thing as a familiar expression. The production-responses involve transformations and
deal with symbolic matter.

To create a correlated-features sequence:

1. Start with a fact.

2. If the fact deals with things or events in the physical world, begin the fact by naming those things that lead
to the outcome or the things that cause the event. Answer the question: “How do you make that outcome?”
or “What must happen?”

3. If the fact deals with an equivalent meaning for a familiar expression, express the fact so that the familiar
expression comes first: e.g., If it is cut out of something, it is excised from that thing. If the fact
involves a transformation, express the fact so that the learner would be required to produce different
responses for different examples. Remember, transformations are sometimes possible; however, they are
never the only choice that is available. The second part of a statement of fact tells if a transformation is
possible.

If the sequence does not involve a transformation, model the ordering of examples after the sensory-discrimination
sequences.

1. Shorten the sequence.

2. Reduce the number of examples modeled.

3. Present two questions with each example. The first question directs the learner to respond to a
discrimination that is not shown (the discrimination mentioned in the second part of the fact). The second
question (“How do you know?”) requires the learner to relate this conclusion to observable features of the
example (the discrimination named in the first part of the fact). Stated simply, the examples show one
thing and the learner uses this information to draw a conclusion about something else.

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 4. If the fact deals with conjunctive criteria (and), the sequence should begin with the modeling of a positive.

5. If the fact deals with disjunctive criteria (or), the sequence should begin with the modeling of a negative.

6. If the learner is familiar with the discrimination named in the first part of the fact, examples may be
described rather than presented as sensory examples. We may describe the position, the temperature, the
color, or whatever discrimination is named first.

7. If the learner’s correct responses are possibly ambiguous, we could change the primary question into an
instruction that requires the learner to produce a stronger response (pointing or producing some other
unambiguous choice response).

Try to avoid wording that requires the learner to relate negative features of one thing to positive features of
another. Avoid fact statements such as: “If it isn’t ___, then it is ___.” or “If it doesn’t ___, then it is ___.” Try to
phrase the fact so the presence of one feature cues the presence of another. “If it is ___, then it is ___.”

A correlated-features transformation sequence is desirable if:

1. The relationship deals with features of symbols.

2. The relationship can readily be expressed in words the learner understands.

3. The fact is not inordinately long and does not require responses that would probably be difficult.

To create a correlated-features transformation sequence:

1. Simply add a second question for each example in a regular transformation sequence. The question that is
used for a single-transformation sequence is the primary question. The second question is “How do you
know?” or “Why?”

2. If the fact used to generate the sequence is true only for a particular subtype, use general instructions to
limit the application of the fact to the examples that are presented. Instead of suggesting that the fact holds
for all examples, start out by telling the learner that the fact applies only to the examples that are
presented. This procedure is usually more effective than qualifying with the word usually or with longer
explanations.

Remember, all discriminations can be treated as variations of basic sensory discriminations and can be processed
through sequences that present choice-responses. Some discriminations may also be presented as transformations.
Some transformations may be capable of being expressed as a fact that deals with all words or mathematical operations
of a particular type, etc. For these, the correlated-features transformation sequence may be appropriate. However, many
options are possible.

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 Section IV

Programs
A program is more than one sequence designed to achieve a given teaching objective. Although a wide variety of
possible program formats exist, there are six basic functions for programs and therefore six types of basic cognitive-
skill programs. These are:

1. Programs designed to introduce coordinate members or skills.

2. Programs designed to introduce higher-order and lower-order members.

3. Programs designed to teach the component of a task that will be presented to the learner.

4. Programs designed to teach the relationship between sets of single-transformation concepts.

5. Programs designed to teach the features of an event or organization of related facts.

6. Programs designed to reduce prompting and stipulation that occur as a function of the initial-teaching
sequences.

1. Programs for Coordinate Members

A member is a stimulus that calls for a particular response or set of responses. When we teach the learner to
identify the letter m, we teach a member: m. When we teach the learner higher than, the concept higher than is a
member. When we teach the learner under, under is a member. The term member assumes that there are coordinate
members that are the same in some ways as a given member. If under is a member, coordinate members would include
on, over, next to, between, and other labels that function in the same way as under with respect to position. The
simplest demonstration that the members are coordinate is that something cannot be both under and on the same target
at the same time. (It is possible for the object to be on one target while being under another target; however, the
concepts are the same with respect to their function in describing position.) Another way to identify members is to start
with a sentence, such as “The ball is under the table.” By replacing the word under with other words that refer to
different positions, we identify various potential members.

If we teach the learner to identify the letter e, we teach a member. Coordinate members would include other letters:
t, m, b, and so forth. These have the same symbolic role as e. All are in the class of things called “letters.”

The programs that are designed to introduce coordinate members are based on objectives such as: “The child
should be able to identify the individual letters;” or, “The learner is to discriminate various prepositions;” or, “The
learner is to identify various parts of speech;” or, “The learner will read single-digit numerals.”

The rules and suggestions for designing a coordinate-member program focus on procedures for separating highly

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similar members and for introducing new members at a reasonable rate. These rules apply to concepts that are
introduced in all program types. The general rule is that highly similar teachings or members should be separated in
time. Note that this rule applies to programs, not to individual sequences. The procedures that have been presented for
sequences are not affected by what happens in a program. If we introduce the letter b as the first member to be
identified in a program, we will not introduce d or p next. If we introduce the discrimination on as the first member, we
will not introduce under next. If we introduce a rule for spelling words in which letters are dropped, we will not
immediately follow it with a rule that presents an exception. Remember, we will not require the learner to learn one set
of behaviors and then immediately learn new, highly similar behaviors. We will separate the instruction for the highly
similar members.

2. Programs for Higher-Order and Lower-Order Members

This program is more complicated than that for the coordinate members because it introduces more than coordinate
members. It also introduces a higher-order relationship. For instance, if we teach (from the beginning) a concept, such
as vehicles, we must do more than introduce the coordinate members—truck, car, bus, etc. We must also introduce
the relationships between these coordinate members and the higher-order label, vehicle. If the learner already knows
many higher-order relationships and their conventions, the teaching of vehicles could be handled with little more effort
than would be implied by the teaching of the coordinate members. If the learner is relatively unpracticed with the
higher-order relationship, the program is fairly complicated.

Some complications result from the verbal conventions that we typically use to describe higher-order relationships.
The convention problem can be seen by referring to a physical object, such as a truck. It is possible to say, “That truck
is a vehicle.” It is also possible to analyze the features of the truck that make it a vehicle (the features that truck shares
with other objects that are vehicles). However, the statement “That truck is a vehicle” is quite different from the
statement “That truck is red,” or “That truck is on the table.” Red and on the table refer to possible variables. Not all
trucks are red, and not all trucks are on the table. The statement, “That truck is a vehicle” implies that is a vehicle is a
variable; however, it is not. All trucks are vehicles. We must convey this fact to the learner.

3. Programs That Derive From Complex Tasks

Many instructional situations begin with the identification of an objective. This objective may be established as a
task, such as, The learner will apply this rule: The closer one is to the equator, the less the sun’s rays will deviate from
the perpendicular during any time of the year. Note that we are neither suggesting that this rule is expressed well or
that it should be introduced without reference to other things that are to be taught. If, for whatever reason, the rule is
identified as something the learner is to be taught, we are provided with a basis for a possible program. The learner may
not know deviates from X, perpendicular during any time, and equator. If the rule is introduced without
preteaching these concepts, we will seriously violate our basic tenet of teaching only one thing at a time. A program is
therefore implied by the task. Before we present the task (the rule), we will teach the meaning of the component
discriminations contained in the task. Each skill identified in the task could be treated roughly as a “coordinate”
member. The program therefore consists of teaching the coordinate members and then teaching the rule that integrates

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them.

4. Programs for Showing the Relationship Between Two Single-Transformations

The single-transformation concept involves an unstated rule that permits the generation of various responses.
Single transformations, however, are often closely related to each other. For example, a single-transformation involves
!
constructing fractions from pictures of pies. For a picture that shows two-thirds of the pie, the learner writes: . Let’s
!

say that we wished the learner to express the same relationship as 3 2 . If the learner knows how to construct fractions

from the picture, the division notation can be taught as a relationship, not as a discrimination that is entirely new. The
reason is that there is an unstated rule for converting any fraction into a division notation. That rule involves tipping the
fraction over in a clockwise direction and modifying the line that separates the two numerals. For instance, if we start
!
with , we rotate it in a clockwise direction: 2 | 6, and modify the line: 2| 6. If the learner already knows one way to
!

express a fraction, the most efficient program involves showing the learner the relationships between the familiar
!
notation ; and the new notation ( 2 3 ).
!

This relationship is referred to as a double-transformation. Any member of the first set (any fraction) can be
converted into a member of the second set (the division notation) by performing the same transformation. Many
relationships of this sort occur in instruction. To show the relationship, we first establish the original transformation
(fractions). We then introduce the second transformation (the division notation) through juxtapositions that show how
the members of the new set are the same as members of the original set (their value) and how they are different (their
notation).

5. Programs for Teaching Systems of Related Facts

To explain the basic organization of the respiratory system, a communication would explain the role of the various
features—providing information about the lungs, the tubes that transport air to the lungs, the movement of the
diaphragm, the blood vessels, etc. This communication teaches a system of facts—not individual discriminations, but a
set of discriminations. Another system might be the organization of a school—the administrator, the relationship of the
school to the central board, the functions of the various departments and teachers within the school, and so forth. The
program for teaching a system of related facts is different from any of the preceding programs because it requires
strong emphasis on communicating information about how the various features of the system are related. The program
involves presenting the preskills that may be needed before the total organization or system is presented. The program
also involves techniques for teaching the system itself—showing how features are relevant to the system’s structure or
function.

6. Programs that Reduce Prompting and Stipulation of Initial Teaching Sequences

Initial-teaching sequences prompt the learner by juxtaposing examples that involve the same response dimension.
These sequences also produce some stipulation. Instruction is therefore not completed when the learner performs

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successfully on an initial-teaching sequence. A program is needed to reduce the juxtaposition prompting that occurs in
the initial teaching sequence and to counteract stipulation. The procedure for reducing juxtaposition prompting is to
“shape the context” in which a given response is produced. A response to a test item in an initial teaching sequence is
highly prompted because it immediately follows demonstration examples that prompt the response. To shape the
context of the response, we change the juxtaposition pattern, so the learner is required to respond when the task is not
immediately preceded by demonstration examples. To reduce stipulation, we present a much broader range of examples
and tasks than the initial-teaching sequence provides. Following the initial teaching of steeper, the learner would be
presented with a variety of tasks involving steeper. These would show objects different from those presented in the
initial-teaching sequence. Also, the wording of these tasks would vary more.

Section IV presents the first five programs listed above. It does not present programs that reduce prompting and
stipulation. The design of these programs is detailed in Section V (Complete Teaching), Section VI (Constructing
Cognitive Routines), and Section VII (Response-Locus Analysis).

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 Introducing Coordinate Members to a Set
New juxtaposition problems arise when we introduce a number of coordinate members to a set (such as the
numerals 1 through 10, the letters A through Z, or the name of seven appliances). Each member has a new label and it
involves new features. Both label and features must be discriminated from all other members that have been taught.
The questions associated with the introduction of coordinate members are:

1. How should we expand or review the members that have been introduced before we introduce a new member?

2. What criterion should we require the learner to meet on discriminating between those members that have
already been introduced before we introduce a new member?

3. Should the order of introduction juxtapose members that differ minimally from each other in either label or
features, or should the order juxtapose members that exhibit large differences?

The Problem of Cumulative Review

When we introduce different members, we must provide a cumulative review. A cumulative review is a test of all
coordinate members that have been taught (or a test limited to those members that have been taught and that might be
confused with the new members on the basis of sameness of label or sameness of features). Let’s say that we have
introduced the reading words hit, pot, pin, and it. If we add the word pit to the set, we must make sure that the learner
can discriminate between pit and all the other words. If the learner is not required to discriminate between all these
words, the learner may not learn that the middle sound distinguishes pit and pot, that the beginning sound
distinguishes hit and pit and it, and that the ending sound distinguishes pin and pit.

The cumulative review must be designed so the learner is required to deal with all the words, not with the words
two at a time. An inappropriate procedure would involve first requiring the learner to discriminate between examples of
pit and hit, then to discriminate between pit and pin, then to discriminate between pit and it, and then to discriminate
between pit and pot. It is quite possible for the learner to learn from such a presentation. The procedure does not
require the learner to discriminate between the features of pit and the cumulative features of the set. Therefore, it does
not assure learning. A test that presents all the words—pit, pot, pin, it, hit—in a random order is much more difficult
than the test involving pairs of them. The test involving pairs requires the learner to attend to any difference between a
pair of words. Any difference will permit the learner to discriminate reliably and successfully. The cumulative test,
which requires the learner to discriminate all words, requires the learner to attend to the differences between the
features of the word pit and the features of all the other words.

The Discrimination Model

Table 10.1 shows a simplified representation of nine members of a set. The table refers to features A through E.
Each member has some of these features. The letters that are not in parentheses stand for features that are possessed by

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a member. Letters in parentheses stand for features absent in the member. Any absent feature is present in some other
members. The features A through E do not exhaust the features of the members. Each member may have hundreds of
additional features. These features are not listed because they are shared by all nine members. The features that are
listed are the only differences that are relevant to the task of discriminating between all members.

Table 10.1
A Hypothetical Set of Members
Set Members

1 2 3 4 5 6 7 8 9

A A A A A A A (A) (A)

B B (B) B B B (B) B (B)

Features C (C) C C C (C) C C (C)

(D) (D) (D) (D) D D (D) D D

(E) (E) (E) E (E) (E) E E E

Assume that the members are introduced one at a time, starting with 1 and proceeding through 9. Unless the learner
attends to all features of member 1 (features A through E), the learner will confuse member 1 with at least one of the
other members when all nine have been introduced. Here is how an awareness of each feature helps the learner
discriminate between member 1 and the other members.

Awareness of feature Rules out

A 9, 8
A+B 9, 8, 7, 3
A+B+C 9, 8, 7, 3, 2
A + B + C + (D) 9, 8, 7, 6, 5, 3, 2
A + B + C + (D) + (E) 9, 8, 7, 6, 5, 4, 3, 2

The circled numbers indicate any additional members ruled out by awareness of the additional features. It is a
logical axiom that if the learner attended to only feature A, the learner would discriminate consistently between
member 1 when it was presented in the context of member 8 and member 9. However, the learner would not
discriminate between 1 and any of the other members (2, 3, 4, 5, 6, 7).

The “concept” of a given member varies according to the composition of the set of members. Within the context of
members 1, 8, and 9, the concept for member 1 can be any combination of features that reliably permits the learner to
discriminate between member 1 and the other members in the set. The learner’s concept could be A or absence of D or
absence of E or any of these features in combination with other features.

As the membership of the set increases, the demands on the learner’s “concept” increase. When all nine members
have been introduced, we can infer that the learner’s concept for member 1 is: A, B, C, (D), (E). This unexpressed

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“rule” for identifying 1 is the only rule that will permit the learner to discriminate consistently between member 1 and
the other members. Therefore, if the learner consistently discriminates between member 1 and the other members, we
may assume that the learner’s concept of member 1 must be: A, B, C, (D), (E).

Perhaps the most difficult notion to grasp is that there is no absolute “concept” of member 1. The concept is
relative to the composition of the group of members that must be discriminated. As the composition of the set changes,
the concept for a given member changes. When the set is small, the learner’s concept may be any of a range of possible
concepts (possible references to different features). The concepts in this range include any combination of features that
permit the learner to reliably discriminate between the members. As the composition of the set changes, the range of
options available to the learner ultimately diminishes until the learner must attend to the only differences that actually
discriminate the members from all the other members.

The Uncontrolled-Feature Misrule

The analysis of introducing members to a set suggests a new type of misrule. When members are added to a set, we
begin with members that are different. They may be different in many ways. Only some of these differences are “valid”
differences, which means that only some will serve the learner when other members are introduced to the set. We do
not have a handy way to point out which features or differences are valid or functional. We can only show which are
critical to the present discriminations.

There are two possible solutions to the problem:

1. Design the order of introducing members so that it juxtaposes minimally-different members.

2. Use procedures to assure that early in the program the learner attends to features other than those that
distinguish members introduced early.

The problem with solution 1 is that it places severe memory demands on the learner. If members are highly similar
in name, the learner must learn two names for highly similar members and must learn which name goes with which
features of the examples. The naive learner may remember the names but may become confused about “associating”
the right name with particular features. If the members are highly similar in form, the chances are increased that the
learner will not attend to the small differences and may have “reversal” problems (confusing n and h, for example).
Because of the problems associated with introducing members that are minimally different, we will not use this
procedure, although it may be the most efficient procedure in some situations. The order of introduction that we will
use juxtaposes greatly different members.

Solution 2, requiring the learner to attend to a wide range of features, can be achieved by following the introduction
of a member with application activities. If the learner “constructs” examples of the early members or engages in
activities that require the learner to use the discriminations in other contexts, the learner will attend to additional details
of the members. If the learner is required to print or copy the letters when they are introduced, the learner will have far
less trouble discriminating between n and h than the learner would if no such activities were introduced. The copying
of the first letter introduced (n) forces the learner to attend to details of the letter that are not needed to discriminate

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from greatly dissimilar members introduced early. The assumption is that if the learner can copy or create an example,
the learner has demonstrated an understanding of all features that are relevant to the member, and the learner should not
have serious trouble discriminating the member from others as the set of members increases.

In summary, the order of introduction that we will use does not juxtapose minimally-different members. (If we
introduce b as the first letter to be identified, we would not introduce d next.) We will use a maximum-difference order,
which shapes the context of each member by increasing the sameness shared by that member and other members. First,
the member is associated with members that differ from it in many ways. Identification of the member is relatively easy
because: (1) any difference or combination of differences between the members serves as a reliable basis for the
discrimination; and (2) the learner will probably attend to some differences. The members s and b differ in many ways.
If the set contains only s and b, any difference between the members may be selected to discriminate between the
members. The learner is not forced to select a particular feature.

As new members are introduced to the set, the context shaping occurs because the number of feature options
available to the learner is reduced.

Following the introduction of each new member is a careful expansion of that member. The expansion calls
attention to features of each member that are not essential and facilitates attention to essential features.

Rules for Ordering Members of a Set

We can arrive at a workable order of introduction by following two ordering rules. These rules take into account
the fact that some members are quite similar to others and that highly similar members must be included in the set at
some time.

Ordering rule 1. Arrange members so that minimally different (highly similar) members are separated by two or
more non-similar members. Here is an order for introducing the letters b and d:

b n s e k m j v d

The letters b and d are separated by seven members. Note also that m is separated from the highly similar n by three
members. This order, therefore, is consistent with the rule: separate highly similar members from each other by two or
more members. The sequence below is not consistent with the rule.

b s e m k n v d

The members m and n are separated by only one dissimilar member.

Ordering rule 2. Separate introductions that involve minimum-differences by at least one introduction that does not
involve minimum differences.

An introduction that involves minimum differences occurs when the second members of a minimum-difference pair
is introduced into the set. In this set:

b m s e k

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no serious minimum differences in features are involved. (There is a letter name minimum difference involved when e
is introduced. The name for e is highly similar to that for b.) In the set below, however, n is added and a minimum
difference in feature results. This minimum difference occurs when the second member of a minimum-difference pair
is introduced.

b m s e k n

Rule 2 indicates that you should avoid successive introductions that involve minimum differences. Accordingly,
the order below would be inappropriate because it introduces d (which is minimally different from b), immediately
after n is introduced.

b m s e k n d

When n is introduced, minimum difference is involved (m-n). The next introduction introduced d and requires
minimum difference labeling of b-d. Note that d is not similar to the preceding member, n. The introduction of d,
however, calls for minimum-difference teaching. On two successive introductions, the learner is engaged in minimum
differences. The learning requirements placed on the learner are reduced if we separate these minimum-difference
introductions and interpolate introductions that do not involve minimum-differences. Here is an acceptable variation:

b m o e s n k d

This sequence violates rule 2:

p n s o k r b e j

A minimum difference introduction is required at r (for n-r). On the next introduction (b), minimum differences are
involved for p-b. Here is one of many possible solutions:

n p s r o k b

The following sequence would not be acceptable, however, because it violates rule 1:

p n s r o k b

Although the introductions that involve minimum differences are separated by two members, n and r are separated by
only a single member (s).

Minimum Differences in Examples and Labels

As we have seen from other minimum-difference sequencing problems, there may be minimum differences in the
features of the examples, in the label, or in both. Members should be separated if they are minimally different in either
features of the examples or in label. The best procedure for ordering members is to:

1. Identify the various minimum-difference pairs, whether they involve labels or features of the examples.

2. Position one member of each minimum-difference pair at the beginning of the sequence (unless the

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 members of different pairs are similar to each other).

Let’s say that we were sequencing some lower-case letters that are to be responded to with the letter sound. And
let’s say that these are the minimum-difference pairs: n-ŭ, ĕ-ĭ, ŭ-ŏ. Note that the last two pairs involve minimum
differences in label or response (the letter sound), not in the features of the examples. The short-e sound is highly
similar to the short-i sound. Also, the short-u sound is highly similar to the short-o sound. Here is a possible sequence:

u i s n t e k o

The introduction begins with u because u is involved in two minimum-difference pairings (n-u and o-u). The next
member, i, is involved in one pairing (e-i). Neither n nor o appear near the beginning of the sequence because they are
minimally different from u. The strategy of identifying the pairs of minimum differences and the placing of one
member of each pair near the beginning of the sequence makes the job of ordering the members much easier.

Identifying minimum differences. Minimum differences are not absolute. They vary considerably from set to set. To
determine which members are relatively minimally different from the others, use the principle of conversion. Begin
with a member. Convert it into another member (convert n into u) and observe the number and type of steps that are
involved in the conversion. Now compare that conversion with a conversion of the original member into a third
member (convert n into l). Does one conversion clearly involve fewer steps or smaller steps? It is not always possible
to specify which of the minimum-difference pairs is “most” minimally different. For instance, is the pair n-u more
minimally different than the pair n-r or n-h? This question cannot be answered because a completely different type of
conversion is involved. If there were three letters: n, r, r, we could specify that the pair n-r is more minimally different
than n-r because fewer conversion steps are involved. Similarly, if there was the letter ɲ, we could specify that the pair
n-ɲ is more minimally different than the pair n-h, again because fewer conversion steps are involved.

The comparison of n-r, n-u, and n-h is difficult because the conversions are generically different. One involves the
position or orientation of the member, while the other two involve line deletion or line extension. Just as we cannot
analytically determine whether n-u is more difficult than n-r or n-h, we cannot analytically determine whether n-u is
more difficult than the short sounds o-u. One conversion involves changes in the example: however, o-u involves
changes in the letter sound. (The short-o sound is more similar to the short-u sound than it is the short-i sound.)

Some members differ minimally from other members in both features and sound (m and n). If possible, design the
order of introduction so that you separate each member from a counterpart that is highly similar. Try not to become
embroiled over the issue of which members are most minimally different. You can compare minimum differences only
if you can place the members on the same continuum of change (such as change in position or change in length of
lines).

Other factors that determine the order of introduction. The similarity of members is not the only criterion for
determining an order for introducing members of a set. Immediate utility is an important consideration. If our goal is to
introduce letters that are most useful in making up regularly-spelled words, utility would be considered first. The letters
x, z, and q would not occur early in the set of letters regardless of how dissimilar they are to other members. These

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letters are not as useful as r or s, d or e, in forming words that are familiar to the learners.

Finally, we may find through empirical investigation that some members are chronically confused, even though
they are not highly similar in either appearance or name. For example, we may discover that many learners confuse the
numerals 9 and 10. We can explain the confusion as a function of their proximity in the counting order. But this
problem may not be obvious from an analysis of the members and their features. In any case, the confusion implies that
the members should be separated.

In practice, it is impossible to specify an order of introduction that successfully meets three or more criteria. (As
the applications above illustrate, it is difficult enough to sequence on the basis of a single criterion.) The instructional
designer must strike a reasonable compromise. The most flagrant similarities in shape must be separated. The
minimum-difference introductions should be distributed and not piled up at the end of the introduction sequence. An
attempt must be made to introduce members that are useful. And the most important empirical information about
members that are most chronically confused must be honored. The order, however, may not fully satisfy rules for
separating members, the criterion of utility, and the mandate suggested by empirical information about specific
members.

Scope of Rules

Although we have illustrated the application of the rules only with letters, the rules hold for anything that is taught.
If we introduce three spelling rules that involve changing y to i, we do not sequence these rules so that one follows the
other. We separate the y-to-i rules with spelling rules that have nothing to do with the letters y or i. We also separate
the introductions of spelling rules that involve pairs of highly similar rules. The teaching of appliances, tree names, or
operations for dealing with fractions are treated in the same manner. Each new thing taught is treated as a member.
Also, introductions that involve minimum-difference pairs are separated by introductions that do not involve minimum
differences.

The idea is to place each new thing taught in a context that is relatively easy for the learner. After the learner
masters the member in this context, we change the context, making it more difficult and requiring attention to
additional details.

Teaching Procedures for Introducing a New Member to the Set

Let’s say that we have arrived at an order of introduction for the members of a set. We now consider the
procedures for introducing each member. The first member must be introduced. Later, the fourth and fifth members are
to be introduced. With the introduction of each member, three activities occur:

1. Initial teaching of the new member.

2. Expansion activities.

3. Cumulative review of the members already introduced.

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 Initial teaching. A new member should not be introduced until the learner is quite firm on all activities involving
members already introduced to the set. The new member is taught through an appropriate sequence; comparative, non-
comparative, noun, transformation, or correlated-feature. If we are teaching prepositions, the introduction of a new
member would be achieved through a non-comparative sequence. If we are teaching names of numerals, each new
member would be introduced through a noun sequence. If we are teaching a set of rules about pressure, we would
introduce the new member (new rule) through a correlated-feature sequence.

Each new member would be taught through variations of the appropriate sequence on two or three consecutive
lessons.

Expansion. Expansion activities begin immediately following the completion of an initial-teaching sequence
(assuming that the learner successfully completes the initial-teaching sequence). These activities introduce new setups
and new response forms. Worksheet activities may be part of this expansion. (These activities are outlined in Section
V, Complete Teaching.)

Cumulative review activities. In its simplest form, the cumulative review is a series of test items containing
examples of the most recently-taught member and some of the other members. The cumulative review also may be
presented through worksheets or as a verbal activity.

If only a few members are to be introduced into the set, the cumulative review is easily achieved. Before the next
member is introduced, the learner is presented with a “test” of all previous members that have been taught. The test is
modeled after an E2 test of the noun sequence, with no predictable pattern to the juxtaposition of items. When the
number of members in the set becomes large, however, the mechanical problems associated with the cumulative review
become exaggerated because including all members becomes impractical.

Designing Reviews

The most difficult part of the procedures for introducing new members to a set is the review, particularly when a
large number of members has already been introduced. To assure that each new member is presented within the context
of the members that have already been taught, the review activities are designed according to these five criteria:

1. The review set should contain no more than six members (If you have taught 3 members before
introducing the new members, the review set will consist of 3 members. If you have already taught 5, the
review set will contain 5. If you have taught 34, the review set will contain 6.)

2. The member most similar to the new member is included in the review set.

3. The two members most recently introduced are included in the set. If the letters l, s, p, m, o, t, e, and r
have been introduced in the order indicated, the two members preceding r are t and e. These members
would be included in the review set that follows the initial teaching of r.

4. The member most frequently missed during previous review exercises is included. Let’s say that learners
most frequently misidentified t. This letter would be included in the review set.

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 5. The two members that have been least frequently presented during the last three or four review exercises
should also be included.

Illustrations of Review Sets

1. The learner has been introduced to these letters: m, o, t, e, r. The next symbol to be taught is v. Here is the
review set: m, o, t, e, r. Why are all members included? The number of members is less than six;
therefore, all are included.

2. The learner has been introduced to these numerals: 1, 6, 4, 7, 5, 9, 8, 10. The new member to be introduced
is 3.

a. The member most highly similar to 3 is either 5 or 8.

b. 10 and 8 will be included in the set because each is a recently introduced member. (5 therefore
becomes the “highly similar” member.)

c. 9 is the most frequently misidentified number, so it is added to the review set.

d. The members 1 and 4 have been introduced least frequently during the last four review exercises.
They are added, bringing the number of members in the review set to six: 10, 8, 5, 9, 1, 4.

(Note that both 8 and 5 are introduced as highly similar members.)

Illustration of Initial Teaching, Expansion, and Review

To summarize, the three major activities involved in the introduction are introducing the new member, expanding
the new member, and presenting a cumulative review that includes the new member.

If the new member is a noun, it is introduced according to the noun-sequence rules. Following the introduction of
the new member comes the expansion activities. Following the expansion activities comes the cumulative review. All
three activities may occur on the same day or lesson.

Initial teaching. Let’s say that the learner has been introduced to these letters:

e s d m t o r

The learner identifies the letters by their sound, not the letter name. The next letter to be taught is c. The sequence used
for initial teaching is a noun sequence. The “negatives” for a noun sequence are those members most highly similar to
the members being taught. The letter e is similar in appearance to c. So is the letter o. The letter t is similar to c in
sound. For the new teaching (to a fairly sophisticated learner), these three letters become the negatives. Here is a
possible sequence:

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 Example Teacher Wording
c This is c.
c What sound?
c What sound?
o What sound?
e What sound?
t What sound?
c What sound?

Expansion Activities. Immediately following the initial teaching are expansion activities. The teacher presents a
group of letters:

c o c t c c e c o t c

“See if I can fool you. When you hear me make a mistake, say ‘stop’.”

Teacher touches c. “This is c.”

Teacher touches o. “This is c. What sound is it?”

Teacher touches c. “This is c.”

Teacher touches t. “This is t.”

Teacher touches c. “This is c. What sound is it?”

The teacher presents a worksheet activity.

“You’re going to copy the new sound. What sound is it?

“Every time you make a c, you have to say the sound out loud. Put your pencil on the ball and follow the dotted
lines for each c in the first row.”

The instructions about saying the sound are important. If the learner does not say it, the connection between the
features of the example and its name may not be established.

Cumulative review. On the next lesson, the initial teaching is repeated (with a sequence containing c, o, e and t). It
is followed by another copying task. The copying task is followed by a cumulative review activity. The members
included in the cumulative review with c are:

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 e and r (the most recently taught)
o (the most similar)
m and d (the least frequently reviewed)
t (the most frequently missed)

Example Teacher Wording

m What sound?
c What sound?
d What sound?
t What sound?
c What sound?
e What sound?
r What sound?
o What sound?
c What sound?

No examples are modeled. The examples are sequenced according to the sameness principle. Note these other
points about the review sequences:

1. The most recently taught member (c) is included more than one time (three times in the sequence above).

2. Each of the other members appears only once; however, if the learner is particularly weak on one of the
other members (such as t), this member could be repeated.

3. The order of examples in the sequence does not correspond to the order in which the members were
originally introduced.

If the learner performs well on the sequence, a new member is introduced. If the learner performs poorly, the
introduction of a new member could be delayed. This assures that the learner is firm on the discrimination of all
members that have been introduced.

The basic procedure outlined above for introducing, expanding, and reviewing would be used for teaching any new
members to a set.

The initial teaching sequence teaches the relevant details of the new member. The expansion activities follow the
initial teaching sequence and introduce an expanded context. The cumulative review assures that the learner learns the
differences between the newly-taught member and the features of all other members in the set.

Minimum-Difference Introductions

When the second member of a pair of minimally-different members is introduced, the learner must discriminate
between minimally-different examples. The circled members in the order of introduction below indicate the places at
which minimum-difference discriminations occur.

b m s e k n j l d

When the letter n is introduced, the learner must discriminate between n and m. When d is introduced, the learner
must discriminate between d and b. If the learner is quite naive and would be expected to have response problems, we

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follow the variations for teaching n that are described in Chapter 6. First n is introduced in an easy context (with highly
dissimilar examples such as s and b as “negatives”). When the learner has demonstrated proficiency with n in this
context, the regular noun sequence is introduced. This sequence consists of examples of m and n.

If no response problems are anticipated, the basic noun sequence can be presented for the initial-teaching sequence.
A possible sequence follows. Note that the sequence is a “straight” noun sequence with only the most minimally-
different member included in the sequence.

Example Teacher Wording

n This sound is n.
n What sound?
n What sound?
m What sound?
n What sound?
m What sound?

Following the introduction of n is the introduction of j. This introduction occurs possibly two days after the first
initial teaching of n. Since the learner has not been taught members that are highly similar to j, we are free to select
negatives from any members that have been taught. The members k and b are more similar to j than the other members
are because k and b, like j, are “taller” letters. Here is the first initial teaching sequence for j.

Example Teacher Wording

j This sound is j.
j What sound?
b What sound?
j What sound?
b What sound?
k What sound?
j What sound?

Possibly two lessons later, after j has been taught, expanded and reviewed, the next sound, l is introduced. Like j, it
is not highly similar to any of the previously taught members; therefore, the initial teaching sequence would include l
and possibly other tall examples—j and k, or k and b.

Following the expansion and review of l, the final member, d, would be introduced. The minimally different
member known to the learner at this time is b. Therefore, b would be included in the initial teaching sequence (unless
response problems were anticipated).

Example Teacher Wording

d This is d.
d What sound?
b What sound?
b What sound?
d What sound?

Charting a Schedule

When we deal with a large set, the integration of various activities—expansion, worksheets, reviews—is simplified
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by charting the introduction of members. When charting, follow these basic rules:

1. Provide initial teaching for at least two consecutive lessons.

2. Immediately follow the teaching with expansion activities.

3. Expansion activities continue after a member has been introduced.

4. Review occurs immediately before we introduce the next member.

By following these rules, we limit the rate at which items are introduced. We assure, however, that the introduction
is careful and that the learner indeed learns each discrimination.

Figure 10.1 shows a schedule for introducing the meaning of new vocabulary words in a language program.

Figure 10.1

obsession

obsession
articulate

articulate
sensitive

sensitive

Initial Teaching X X X X X X

Expansion X X X X X X X X X X

Cumulative Review X X X X

Lesson 50 51 52 53 54 55 56 57 58 59

The first word is introduced on lesson 50, at which time the learner has already been taught fourteen vocabulary
words.

The cumulative reviews occur immediately before the introduction of the next word (lessons 52, 55, 56, and 59).
Note, however, that lesson 56 does not introduce a new word. It is a consolidation lesson (like 55) that involves
expansion activities and review.

With facile learners it is possible to introduce the words faster, perhaps at the rate of one word a day. After ten
lessons, however, we would have to provide a review before introducing the next block of words. The rate that is most
appropriate for a group of learners can be determined by observing performance on the activities that are presented. A
rule of thumb, however, would limit the rate of introduction to an average of one new item every two lessons.

The activities above do not comprise an entire lesson, merely a part. The time required by each activity should
average no more than two minutes, which means that the vocabulary should require no more than six minutes a lesson.

Applications

As noted earlier, the principles for introducing members to a set apply to any set of coordinate or associated

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concepts, discriminations, or responses. For example, if we teach different hierarchical classes, each class is coordinate.
Let’s say that we plan to introduce: tools, household appliances, furniture, articles of clothing, food, names of
rooms in a house, animals, plants, things to read, vehicles and containers. For each class, the learner would be
required to use the class name and also to identify the various subclass members. (“This piece of furniture is a couch . .
. this tool is a hammer.”) Since each class is coordinate, each is treated as a member. The members are introduced
according to the same procedures used for simple members. If tools is introduced first, household appliances would
not follow, because household appliances are minimally different from tools. Tools would have to be separated from
household appliances by some members that are not similar to tools—possibly animals or plants. Also, we would
separate the teaching of plants and food. The teaching involved in the introduction, review, and expansion for the
hierarchical classes is more involved than the teaching for letter names. Also, the discrimination teaching for similar
members is more involved. However, the introduction sequence follows the same rules used in introducing members in
a set.

Other types of coordinate members would be facts or rules. Let’s say that we wish to teach the learner facts about
different people and different dates. For each of these facts we ask a question, and the learner produces a verbal
response. Like other members, facts may be similar to each other. These facts should be separated. The ordering of the
different facts should follow the procedures outlined above.

Another type of coordinate members might be responses to different commands. You tell the learner, “Touch your
nose,” and the learner responds. The teaching involved for these responses is different from concept teaching.
However, if we are going to teach more than one response, we should consider each task to be taught a member, and
we should order these members so that similar commands are separated. For example, we would not teach: “Touch
your nose,” “Touch your mouth,” and “Touch your knees,” in that order. We would separate the commands because
they are similar to each other either with respect to the features of the command or the features of the response.
Touching the nose and touching the mouth are similar responses. “Touch your nose” and “Touch your knees” are
similar sounding commands. To separate these, we would interpolate commands that are quite dissimilar from them.
Perhaps we would start with “Touch your nose.” Follow it with the commands “Clap,” “Smile,” and “Touch the floor.”
Next, we could introduce “Touch your mouth.”

Even complicated operations should be treated as members. Let’s say that we wished to teach clearing the
denominator and finding a common denominator. We treat each of these operations as members, and we separate them
by interpolating members that are not similar (such as dealing with exponents and solving ratio problems).

Remember, if teachings are coordinate or associated, treat the teachings as members and separate them so that
similar members are not juxtaposed in the order of introduction, and so that no juxtaposed members involve minimum-
difference teachings.

Introducing Related Subtypes

The transformation chapter (Chapter 8) presented a procedure for analyzing subtypes. These subtypes are like
unrelated members (such as s and e) in that the examples of a subtype are different from the examples of another
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subtype. However, subtypes are related. A variation of the same transformation applies to all regularly-formed
subtypes.

To deal with transformations involving related subtypes, we use an order of introduction that is slightly different
from that described for introducing unrelated members. We use two basic approaches: the A-Z integration approach
and the subtype-difference approach.

We use the A-Z integration approach if we introduce one subtype at a time. We use the subtype-difference
approach if we introduce the transformation through a sequence that shows all subtypes when these are presented
within the context of examples that have many shared features.

The A-Z Approach

The letters A and Z refer to two different subtypes. Each subtype is processed through a sequence. All examples in
the A sequence are from a single subtype. They share many common features. The sequence is therefore relatively easy
(far easier than it would be if it showed the response variation of two or three subtypes). The A sequence, however, is
guilty of stipulation. It implies that the transformation is limited to the subtype presented in the A sequence. To counter
this stipulation, follow the A sequence with the Z sequence. The Z sequence is specifically designed to counteract the
major stipulation implied by the A sequence. Examples in the Z sequence do not share the features that are stipulated
by the A sequence; therefore, the Z sequence presents the transformation in a context that is greatly different from that
of the A sequence. By following the A sequence with the Z sequence, we order the sequences to show sameness of the
transformation across two greatly different contexts.

To assure that the learner masters this sameness, we follow the Z sequence with an integration sequence. This
sequence presents a mix of examples from the A subtype and the Z subtypes.

The procedure is repeated by introducing a sequence that counteracts any possible stipulation created by the
combination of A and Z. A third sequence (B) is introduced that counters this stipulation. This sequence is followed by
a review. The cycle repeats until all subtypes have been introduced.

Table 10.2 presents a schedule for introducing four subtypes: A, Z, B, Y. Subtypes Z, B, and Y are designed to
counteract whatever stipulation has been created by the subtypes that have been previously introduced.

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 Table 10.2
Sequences Lessons

1 2 3 4 5 6 7

Subtype A X X

Subtype Z X X

Subtype B X X

Subtype Y X

Integration A-Z A-Z A-Z-B A-Z-B-Y A-Z-B-Y

Each subtype is scheduled in the same basic manner used for any other group of coordinate members. Subtypes A,
Z and B are each introduced on two successive lessons. (Subtype Y is not because the preceding integrations have most
probably overcome any possible stipulation.) The schedule presented in Table 10.2 may provide a too-careful
introduction for many transformations; however, for those that involve complicated behaviors or fine discriminations
(such as a transformation involving subject of a sentence) the schedule is reasonable. If we added expansion activities
to the schedule, it would almost be identical to one for a group of coordinate members. The difference has to do with
the procedure for determining which member should be introduced next.

Illustration of A-Z integration. Let’s say that the learner is first presented with a sequence for say-it-fast in which
word parts are presented with a pause between them: motor (pause) boat. This is the subtype A sequence and it is
appropriate as the first sequence because it presents a large, but relatively easy, subtype. Another subtype of say-it-fast
tasks presents single continuous sounds. (To present the sound mmm, simply say the sound for about two seconds.)
This subtype counteracts the stipulation created by the A sequence (that all examples of saying words fast involve
pauses between the word parts). This subtype becomes the Z subtype.

My turn to say it fast. Listen. aaa. Say it fast. a.

My turn again: ooo. Say it fast. o.

Your turn. ooo. Say it fast.

eee. Say it fast.

mmm. Say it fast.

sss. Say it fast.

rrr. Say it fast.

shshsh. Say it fast.

lllll. Say it fast.

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The initial sequence (A) implies that say-it-fast is achieved by eliminating a pause. The Z sequence shows that say-it-
fast may involve increasing the rate of saying the sound. Following the presentation of the Z sequence, the integration
sequence is presented. The integration sequence shown in Figure 10.2 follows the basic juxtaposition pattern used for
single-transformation sequences; however, there are no modeled examples, simply test examples. An alternative
integration sequence would simply present examples of the two subtypes, juxtaposed to show sameness.

Figure 10.2
Example Wording
mmmm Say it fast.
C1
rrr Say it fast.

row (pause) boat Say it fast.

C2
boat (pause) house Say it fast.

A ssss Say it fast. E

drug (pause) store Say it fast.

eeee Say it fast. D

dog (pause) collar Say it fast.

fffff Say it fast.

Both subtypes appear in the minimum-difference segment of the sequence (C1 and C2). The transition from one
type to the other is fairly arbitrary. The word rowboat begins with the same sound that appeared in the preceding
example. When constructing these transitions, you may not find a handy way to do it. In this case, just create an abrupt
switch from progressively arranged examples of one subtype (C1) to progressively arranged examples of the next (C2).

The D segment of the integration series requires the learner to treat juxtaposed examples that are quite different
(subtypes A and Z items) in the same way (operating on them by saying them fast).

Some slight stipulation occurs following the integration of A and Z, which is that only single sounds (not words)
are speeded up. To counteract this stipulation, we would follow the integration of A and Z with a sequence that
counteracts this stipulation (the B sequence). It would consist of words that are presented continuously (no pauses), but
slowly. (Listen: Mmmmeeee. Say it fast.) After this subtype is taught, subtypes A, Z and B are integrated.

Listen: Motor (pause) boat. Say it fast.

Listen: eee. Say it fast.
Listen: mmmmeee. Say it fast.
Listen: eeeet. Say it fast.
Listen: Out (pause) fit. Say it fast.
Listen: Sad (pause) ness. Say it fast.
Listen: ooo. Say it fast.
Listen: O (pause) pen. Say it fast.
Listen: O (pause) ther. Say it fast.
Listen: mmmmaaaannnn. Say it fast.

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 If the final subtype consists of words presented one sound at a time with pauses between the sounds, it would be
presented as the final subtype (Y). (Listen: mmm (pause) aaa (pause) t. Say it fast.) This subtype would be delayed
because it is a more difficult subtype. The goal of the earlier subtypes is to counteract the stipulation with the relatively
easiest subtype that is available. Difficult subtypes should always be delayed until the easier subtypes have been
practiced. Note that a subtype may be difficult because it involves a combination of behaviors (as in the case of words
presented a sound at a time with pauses between the sounds) or because it is irregular.

Following the presentation of the Y subtype would be a sequence that integrates all types. Included in this
sequence would be words separated by pauses, single sounds, words presented continuously with each sound held, and
words presented a sound at a time with pauses between the sounds.

The order of introduction for the say-it-fast subtypes may be far more elaborate than we would need in many
settings; however, if a careful introduction is required (for very naive learners), the A-Z integration procedure would
assure that all subtypes are introduced and cumulatively integrated with the other subtypes.

Identifying a Subtype that Counteracts Stipulation

The simplest way to identify a sequence (or to design one) that counteracts the stipulation created by an initial
sequence that presents a fairly easy subtype is:

1. Make up a list of the features that are shared by all examples in the original sequence.

2. Determine which of these features are unique to the subtype presented through the sequence.

3. Design a sequence with a subtype that has none of these subtype features (the ones that are shared by all
examples within the sequence, but that are not shared by all examples of the transformation).

4. Make sure that the sequence that counteracts the stipulation presents a relatively easy subtype (one that
involves neither complicated behaviors nor irregularly-formed examples).

Illustration. We can illustrate the procedures with a series of transformation sequences that teach adjectives. The
first sequence (the A subtype) assumes that adjectives will be taught within the context of sentences or phrases. This
sequence is relatively easy, presenting examples that consist of three-word phrases, the last word of which is always a
noun. (See Figure 10.3.)

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 Figure 10.3
Subtype A
Example Teacher Wording
1. The sad boy My turn to name the adjectives: the and sad.
2. Those sad boys My turn to name the adjectives: those and sad.
3. Five sad boys Your turn: Name the adjectives.
4. Five sad dogs Name the adjectives.
5. Five big dogs Name the adjectives.
6. Our big dog Name the adjectives.
7. A strong farmer Name the adjectives.
8. My little brother Name the adjectives.
9. A singing cat Name the adjectives.
10. Those happy hours Name the adjectives.
11. An unfortunate event Name the adjectives.

The potential stipulation problems of the sequences are obvious. The learner may suppose that adjectives always
occur in pairs or that they include all but the last word of the utterance. Subtype Z should be designed to counteract this
possible stipulation.

Many possible sequences may be designed to counteract the stipulation. The sequence in Figure 10.4 (the sequence
for the Z-subtype) does not share the most highly stipulated features with the original sequence. The examples in the
sequence do not end with a noun, and do not present a pair of adjectives.

Figure 10.4
Subtype Z
Example Teacher Wording
1. The dog was sitting. My turn: Name any adjectives: the.
2. The sitting dog was singing. My turn: Name any adjectives: the and sitting.
3. The singing dog was sitting. Your turn: Name the adjectives.
4. The sitting dog was panting. Name any adjectives.
5. The dog was panting. Name any adjectives.
6. A panting dog was sitting. Name any adjectives.
7. Our dogs were panting and sitting. Name any adjectives.
8. Her cat was sitting and panting. Name any adjectives.
9. Her singing cat was panting. Name any adjectives.

The sequence shows that a word is an adjective only if it occupies a particular position in the sentence. The word
sitting is not an adjective in example 1. Sitting is an adjective in example 2.

Following successful completion of the preceding sequences is the integration sequence that requires the learner to
respond to both types A and Z. Figure 10.5 shows the examples for the first part of the integration sequence.

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 Figure 10.5
Example Teacher Wording
1. Five walking cats Name any adjectives.
were crying.
2. Five cats were Name any adjectives.
walking.
3. The silly cat Name any adjectives.
4. My silly brother Name any adjectives.
5. My brother was Name any adjectives.
walking.
6. That unusual song Name any adjectives.
7. A dog was snoring. Name any adjectives.
Etc.

The integration sequence counteracts some stipulation. The learner responds to adjectives in sentences and
adjectives in phrases. The integration sequence is still guilty of stipulation, however. Adjectives are shown to occur
only at the beginning of the utterance. To counteract this stipulation, we introduce subtype B (see Figure 10.6). This
subtype presents adjectives in the predicate only. Note that this sequence presents a narrow subtype in which the first
words are the same (she met) and the last word is a noun that is preceded by adjectives. The purpose of the subtype is
to show the learner that adjectives are not limited to the beginning of an utterance.

Figure 10.6
Subtype B
Example Teacher Wording
1. She met a happy man. My turn to say any adjectives: a and happy.
2. She met five happy men. My turn to say any adjectives: five and happy.
3. She met five old men. Your turn: Say any adjectives.
4. She met that old man. Your turn: Say any adjectives.
5. She met my old dog. Say any adjectives.
6. She met a tired mailman. Say any adjectives.
7. She met a yellow moop. Say any adjectives.
8. She met an unusual problem. Say any adjectives.
9. She met four strange people. Say any adjectives.
10. She met an unhappy ending. Say any adjectives.
11. She met that silly hunter. Say any adjectives.

Follow this sequence with another integration sequence with all subtypes—A, Z, and B. Figure 10.7 shows the first
part of a possible sequence.

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 Figure 10.7
Integration Sequence
Example Teacher Wording
1. I read about an old Say any adjectives.
goat.
2. I read about those Say any adjectives.
old goats.
3. Those silly goats Say any adjectives.
4. That silly cat was Say any adjectives.
sleeping.
5. That sleeping cat was Say any adjectives.
running.
6. I read about that Say any adjectives.
sleeping cat.
Etc.

The integration of A, Z, and B fairly well conveys the different subtypes or contexts in which adjectives occur. The
various examples presented so far stipulate that adjectives always precede the name of something. There is, however,
the type of adjective that occurs in the predicate: “Johnny is happy.” No noun follows happy. This type is irregular.

The problem that we may create if the irregular subtype is introduced too early is confusion of adjectives and
verbs. Consider the sentence: The running bear was happy. The word running is an adjective, and happy is an
adjective. We now reverse the two adjectives: The happy bear was running. The word happy is an adjective, but the
word running is now a verb. A solution to the problem would to be: (1) teach the regular forms of adjectives; (2) teach
the discrimination verb; and (3) introduce the irregular adjectives in the predicate. They would be easily identified as
not-verbs and not-nouns. Therefore, they must be adjectives (with respect to what the learner has been taught).

Another possible solution would be to ignore the verb problem and use a different type of transformation to teach
the irregulars after the learner has been firmed on the integration of A, Z, and B. Figure 10.8 shows the beginning of a
possible sequence. Note that the sequence is preceded by a brief verbal explanation and that it does not deal with
identifying adjectives, merely with restating the sentence.

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 Figure 10.8
Example Teacher Wording
Some adjectives are funny. Before you can figure them out, you
have to say the sentence another way.
1. The boy was happy. My turn to say it another way: The boy was a happy boy.
2. The dog was happy. My turn to say it another way: The dog was a happy dog.
3. The dog was big and frisky. My turn to say it another way: The dog was a big and frisky dog.
4. The dog was big and frisky. Your turn. Say it another way.
5. The dog was old and fat. Say it another way.
6. The man was old and fat. Say it another way.
Etc.

After the learner has successfully completed the sequence, the sequence is repeated, this time with an additional
task to be added to each example. Instead of simply directing the learner to say it another way, the teacher also directs
the learner to say any adjectives at the end of the sentence. For example:

Example Teacher Wording

The dog was big and
frisky. Say it another way.
Now say the adjectives
that are near the end
of the sentence.

In summary, the A-Z integration strategy for dealing with subtypes is to introduce new subtypes one at a time and
integrate each new type with all subtypes that have been taught.

1. Begin with a subtype (A) that has a relatively large membership.

2. Analyze the sequence to determine the stipulation that is implied.

3. Introduce a sequence involving a subtype (Z) that does not have the features stipulated by the original
sequence. This sequence should be quite different from the original.

4. Create an integration sequence that begins with progressive minimum differences (which are tested) and
that terminates in sameness juxtapositions of the two subtypes (A and Z).

5. Analyze the integration sequence for possible stipulation, and introduce a subtype (B) that does not have
the features stipulated by the integration sequence.

6. Create an integration sequence that begins with progressive minimum differences and that terminates in
sameness juxtapositions of the three subtypes (A, Z, B).

7. Repeat this procedure until all “regular” subtypes have been introduced.

8. Delay the introduction of irregular or contradictory subtypes until the learner is firm on the regulars. The
irregularities may then be taught in a sequence followed by an integration sequence that processes all

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 regular types that have been taught.

The Subtype-Difference Procedure

If we are required to introduce many subtypes for a given discrimination, we may elect to show all (or nearly all) of
the regular subtypes in a single introductory sequence. We design this sequence so that the only differences between
examples is a subtype difference. This type of sequence is possible for transformation concepts that have many
subtypes.

The subtype-difference procedure is basically the same as the A-Z integration procedure except that the first
sequence shows all subtypes. Subsequent sequences expand the range of application for the various types presented in
the initial sequence. The schedule in Table 10.3 presents a typical pattern of subtype-difference introductions.

Table 10.3

Sequences Lessons

1 2 3 4 5 6

Initial Sequence (I) X X

Subtype Z X X

Subtype B X X

Integration I-Z I-Z Z-B Z-B

Note that initial sequence (I) is not contained in the integration after Z and B become integrated. The reason is that
the examples presented through Z and B present enough variation in examples to assure that the learner is attending to
the sameness across the various possible subtypes.

Illustration. Let’s say that the first sequence used to teach verbs is the one presented in Chapter 8. Figure 10.9
below is the first part of that sequence.

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 Figure 10.9

Example Teacher Wording

1. She runs in the park. The verb is runs.

2. She ran in the park. What’s the verb?
3. She will run in the What’s the verb?
park.
4. She would run in the What’s the verb?
park.
5. She was running in What’s the verb?
the park.
Etc.

Although this sequence introduces a broad range of subtypes, it does not show any range of variation for each of
the subtypes (such as the one-word verbs). The words “She ___ in the park” is common to all examples, and a form of
the word run appears in all examples.

To counteract this sequence, we would introduce the initial sequence with one that expands a particular subtype. A
possible Z sequence that presents one-word verbs and that tends to counteract the stipulation created by the original
sequence (which is that verbs must have to deal with running) is as follows:

They ran in the park.

She sat in the park.

She is in the park.

She was in the park.

They were in the park.

They sat in a tree.

They sat.

They ate.

They climbed a tree.

They have a tree.

Etc.

The sequence may be attempting to introduce too many subtypes (suggested by the number of progressively
minimum-difference examples that are required to cover the variation in one-word verbs). However, a single sequence
or more than one sequence of this type would function well as a Z sequence to counteract the stipulation of the initial

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sequence. Following this sequence would be an integration of the type of examples presented in the initial sequence
and the subtype that is elaborated in the Z sequence. Following this integration would be another sequence that
introduced two-word verbs. Note that the type of examples presented in the initial sequence would be dropped
following the introduction of the next subtype. This sequence would introduce three-word verbs. Another sequence
might follow to introduce verbs when sentences are not in their regular subject-predicate order.

Perspective

We illustrated the integration of subtypes with transformation sequences. The same pattern of integration and the
same options exist when we deal with different subtypes of examples that are processed through a cognitive routine or
different subtypes of a physical operation (such as using a screwdriver with different types of blades—slotted, phillips,
etc.). Our two major options are the A-Z integration or the subtype-difference approach. In both cases, the initial
sequence introduces stipulation. It may show the range of subtypes but does so in a very restricted setup, or it may
show the range of a single subtype with- out showing other subtypes. In both cases, subsequent sequences are needed to
counteract the stipulation.

Summary

Members of a set are coordinate discriminations that are potentially confusing. To schedule the introduction for a
group of members, we follow two basic ordering rules:

1. Arrange members so that minimally-different members are separated by two or more non-similar
members.

2. Separate introductions that involve minimum-differences by at least one introduction that does not involve
minimum differences.

Those members that are designed as “minimally different” are the most similar in features of the examples or in the
labels.

The order of introduction is not wholly determined by the similarity of members. The criterion of immediate utility
strongly influences selection. We introduce things because they are immediately useful to the learner. The greater the
potential application, the stronger the argument for introducing it early.

Empirical investigation may disclose that some members are quite difficult for the learner and should perhaps be
scheduled for late in the sequence. For instance, unvoiced sounds f, s, and p are more difficult for learners than their
voiced counterparts v, z, and b. Also, stop sounds are more difficult than continuous sounds.

To introduce a member to a set, design three basic activities: initial teaching, expansion, and cumulative review.

When the set becomes large, do not review all members. Review the most similar, the most recently introduced, the
most frequently misidentified, and the least frequently reviewed.

Limit the number of examples in the review set to six.

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 The initial teaching follows the procedures outlined in Chapters 4 to 9. If the set consists of nouns (which is
characteristic of most large sets), a noun-sequence is used for the initial teaching of each member.

The expansion activities include foolers, manipulation activities, and tags. Also, worksheet activities are used to
expand the newly-taught member. (See Section V.)

When designing activities to introduce each member, chart the schedule, allotting about two lessons for the
introduction of each new member and providing periodic consolidations during which no new members are introduced.

The procedures for ordering members assures that the learner practices the earlier-taught members before the later,
minimally-different, members are introduced. Hopefully, the amount of learning demanded when the second member is
introduced is reduced because the learner is quite firm on the earlier-taught member. Confusions do not become chronic
(the biggest single problem associated with members that are highly similar).

Modify the procedures for quite facile learners. The number and types of expansion activities can be reduced. The
regularity of cumulative reviews can be disregarded somewhat. Intuition may suggest that far less teaching and
expansion are needed to maintain new concepts; however, empirical investigation may disclose that the optimum
amount of reduction is not as great as intuition suggests.

Procedures for introducing subtypes of transformations follow the same basic pattern as the introduction of
coordinate members. Because subtypes are related, however, the procedure for identifying an order for the introduction
of subtypes is different from that for ordering coordinate members.

Begin with a relatively easy sequence that either shows the full range of subtype variation in a restricted setup or
presents a relatively large and easy subtype.

Follow with a subtype that counteracts the stipulation created by the initial sequence.

Integrate the subtypes.

Continue in this manner until all subtypes are introduced and integrated.

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 Hierarchical Class Programs
A higher-order class is a group of objects or events that: (1) have a higher-order label, and (2) have different
subtype labels.

The program for a higher-order or hierarchical class is more complicated than the program for introducing
coordinate or associated members, because more teaching is involved. To teach various objects in the class tool, we
must introduce a variety of tools. We must teach the name for the various members that are taught: hammer, saw,
pliers, etc. This part of the program has the same requirements as the basic programs for ordering members of a set.
Members are introduced one at a time. Some are minimally different from others and must be separated in time. For
each member that is introduced, there should be initial teaching, expansion-activities, and some sort of review to assure
that the member is being discriminated from all other members in the set. Introductions of members that require
minimum-difference pairings of examples should be separated from other introductions that involve minimum
differences.

All these features for higher-order teaching are shared by the basic program for teaching coordinate or associated
members of a set. In addition to the basic teaching, however, we must provide for teaching of each member’s second
name, the higher-order designation. In addition to teaching the learner that the car is called “car,” we must provide for
the teaching of vehicle. This higher-order teaching suggests that we must show the learner how one label of the object
(car) is based on some features, while the other label (vehicle) is based on other features. Also, we must show the
learner that although all cars are vehicles, not all vehicles are cars. Finally, we must show the learner that not all
things are vehicles. One of the biggest practical problems associated with the higher-order program is that of
stipulation. If we teach the learner that an object is a car and we continue to reinforce the learner for responding to it
only as a “car,” we induce strong stipulation. The learner will later resist classifying the object as vehicle.

Each problem associated with the structure of higher- order class can be solved by recognizing it and by providing
the appropriate instructional remedy.

Structure of Higher-Order Relationships

The structure of higher-order relationships is intriguing. An understanding of this structure helps clarify the
discriminations that must be taught. We will look at the structure two ways. First we will view the various hierarchical
arrangements that are implied by a single concrete example. Then we will look at a series of progressively more
inclusive higher-order classes.

Every concrete example has an indefinitely large number of features. The features (either individually or in groups)
occur in other examples. Therefore, the features (either individually or in groups) serve as the basis for creating
different groups in which the particular object would be placed on the basis of sameness of feature. Consider a
particular pencil. It could be classified as something that has erasers, something that is yellow, something longer

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than two inches, something with a point, something made of more than one material, something with wooden
parts, and so forth. Some of these classes are more inclusive than others. For instance, the pencil may be in the class of
things made of cedar wood. It would also be in the class of things made of wood. All other members of the class,
things made of cedar wood, would also be in the class things made of wood. Therefore, the class things made of
wood includes the class things made of cedar wood. When all members of one class are included in another class, we
say that a hierarchical or higher-order class relationship exists.

If we start with the concrete example described above (the particular pencil) and create higher-order classes based
on color, we may identify the following classes: middle-yellow objects, yellow objects, brightly-colored objects, and
objects. We can repeat the procedure for other classifications based on the hardness, weight, length, or any other
discriminable characteristic of the object. The same pattern will emerge, however, for all the higher-order classes we
specify. The ultimate class that is named is objects. The “objectness” is common to all possible higher-order
arrangements that we can specify for something that is an object.

The diagram below shows the relationship between the features of a concrete object and various higher-order
groups.

A A
AB AC

ABC
DE

AE AD
A A

Each feature relevant to the various classifications is indicated by a letter: A, B, C, D, E. The center of the “flower”
is the concrete object that is being classified in different ways. Each “petal” of the flower shows a simple higher-order
program that consists of only two inclusive classes (such as writing object and object, or yellow object and object).
(Letter A refers to object.)

As the diagram indicates, the basis for all possible higher-order classifications is the set of features possessed by
the concrete object (the center of the flower). Different features are selected to create different higher-order
arrangements. If feature B is “yellowness,” the yellowness may be used as the basis for creating a higher-order class.
The class would be yellow objects. The entities that would be included in this class would be the concrete object in the
illustration and all other objects that have the feature of yellowness. Another feature of the concrete object is
“pencilness.” Therefore, pencil objects is a possible higher-order classification.

Although there is a progression of higher-order classes for any feature of the object (yellow objects and objects,
for instance), each progression (petal) is independent of the other progressions (petals). The single concrete object that

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we start with, in other words, may be included as an example in any of the higher-order progressions that are identified.
(The pencil is an example of pencils, of yellow objects, of objects with points, etc.) The membership of each of these
categories, however, is different. (Many objects in the class pencils would not be in the class yellow objects or in the
class objects with points.) This point is very important. For every feature of the concrete object, there is a possible
progression of higher-order classes. Each progression, however, is independent of the others and describes a different
criterion for grouping objects on the basis of common features.

The final point about the petal diagram is that as we proceed from the center to the end of any petal, we are using
classification criteria that are increasingly inclusive. (All yellow objects are objects.) Therefore, the membership of
each class is based on fewer and fewer common features as we progress toward the end of the petal. The objects in the
class that has features A, B, C, D, and E have at least five common features. The objects in AB have only two common
features. The objects in A have only one—“objectness.”

Higher-Order Progressions

A single higher-order progression is represented as a petal in the flower diagram. Each petal in the diagram has
only two divisions, or shows only two classes in the higher-order progression. Some progressions may consist of many
classes, however. The diagram in Figure 11.1 presents a different perspective for higher-order progression. This
diagram is a ladder, showing a progression of some of the possible classes for a particular object—an unripe Bing
cherry. The value of the diagram is that it shows the three major relationships that characterize a higher-order
progression. These relationships are:

• Relations of a shaded part to the row below it.

• The relationship of a shaded part to the other part of the same row.

• The relationship of a row to the shaded part in the row above.

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 Figure 11.1
Higher-Order Progression Classes of an Unripe Bing Cherry

1. OBJECTS NON-OBJECTS

ALL OTHER
2. FOOD OBJECTS

ALL OTHER
3. CHERRIES
FOOD

ALL OTHER
4. BING CHERRIES CHERRIES

5. UNRIPE BING ALL OTHER

CHERRIES BING
CHERRIES

Each row is divided into two parts. The first part names the positive (+) examples. The second part names the
coordinate negative examples (–). For row 4 the positive examples are “Bing Cherries” and negative examples are “All
Other Cherries.” The negatives on each row suggest the positive classification for the row above. The positives on a
row imply the row below. “Bing Cherries” implies different subtypes of Bing Cherries. The positives on row 5 present
a possible subtype of Bing Cherries (“Unripe Bing Cherries”).

Row 5 shows that Bing Cherries are divided into two types—“Unripe Bing Cherries” and “All Other Bing
Cherries.” (There may be two other types or ten. We do not know the number. We only know that they are coordinate
with Unripe Bing Cherries, which means that an object cannot be both an Unripe Bing Cherry and another subtype
of Bing Cherries.)

Row 4 shows that there is a coordinate relationship between “Bing Cherries” and “All Other Cherries.” This
relationship suggests that we can divide cherries (arbitrarily) into two groups: the Bing cherries and the non-Bing
cherries. This division shows that we can create a discrimination between cherries that are Bings and cherries that are
not-Bings.

The relationship between “Bing Cherries” and the part above (“Cherries”) provides a restatement of the fact that
Bing cherries and all other cherries make up the class cherries, and that cherries are one type of food.

The structure of these discriminations suggests the teaching that is needed to introduce simple higher-order groups,
such as various cherries, types of vehicles, or chemical elements.

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 Programs for Higher-Order Relationships

If we were to teach the higher-order name cherry and the lower-order membership Bing, Queen Ann, and
Lambert, we would be responsible for teaching a total of four names, requiring three types of sequences.

1. A sequence for introducing the higher-order class discrimination (usually a variation of a non-comparative
sequence).

2. A sequence for introducing each lower-class member (usually noun sequences).

3. Cumulative review sequences that begin after the second lower-class member has been introduced and that
continue periodically throughout the program.

We teach the higher-order class discriminations first, then the various lower-order class names. There are many
reasons for introducing the higher-order name first. Perhaps the most important is that if the higher-order name (vehicle
or cherries) is taught, we can use the higher-order name in questions about the lower-order members. We can ask, “Is
this cherry a Bing cherry?” “Is the vehicle a truck?” or, “What kind of vehicle is this?” These questions tell the
learner that the object involved is a cherry or a vehicle, but is also a particular type. If we do not teach the higher-order
discrimination first, we cannot express the relationship between higher-order and lower-order names. We must ask, “Is
this a truck?” or “What is this?” As noted earlier, if the learner works with the lower-order labels for a long time before
their higher-order name is taught, the presentation stipulates that the object identified as boat or car has only one noun-
type label. The learner may resist the idea that these objects can also be called vehicles. In effect, the learner will insist
that it is not a vehicle because it is a boat or because it is a car.

The problem is not easily remedied by pointing out that it is also a vehicle. The sentence, “This boat is a vehicle,”
does not convey the idea that “All boats are vehicles.” The somewhat tricky relationship between vehicle and car can
be taught quite simply and without serious stipulation by teaching the learner to identify all objects as vehicles and then
by showing the types of vehicles. The statement, “This vehicle is a boat,” clearly makes the point that the object has
two labels.

Following the teaching of the higher-order discriminations, the various members are taught one at a time. The
conventions used to introduce each member are basically the same as those specified in Chapter 10. Each introduction
involves initial teaching, expansion, and review. A cumulative review appears periodically, starting before the
introduction of the third member and continuing through the program.

The Sequences

1. The first initial-teaching sequence in the program is a non-comparative sequence that teaches the higher-
order class name. If the higher-order name is cherry, the non-comparative sequence teaches cherry, not-
cherry. If the higher-order name is food, the sequence teaches food, not-food. This first step would be
used for any higher-order name: dream, not-dream; gymnosperm, not-gymnosperm; condominium,
not-condominium; vehicle, not-vehicle.

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 2. The next group of sequences introduce the various lower-class members, usually through noun sequences.
If the higher-order name is vehicle, different noun sequences would be used to introduce the subclasses
boat, car, truck, etc. For each vehicle, a variation of the initial-teaching sequence would be provided on
two consecutive lessons. Only one new member would be introduced at a time.

Note that the lower-order members introduced are ordered according to the rules for ordering
coordinate members (specified in Chapter 10). Members that are highly similar in name or features are
separated by dissimilar members. Also, introductions that involve minimum-difference introductions are
separated.

3. Before the third lower-class member is introduced (usually on the second day of the initial teaching
sequence for the second member), the cumulative review sequences begin. At least one cumulative
sequence is introduced following the initial teaching of a new member.

The Program

Figure 11.2 shows a schedule for teaching the higher-order name and the name of five members. The first sequence
teaches the higher-order name vehicle. Subsequent sequences introduce the various members, each being presented on
two consecutive days before the next member is introduced.

Figure 11.2

Concepts Lessons
1 2 3 4 5 6 7 8 9 10 11 12

A. Vehicle (non-comparative) X X

B. Truck (noun) X X

C. Boat (noun) X X

D. Wagon (noun) X X

E. Train (noun) X X

F. Car (noun) X X

Expansion A AB C A D BCD E F A BCDEF

Cumulative Review BCD BCDE BCDEF

The numbers at the top indicate the lessons or days of instruction. If the initial teaching of a member occurs on a
particular lesson, an X appears on that lesson. As the pattern of Xs shows, initial teaching is repeated on two
consecutive lessons. On only two lessons (2 and 3) is there more than one initial-teaching sequence, and there is no
initial teaching provided on lessons 7 and 12. More than one member is being taught on the early lessons (2 and 3),
because the learner is not required to remember many new names on these lessons.

Below the heavy bar on the time-table are the schedules for expansion activities and cumulative review. The letters

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in the cells indicate which members are involved in activities. The expansion starts on lesson 2 (dealing with A, the
vehicle—not-vehicle discrimination) and continues on a periodic schedule. The cumulative review begins on lesson 6
(after three lower-order members have been taught) and is presented before each new member is taught.

Gaps in the Schedule

It is much more difficult to predict the difficulty of various responses when we deal with a program rather than a
simple sequence. If one discrimination presents problems, a domino effect is sometimes created, with the learner
confusing the difficult member with the one that is introduced next (or immediately before). This effect continues as
subsequent members are introduced.

To avoid possible domino effects, we put “gaps” in the program. The program in Figure 11.2 above has one gap,
which occurs on lesson 7. No new discrimination is taught on this day. The learner engages in expansions of the
previously taught discriminations (BCD).

If we suspect problems, we space out the introduction so that there are more gaps or larger gaps. During the gap,
we present either expansion activities or cumulative review sequences. We also provide a consolidation at the end of
the sequence, which involves all the members taught in the sequence (lesson 12).

The cumulative review presents only the lower-order members (not the higher-order discrimination members)
because the task form used for all items is, “What kind of vehicle is this?” This task does not accommodate non-
vehicles and does not permit discrimination of vehicles and not-vehicles.

Before a new member is introduced, the learner performs on a cumulative sequence consisting of all the lower-
class members that have been taught. If the learner does not perform well on the cumulative review, the next member
should be delayed.

The Higher-Order Sequence

The higher-order sequence is the first taught in the program. It is a non-comparative sequence. The negative
examples should be “minimally different;” however, we cannot easily present examples of most higher-order labels
(such as vehicle, not-vehicle) through continuous conversion. We therefore use non-continuous examples and use the
test instruction, “Tell me vehicle or not-vehicle.” Figure 11.3 shows a possible sequence.

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 Figure 11.3

Example Teacher Wording

Swing This is not a vehicle.

Rocking horse This is not a vehicle.
Bicycle This is a vehicle.
Boat This is a vehicle.
Trailer truck Tell me vehicle or not-vehicle.
Large shed Tell me.
Motorcycle Tell me.
Car Tell me.
Chair Tell me.
Train Tell me.
Swing Tell me.
Wagon Tell me.
Dog Tell me.
Sled Tell me.

The sequence above presents two negatives followed by three positives and then a series of test examples. Note
that both opening negatives are objects that move; however, they are not vehicles. We could have presented an
“exercycle” type of machine as one of the negatives. The sequence is a higher-order non-continuous sequence.
Examples are sequenced like non-comparatives, and a choice-response task is presented with each test item.

When the sequence for vehicle is presented on the second day, a variation could be introduced. Instead of opening
with negatives, it could open with positives. The higher-order sequence should skip the modeled examples on the
second day. Instead of containing 13 examples, it could contain 8 to 10 examples.

The strategy for identifying minimum-difference examples was discussed in Chapter 6. In review:

1. First consider the various misrules that would be implied by a presentation of positive examples. (All
positives share the property of moving and the property of accommodating a passenger in a sitting
position, etc.)

2. Identify negatives that have the particular feature involved in the misrule. (All vehicles move. The
negative, therefore, should be something that moves but is not a vehicle.)

3. Eliminate questionable examples. What is a toy or a model? Are they vehicles? If the examples are not
clear negatives, do not use them.

The Lower-Order Sequence for the First Member

The lower-order members are generally introduced through noun sequences. The question that accompanies each

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example tested is: “What kind of vehicle is this?” Note that the question contains the higher-order name.

There is one exception to the rule that lower-order members are introduced through noun sequences. The first
lower-order member to be taught should be introduced through a non-comparative sequence. This sequence counters
any possible stipulation created by the higher-order discrimination. The learner now responds to the various positive
examples of vehicles as “truck” or “not-truck.” Note that all examples are vehicles. Figure 11.4 gives a possible
sequence for the initial teaching of the first lower-class member, truck.

Figure 11.4

Example Teacher Wording

These are vehicles.

1 This vehicle is a truck.

2 This vehicle is a truck.

3 This vehicle is not a truck.

4 This vehicle is not a truck.

5 Your turn. Tell me if each vehicle is a

truck or not a truck.

6 Tell me about this vehicle.

7 Tell me about this vehicle.

8 Tell me about this vehicle.

9 Tell me about this vehicle.

The sequence is similar to the higher-order sequence. It is a non-comparative non-continuous sequence with
“minimum-difference” negatives. The negatives are vehicles that have not been taught yet. They are the vehicles that
are most highly similar to truck: train and car.

Note that the sequence uses the test wording, “Tell me about this vehicle.” The label for the higher-order
discrimination, vehicle, must appear in the task. Also, the test wording should call for a labeling response, not merely
for a “yes” or “no” answer. We want to make sure that the learner uses the lower-class name in response to objects that
have previously been labeled with the higher-order name. (The learner responds: “This vehicle is a truck,” or “This

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vehicle is not a truck.”)

Lower-Order Sequences for Other Members

After the first member is taught through a non-comparative sequence, all other members are taught through noun
sequences. Here are the procedures for constructing these sequences:

• Limit the examples to the new member and to minimally different members that have been previously
taught.

• Use non-continuous conversions.

• Use test wording that requires the learner to label all examples with the lower-class names (“What kind of
vehicle is this?” or “Tell me about this vehicle.”)

Figure 11.5 gives a sequence for introducing boat, the second lower-class member. Note that all not-boats are
trucks. Truck is the only minimally-different member that has been taught. The sequence is fairly short because boats
are greatly different from trucks, and because there should be no problems with responses.

Figure 11.5
Example Teacher Wording

1 My turn. What kind of

vehicle is this? A boat.

2 My turn. What kind of

vehicle is this? A boat.

Your turn. What kind

3 of vehicle is this?

What kind of vehicle

4
is this?

What kind of vehicle

5
is this?

What kind of vehicle

6
is this?

What kind of vehicle

7
is this?

What kind of vehicle

8
is this?

The initial-teaching sequence on the second day of teaching boat could be shorter and could have no modeled
examples. Like the original sequence, it would begin with examples of the new member.

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 The next member to be introduced is wagon. The sequence for wagon is the same as that for boat. It presents
examples of the new member, wagon, and examples of previously taught minimally-different members. The only two
members that are known to the learner are truck and boat; therefore, these must serve as the minimally-different
familiar members. Figure 11.6 gives a possible sequence.

Figure 11.6

Example Teacher Wording

Here are some vehicles.

1 This vehicle is a wagon.

2 This vehicle is a wagon.

3 Tell me about this vehicle.

4 Tell me about this vehicle.

5 Tell me about this vehicle.

6 Tell me about this vehicle.

7 Tell me about this vehicle.

8 Tell me about this vehicle.

9 Tell me about this vehicle.

10 Tell me about this vehicle.

For each subsequent noun sequence, not all known vehicles would be included in the sequence, only those that are
minimally different from the new member that is being taught. As a rule, only two familiar members should be
presented in the noun sequence. For the first three members that are taught, the decisions are easy. The first member is
taught through a non-comparative sequence, so we do not have to worry about which minimally-different members to
include. (We select those that are minimally different in features.) The next member is taught through a noun sequence;
however, there is only one known member, which automatically becomes the negative in the sequence. The same
situation exists for the third member. Two members are known; therefore, they become the two negatives introduced in
the sequence. After the third member has been taught, only two negatives will be included as familiar members. These
two are the most similar (in name, shape, or name and shape) to the member that is being taught.

Cumulative Review Sequences

The schedule for teaching vehicles (Figure 11.2) shows that on lesson 6, three different activities are presented.

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The new member wagon is being taught for the second day. At some other time during the lesson, the learner receives
expansion activities that involve wagon. At still another time, the cumulative review involving truck, boat, and wagon
is presented. Note that these activities do not necessarily follow each other. The idea is to separate them in time. This
separation achieves the following:

1. It helps distribute the learning by providing a “change up” or break between activities that involve the
discriminations being taught. This break militates somewhat against possible fatigue or boredom.

2. The break also makes the juxtapositions more difficult by reducing juxtaposition prompting. The learner
might be able to remember the name wagon when working within a sequence, because the learner has just
heard the name and possibly just said it. This performance does not imply that the learner will remember
the name one minute or ten minutes later. The performance within a sequence is easy because the
sequence is designed to juxtapose examples in which the learner does the same thing (such as respond to,
“Tell me about this vehicle.”) When we present a break in the activity and later come back to the
discrimination, the first examples in the cumulative review sequences are relatively difficult because the
learner is not prompted by the same thing on preceding tasks.

3. The distribution of practice demonstrates that the learner will be expected to remember the names being
taught. If the sequence always models the correct responses or immediately follows a sequence that
models the correct responses, the learner may not understand that these names are to be remembered.

Figure 11.7 shows a cumulative review sequence that might be presented after truck, boat, and wagon have been
introduced (lesson 6).

Figure 11.7

Example Teacher Wording

Here are some vehicles.
1 What kind of vehicle is
this?
2 What kind of vehicle is
this?
3 What kind of vehicle is
this?

4 What kind of vehicle is

this?

5 What kind of vehicle is

this?

6 What kind of vehicle is

this?

All the lower-order discriminations that have been taught are presented in the sequence. This procedure is used for

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all cumulative reviews.

Summary of Program Events for Hierarchical-Class Programs

In its simplest form, the hierarchical-class program is presented so that:

1. The higher-order class is taught as a higher-order noun (a non-cognitive sequence).

2. Each lower-order member is introduced, the first through a non-comparative sequence, all others through
noun sequences.

3. The expansion activities for each member begin as soon as a member is taught (on either the first day or
the second day of the discriminations’ initial teaching sequence).

4. Cumulative review begins before the fourth member is introduced. An additional review sequence is
presented before each subsequent member is introduced.

Advanced Applications

Three changes that occur in the basic procedures when we deal with advanced applications are:

1. Statements of facts or rules are sometimes used to indicate the basis for classifying members (implying the
use of correlated-feature sequences).

2. The rate of introduction is accelerated.

3. The number and type of expansion and review activities is reduced.

If we were to teach vehicles as a more advanced application, we could possibly present a rule such as, “If it is
made to take things from one place to another, it is a vehicle.”

Figure 11.8 shows a possible sequence for the higher-order label. The sequence is a correlated-feature sequence.
The first test question in each pair is designed so the learner responds to whether the objects are vehicles (the new
name). Then the learner justifies the conclusion by reference to whether the objects are designed “to take things from
one place to another.”

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 Figure 11.8

Example Teacher Wording

1 My turn. Is this a vehicle? No. How do I know? Because it’s not made to take
things from one place to another.

2 My turn. Is this a vehicle? No. How do I know? Because it’s not made to take
things from one place to another.
3 My turn. Is this a vehicle? Yes. How do I know? Because it is made to take
things from one place to another.

4 Your turn. Is this a vehicle? How do you know?

5 Is this a vehicle? How do you know?

6 Is this a vehicle? How do you know?

7 Is this a vehicle? How do you know?

8 Is this a vehicle? How do you know?

Correlated-feature sequences may be used for teaching lower-order members. For instance, in teaching varieties of
gymnosperm trees, a correlated-features sequence might be used to show the difference between highly similar
members such as Sitka spruce and blue spruce. Figure 11.9 shows a possible sequence that could be used when blue
spruce is introduced. The sequence is a correlated-feature sequence. The first question in each pair requires the learner
to identify the tree. The second question requires the learner to relate the conclusion to facts about whether the needle
rolls.

Figure 11.9

Example Teacher Wording

1 My turn. This is a blue spruce. How do I know? A needle rolls through my fingers.

2 My turn. This is from a Sitka spruce. How do I know? A needle does not roll between my fingers.

3 Your turn. Is this a blue spruce or a Sitka spruce? How do you know?

4 Is this a blue spruce or a Sitka spruce? How do you know?

Note that the test wording violates the rule about naming the higher-order class. The wording could be: “Is this

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gymnosperm needle from a Sitka spruce or a blue spruce?” However, the additional wording simply weakens the task,
which clearly focuses on the type of spruce. With advanced applications, we do not have to be as rigid about naming
the higher-order class in the test question as we are when dealing with the relatively naive learner.

Adjusting the Sequence for the Learner’s Prior Knowledge

Often, the learner is familiar with either the higher-order name or the names of some members. If the learner
knows the higher-order name, but not the name of members, simply drop the teaching of the higher-order name and
begin with the introduction of members. The introduction of members would follow the schedule for the scope-and-
sequence for vehicles, except the sequence for teaching the higher-order name vehicle would be dropped.

If the learner knows some lower-order names, but does not know the higher-order name, a potentially awkward
situation arises. A lower-order statement should mention the higher-order name first: “This tree is a larch.” If the
learner already knows lower-order names, we might be tempted to use a statement that reverses the order: “This Larch
is a tree.” This statement treats the “treeness” as a variable, which it is not. “Treeness,” unlike “tallness,” is common to
all larches.

The simplest solution is to teach the higher-order name and then relate the higher-order name to the familiar lower-
order names. If the learner knows the lower-order name collie, but not the higher-order name dog, we could start with
the higher-order noun sequence given in Figure 11.10.

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 Figure 11.10

Example Teacher Wording

1 This is not a dog.

2 This is not a dog.

3 This is a dog.

4 This is a dog.

5 Is this a dog?

6 Is this a dog?

7 Is this a dog?

8 Is this a dog?

9 Is this a dog?

10 Is this a dog?

Initially, the learner may not accept the idea that a collie is a dog, saying, “No, it’s not a dog. It’s a collie.” The
teacher should respond to this objection by saying, “It’s a dog,” and repeating the task. (Long explanations will not
help.)

Note that collie is not the first positive that is presented. The reason is that the sequence must establish a basis for
grouping collie with other dogs on the basis of common features. By first presenting two other dogs that are obviously
not collies, the sequence makes the basis for giving collie a new name more obvious.

After the learner has successfully responded to all examples in the higher-order sequence, collie is introduced as a
type of dog. The teacher points to the collie and asks, “Is this a dog?” (“Yes.”) “You already know what kind of dog
this is. This dog is a collie. What kind of dog is this?” Following this step, the teacher would proceed to introduce
names of other dogs using noun sequences for each.

Collie is not taught through a sequence. The teaching occurs as the last part of the initial teaching for the higher-
order category. A variation of this procedure would be used if the learner knew names of more than one member.

Other Hierarchical Classes

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 Variations of the procedures outlined above can be used to teach any higher-order relationships. To teach
successive and simultaneous events, begin with the higher-order class events. (See Figure 11.11.)

Figure 11.11
Example Teacher Wording

A ride on a pony Is that an event? Yes.

Sitting on a pony Is that an event? Yes.
A pony Is that an event? No.
A room Is that an event?
Walking through a room Is that an event?
A room Is that an event?
A party Is that an event?
A party hat Is that an event?
A walk in the park Is that an event?
A park near the river Is that an event?
A bullet and a piece of Is that an event?
wood
A bullet going through a Is that an event?
piece of wood

After the learner has mastered the higher-order discrimination of event, not-event, the lower-order members
(successive and simultaneous events) would be introduced. Figure 11.12 gives a sequence for successive events.

Figure 11.12

Example Teacher Wording

A man presses on the Those are successive events.

brake. Then the car
stops.
A woman runs and then Those are successive events.
jumps over the fence.
A stone goes up and then Those are successive events.
comes down.
The woman presses on the Tell me about those events.
brake. At the same time,
she hits the horn.
As the sun moved one Tell me about those events.
way, her shadow moved
the other way.
Before he moved to the Tell me about those events.
city, he bought five hats.
While standing on one Tell me about those events.
foot, she scratched her
nose.
He sneezed and blinked Tell me about those events.
at the same time.
After carrying the Tell me about those events.
refrigerator down the
stairs, he collapsed.

The next sequence would teach simultaneous events. The sequence could present the discrimination as a

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correlated feature (“If the events are not successive, they are simultaneous”). Figure 11.13 gives the beginning of a
possible sequence. Or it could begin with an explanation of the relationship followed by a set of examples.

Figure 11.13

Example Teacher Wording

Events that are not

successive are
simultaneous. They
happen at the same
time.
The woman shouts What kind of events?
and claps at the
same time
The top was spinning What kind of events?
and moving at the
same time.
The man stood up What kind of events?
and then stretched.
They sang the words What kind of events?
at exactly the same
time.

Figure 11.14 shows a possible sequence preceded by an explanation.

Figure 11.14

Example Teacher Wording

She fell, then skidded These events are successive.

6 meters. How do I know? One
event followed the other.
He talked as he ate. These events are not suc-
cessive. How do I know?
One event does not
follow the other.

Teaching the lower-order members successive and simultaneous involves less teaching because there are only two
possible lower-class members. Therefore if a member is not one type, it must be the other. The learner is not required to
look for new features to identify the second type; merely to substitute the designation “not-successive” for the new
word, “simultaneous.” For this introduction to be effective, the learner must be quite firm on successive, not-
successive. If the learner is weak on this discrimination, serious problems will arise when simultaneous is taught.
Figure 11.15 gives a schedule. Note the large gap between the teaching of successive and simultaneous.

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 Figure 11.15

1 2 3 4 5 6 7

A Event, not-event A A

B Successive event B B

C Simultaneous event C C

Expansion activities AB AB AB C

Cumulative Review BC

Successive events and simultaneous events are treated as minimally different members. According to the rules for
ordering members, these members cannot be juxtaposed in the introduction. A gap in the sequence following the
introduction of successive events prevents them from being closely juxtaposed. During this gap, the learner practices
the successive-event discrimination through expansion activities. On lesson 5, simultaneous events is taught.

Coordinate Higher-Order Classes

When we teach different higher-order classes, we frequently find ourselves teaching two coordinate higher-order
classes, such as gymnosperms and angiosperms, or vehicles and clothing, or nouns and adjectives. Even if the
learner knows the names of members of the classes, there may be some question about whether the learner can
correctly classify trees as gymnosperm trees or angiosperm trees.

To firm the higher-order classes, we introduce a sequence after two classes have been taught and the learner has
practiced working with each. (A class would include the higher-order label and the names of members.) The sequence
takes the form of a cumulative-review sequence (no examples are modeled). Different trees are presented in an
unpredictable order. The learner indicates the class to which each belongs.

Figure 11.16 gives the first part of a possible sequence that could be presented after the learner has been taught
vehicles and clothing.

Figure 11.16
Example Teacher Wording
I’ll show you things that are
either in the class of
clothing or the class of
vehicles.
What class is this in?

What class is this in?

What class is this in?

Etc.

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If the learner has been taught correlated features for the different classes, the firming sequence for the higher-order
discrimination would be modeled after a correlated-features test segment (see Figure 11.17).

Figure 11.17

Example Teacher Wording

I’ll show you seeds from gymno-

sperm trees and angiosperm trees.
(White spruce) What type of tree? How do you
know? (because the seeds are
naked.)
(Beech) What type of tree? How do you
know? (because the seeds are
not naked.)
Etc.

Variations of this higher-order firming sequence would be repeated after three coordinate classes have been
introduced, and after each subsequent higher-order class has been taught.

Variations of the basic higher-order discrimination sequence are possible, such as a “fooler.” (Fooler procedures
are more fully discussed in Chapter 15.) Here’s an example of how a fooler would be used after the learner had been
taught vehicles, appliances, clothing, and containers.

Figure 11.18

Example Teacher Wording

My turn to name some vehicles.

When you hear me make a
mistake, say “stop.” Here I go,
naming vehicles.
Truck . . . car . . . Why did you say stop? (Because a
motorboat . . . washing machine is not a vehicle.)
washing machine What is a washing machine? (An
appliance.)
Motorboat . . . Why did you say stop? What is a
motorcycle . . . shirt?
shirt
Motorhome . . . Why did you say stop? What is a
car . . . truck . . . suitcase?
suitcase
Blender Why did you say stop? What is a
blender?

If vehicles had been the last-taught class, the sequence above would be well-designed to firm the higher-order
discriminations. If the last-taught class had been containers, the format should change to: “I’ll name some containers.
When you hear me make a mistake, say ‘stop.’”

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 The schedule in Figure 11.19 shows the pattern for higher-order firmings. Note that the schedule does not show all
the sequences used to teach the members of the various classes and does not show the time interval between the various
higher-order introductions. (Possibly, 15 intervene between the firming of appliances and the firming of containers.)

Figure 11.19
Skills Lessons

A Vehicles X

B Clothing X

C Appliances X

D Containers X

Higher-Order Firming AB ABC ABCD

The schedule shows that the firming activities first appear after the learner has completed the second class (higher-
order label and members). The higher-order firming could begin as the class is still being taught. This decision and the
frequency of the higher-order firming activities depend on other applications that are presented to the learner. Once
taught, however, a discrimination should be used.

Alternative Approaches to Teaching Higher-Order Relationships

The procedures presented for teaching higher-order relationships is quite thorough. It assumes that the new
members and the higher-order discriminations must be taught carefully. In many cases, this caution is not needed. The
learner could easily learn the information about simultaneous events or gymnosperms far faster than the procedures
we have presented would permit. For the sophisticated learner, the discriminations can be described and the description
can even tell about the variation of features observed in the examples. The goal of instruction for these learners is not
so much to teach the discriminations as it is to teach the various labels and the relationships of the parts within the
higher-order system. Procedures that are appropriate for these situations are specified in Chapter 14 (Programs for
Teaching Fact Systems).

Summary

If the goal is to provide complete teaching for a hierarchical or higher-order relationship, a program is needed. The
program for the members follows the same rules as the program for introducing coordinate members to a set. In
addition, a sequence for the higher-order class name is needed.

Follow these steps:

1. Begin the program with the sequence for the higher-order name.

• Use a higher-order noun sequence (test question: “Is this a ___?”).

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 • Present non-continuous examples.

• Use minimum-difference negatives that are safe.

2. Follow the teaching of the higher-order class name with a sequence for the first lower-order member.

• Use a higher-order noun sequence, non-continuous examples, and minimum-differences selected from
the set of members that will be taught later.

• The test question is: “Is this (higher-order noun) a (lower-order noun)?”

3. Follow the first member with noun sequences for the teaching of each subsequent member. (The test task
is: “What kind of ___ is this?” or, “Tell me about this ___.”)

4. Process each member through an initial-teaching sequence, expansion activities, and cumulative review.

5. If more than one class is introduced, present a higher-order firming sequence that requires the learner to
identify examples from different classes by their higher-order name. (“Tell me vehicle or clothing.”)

6. For the more facile learner, use correlated-feature sequences to express the basis or rule for classifying
members in a particular class.

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 Programs Derived from Tasks
The most elementary program is one that simply introduces members into the set. A variation of this program is the
higher-order class program that is more complicated because it teaches both higher-order discriminations and
discriminations for various members. This chapter presents a third basic program. It derives from a task. A task
consists of some sort of directions to the learner and some requirement for the learner to respond. “Put the ball on the
table,” is a task. So is “Write the answers to all questions on Part 1 of your worksheet.” This is not a task: “The learner
will respond to questions that involve prepositions.” It is not a task because we no not know the words that occur in
these questions. Unless we have the exact instruction that the learner is to follow and the exact behavior that the learner
is to produce, we do not have a task. Conversely, any specific instruction to the learner (any question or direction) for
which a specific response (or set of responses) is called for, is a task.

Programs that derive from tasks share many features with the other programs. The major difference has to do with
the procedures by which the members or component discriminations to be taught are identified. Programs that derive
from tasks are the product of task analysis. The process of task analysis consists of identifying the component
discriminations and responses that appear in the task. Some components do not have to be taught at all, because they do
not imply any new or difficult behavior on the part of the learner. Most components, however, should be pretaught,
before the complete task is taught. These components are treated as members that we must introduce into a set. We
follow the same rules for processing these members that we used for members in the other basic programs—the rules
about separating members that are minimally different and separating introductions that involve minimum-difference
discriminations. We cycle each member through these phases: initial teaching, expansion, and cumulative review.

Task Analysis

A task analysis includes three steps:

1. Identifying the component discriminations of a task that should be pretaught. The test for preteaching is:
Could the learner who does not understand a discrimination that appears in the task possibly fail the task
because of this deficiency? If the answer is “Yes,” the component discrimination should be pretaught.

2. Classifying each component discrimination identified for preteaching. A discrimination is classified as

either a single-dimension concept, a noun, a transformation, or a correlated-feature concept. This
classification indicates how we would teach each discrimination identified.

3. Scheduling the component discriminations. Each discrimination is treated as a member of a set to be

scheduled for initial teaching, expansion, and cumulative review. As noted earlier, the scheduling is
governed by the basic rules for introducing members to a set.

Strict Task Analysis

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 A strict task analysis does not look beyond a specific task. It does not take into account the fact that the learner will
be expected to do things that are closely related to those specified by the task. The analysis assumes only that the task is
to be taught and that the components of the task can be identified and pretaught.

Let’s say that we conducted a strict analysis on the task: “Put the ball on the table.” By inspecting the task, we
identify the components: ball, table, and put (or put the ball). By performing the first steps suggested for the task
analysis, we arrive at judgments about the necessity of preteaching these components. We do not have to preteach the
components if our objective is to teach the task and only that task. If the learner is simply going to put the ball on the
table in response to repeated presentations of the direction, “Put the ball on the table,” no preteaching is necessary
because ball is not discriminated from any other objects that could be put on the table, and table is not discriminated
from any other object that could receive the ball. Therefore, the learner does not have to attend to these words. After a
naive learner has performed on the task a dozen times, the teacher could say, “Blop the tober umuck an elephant,” and
the learner would produce the behavior of putting the ball on the table.

This situation points out a serious problem with a strict task analysis. The analysis is not appropriate for cognitive
skills because it creates generalization problems. The strict analysis is more appropriate for physical skills (shoe-tying,
teeth brushing, downhill skiing, etc.). But even for these tasks, we are faced with the problem of specifying the task so
that the teaching will account for generalization, not merely for performance on a single task.

Transformed-Task Analysis

The analysis of cognitive skills is more reasonable if we deal not only with the task that is presented, but also with
a set of related tasks or with a transformed task. The transformed task is created by indicating substitutions for parts of
the task. The task, “Put the ball on the table,” is a member of the set, “Put the A on the B,” with the letters referring to
any familiar objects. By transforming the task in this way, we indicate the generalizations that are to be taught. Now
the teaching of various components that we identify becomes functional because the learner must discriminate the
names of different objects. (The learner must discriminate between “Put the book on the table,” and “Put the ball on the
book.”) Note that we could transform the task even more by creating the transformed task, “Put A (position) B.”

The component skills also change when we use a transformed-task analysis. Now it is important to teach
components because the learner could possibly fail the original task by not knowing the component names.

The teaching of components for “Put A on B” is fairly straightforward. We teach a set of objects (or we assume
that the learner has been taught a set of objects). We assure that table and ball are in the set before they are named in
the original task; however, we do not limit the set to these objects. We then teach the discrimination on using a non-
comparative sequence. Next we present a series of tasks that are described by our transformed objective, “Put A on B.”
These tasks would include: “Put the ball on the table,” “Put the ball on the floor,” “Put the spoon on the ball,” and so
forth. Note that the program is designed to include the original task (“Put the ball on the table”); however, it is not
limited to this task. It includes a range of tasks that have the same form (“Put A on B”).

Correlated-Feature Sentences

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 In later chapters, we will use transformed-task analysis to derive programs for a variety of skills, such as routines
for solving column multiplication problems. The applications that we will present in the remainder of this chapter,
however, involve simpler tasks and simpler programs. These tasks are simpler because they have already been
transformed. All take this form: “The learner will respond appropriately to applications of this rule: ‘When air gets
hotter, it expands.’” The responses the learner will produce are not specified. The specific tasks that will be presented
are not specified. However, the transformed task indicates that we must provide a good test of the learner’s ability to
handle applications that derive from the rule, “When air gets hotter, it expands.” The sentence expresses a correlated-
feature relationship; therefore, we would teach the relationship through a correlated-features sequence. This sequence
would provide us with the specification of responses the learner is to produce. Most important for our present purpose,
the sentence itself serves as a part of the task sequence that is given. If the learner is to deal with applications based on
the sentence, we must teach any component discriminations named in the sentence that would serve as possible causes
of confusion for the naive learner.

For the remainder of this chapter, we will deal only with sentences that express correlated-feature relationships. We
will identify the component skills expressed in the sentences. We will indicate how the component skills could be
taught and scheduled.

Correlated-feature sentences are well suited to an analysis of component skills because they express one particular
meaning of the words and they admit to a relatively straightforward analysis.

Meanings and Words

Different sentences generate different meanings of words. “This picture should be hung on the west wall,” uses the
any-surface meaning of on. “Pillows belong on the bed,” assumes a different meaning of on. The particular meaning
that we will teach depends on the sentence we start with. When analyzing components of sentences, we are concerned
with only that meaning. (Without this stipulation, the designer’s quest becomes one of trying to teach all meanings. The
task does not require the learner to know every meaning of a word like on.)

Wording. A task indicates the meaning of specific words that are to be taught and specifies the wording that must
be used to teach particular discriminations. If the task that we begin with contains the words in contact with, we would
use these words and only these words in teaching the idea in contact with. We would not substitute the words in
contact with for on. Similarly, if the task refers to a living organism, we would design sequences that teach the idea
expressed by this component in the task, and we would use the words living organism (and only these words) when we
taught this component.

In summary, the task provides specific directions for both the meaning that is to be taught and the exact wording to
be used in conveying the meaning.

Analysis of Sentences

The sentence we start with specifies an entity and then tells about one feature correlated with that entity. Since the
sentence is divided into two parts (the specifying of the entity and the telling of the correlated feature), the sentence

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contains at least two component discriminations. The naming of the entity (the first part of the statement) may be made
up of component discriminations. The second part of the statement may also contain component discriminations. In a
careful program, components of each part would be taught first, the entire part would be taught next, and the correlated-
feature relationship would be taught last.

Figure 12.1 shows a diagram of how the teaching might occur for a sentence made up of two parts, each of which
contains three component discriminations (labeled A through F).

Figure 12.1
Sentence: ABC/DEF

1. First taught:
components of parts. ABC DEF
2. Next taught: parts. ABC DEF
3. Last taught:
correlated-feature ABC/DEF
relationship.

For some sentences, the pattern might look quite different because all the discriminations that must be pretaught
would be in the first part of the sentence. For other sentences, some discriminations occur in the first part and perhaps
one occurs in the second. Regardless of the pattern, the general strategy is suggested by the diagram above. First the
components are taught—one at a time. Then these are integrated to form parts of the sentence. The last step is the
correlated-feature relationship.

The Program

Suppose we start with a correlated-feature relationship expressed by the sentence: If the leaf is pinnately venated
and has pointed lobes, it is a black oak leaf. The first part tells the conditions that must be met for the oak leaf to be a
black oak leaf. The second part specifies the label that we will use to refer to objects that meet the conditions.

Identifying components. The discriminations that should be pretaught appear in the first part of the sentence: leaf,
pinnately venated, and pointed lobes. Although we can test the learner with a sequence that involves leaf, not-leaf,
we can combine the teaching of this discrimination with that of other components.

Classifying components. To teach pinnately venated, use a non-comparative sequence. To teach pointed, use a
non-comparative sequence. The correlated-feature sequence would be used to teach the relationship expressed in the
sentence.

Schedule. Figure 12.2 shows a possible schedule for the components.

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 Figure 12.2
Components Days

1 2 3 4

A. Pinnately venated X

B. Pointed lobes X

C. Correlated-feature
sequence X X

Expansion C

Review AB

Note that the initial teaching is presented on only one day for each component. The correlated-features sequence
appears on two days. On the first day it appears, it would be the last teaching event, occurring after the review and
work on pointed lobes.

Figure 12.3 shows a sequence for teaching pinnate, not-pinnate.

Figure 12.3
Example Teacher Wording

Here’s a leaf with no vein

pattern on it. I’ll put a
pattern on it and tell you if
it’s a pinnately-venated
pattern.

1 This pattern is not pinnately

venated.

2 This pattern is pinnately

venated.

3 Tell me about this pattern.

4 Tell me about this pattern.

The various patterns could be created by placing two different transparencies over the leaf. One would show the
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pinnately venated pattern; the other would show the other vein pattern. If the teacher were dealing with sophisticated
learners, the teacher could require labels for both patterns. In this case, the introduction would be changed to: “Here’s a
leaf with no vein pattern on it. I’ll put patterns on and tell you if the pattern is a pinnately-venated pattern or a
palmately-venated pattern.” The teacher might require the learner to repeat the names before proceeding. The
sequences could also call attention to the difference between the patterns. “If the veins don’t come together at one
point, the pattern is pinnately venated.” The essential aspects of the sequence are the presentation of positive and
negative examples and the use of the label that will be used in the correlated-feature sequence. The directions for each
test example, “Tell me about this pattern,” require the learner to say the word rather than responding by saying “Yes”
or “No.”

The next discrimination to be taught is pointed lobes (B). We may decide first to teach lobes, then pointed lobes.
Decisions of this type are frequently required when we analyze sentences for component discriminations. It is usually
safe to deal with the conjunctive concept (pointed lobes) because the negative examples that are implied by the
sentence are lobes that are not pointed. (The leaves that are most similar to black oak leaves have lobes. Whether the
lobes are pointed becomes the critical basis for discriminating black oaks from these leaves.) As a general rule, you can
deal with the larger phrases that appear in a sentence.

Figure 12.4 shows a possible sequence for teaching pointed lobes.

Figure 12.4
Example Teacher Wording

Here’s an outline of a leaf

with no lobes on it. I’ll put
lobes on it. Tell me if the
lobes are pointed or not
pointed.

1 Tell me about the lobes.

2 Tell me about the lobes.

3 Tell me about the lobes.

4 Tell me about the lobes.

The examples above could be created by placing transparent overlays over the original leaf form. No models are
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presented in this sequence because we assume that the learner already knows the difference between pointed and not-
pointed. We could expand the sequence by introducing a wider variety of lobe examples. We could also change the test
instructions so that the learner responded by saying “pointed lobes” or “rounded lobes.”

After we have taught the discrimination of pinnately-venated and pointed lobes, we are ready to teach the
correlated-feature relationship: “If the leaf is pinnately venated and has pointed lobes, it is a black oak leaf.” The
sequence will show what is mentioned first in the sentence-leaves that have (or do not have) pinnately venated patterns
and pointed lobes. The learner will respond by identifying it either as a black oak or a not-black oak. A second question
will ask, “How do you know?” Figure 12.5 shows the sequence.

Figure 12.5
Example Teacher Wording

This is a black oak leaf. How

do I know? Because it is
1 pinnately venated and it has
pointed lobes.

Your turn: Tell me black oak

2 or not black oak. How do
you know?

Tell me black oak or not-black

3 oak. How do you know?

Tell me black oak or not-black

4 oak. How do you know?

Tell me black oak or not-black

5
oak. How do you know?

Tell me black oak or not-black

6 oak. How do you know?

The sequence of examples shows the following variations:

pinnately venated and pointed lobes

pinnately venated and not-pointed lobes

not-pinnately venated and not-pointed lobes

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 not-pinnately venated and pointed lobes

We could construct a sequence that has a larger number of examples; however, the crucial aspects of any
acceptable sequence are that it must show minimum negatives and that it must require the learner to conclude whether
or not each example meets the criterion for black oak.

The program above derives from a sentence that is complicated; however, there is nothing really new in the
program. It is a simple extension of the sequences and rules presented in the preceding chapters.

Validity. There is always the question of truth or validity of the statement that is being taught. The statement about
black oak leaves would not permit the learner to discriminate between black oak and all other leaves. A qualification
could be presented to resolve the problem. “I’ll show you leaves that are black oak leaves and leaves that are not-black
oak leaves.” This qualification permits us to use the sequence.

Illustration

Here’s a different correlated-feature sequence that could serve as the objective for a task-analysis program: When
solids are heated enough, they turn into liquids.

Component discriminations. The discriminations are solids, heated, and liquids. All these discriminations would
be taught before the correlated-feature relationship is taught. Heated is a comparative (getting hotter). Solid and liquid
are coordinate members of the same set. They would be taught as members of a set with the first member taught
through a higher-order noun sequence that presents the liquids as minimum-difference negatives. The sequence for the
second member is a regular noun sequence.

The schedule in Figure 12.6 shows a fast introduction. No problems would result if the discriminations were
presented with time gaps between them, so long as the reviews were adequate.

Figure 12.6
Components Days

1 2 3 4

A. Solid X

B. Liquid X X

C. Heated X

D. Correlated-features X X

Review A A

For the teaching of solids, the negative examples would be liquids and possibly gases. The range of solids includes
things that are pliable, such as cloth, and those that are stiff, such as ice. Figure 12.7 shows a possible sequence.

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 Figure 12.7
Example Teacher Wording

Ice cube This is a solid.

Plastic bag This is a solid.
Paper bag This is a solid.
Plastic bag Tell me solid or not-solid.
Brown rock What’s this?
Molasses What’s this?
Water What’s this?
Wood chips What’s this?
Ice cream frozen What’s this?
Block of glass What’s this?
Melted ice cream What’s this?
Blanket What’s this?

The sequence is a higher-order noun sequence that opens with three positives followed by a minimum difference
and mix of positives and negatives. The learner identifies each example as “solid” or “not-solid.”

Heated is a comparative. To teach heated as a basic sensory quality, we could present the learner with water from
a mixer faucet. Note that we would teach the wording heated if that wording appeared in the sentence that was to be
taught. If the sentence referred to getting hotter, we would use the same basic sequence of examples with the wording
“getting hotter.”

Figure 12.8 shows a sequence that is created by referring to degrees. This sequence assumes that the learner has
already been taught the basic sensory discrimination heated.

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 Figure 12.8
Example Teacher Wording
Let’s say an object is 30
degrees. I’ll tell you
if it gets heated.
Temperature changes to 31°. Did it get heated? Yes.
Temperature changes to 85°. Did it get heated? Yes.
Temperature changes to 84°. Did it get heated? No.
Temperature changes to 83°. Did it get heated?
Temperature changes to 84°. Did it get heated?
Temperature changes to 12°. Did it get heated?
Temperature changes to 15°. Did it get heated?
Temperature changes to -15°. Did it get heated?
Temperature changes to
2081°. Did it get heated?
Temperature changes to
2029°. Did it get heated?
Temperature changes to 5°. Did it get heated?
Temperature changes to 28°. Did it get heated?
Temperature changes to -3°. Did it get heated?
Temperature changes to 11°. Did it get heated?

Examples are created verbally in this sequence. The sequence is non-continuous. It follows the juxtaposition
pattern for the comparative sequence. It begins with a starting point example, followed by two positives, two negatives,
and a series of unpredictable examples. There are no no-change negatives in the sequence.

The third component of the task is liquids (see Figure 12.9).

Figure 12.9
Example Teacher Wording

Water My turn. This is a liquid.

Molasses This is a liquid.
Water What’s this?
Ice What’s this?
Brown rock What’s this?
Gasoline What’s this?
Plastic bag What’s this?
Mercury What’s this?
Wood What’s this?

Two positives are followed by an E1 test and then an E2 test. The learner responds to the test examples by saying

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either “liquid” or “solid.”

After the learner has been taught the concepts solid, heated, and liquid, the correlated-feature relationship is
taught. We present examples referred to in the first part of the sentence (solids being heated or not-being heated), and
the learner tells whether this change would make the solid into a liquid. (See Figure 12.10.)

Figure 12.10
Example Teacher Wording

Here’s what happens. A solid object goes What’s going to happen if this change keeps going? The
from 45 degrees to 46 degrees. solid will turn into a liquid. How do I know? Because
the solid is being heated.
Now it goes to 312 degrees. My turn. What’s going to happen if this change keeps
going? The solid will turn into a liquid. How do I know?
Because the solid is being heated.
Now it goes to 311 degrees. What’s going to happen if this change keeps going?
How do you know?
Now it goes to 56 degrees. What’s going to happen if this change keeps going?
How do you know?
Now it goes to 300 degrees. What’s going to happen if this change keeps going?
How do you know?

Note that the correlated feature expressed in this sequence is not the strongest one possible. However, if it is the
one selected for instruction, it serves as a task that implies the component sequences (or their equivalent as preskills).

Illustration

A final example is the sentence: If the word ends in CVC (consonant-vowel-consonant), and the next part begins
with a V (vowel), you double the last C (consonant).

Components. The learner should be taught whether letters are C (consonant letters) or V (vowel letters). The
learner should discriminate whether a word ends in CVC. The learner should discriminate whether the next part begins
with a V.

Sequences. The discrimination of vowel would be taught first. The strategy would be to teach the members that are
vowels: “Name the vowels.” This is a memorization task. After it has been taught, the learner would be presented with
different examples of letters. The learner would identify each as either “vowel” or “not-vowel.” The sequence would be
a higher-order noun sequence.

The discrimination consonant would be taught through a noun sequence. Those letters that are not vowels are
consonants. The learner identifies each example as “vowel” or “consonant.”

The discrimination ends in CVC implies a non-comparative sequence or a transformation. The discrimination,
next part begins with a vowel implies a non-comparative sequence or transformation. Figure 12.11 shows a possible
program schedule.

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 Figure 12.11
Components Days
1 2 3 4 5 6 7 8 9 10 11

A. Vowel X X X

B. Consonant X X

C. Ends CVC X X

D. Part begins V X X

E. Correlated-feature X X

Review and expansion A A A A AB AB AB AB

After vowels have been taught, the teacher might list them, then present a sequence that requires the learner to
identify whether each letter is a vowel or a not-vowel (lesson 5). Figure 12.12 shows the first part of a possible
sequence.

Figure 12.12

Example Teacher Wording

Letters that are not vowels are

consonants.

c My turn. What’s this? A consonant.

d My turn. What’s this? A consonant.
v My turn. What’s this? A consonant.
d Your turn. What’s this?
h Your turn. What’s this?
a What’s this?
j What’s this?
r What’s this?
p What’s this?
e What’s this?

The sequence begins with a rule. Note that the discrimination could be taught without this rule; however, the use of
the rule makes the discrimination more obvious to the learner.

After the learner has been taught to discriminate consonants and vowels, we would teach the learner to discriminate
what words end in. Perhaps the best sequence for this teaching is a single-transformation sequence (see Figure 12.13).
The single-transformation juxtapositions permit a precise demonstration of how each word ends.

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 Figure 12.13

Example Teacher Wording

band I’m going to call any vowel V and

any consonant C.
ban My turn. What type are the last
three letters in this word: VCC.
bran My turn. What type are the last
three letters? CVC.
brad Your turn. What type are the last
three letters?
beak What type are the last three letters?
smoke What type are the last three letters?
tree What type are the last three letters?
sent What type are the last three letters?
rot What type are the last three letters?
strip What type are the last three letters?
Etc. What type are the last three letters?

The sequence follows the pattern of other single-transformation sequences. A series of progressive minimum-
difference examples is followed by greater-difference examples.

A similar sequence would be used to teach the beginning of parts. Figure 12.14 shows some test examples.

Figure 12.14
Example Teacher Wording
er This part begins with a vowel (V).
ed What type of letter does this part
begin with?
red What type of letter does this part
begin with?
s What type of letter does this part
begin with?
ing What type of letter does this part
begin with?

Only one example is modeled because the learner already can identify vowels and consonants. We assume that the
learner also knows word (or part), beginning and end. (If the learner does not know beginning or end, these would be
taught through a non-comparative sequence. The teacher would say, “I’ll touch a part of a word. I’ll tell you if it’s the
beginning.” The teacher would then touch either the beginning or end. The teacher would model several examples and
then test on others. Only one discrimination would be taught initially, either beginning, not-beginning; or end, not-
end.)

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 To teach the discrimination double the last C, we could present a single-transformation sequence. Figure 12.15
shows the first part of such a sequence.

Figure 12.15
Example Teacher Wording
bent My turn to double the last C in
this work: t, t.
bend Your turn. Double the last C.
bena. Double the last C.

After the learner has been taught to identify the last three letters of a word, whether a part begins with a vowel or a
consonant, and how to double the last C, the final relationship would be taught through a correlated-feature sequence.
The relationship being taught is: If the word ends in CVC and the next part begins with a vowel, you double the last C
(see Figure 12.16).

Figure 12.16
Example Teacher Wording

sit + ing My turn. Do I double the last C in sit? Yes. How do I know? Because sit ends in CVC
and the last part begins with a vowel.
sit + s My turn. Do I double the last C in sit? No. How do I know? The last part doesn’t
begin with a vowel.
sleep + ing My turn. Do I double the last C in sit? No. How do I know? Because sleep does not
end in CVC.
slip + ing Your turn. Do you double the last C in slipping? How do you know?
drop + s Do you double the last C in drop? How do you know?
drift + ing Do you double the last C in drift? How do you know?
slip + s Do you double the last C in slip? How do you know?
treat + ing Do you double the last C in treat? How do you know?
look + ing Do you double the last C in look? How do you know?
ask + er Do you double the last C in ask? How do you know?
flat + en Do you double the last C in flat? How do you know?
slit + s Do you double the last C in slit? How do you know?

The rule is complicated, on the verge of being unmanageable. To make it more manageable, we might make more
of the steps overt. However, the same basic relationship shown in the sequence above would remain in the more
overtized procedure. There would be a series of examples, at first those that show minimum difference, then those that
show greater difference.

This description of the program for the CVC rule is quite incomplete. The point, however, is not to articulate the
development of the program, but to show that even very complicated rules are composed totally of sensory
discriminations, correlated-feature relationships, and transformations. These components are contained in the facts that
serve as tasks for analysis. The facts provide the specific wording for all the necessary discriminations and

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relationships. Also, the facts provide the precise meanings that are to be taught.

Analyzing Preskills Contained in Facts

We do not have to preteach all discriminations presented in a correlated feature sentence—merely those that pose
possible confusion. Sometimes it is easier to see what should be taught by rephrasing the sentence as an if-then
sentence. Here are some rephrased items:

• The faster something moves, the more inertia it has.

Rephrased: If it goes faster, it has more inertia.

• Sedimentary rocks include limestone, sandstone, and shale.

Rephrased: If the rock is limestone, sandstone, or shale, it is a sedimentary rock.

• Ornithischian dinosaurs have a pelvis structure like birds’.

Rephrased: If the dinosaur has a pelvic structure like a bird’s, the dinosaur is an ornithischian dinosaur.

• Heated objects expand.

Rephrased: If the object is heated, it expands.

• Indolent means lazy.

Rephrased: If a person is lazy, a person is indolent.

When the sentences are set up this way, the first part of the sentence contains names that must be pretaught.
Perhaps the last part does, too. However, you must decide whether the last part tells about something that will also
happen or simply provides a new name for the events described in the first part of the sentence. The sentence: If the
animal has a backbone, the animal is a vertebrate, does not mean that the animal changes into a vertebrate. It means
that if the animal meets the classification criterion of having a backbone, the animal is called a vertebrate. The situation
is quite different from this sentence: If the object is heated, it expands. Expanding is not simply another way of
saying that the object is heated.

One test of whether the last-mentioned words should be pretaught is this: if you can add the word called to the rule,
you are dealing with an equivalent-meaning sentence, and you should not teach the last-mentioned words. The word
called has been added to the following sentences:

• If the pubis bone of a dinosaur’s pelvis is forward, the dinosaur is called saurischian.

• If the organism lives in an environment that has no free oxygen, the organism is called anaerobic.

• If someone tends to be irritable, that person is called irascible.

For all these sentences of equivalent meaning, we would not preteach the new word. The correlated-feature

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sequence would provide that teaching.

If the word called cannot be added, the discrimination referred to in the last part of the sentence should be
pretaught.

For the sentence, If molecules are heated, they move faster; molecules, heated, and moving faster should be
pretaught.

For the sentence, If something moves farther from the surface of the earth, it weighs less; moving farther
from, the surface of the earth, and weighing less should be pretaught.

These and similar sentences tell about the relationship of one event to another.

Prescriptive Applications of Programs

The basic task-analysis procedure applies to prescriptive, or remedial instruction. In these situations, learners have
trouble because they have not been taught some necessary prerequisite skills or discriminations.

This situation is very common. The teacher tries to teach how rain is formed, the difference between adjectives and
adverbs, the spelling of a particular class of words, or how expansion works. Often, students do not understand what
the teacher is trying to communicate.

Instead of providing an entire program, the remedy involves probing (to determine precisely what the students do
know) and determining an appropriate sequence of skills. The procedure for probing is not the random exploration that
is sometimes called probing. It is the identification and testing of the component basic discriminations that are implied
by the correlated-feature relationship. The teaching of those skills follows from the probe.

Testing Skills

We begin with the test of the correlated-feature relationship. Let’s say that the teacher is trying to teach students
how rain is formed. As part of the instruction, the teacher explains that “When the water-laden air rises over the
mountains, it cools and can no longer hold all the water. The result is rain.” We suspect that the students do not
understand the explanation. This assumption is reasonable, because the explanation is mystical, suggesting that the
inability of the cooler air to hold water is related to mountains. Also, the sentence contains far too much information.
The critical relationship is: “When air rises, it can’t hold as much water.” Figure 12.17 illustrates a test.

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 Figure 12.17
Example Teacher Wording

(Teacher touches spot on

chalkboard.) “Pretend that
my finger is a mass of air.
Watch what the air does
and tell me about the
amount of water it can
hold.”
Finger goes up “Can it hold more water
one foot. now, or less water?
How do you know?”
Finger goes down “Can it hold more water
six inches. now, or less water?
How do you know?”
Finger goes up “Can it hold more water
slightly. now, or less water?
How do you know?”
Finger goes way “Can it hold more water
up. now, or less water?
How do you know?”

Another correlated-feature probe could be used to test the learner’s understanding of the temperature of the air as
illustrated in Figure 12.18.

Figure 12.18
Example Teacher Wording

Finger goes up “This time, I want you to

one foot. tell me whether the air
cools or becomes hotter.
Watch my finger.”
Finger goes up “Tell me what happens to
a few inches. the temperature . . .
How do you know?”
Finger goes up “Tell me what happens to
a foot. the temperature . . .
How do you know?”
Finger goes down “Tell me what happens to
a few inches. the temperature . . .
How do you know?”

If the learners performed well on the sequences, the teacher could proceed with the original explanation and could
ask questions that require students to predict what will happen.

“The air moves up the mountain. Is this air rising or falling? . . . So what’s going to happen to the temperature of
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the air? . . . And what’s going to happen to the amount of water the air can hold? . . . If the air can’t hold the water,
where will the water go? . . . And that’s how rain is formed.”

In later chapters, we will provide more information about how to construct applications and how to test preskills.

Summary

The programs formulated in this chapter derive from a task analysis. There are two basic approaches to task
analysis. The first is to adhere rigidly to the wording and behaviors that are called for in the task. This approach is more
reasonable when the instruction involves teaching motor behavior rather than discriminations. The second type of task
analysis begins by transforming the original task into a form that suggests the range of discriminations the learner is to
make when dealing with the tasks of that type. The transformed task indicates the substitutions that can be made to
create the crucial discriminations.

Task analysis applies to any complex task. This chapter applied the analysis to sentences of correlated-feature
relationships. These sentences are usually good starting points for developing small programs because they usually do
not require transformations.

Deriving a program from a correlated-feature sentence (or from other tasks) involves three steps:

1. Identifying the component discriminations presented in the task.

2. Classifying each component discrimination in a way that indicates how to teach the discrimination.

3. Scheduling the components that are to be included in the program.

When we use the task analysis, we assume that the task is given. That means the wording in the task is accepted as
the wording that the learner will respond to. It also means that the particular meaning of a word or phrase presented in
the sentence is the meaning that we accept for the program.

• The sentence contains the wording that should be used in the program for teaching component skills.

• The sentence contains the meanings that should be used with each component.

• Each sentence consists of two major discriminations: the entity named first in the sentence and the features
attributed to the entity.

• Possibly, both the entity and the feature can be broken into component discriminations.

• The test of whether a discrimination should be taught is: Would it be possible for the learner to fail to
understand the relationship expressed by the sentence without knowledge of individual components?

• As a rule, the most relevant components are composed of a group of words. If you are in doubt, however,
teach some of the smaller components.

• Classify each component as a single-dimension discrimination, a noun, or as one of the joining forms. This

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 classification suggests how we will teach the component.

• The scheduling of components follows the procedures for introducing members to a set. Highly similar
discriminations are separated in time. So are introductions that involve minimum-differences.

The biggest point this chapter makes is that if we accept a task, such as a correlated-feature sentence, we can
identify the preskills that the learner should be taught.

Section VI will show how the analysis applies to complex cognitive routines, tasks far more complicated than
sentences. Although the task analysis is more complicated when applied to these routines, it involves the same
procedures outlined in this chapter—identifying the component discriminations and specifying the sequences and
examples needed to teach each discrimination. The building blocks, or component discriminations, always imply
sequences for teaching comparatives, non-comparatives, nouns, transformations, or correlated features. The wording
and meaning presented in the complex task directs which sequences, which wording, and which meaning are implied
for each discrimination.

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 Double Transformation Programs
Although the structure of the double-transformation relationship is complicated, the strategy for the double-
transformation program is not. The double-transformation relationship always involves two sets of single-
transformation examples that share some relevant features. The strategy for teaching the relationship involves first
teaching one set using the standard procedures for introducing single-transformations. The second set, however, is
taught in a new way. The learner is shown the difference between the first and second sets. Since the learner is required
to learn only the difference, the learning demands are substantially reduced for the second set; there- fore, the amount
of teaching time is decreased. If we arbitrarily say that concept X and concept Y each have ten features, and eight of
these are common to both discriminations, we could reduce the amount of teaching for Y by eight features if we
showed Y as being a variation of X. In effect, teaching the difference is designed to induce this type of awareness for
the new concept, “Oh, that’s just like X except that it . . .”

The strategy for teaching a double-transformation relationship has two advantages:

1. It achieves savings in teaching time.

2. It assures that the learner will learn the relationship between discriminations that are related in significant
ways.

We have used a variation of this strategy for the teaching of polars. We do not try to teach the polars hot and cold
at the same time. Instead, we teach one of the discriminations (hot, not-hot). Later, when the learner has mastered this
discrimination, we teach the difference between not-hot and cold. (The only difference is the label.)

The double transformation relationship is more complicated than the relationship between not-hot and cold;
however, the strategy is basically the same.

Double-Transformation Structure

Every double-transformation relationship involves two single-transformation sets of examples. Arithmetic provides
many examples of double-transformation sets; however, the double-transformation relationship is not limited to
arithmetic. To find a double-transformation relationship, start with a set of single-transformation examples. For
instance, start with the arithmetic facts 0+1 through 19+1.

Now think of a set of examples that “parallels” the plus-one set. This second set parallels the plus-one set if you
can create all members of the second set by applying the same transformation to any member of the first set. A parallel
set would be plus-ten facts. For every plus-one fact, a corresponding plus-ten fact is created by applying exactly the
same transformation (adding a zero to each numeral): 4+1=5 changes into 40+10=50; 5+1=6 changes to 50+10=60; etc.
By following the same transformation rule, we can create all members of the plus-ten set.

The plus-ten set is not the only set that bears a double transformation relationship with the plus-one set. For every

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member of the plus-one set, there is a member of the minus-one set. Every member of the minus-one can be created by
applying the same transformation to a member of the plus-one set. By applying the transformation to 4+1=5, we create
4–1=3. By applying the transformation to 19+1=20, we create 19–1=18, and so forth.

Still another set that bears a double transformation relationship with the plus-one set is the plus-two set. By
applying the same transformation to different members of the plus-one set, we change 7+1=8 into 7+2=9, and change
4+1=5 into 4+2=6, etc.

Figure 13.1 shows the double-transformation associated with plus-one facts and corresponding plus-two facts. The
left column shows the plus-one set. There are minimal differences within the plus-one set (the plus-one column). The
horizontal arrows show that the same transformation converts any plus-one problem into a corresponding plus-two
problem. The right column shows the corresponding minimum differences that are generated within the plus-two set.

Figure 13.1
Plus-One Set Standard Transformation Plus-Two Set

Add 1 more

Minimum differences
6+1 6+2
Minimum differences

to second number

Add 1 more
5+1 5+2
to second number

Add 1 more
4+1 4+2
to second number

Note that the same transformation (horizontal arrows) accounts for the creation of every example in the plus-two
set.

Criteria for Double Transformations

The test of whether a concept has double-transformation features involves two questions. If the answer to both
questions is “Yes,” the concept is a double-transformation concept.

1. Are different within-set responses created by applying the same operation to different examples? For
example, different examples of words said loudly are created by applying the same operation—“Say it
loud”—to different words of two syllables. “Meeting. Say it loud.” “Funny. Say it loud.” Etc.

2. Can you create a corresponding member of another set by applying the same transformation to any
member of the original set? We can convert any member of the set of words that is said loudly into a word
that has only the last part pronounced loudly. We apply the same transformation to every word in the said-
loudly set. This transformation generates all members of the second set we refer to and is therefore the
standard transformation of the across-set transformation.2

                                                                                                                         
2
We frequently do not have a tidy vocabulary that expresses these across-set transformations. Most of our standard
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 Criterion 1 is the test for within-set differences. If the learner produces different responses while the same
instructions are presented with various examples, the set has the essential features of a single-transformation concept.
Criterion 2 indicates whether a double-transformation is involved. If it is possible to convert every member of the
single-transformation set into a member of another set by performing a standard transformation, the concept is a double
transformation.

There are two kinds of double transformations. One involves a difference in the examples (with the same
instructions used for both sets); the other involves changes in the instructions (with the same examples used for both
sets). The second type is the easiest to recognize. If we change the instructions so that a different instruction creates all
members of the transformed set (“Say it in a whisper,” “Tell the percent the fraction equals,” “Reverse the notation”),
we are dealing with a double transformation. The instructions indicate that a variation of a familiar operation is to be
performed. The type of double-transformation that involves the same instructions, but different examples, is more
difficult to recognize and is more arbitrary. For instance, we can identify many subsets of written words that are
irregular from a decoding standpoint. For instance, words that contain al (tall, malt, also, etc.) form such a subset. The
set could be taught as a double-transformation set. The examples in this set are different from words in which the a
sound is “regular” (at, man, back, etc.); however, the instructions for reading al words is the same as that for reading
any other word: “Read it.” Theoretically, any large subtype of examples (examples that require a variation of the
operation used for other examples) can be treated as a double-transformation set.

In summary, the two types of double-transformation sets are:

• Those in which there are different instructions for the same example that had been processed with a
familiar operation.

• Those in which the instructions are the same, but the examples are a subset requiring a different operation.

Note that if the examples are the same for the familiar set of examples and for the transformed set, the instructions
are different. If the examples are different for the transformed set, the instructions are the same.

Juxtapositions

Our objective for the double-transformation program is the same as the objective for more basic sequences—to go
from the simplest juxtapositions to the most difficult or unpredictable ones.

Below are four examples of word-saying tasks (two from the familiar set and their counterparts from the
transformed set).

                                                                                                                                                                                                                                                                                                                                                                                                                                     
vocabulary focuses on within-set features. We can take any word and “say it louder” or “say it faster.” But we may have
trouble describing precisely how to change some word to make it “louder” or make it “faster.” An explanation is possible.
However, it would be very elaborate and technical.
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 1. Map. Say it in a 3. Map. Say it loud.
whisper.

2. Alligator. Say it in a 4. Alligator. Say it

whisper. loud.

The juxtaposition of 1 and 2 is the easiest juxtaposition because it permits the learner to do the same thing with
juxtaposed examples.

The 1, 3 juxtaposition is an across-set difference. The same example is presented with different instructions. Only
two examples of this type can be juxtaposed at a time; therefore, the learner does not receive the opportunity to work on
a series of examples in which the instructions are different, but the examples are the same. This type of juxtaposition is
more difficult than the 1, 2 juxtaposition because it does not provide a series of examples of the same type.

The juxtaposition of 1, 4 is the most difficult type because it calls for the greatest difference between examples.
The example changes (from map to alligator). The instructions change (from “Say it in a whisper,” to “Say it loud”).
The amount of processing involved is therefore greater than that required by the other juxtapositions.

Based on the relative difficulty of the patterns, the double-transformation program will proceed from 1, 2
juxtapositions (within-set) to 1, 3 (across-set minimum differences) to 1, 4 (mix of the two sets).

The Double-Transformation Program

The double transformation involves two sets of examples: examples that are in the familiar set, and examples that
are in the transformed set. The familiar set has therefore been taught. If the learner knows plus-one facts, selected plus-
one examples comprise the familiar set. The transformed set would consist of the plus-two counterpart for each
member of the familiar set presented in the program. (If 7+1 is a member of the familiar plus-one set, 7+2 would be a
member of the transformed, plus-two set.) The transformed set is the one we teach through the double-transformation
program.

No within-set minimum difference examples are used in the program. The only minimum-difference juxtapositions
are across-set differences, achieved by juxtapositioning a member of the familiar set with a corresponding member of
the transformed set. (7+1 juxtaposed with 7+2, for instance.)

We will describe the double-transformation program as a very elaborate sequence, although we probably would not
present the entire sequence at one time. (The relationships among examples are more apparent when the program is
viewed as a sequence.) After we have discussed these relationships, we will examine ways of distributing parts of the
elaborate sequence to make a variety of possible programs.

The examples in a double-transformation program are grouped into four parts or cycles.

1. Familiar Set

The double-transformation begins with a set of four or five members of the familiar set that are sequenced to show

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sameness. Juxtaposed examples are different from each other.) The familiar-set examples appear first to assure that the
learner is firm on what has already been taught. We refer to the familiar-set members as examples 1, 2, 3, 4, 5.

2. Transformed Set (Within-Set Juxtapositions)

Immediately following example 5 of the familiar set, we present corresponding members of the transformation set,
sequenced in an order different from that of the familiar set. The transformed set is sequenced so that the last member
of the familiar set (5) is juxtaposed to a minimally different member of the transformed set (5’). The transformed-set
members are referred to as 1’, 2’, 3’, 4’, and 5’.

Thus: 1, 2, 3, 4, 5, 5’, 3’, 1’, 4’, 2’

The juxtapositions presented so far in the sequence are: sameness juxtapositions for the familiar set (1, 2, 3, 4, 5);
minimum difference juxtaposition across-set (5, 5’), sameness juxtapositions for transformed set (5’, 3’, 1’, 4’, 2’).

3. Partial Cycle (Across-Set Juxtapositions)

The presentation of the familiar set followed by the transformed set constitutes a full cycle. Following this full
cycle is a partial cycle. Only two familiar and two transformed members are included in the partial cycle.

Full Cycle Partial

Cycle
1, 2, 3, 4, 5, 5’, 3’, 1’, 4’, 2’ 2, 1, 1’, 3’

The partial cycle has two familiar-set examples and two transformed-set examples. The partial cycle involves two
minimum-difference across-set juxtapositions (2’ and 2, 1 and 1’).

The full cycle emphasizes the within-set sameness of the transformed-set members. The learner does the same
thing with five juxtaposed examples of the transformed set (within-set juxtapositions). The partial cycle does not stress
the sameness of the transformed set members, but rather the difference between transformed-set members and familiar-
set members. This change in emphasis is achieved by using more frequent across-set juxtapositions. By the time the
learner has completed the partial cycle, across-set juxtaposition have been provided for three of the five pairs of
examples. (5, 5’, 2, 2’, 1, 1’).

4. Integration of Familiar Set and Transformed Set

Following the partial cycle are unpredictably ordered examples. The examples in this part include:

• All familiar-set members 1-5.

• All transformed-set members 1’-5’.

• Additional transformed-set members 6’-8’.

The juxtaposition of examples is unpredictable with the familiar set and the transformed-set items “mixed up.”

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 Full Cycle Partial Integration
Cycle
1 2 3 4 5 5’ 3’ 1’ 4’ 2’ 2 1 1’ 3’ 5 4’ 2’ 3 5’ 6’ 1 4 7’ 1’ 2 8’ 3’

Note the integration segment presents the most difficult juxtapositions. Juxtaposed examples differ in both
within-set and across-set features. Example 5’ is not juxtaposed to either 4’ or 5. It is juxtaposed to 3. The difference
between 3 and 5’ is a combination of within-set and across-set differences.3 Also the integration segment introduces
new examples of the transformed set (6’, 7’, and 8’).

Illustration 1

Figure 13.2 shows the double transformation sequence that conveys the relationship of plus-one facts and plus-two
facts. The familiar set consists of plus-one problems. (The learner is assumed to know a wide variety of these.) The
transformed set consists of plus-two problems. (These are not known to the learner.) Note that there is no difference in
test wording in the familiar set and the transformed set. The same question (“What’s the answer?”) is presented for all
test examples. The only change from the familiar set to the transformed set, therefore, is a change in example—not in
wording.

                                                                                                                         
3
The integration segment also provides the least specific diagnostic information about the specific causes of errors. If the
learner makes a mistake on an item that is the same as the preceding item except for a single detail, we know that the learner
is not processing that detail. The cause of failure is quite specific. If the learner makes a mistake on an item that differs from
the preceding item in many ways, however, the error provides us with very little specific information about the cause of the
error. We know only that when the task that was failed is presented in the context in which it appeared, the learner cannot
process it.
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 Figure 13.2
Example Teacher Wording

1 3+1 What’s the answer?

2 15 + 1 What’s the answer?
Familiar Set 3 6+1 What’s the answer?
4 1+1 What’s the answer?
5 37 + 1 Across-set What’s the answer?
minimum
5’ 37 + 2 difference My turn. What’s the answer? Thirty-nine.
3’ 6+2 My turn. What’s the answer? Eight.
Transformed Set 1’ 3+2 Your turn. What’s the answer?
4’ 1+2 What’s the answer?
2’ 15 + 2 Across-set What’s the answer?
minimum
2 15 + 1 difference What’s the answer?
Partial Cycle 1 3+1 Across-set What’s the answer?
minimum
1’ 3+2 difference What’s the answer?
3’ 6+2 What’s the answer?
5 37 + 1 What’s the answer?
4’ 15 + 2 What’s the answer?
3 6+1 What’s the answer?
2’ 15 + 2 What’s the answer?
3 6+1 What’s the answer?
Integration 4’ 1+2 What’s the answer?
6’ 72 + 2 What’s the answer?
1 3+1 What’s the answer?
5’ 37 + 2 What’s the answer?
4 1+1 What’s the answer?
7’ 100 + 2 What’s the answer?
1’ 3+2 What’s the answer?
2 15 + 1 What’s the answer?
8’ 24 + 2 What’s the answer?
3’ 6+2 What’s the answer?

Illustration 2

Figure 13.3 shows the double-transformation sequence applied to the relationship between reading words spelled

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with all and their counterparts spelled with ar.

Figure 13.3
Example Teacher Wording

1 call What word?

2 ball What word?
Familiar Set 3 fall What word?
4 mall What word?
5 tall Minimum-difference What word?
5’ tar My turn. What word? Tar.
3’ far My turn again. What word? Far.
Transformed Set 1’ car Your turn. What word?
4’ mar What word?
2’ bar Minimum-difference What word?
2 ball What word?
1 call Minimum-difference What word?
Partial Cycle 1’ car What word?
3’ far What word?
5 tall What word?
2’ bar What word?
3 fall What word?
4’ mar What word?
6’ par What word?
Integration 1 call What word?
5’ tar What word?
4 mall What word?
7’ jar What word?
1’ car What word?
2 ball What word?
8’ star What word?
3’ far What word?

Like the illustration of the relationship between plus- one and plus-two problems, the all-ar sequence uses the
same test wording for members of both sets. The difference between the familiar set (words that end in all) and the
transformed set (words that end in ar) has to do with the type of examples.

Teacher Models

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 In the double-transformations that we have shown, test wording is presented on all examples except the first two
members of the transformed set (5’, 3’). The teacher models these items and then tests on the remaining items. Other
modeling conventions could be used for transformations that are more difficult to demonstrate or that involve responses
that are more difficult. The teacher may model the first two examples and then test on these examples before testing on
the remaining items in the transformed set (modeling examples 5’ and 3’, then testing on 5’, 3’, 1’, 4’, 2’, etc.). The
teacher may model all examples of the transformed set, then return to the beginning of the transformed set and test on
all examples. This approach is justified if each example is elaborate. For instance, if the double transformation presents
the relationship between predicates in regular-order statements and predicates in their inverted counterparts (“He went
to the store after supper; After supper he went to the store”), the teacher might model all five transformed-set items.
Figure 13.4 illustrates the transformed set that would follow the familiar set.

Figure 13.4
Example Teacher Wording

To get to the roof, she climbed the gutter. My turn. What’s the predicate?
Climbed the gutter to get to the roof.

When the game was finished, the crowd cheered. My turn. What’s the predicate?
Cheered when the game was over.

Next to the large box, a leaf nestled. My turn. What’s the predicate?
Nestled near a large box.

Shortly before noon, the train started to move. My turn. What’s the predicate?
Started to move shortly before noon.

Just so he could boast, Henry climbed the oak. My turn. What’s the predicate?
Climbed the oak just so he could boast.

To get to the roof, she climbed the gutter. Your turn. What’s the predicate?
Etc.

Usually, two models of the transformed set are sufficient to prompt the operation that is common to all
transformed-set members.

One-Difference Across Sets

An important feature of appropriately designed double-transformation sequences is that there is only one difference
between any member of the familiar set and the corresponding member of the transformed set. As noted earlier, that
difference can either be a feature of the instructions or of the example presented—but not of both. Let’s say that this is
the familiar-set item:

Example Teacher Wording

mat Read it.

A transformed-set counterpart has the same examples or the same wording. Here’s an illustration of an item that
involves the same example, but different wording:

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 Example Teacher Wording
mat Read it. (Familiar item)
mat Spell it. (Transformed to
spelling instructions)

Here’s an illustration of an item that involves different examples, but the same wording as the familiar item:

Example Teacher Wording

mat Read it. (Familiar item)
mate Read it. (Transformed to a
long-a example)

If there is a difference in both wording and in the example, the learner may perform on the sequence without attending
to relevant details of the examples or the instructions.

The following is the first part of an inappropriate sequence of items that presents differences in both the example
and the wording.

Example Teacher Wording

mat Read it.
fat Read it.
hat Read it.
hate Spell it.
mate Spell it.
fate Spell it.

The presentation generates two serious misrules:

• If the word does not end in e, read it.

• If the word ends in e spell it.

The misrules are serious because the learner can correctly respond to all examples without attending to the
instructions.

To avoid the misrules, we must design the sets so there is only one difference between the familiar and the
transformed sets.

Here are two ways to correct the sequence. Part of the full cycle is shown for each sequence. The first sequence
shows the difference between reading and spelling. The examples are the same across sets, while the instructions vary
across sets.

Example Teacher Wording

mat Read it.
fat Read it.
hat Read it.
hat Spell it.
mat Spell it.
fat Spell it.

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The next sequence shows the difference between short-a words and their long-a counterparts. The instructions are the
same for all examples.

Example Teacher Wording

mat Read it.
fat Read it.
hat Read it.
hate Read it.
mate Read it.
fate Read it.

Across-Set Differences in Wording

The two full, double-transformation sequences that we presented use the same wording for the familiar and
transformed sets. The across-set difference is a change in the examples. Figure 13.5 shows a full sequence of a different
type. The only across-set difference is in the wording. The same set of examples is presented for the familiar and the
transformed set.

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 Figure 13.5
Example Teacher Wording

You are going to make up sentences about the

person in each picture.

1 Picture of boy climbing tree Make up an active voice sentence.

2 Picture of boy washing car Make up an active voice sentence.
Familiar Set 3 Picture of boy brushing his teeth Make up an active voice sentence.
4 Picture of boy kissing a girl Make up an active voice sentence.
5 Picture of boy eating a hamburger Make up an active voice sentence.
5’ Picture of boy eating a hamburger My turn. Make up a passive voice sentence.
The hamburger is being eaten by the boy.
Say that passive-voice sentence.

Transformed 3’ Picture of boy brushing his teeth My turn. Make up a passive voice sentence.
Set The boy’s teeth are being brushed by the boy.
Say that passive-voice sentence.

1’ Picture of boy climbing tree Make up a passive voice sentence.

4’ Picture of boy kissing girl Make up a passive voice sentence.
2’ Picture of boy washing car Make up a passive voice sentence.
2 Picture of boy washing car Make up an active voice sentence.
Partial 1 Picture of boy climbing tree Make up an active voice sentence.
Cycle
1’ Picture of boy climbing tree Make up a passive voice sentence.
3’ Picture of boy brushing his teeth Make up a passive voice sentence.
5 Picture of boy eating hamburger Make up an active voice sentence.
2’ Picture of boy washing car Make up a passive voice sentence.
3 Picture of boy brushing teeth Make up an active voice sentence.
4’ Picture of boy kissing girl Make up a passive voice sentence.
6’ Picture of boy riding a horse Make up a passive voice sentence.
Integration 5’ Picture of boy eating hamburger Make up a passive voice sentence.
4 Picture of boy kissing girl Make up an active voice sentence.
7’ Picture of girl throwing ball Make up a passive voice sentence.
1’ Picture of boy climbing tree Make up a passive voice sentence.
2 Picture of boy washing a car Make up an active voice sentence.
8’ Picture of girl sitting on fence Make up a passive voice sentence.
3’ Picture of boy brushing teeth Make up a passive voice sentence.

The progression of juxtaposition follows the same pattern as that for transformations that involve the same wording
for both sets.

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 During the full cycle of the transformed set, the learner does not have to attend to the instructions about “active” or
“passive” because the learner can follow the model provided by the teacher at the beginning of the transformed set.

The difficulty in the partial cycle increases because the learner must now attend to the words active and passive.

In the integration segment, the difficulty is further increased because the learner must now: (1) listen to the
instructions, and (2) attend to the relevant details of each picture. Also, the learner must apply knowledge of passive-
voice constructions to new sentences (6’, 7’, and 8’). Two of these involve a new actor, which means that the items are
more difficult than those involving “the boy.” The addition of these new examples should not be too difficult for the
learner, if the learner has successfully performed in the sequence to the point at which they are introduced. If the
learner understands a wide range of examples from the familiar set and if the learner has learned the standard
transformation for the transformed members presented in the full cycle, the learner should be able to apply the standard
transformation to other members of the familiar set.

Other Applications

1. If we wanted the learner to respond with percent notations for the familiar-set examples and with decimal
notations for the transformed set, we could use an across-set difference in wording. We could present the
learner with a set of fractions that have 100 as the denominator. The same examples would be presented
for both the familiar set and the transformed set. The instructions would differ across sets, however. The
learner would be instructed to “Tell me the percent” for the familiar-set items, and to “Tell me the
decimal” for the transformed-set items.

2. If the learner knows consonants, we could show their relationship to vowels by presenting the learner with
short, written words. For the familiar-set items, the learner would be told to “Name all the consonants.”
For the transformed set the instruction would be, “Name all the vowels.”

3. By creating across-set difference in instructions we can firm the learner’s understanding of item types that
are often confused. (See Figure 13.6.)

Figure 13.6
Example Teacher Wording

53 Read the digits in the

75 ones column.
81 (Familiar)
53 Read the digits in the
75 tens column.
81 (Transformed)

Example Teacher Wording

The girl said, “Let’s go.” Say the sentence.

(Familiar)
The girl said, “Let’s go.” Say what the girl said.
(Transformed)

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The two pairings in Figure 13.6 present instructions that the learner may confuse. By first firming one of the sets (so
that it becomes familiar) and then processing the discrimination through the double-transformation program, the learner
will be shown the difference between the two instructions. As a rule, if the learner is confused on instructions, we
create a sequence in which the only difference between sets is a change in instructions.

Designing the Double-Transformation Program

We usually present the double-transformation relationship in three different parts or sections. The first establishes
the familiar set. The second introduces the transformed set. The third integrates familiar and transformed sets.

The Familiar Set

The double-transformation relationship assumes that one single-transformation discrimination has been taught.
Once the initial single-transformation discrimination has been identified, it is taught through initial-teaching sequences.
These are followed by expansion activities.

The transformed set does not have to be introduced soon after the teaching of the familiar-set concept. The learner
will not learn serious misrules if the transformed set is delayed.

Rather, a firm knowledge of the familiar-set discrimination facilitates mastery of the transformed set, because it
reduces the possibility of the learner confusing transformed-set items or instructions with the familiar-set counterparts.

The initial teaching of the familiar set is achieved through a single-transformation sequence. The expansion and
review activities continue until the transformed set is introduced. (See Figure 13.7.)

Figure 13.7
Components Familiar Set Lessons Transformed
Set Begins
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

Initial Teaching X X

Expansion X X X X X X X X

Review X X X

The number of days specified on the schedule is arbitrary. In calendar time, the familiar set illustrated above would
be introduced about one month before the transformed set begins. The procedures for the initial teaching are those used
for single-transformation concepts (Chapter 10). The expansion activities require the learner to use the discrimination
in new settings and contexts. The schedule show more expansions than reviews, because the expansions permit the
learner to use the discrimination in relevant applications.

As a rule of thumb, the learner should work with the familiar-set items for a minimum of six to ten days before the
transformed set is introduced. This period is usually enough to assure that the familiar discriminations are familiar. For
many discriminations, however, the transformed set may not be introduced for months after the familiar set has been

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introduced. This situation presents no problem so long as the learner is not engaged in any activities that require the
transformed-set discrimination.

The Transformed Set

There are two approaches to the introduction of the transformed set. The first approach is to introduce it in the way
suggested by the double-transformation sequence. The other, called the separate-set approach, is to teach the
transformed set apart from the familiar set, then integrate it with items from the familiar set. The schedule in Figure
13.8 shows the strategy for introducing the transformed set that is based on the events of the double-transformation
sequence.

Figure 13.8
Components Transformed-Set Lessons
1 2 3 4 5 6 7 8

Familiar Set X X

Transformed Set X X X X

Partial Cycle X X X

Integration X X X X X

Expansion X X X X

On the first lesson, the full cycle is presented (the familiar and transformed set presented in sequence). On lesson 2,
the same sequence is presented, and the partial cycle is added to the sequence. On lessons 3 and 4, the familiar set is
dropped; only the transformed set and partial cycles are presented. The assumption is that the learner is familiar with
the familiar-set discrimination. Also, the partial cycle and the integration require the learner to use familiar-set
discriminations. The integration continues after lesson 4. It functions as a cumulative review, testing the learner on the
familiar and the transformed discriminations.

The expansion activities that occur on the first three lessons involve only the transformed-set discrimination. (They
require only the within-set understanding of the concept, not an ability to discriminate between transformed-set items
and familiar-set items.) The idea is to give the learner some familiarity with the transformed-set examples before the
learner is required to discriminate between these and their familiar-set counterparts. The expansion on lesson 7 involves
both sets.

The rate of introduction for the different parts of the program can be modified substantially, depending on the
difficulty of the discrimination being taught. Possibly, lesson 1 could be eliminated from the introduction. If the learner
is quite facile, the entire sequence (full cycle, partial cycle, and integration) can be presented in the first lesson,
repeated on the second lesson, and reviewed on a few subsequent lessons. If the learner is not as facile, a more careful
introduction is needed. Think of the difficulty of this introduction in terms of the types of juxtapositions that are
presented. Lesson 1 presents the easiest juxtaposition (within-set juxtapositions for the transformed set). Lesson 2 adds

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a more difficult type of juxtaposition (across-set juxtapositions). Lessons 3 and 4 add a still more difficult type—
within-set and across-set discriminations. The remainder of the program presents additional reviews of the difficult
juxtaposition. Without changing the general direction of the program, we can introduce variation in the rate at which
the different juxtapositions are introduced.

Separate Sets

The strategy above is a phased introduction of the double-transformation sequence. This strategy is appropriate for
most discriminations. However, if the responses involved in the program are difficult for the learner, the separate-sets
strategy should be employed. This strategy assures greater practice with the transformed-set responses before they are
presented in the same context as the familiar-set responses.

The separated-sets strategy involves first working on the within-set discrimination for the transformed set, then
teaching the relationship.

This strategy potentially increases the danger of stipulation; however, the strategy frequently is desirable,
particularly when the discriminations involve difficult responses. The schedule in Figure 13.9 shows the separate-sets
strategy. Note that this schedule begins after the familiar set has been taught.

Figure 13.9
Components Separate-Sets Lessons
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

Familiar Set A X X

Transformed Set B X X X X X X X

Partial Cycle X X X

Integration X X X X X X

Expansion & Review A A B B A B AB AB A B A B B B A B

The transformed set is taught for three lessons. The teaching of the transformed set follows juxtaposition rules for
introducing single-transformation concepts. The number of items introduced is not limited to five. The same type of
minimum-difference juxtapositions used for teaching the familiar set is used for teaching the transformed set. The
transformed set is expanded starting with the third lesson. Note that expansion activities for the familiar set (A) occur
on lessons 1 and 2. These activities would not be juxtaposed to the initial teaching of the transformed set in the lesson.
They would occur at some other time. Review and expansion continue on lessons 4, 5 and 6.

Starting with lesson 7, both the familiar set and the transformed set are reviewed. This review consists of examples
of each set that are fairly different from each other (the same type of examples that will be used in the sequence).
Presentation of these sets would not be juxtaposed in the lesson. The idea is to make the learner facile with each
discrimination, but not to require the learner to discriminate between the sets. Starting with lesson 9, the parts of the full
cycle are juxtaposed as they are in the double-transformation sequence. First, the familiar-set items are presented,

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followed by the transformed set. There are five items in each set, and the sets are designed according to the
specifications of the double-transformation sequence. The teacher models the first two examples of the transformed set
(as in presenting the double-transformation for the first time).

On lesson 10, the parts are again juxtaposed, with the partial cycle following the transformed set. The remainder of
the schedule is similar to that for the program based on the parts of the double-transformation sequences; however,
there is a more careful scheduling of expansion activities. The reason for this elaboration is that the responses are
relatively difficult for the learner (either because the learner is relatively naive, or the responses are difficult even for a
sophisticated learner). Think of the separate-sets program as two sets that remain separate in time until both are firm.
They are separated in time, but both receive attention. Lessons 1-8 of the program teach the within-set discriminations
for the transformed set and assure that the learner also retains the within-set discriminations for the familiar set.
Starting with lesson 9, the teaching begins for the across-set discrimination. The first step in this teaching involves
bringing the sets closer together in time. The next steps are identical to those in the sequence-derived program.

Illustration

Arithmetic skills often require the separate-set approach. Assume that the learner has been taught multiplication
(familiar set) and is to be taught algebra multiplication (transformed set).

The teaching for the transformed set begins when the learner is firm on the familiar-set examples. For three days,
the learner works on transformed set problems while engaging in review activities that involve the familiar set. Figure
13.10 shows a possible sequence for lesson 1.

Figure 13.10
Example Teacher Wording

5 = 20 My turn. Five times how many equals twenty? Five times four equals twenty.
5 = 15 My turn. Five times how many equals fifteen? Five times three equals fifteen.
5 = 25 Your turn: Five times how many equals twenty-five?
5 = 5 Five times how many equals five?
5 = 30 Five times how many equals thirty?
5 = 45 Five times how many equals forty-five?
5 = 15 Five times how many equals fifteen?
5 = 50 Five times how many equals fifty?

The teacher may present more information about how to figure out the answer to each problem. (Strategies for such
routines are specified in Section VI.)

Starting with lesson 7, the teacher would present review examples of algebra multiplication at one time during the
lesson and review examples involving regular multiplication at another. The juxtaposition of problems of the same type
serves as a prompt to the learner to do the same thing on juxtaposed examples. The following is a possible group of

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transformed-set problems that might be presented on lesson 7.

5 = 15
5 = 30
5 = 0
5 = 20
5 = 45
5 = 10
5 = 50
5 = 25
5 = 5

Note that the juxtaposed examples in the sequence are quite different from each other, forcing the learner to deal
with the sameness of the transformed-set members. This juxtaposition is the same type that will be used on lesson 9
when the familiar and transformed sets are linked together to form the first part of the sequence.

Figure 13.11 shows the examples that might be presented on lesson 9. The familiar and transformed sets are
juxtaposed to form a full cycle. Note that the set of examples is not the same as the set presented on preceding lessons.

Figure 13.11
Example Teacher Wording

1 5 2= What’s the answer?

2 5 10 = What’s the answer?
Familiar Set 3 5 7= What’s the answer?
4 5 4= What’s the answer?
5 5 3= What’s the answer?
5’ 5 = 15 My turn. What’s the answer? Three.
2’ 5 = 50 My turn. What’s the answer? Ten.
Transformed Set 1’ 5 = 10 Your turn. What’s the answer?
3’ 5 = 35 What’s the answer?
4’ 5 = 20 What’s the answer?

On subsequent lessons, the teaching at the beginning of the transformed set is dropped. Note that the sequence
above presents very abbreviated wording. To reduce the possibility that the learner’s errors are caused by misreading
the problem, the teacher could present two tasks with each problem that is tested: “Read it . . . .What’s the answer?”
The introduction of the first question would make the learner’s perception of the problem overt. At the same time, the
presentation of the first question does not create a misrule because the same wording is used for each example.

Lessons 13 through 19 assure that the learner practices the integration part of the sequence with different examples.
This part presents the most difficult discriminations. There are no juxtaposition prompts to tell the learner whether the
next example will be from the familiar or from the transformed set. Figure 13.12 shows a possible integration sequence.

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 Figure 13.12
Example Teacher Wording

5 = 35 Read it. What’s the answer?

5 1= Read it. What’s the answer?

5 6= Read it. What’s the answer?
5 = 0 Read it. What’s the answer?
5 = 30 Read it. What’s the answer?
5 2= Read it. What’s the answer?
5 = 15 Read it. What’s the answer?
5 9= Read it. What’s the answer?
5 = 20 Read it. What’s the answer?

Summary

To identify double transformation items, we start with a series of single-transformation tasks, such as those
involving the directions, “Add those numbers.” We ask, “Can you create a parallel set by changing either the examples
or the instructions?” If we can, we are dealing with a double-transformation relationship.

The change may be in the examples:

Examples Wording

Familiar set: 4+2 Add these numbers.

Transformed set: 400+200 Add these numbers.

The change may be in instructions.

Examples Wording

Familiar set: 4 2 Add.

Transformed set: 4 2 Subtract.

The double-transformation program consists of four parts:

1. The familiar set: five familiar examples that come from different subsets and are ordered to communicate
sameness.

2. The transformed set: five counterparts for each of the familiar-set examples.

3. The partial cycle: juxtapositions that show across-set differences.

4. The integration: a near-random juxtaposition of items from either set, and additional items for the
transformed set.

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 The program teaches both the transformed set and the relationship between the familiar and transformed sets.

As a sequence, the double-transformation is long; therefore, the probability is great that the learner will become
confused, will forget, and will make errors if the entire sequence is presented at one time. To avoid these possibilities,
we expand double-transformation sequences into double-transformation programs.

There are two basic types of double-transformation programs. The sequence-derived program introduces the parts
of the program in roughly the same order they occur in the double-transformation sequence. This order of introduction
quickly teaches both the within-set discrimination for the transformed set and the across-set difference between sets.

The other approach is the separate-sets program, which first teaches the within-set discrimination for the
transformed set, while reviewing the within-set discrimination for the familiar set. After both sets have been mastered
in isolation (or in blocks of items), the across-set discrimination is taught through a series of events modeled after the
double-transformation sequence.

The assumption behind both double-transformation programs is that the work of the transformed set does not begin
until the familiar set has been thoroughly mastered. However, the learner may have trouble with the “familiar” items
when they become juxtaposed with the transformed-set items. The familiar items were firm in one context; however,
the context presented by the double-transformation sequence implies attention to a much wider range of detail than the
earlier contexts required.

The same expansion-review procedures used for other programs are employed in the double-transformation
programs. Many details of a schedule for introducing a double-transformation concept are arbitrary. The non-arbitrary
aspects of the program are that the learner must work on newly-taught skills for at least two consecutive lessons, must
review skills periodically, and must use newly-taught skills in a context other than those provided by the review
activities.

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 Programs for Teaching Fact Systems
In the teaching sequences that we have dealt with, the same response dimension is involved in juxtaposed
examples. If the sequence deals with words that rhyme with at, all items deal with the same response dimension,
rhyming with at. If the sequence deals with getting hotter, all examples deal with whether the objects gets hotter.
Some double-transformation programs use two different response dimensions (indicated by the two different directions
presented in the sequence); however, the double-transformation program presents two sets of discriminations that are
related through a single transformation.

The programs that we will examine in this chapter are completely different. Each fact-system program calls for a
variety of different responses. The various response dimensions of the fact system are the features of the system, the
details that distinguish the system from other systems. The juxtaposed items for this type of program virtually never
deal with the same response dimension. Furthermore, the goal of the instruction is not to teach a particular response
dimension and how it works (which is the goal of the preceding programs). Rather, the goal is to teach a complex
system by teaching the various features of the system. The communication that teaches the system is a complex
communication because it must convey information about various features of the system, each of which involves a
different fact or response dimension.

The common characteristic of these programs is that they present a visual-spatial display of some type. The display
shows the whole or the system that is to be learned. Parts of the system are also shown or labeled. These details provide
information about particular aspects of the whole and about relationships among the parts.

Fact systems that can be presented as visual-spatial displays include virtually any topic that can be outlined or
diagrammed, including:

• John Stuart Mill’s views on personal liberty.

• The classification of cherries.

• The workings of an internal combustion engine.

• The human circulatory system.

Types of Fact Systems

There are three different types of fact systems.

1. The Natural-Part, Natural-Whole System

This type of system is a representation of something that can be found in the environment. The fact system shows
the parts and relationships of the object. For instance, a cut-away view of a volcano is something that would be
observed if we could make a vertical slice to create a cross-section of a volcano. Similarly, a diagram of the human’s
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breathing apparatus, the workings of a carburetor, the parts of a typewriter—all are systems that may be found in the
environment. The presentation may make some parts more prominent than they would be in reality; however, the
presentation deals with something that exists in basically the same form that it appears in the fact system. The parts are
“natural” parts (the core of the volcano, for instance) and the whole is a “natural whole” (the entire volcano).

2. The Natural-Part, Created-Whole System

For this type of system, the parts exist in nature or in the environment; however, the parts do not occur in the type
of “whole” or organization presented by the fact system. For example, the game birds of North America is a system that
is made up of parts. The parts—pictures of various game birds—are representations of things that exist in nature. The
whole, or system that is created when these birds are organized and displayed, however, does not exist in nature. It
would be highly unusual if one observed a quail next to a ringneck pheasant, next to the various ducks and geese, etc. A
chart showing various landforms is also made up of parts that exist in the environment; however, the organization does
not exist in nature. Another example of this type would be a classification of currency. The particular arrangement
shows the relationship of the various bills and coins. Although this arrangement is a fabrication designed to
communicate the system, the parts—the coins and bills—exist in the environment.

3. The Created-Part, Created-Whole System

Both the parts and the whole of this system are representations that do not actually occur in the environment. If we
showed the organization of a business, with the president, the various vice-presidents, etc., we would represent the
positions by words such as “director of marketing.” The diagram would not show a picture of a director. Neither would
it show a system of relationships that exists physically. The relationships would be visual representations of a
hierarchy. Other fact systems of the created-part, created-whole form include all the systems that could be put into an
“outline.” Descartes’ argument about the existence of God, the history of events in Napoleon Bonaparte’s career, etc.
Systems of this type also include things that are diagrammed or symbolized, such as the horsepower and torque curves
for an engine at different speeds.

The distinction of the three types becomes important when we consider what kind of prompts the display should
give them. However, the general rules for presentation and design hold for all three types of systems.

Fact System Teaching

To teach a learner the fact system, we must convey the individual facts and relationships. Note that within the
system, the facts (and their arrangement) are the features that distinguish the system from other possible fact systems.
The primary objective of the instruction is not to teach the meaning of individual facts, but to teach the particular fact
system as the sum of the component facts.

For showing the relationship between the facts we will use codes. For example, if something is coordinate with
another thing, we will show the two things occupying the coordinate space. If something happens before something
else, we will show a linear relationship, with the events arranged spatially. If something includes something else, we

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will show the larger thing as if it physically includes the smaller thing.

The conventions that we use for designing a fact system suggest whether we would use a specific program with a
given learner. For the learner to qualify for entry, the learner must possess generalized understanding of the coding that
we will use to create the event. Our system may consist of words and spatial relationships to show exclusion,
coordination, seriation, or temporal relationships. It may contain arrows, diagrams, and other symbols. If the learner
does not understand each of these code functions, the system should not be presented through a visual-spatial display. If
the learner cannot read, the learner should not be exposed to a system that contains words. If the learner does not know
that things occurring before other things are positioned to the left of other things, the system that assumes this
knowledge should not be presented.

The learner, in other words, should know enough that misrules are not implied by the presentation of the fact
system. (We do not want the learner to think that the boxes on a diagram exist in nature.)

The Presentation Format

Although there are many ways to convey the information presented on a visual-spatial display, we will deal with a
three-step format:

1. Prompted chart and script. The teacher presents a visual-spatial display and rehearses the learner on the
words that go in the various cells of the display. The teacher reads a script that refers to the parts of the
chart and also possibly refers to information not directly displayed.

2. The unprompted chart and test. The teacher presents an unprompted version of the chart. This chart does
not have the words written in the various cells. The learner must identify what words go in the cells.

3. The game. A group of learners play a game in which they roll dice and answer the question that
corresponds to the number on the dice. Most of the questions ask about the wording in various cells of the
visual-spatial display.

The three phases of the presentation correspond roughly with the progression of events in the initial-teaching
sequences we have examined. The first step is the teaching, the communication of information that will be called on for
the test step.

The second step tests what was demonstrated in step 1, using a direct, unembellished test.

The third step functions as an expansion activity, permitting the learner to perform in a different context, to tasks
that are somewhat different from those presented in the test. At the same time, the game (third step) provides for
sufficient repetition to firm the information to firm the information (in a reinforcing manner).

Illustration

Figure 14.1 gives a complete presentation for a visual-spatial fact system dealing with basic information about
teeth. The display is a natural-part, natural-whole type, with the different types of teeth labeled as parts and a diagram

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of the teeth showing the whole.

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 Figure 14.1 6. Everybody, touch E. Wait. Incisors cut food. Say it.
Signal. “Incisors cut food.”
Chart 3 Remember, incisors cut food.
A. Teeth 7. Everybody, touch F. Wait. Four canines are pointed like
dogs’ teeth. Say it. Signal. “Four canines are pointed like
dogs’ teeth.”
Your canines are next to the incisors.
B. You have 28 teeth C. Adults have 32 teeth You have four canines–two on the top and two on the
bottom.
Everybody, touch a canine on the bottom of your
mouth. Check.
Everybody, touch a canine on the top of your mouth.
Check.
Remember, four canines are pointed like dogs’ teeth.

8. Everybody, touch G. Wait. Canines tear food. Say it.

Signal. “Canines tear food.”
D. H. Remember, canines tear food.
Incisors Molars are
are flat and F. 9. Everybody, touch H. Wait. Molars are thick. Say it.
thick Signal. “Molars are thick.”
sharp Four canines
are pointed Molars are in the back of your mouth.
like dogs’ Everybody, touch a molar on the bottom of your mouth.
E. teeth Check.
Incisors I. Everybody, touch a molar on the top of your mouth.
cut food Molars Check.
grind food You’ll get four more molars when you get older.
Remember, molars are thick.
G.
Canines 10. Everybody, touch I. Wait. Molars grind food. Say it.
tear food Signal. “Molars grind food.”
Remember, molars grind food.
Chart 3 3. Teeth–Introduce
11. Let’s go over the facts about teeth one more time.
1. Everybody, open your student book to page 3. Check.
Remember the exact wording for each space. 12. Everybody, touch A. Wait. Tell me the words. Signal.
(Students respond.)
2. Everybody, touch A. Wait. Teeth. Say it. Signal. “Teeth.”
I’m going to tell you about teeth. You have different 13. Repeat step 12 with B, C, D, E, F, G, H, I.
kinds of teeth to do different jobs.
Remember, teeth. 3. Teeth–Firm-up
3. Everybody, touch B. Wait. You have 28 teeth. Say it. 1. Now you’re ready to play the starter’s game for Teeth.
Signal. “You have 28 teeth.” I’ll tell you who the monitors are for today.
You don’t have all your teeth yet. You’ll get more as you Identify a monitor for each group.
get older. Monitors, open your student book to answer sheet 3 for
Remember, you have 28 teeth. Teeth on page 60. Check.
Monitors, give out a scorecard to each player, including
4. Everybody, touch C. Wait. Adults have 32 teeth. Say it. yourself. Wait.
Signal. “Adults have 32 teeth.”
Adults have four more teeth than you have. These teeth 2. Write on board: 3. Teeth.
are in the back of their mouth. Everybody, write your name and the name of this game
Remember, adults have 32 teeth. at the top of your scorecard. Check.
5. Everybody, touch D. Wait. Incisors are flat and sharp. 3. Remember the rules: You read each question out loud.
Say it. Signal. “Incisors are flat and sharp.” Then tell the exact words. Remember, you must say the
Incisors are in the front of your mouth. exact words for each space to get a point. Also, if you
Everybody, touch an incisor on the top of your mouth. argue with the monitor, you lose one turn. If you argue
Check. again with the monitor, you’re out of the game. If you’re
Now touch an incisor on the bottom of your mouth. a monitor, you have to be careful not to show anybody
Check. the answers. Keep the game moving quickly. But don’t
You can feel the edge of your incisors. let the players roll the dice before the last player has
Remember, incisors are flat and sharp. answered the question.

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 4. Tell the monitors to start the game. Note the time.
The game is to continue for ten minutes.
Observe games. Give feedback to groups that are playing well.
5. After players have played for ten minutes, stop the games.
Monitors, give yourselves twenty points for monitoring.
For each game that ran smoothly, tell the monitors to award
five bonus points to themselves and the players.
6. Instruct students to return their scorecard to their
notebook or folder.

The game. When students play the game, they use the score card illustrated in Figure 14.2. The scorecard displays
a picture of the unprompted display. The student rolls the dice, orally reads the question that corresponds to the number
shown on the dice, and answers the question. If the answer is correct, the monitor makes a check mark in one box.

Figure 14.2

2. What words go in space B?

3. What words go in space E?
4. What words go in space D?
5. What words go in space H?
6. Which teeth can cut food?
7. What words go in space F?
8. Which teeth grind food?
9. What words go in space I?
10. What words go in space C?
11. What words go in space G?
12. What words go in space A?

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 A.

B. C.

D. H.
F.

E.
I.

G.

Name
Scorecard
Superstar Scorebox
Starter’s Scorebox

Expert’s Game

Better’s Scorebox

Here’s How Much I Bet

Here’s How Much I Won

Game Points
Starter’s Challenge Box
Superstar
30 15 c.o.
Better’s
Challenge

Total Initialed

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 Note that the game is played on more than one day. On subsequent lessons, the teacher provides either a brief
review of the chart or no review before the game begins.

The teaching. As the script given as part of Figure 14.1 indicates, the teacher teaches new names and new facts.
The initial teaching for incisors, canines, and molars is presented through the visual-spatial presentation. The script
also introduces the facts about each type of tooth (how they look, where they are, and what they do) and the facts about
the number of teeth children and adults have.

Although a great deal of teaching is presented through a single communication, the teaching is not extremely
difficult because:

1. The visual-spatial display provides prompts about the position of each fact.

2. The new names are incorporated in sentences that provide information about the names.

3. No serious discrimination problems are involved for the learner who is sophisticated enough to qualify for
a visual-spatial presentation.

If the learner learns the fact statements about molars (“molars are thick”), the learner is provided with a mnemonic
for remembering molars. The primary communication problem for teaching molars is not to teach subtle differences
between examples of molars and other teeth. These are the teeth that are quite familiar to the learner as teeth in the back
of the mouth. The learner must learn the name for these teeth and have some method of associating the name with
information about function, shape, and position. The wording in the cell provides information about the relationship of
name and function. The visual display shows the relationship of position and name. Therefore, the display nicely
provides the essential information that the more sophisticated learner would need to understand molars as part of the
system of teeth.

The contrast of the visual-spatial display approach to teaching fact systems with traditional procedures is
instructive. Traditional procedures that are supposed to teach fact systems usually require the learner to extract the
information from material that is read. The learner’s first exposure to molars may be in a health book. The assumption
of the book is that the explanations, which are frequently elaborate, will induce knowledge of the discrimination and
knowledge of the word. This assumption is unfounded. For the student to become firm on the facts presented on the
teeth display, the students would have to practice these names in an unprompted setting. No such practice is provided.
An even more serious problem is that a chapter of the student text typically deals with far more information than the
display of teeth presents. The overload of information is typically so great that the student learns very little from
reading the text. After reading the text, the teacher may explain what the student has read. This juxtaposition of events
suggests that the text is not a purveyor of information. A far more reasonable order of events would introduce the basic
facts first (through a visual-spatial display). Following the firming on the basic facts, the student reads the text. Now
the text is perfectly comprehensible, even if it introduces some information that is not related to the facts presented on
the visual-spatial display. The reason is that the student has well-grounded facts to which the new information relates.
Without the well-grounded facts, however, the new information is unrelated, difficult to classify or associate, and
therefore difficult to remember.
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 Designing Displays

It would certainly be possible to include more information in a display than the example in Figure 14.1 provides.
As a rule, the learner should easily be able to handle a display with 11 cells, but not more than 16. (Figure 14.1 has only
9.)

Note that in the teeth display, coordinate wording is contained in cells that are spatially coordinate. Cells B and C
are spatially coordinate. The wording in these cells is coordinate. “You have 28,” “Adults have 32.” Both cell B and C
are subordinated to cell A.

The other cells occur in pairs. The top cell in the pair tells: (1) the name of the tooth, and (2) how it looks. The
bottom cell tells what the tooth does (cuts food, tears food, grinds food).

One important consideration in designing a display is the configuration and content of the other displays the learner
is being taught. Each display is a member of a set. If the learner is presented with a series of visual-spatial displays,
highly similar members should not be juxtaposed, which means that displays that are highly similar in either content or
form should be separated in their order of introduction.

We have many options in designing a display so that it is highly dissimilar from the preceding display. We can
make a given organization linear, round, or angular. We can make the cells boxes, circles, or some other shape.
Therefore, if the preceding display is angular, we can make one that is vertical or round. Figure 14.3 presents a few
options that are available for showing the higher-order relationship of five vehicles. Note that on all displays, the
higher-order label (vehicles) is treated differently from the other labels; however, each of the lower-order cells is
related in the same way to the higher-order label. All five displays treat the information as a created-part, created-whole
system (although it would be possible to treat it as a natural-part, created-whole system by showing pictures of the
various vehicles).

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 Figure 14.3

car
boat car truck train bus

boat
Vehicles
Vehicles bus

train
train
bus
truck
truck

Vehicles
bus train boat
car
Vehicles

car Vehicles
truck
boat
boat car truck train bus

Designing Individual Cells

The basic rule for designing cells for coordinate cells is that there should not be more than four coordinate cells
that are the same, unless the cells are always approached in the same order. According to this criterion, the arrangement
of coordinate cells for both the vertical display and one horizontal display above are inappropriate. The reason is that if
all cells are empty, they are hard to discriminate. We will easily remember the end cells and possibly some of the
others, but the coordinate cells in the middle share too many features.

There are three solutions to this problem:

1. Remove at least one of the cells.

2. Change the shape of at least one coordinate cell (preferably the middle cell).

3. Always require the learner to respond to the coordinate cells in the same order.

One horizontal display presents a unique shape for the middle cell (truck). Some of the other displays adequately
deal with this problem in different ways. For instance, the round display presents different shapes for each cell.

Another way to make the cells more distinguishable is to change the display so that it is a natural-part, created-
whole type. When the actual pictures (or representations of the different vehicles) are presented in each cell, the

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coordinate cells are no longer similar.

For other situations in which there are more than four coordinate cells, one of the other solutions may be
reasonable—removing a cell or always presenting the cells in the same order.

If we were teaching the five types of vertebrates, we might arrange the five highly similar cells in order:

mammals

birds

reptiles

amphibians

fish

And we might always refer to the cells in order, starting with fish. The rationale for this approach is that the
arrangement of vertebrates corresponds to their appearance geologically. By presenting the cells in the same order, we
prompt the learner to understand the evolutionary changes in vertebrates. Later, we can add geological information:

Later
Cenozoic { mammals

Mesozoic
{ birds

reptiles

{
amphibians
Paleozoic
fish
Earlier

We now have a timeline that corresponds to rock strata, evolutionary development, and to geological eras. The
potential economy of this approach justifies the use of five cells that are highly similar, although it could be possible to
modify the shape of the middle cell, perhaps by making a heavy outline around it.

Spurious Prompts

A prompt is spurious if it permits the learner to produce the appropriate response for the wrong reason. If the
learner is able to tell that the next example in a teaching sequence is positive because the examples alternate from
positive to negative to positive, the alternation serves as a spurious prompt for the correct response. Here are rules for
avoiding spurious prompts in a visual-spatial display:

1. The shape of the cell should never be designed to give the learner “decoding” information about how to
pronounce the name. If the response for a cell is “mammal,” and the cell is shaped like an M, the shape of

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 the cell provides a spurious prompt.

2. If the learner is expected to learn a group of cells in a particular order, the cells should be as similar as it is
practical to make them. The cells for the five classes of vertebrates are identical. Figure 14.4 shows
spuriously-prompted cells for vertebrates. Although the cells are not in the appropriate order, the learner is
able to respond appropriately to every name (mammal, fish, reptile, amphibian, and bird). The learner is
not required to learn a particular order because the features of the individual cells provide information
about the responses for each cell. Therefore, the learner may attend to the individual features of the cells,
regardless of their order.

3. Closely related to rule 2 above, if the learner is expected to learn a particular spatial arrangement of parts,
the display should be designed so that the learner must attend to the spatial details of the display. If
individual cells are designed to “tell” the learner what to say, the cell does not reinforce attention to the
spatial arrangement.

4. If familiar names are called for by cells, the shape of the cell should not be designed to prompt the name.
The spurious display in Figure 14.4 shows an outline of a fish and a bird. The responses “fish” and “bird”
are familiar responses of the learner. The pictures, therefore, prompt these names spuriously. The outline
of the frog is not particularly spurious, however, because the name “amphibian” is called for, a name that
may not be familiar to the learner.

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 Figure 14.4
Examples of Spurious Prompts

As a general guideline, a created-part, created-whole display should not prompt responses for the individual cells.
For this type of display, the learner is expected to learn both what goes in the cells and the organization of cells. The
objective is most easily achieved if the learner must respond to where the cell is, not to its shape or to features that
“tell” the learner what to say. The test for cells of a created-part, created-whole display is: If the learner could respond
to a cell no matter where we might move it, the cells are probably providing some spurious cues.

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 For the natural-part, natural-whole display, the organization of parts is not a spurious problem, because the
organization is already given. We simply require the learner to label the parts and label the whole (such as the parts of
the respiratory system).

The natural-part, created-whole display presents the most serious problems because it is an organization of natural
parts. If we want the learner to respond to the organization, however, we should not prompt the individual cells by
presenting pictures in them. If we want the learner to identify the various parts, however (and if this learning is new for
the learner), we should present the pictures that are to be identified. The solution to the problem of the natural-part,
created-whole display is to present pictures and to add questions that require the learner to respond to the organization
of the parts. We can illustrate both the problem and the solution with a display designed to teach birds of prey. First
we must decide whether we should present the information through a natural-part or a created-part display.

If the learner can already identify all the birds, we could teach the organization of the various names through a
created-part, created-whole presentation. If the learner knows only some individual birds but not others and does not
know the organization, we could present a natural-part, created-whole display, such as the one in Figure 14.5.

Figure 14.5

Chart 25
D. Very
A. Birds of Prey
good eyes
B. Strong
beaks

C. Strong
talons

E. Eagles H. Hawks M. Falcons O. Owls Q. Vultures

I. Buteo hawks K. Accipiter
F.
hunt in the hawks hunt in
open the forests

J. L.
N. Falcons
Golden eagle are the P. Owls
fastest birds hunt at
G. of prey night

Cooper hawk
Red-tailed
hawk
Bald eagle

After the information has been introduced, the learner is tested on a variation of the chart that is identical to the one
in Figure 14.5 except that all wording is removed. The pictures remain. This test display is relatively weak at teaching

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the coordinate names because the names are always prompted by pictures. (The learner can look at the pictures of the
eagles, for example, and identify E as “eagles.”)

The possible shortcomings of this display does not imply that the display is unreasonable. The display teaches the
learner to identify the various birds. It also teaches the information in cells B, C, D, I, K, N and P. It requires the
earning of distinguishing features and new names (particularly in the hawk family). It is simply relatively weak in not
requiring the learner to “list” the various types of birds of prey without operating from picture prompts (or to name the
two types of eagles without looking at the pictures). To compensate for these possible weaknesses, we could simply
add activities to the game that follows the presentation of the display. One item might instruct the learner to: “Name the
five types of birds of prey without looking at the picture.” Another might be: “Name the two hawk families without
looking at the pictures.” The learner would not be required to name the entries in a particular order, but merely list
them.

In summary, visual-spatial displays that retain pictures tend to provide prompts that may be spurious, depending on
what kind of organizational information we expect the learner to remember. The displays that are most susceptible to
this problem are the natural-part, created-whole displays because we are creating an organization. The extent to which
we expect the learner to remember this organization determines the extent to which we supplement the display with
questions that require the learner to respond to organizational features without looking at the pictures on the chart.

Choice of Displays

Different fact systems are appropriate for different objectives. If the system to be tested is a natural-whole system,
such as a carburetor, teeth, a mountain, or a window and its frame, the natural-part, natural-whole display is clearly
implied. The reason is that the learner will always encounter the parts within the context of the whole; therefore, the
objective of instruction should be simply to teach the learner to label the parts and the whole.

For many systems, the same information that can be presented in a created-part, created-whole display can also be
incorporated into a natural-part, created-whole display. The choice of displays depends on whether the primary
objective is to teach new parts or the system of organizing familiar parts into a new system. If the objective is to teach
new parts, the natural-part, created-whole display is most appropriate. This display shows the parts that are to be
identified. If the objective is to teach a system of organizing familiar parts, the created-part, created-whole display is
most appropriate.

Decisions about created-whole displays are not as easy to make as those about natural-whole systems. The reason
is that the learner will never see the created-whole in the environment. If the parts are presented as unprompted cells
with wording in them, the learner will be forced to learn the whole or the system of relationships between the parts
because these relationships provide the only clues about which wording goes in which cell. To teach any system of
classification, the created-part, created-whole display is the strongest.

If the learner does not know the parts, the presentation of a created-part, created-whole display would be a word
game. If we change the display into a natural-part, created-whole display, however, the communication is not as

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emphatic in requiring the learner to learn the whole, because the parts are clearly identifiable. The display reinforces
more attention to the individual differences in the cells, and not as much to their arrangement.

Logical Interpolation or Extrapolation

If designed properly, the visual-spatial display will prompt interpolation or extrapolation. For example, if we
present the display of information about metals shown in Figure 14.6, we prompt the learner to organize the metals
according to their weight.

Figure 14.6

B. Heavy C. Light
A. Metals

J.
Aluminum
is very light

G.
Iron I.
Metals
F.
H. that stick
Copper
Steel is to
is the color
E. strong magnets
of a penny
D. Lead is
Gold is almost as
seven times heavy as
heavier than gold
aluminum

The display uses two devices to assure interpolation and extrapolation:

• The left-to-right arrangement, with the heavier metals to the left of the lighter ones.

• The length of the stems, with longer stems on heavier metals.

Neither prompt is spurious. Together, they prompt an organization of metals in a seriated arrangement. With the
understanding of this arrangement, the learner can interpolate new metals and can also extrapolate.

At some later time, we may be able to add new entries by relating them to entries already on the display. To do
this, we would describe the relative weight of each new entry. For instance, we present a test chart (no wording in the
cells) and describe a new metal this way: “Mercury is heavier than lead, but not as heavy as gold. Show me where
mercury would go on the chart.” “Magnesium is lighter than aluminum. Show me where it goes on the chart.”

Add-On Information

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 The design of visual-spatial displays should be approached with the understanding that it is not necessary to teach
everything through a single chart. If charts are properly designed, we can later add on additional information.

The strategy for add-on information should be that what had been taught earlier remains intact. For instance,
information about geological time can be added on to the simple display of the five types of vertebrates. The
illustration in Figure 14.7 shows five vertebrates with a different type of add-on information.

Figure 14.7

A. Vertebrates are animals

with backbones

B. Five types
D. Latest

5. Mammals

4. Birds
3. Reptiles
2. Amphibians
C. Earliest

1. Fish

E. 25 thousand types

F. Cartilage G. Bony Skeleton

Skeleton

H. Sharks J.
Eel
I. Largest fish
K.
is the whale
Perch
shark
L.
Tuna

M.
Catfish

Information about fish is added to the basic five-cell display for vertebrates. Additional add-ons can be made for
each of the other vertebrates. These add-ons would be positioned to the left of the basic display.

Add-on displays are presented one at a time. Each add-on would involve between 8 to 16 cells. Each would be
firmed and reviewed in the same manner as the original display is introduced and reviewed. A cumulative review
would follow in the form of a “super display,” showing both the original and all add-on displays. The students would

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play a game that requires identifying the wording for all empty cells in this super display.

Showing Variation in Coordinate Entries

If there are four coordinate entries and one of these differs from the others with respect to an important feature, we
can visually show both that the variant entry is coordinate with the others and that it is different. The display in Figure
14.8 illustrates the procedure. Four western cedar trees are presented. The fact that they are coordinate is shown by the
fact that all are subordinated under the same heading. One cedar is visually different from the others, however. The
Alaskan cedar occupies a unique position. This position is intended to reflect the fact that the leaves of the Alaskan
cedar are different from those of the other cedars. The leaves of all others have stomata patterns (distinctive patterns of
white lines) on the underside. The Alaskan cedar leaf has no stomata pattern.

Figure 14.8

Cedars in the
Western U.S.

4 types

Port Orford Cedar

Red Cedar Alaskan Cedar

Incense Cedar

This technique is appropriate if only one entry is different from all others in an important way. It is not appropriate
to show that each coordinate entry has some unique feature. If all coordinate cells are different from the others, the fact
that they are coordinate becomes obscured.

In summary, if only one cell is different from other coordinate members, show: (1) that it is coordinate by giving it
coordinate position or coordinate shape, or other features that are possessed by the coordinate members; and (2) that it
is different by creating a salient (obvious) difference between it and any other coordinate member.

Do not use this technique if there are only two coordinate entries or if each coordinate entry in a larger group has
an important distinguishing feature. For the latter situation, add a cell below each entry that tells about the feature.

Selecting Information for Displays

The amount of information included on the display should not exceed 16 cells, with each cell containing either a

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single label, phrase or sentence. Selecting information is easy for some displays. If we are dealing with a simple higher-
order classification system such as vehicles or types of trees, the information that goes on the visual display has been
clearly specified and ordered for us. We simply translate it into visual relationships that show which classes are higher
and which are lower.

When we deal with more complex events, however, the design becomes less clear. A recurring guideline in a
logical approach to instructional design is that we should not attenuate what we teach, but we should teach no more
than is necessary. When we teach a noun, such as shoe, we do not try to find the most precise “definition” of shoe. We
teach the discriminations that are suggested by what the learner already knows. Similarly, when we select information
to include on a visual-spatial display, we select only the most salient information. We do not try to exhaust the subject,
but merely to provide the learner with information needed for the immediate applications. The amount of information
that should be provided for a learner will vary from one learning situation to another, just as the specificity of a noun
changes from situation to situation, depending on the context in which the noun is to be used. Here are procedures for
selecting information:

1. Include the information that would be included in a good outline of the topic. An outline specifies the main
ideas and provides some supporting detail. The outline is not exhaustive with respect to the supporting
detail; however, the main ideas imply how supporting detail that is not mentioned would be included in the
outline. For instance, an outline of general tree classifications might mention oaks and maples as shade
trees (the entries for the tree-classification chart). Although other shade trees are not named (sycamore,
cottonwoods, etc.), these are implied by the outline (given that the learner has information about how
these trees look). By following the same basic conventions for developing a good outline, we can create a
visual display far more powerful than the outline because it is not limited to words. The various
relationships can be shown through visual analogues, instead of mere indentations under a heading or
through parallel structure.

2. Design the wording of cells so the wording for coordinate cells is parallel. If we use a sentence in one cell
of a particular type, we use a sentence in other coordinate cells. This requirement may seem trivial, but it
is quite important. When the teacher asks a group of students which words go in various cells, the activity
will quickly reduce to frustration if the wording for one cell is a phrase and the wording in a coordinate
cell is a sentence. The strong tendency of the students will be to say phrases for both cells or to say
sentences for both. Much hardship may be avoided by designing the cells so that parallel structure suggests
parallel wording.

3. Try to limit information to that which is relevant. Many facts about a particular topic are not relevant to the
understanding of it. The history of the internal combustion engine is not relevant to how the engine works.
If the goal of the topic is to provide information about how the engine works, the display should be limited
to information about how it works. If the topic is the history of the internal combustion engine, on the
other hand, the decisions about which information to include become far more difficult. We do not know
the value of various contributions unless we understand how the machine works. Ideally, our approach to

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 history would be to frame specific problems and then present solutions. For instance, a problem would be
that starting cars by cranking was bothersome and sometimes dangerous (broken arms). The solution:
Kettering invents the electric starter. If a topic is new for the learner and presents many new names, try to
limit the information to 11 cells with the understanding that subsequent add-ons may be introduced.

Designing the Script

The script is designed by: (1) expanding the information in the cells of the visual-spatial display; and (2) adding
information that is related to some cells.

The spatial relationships of the cells imply statements that should appear in the script. The words that appear in
each cell should also appear in the script. The script should present the displayed information in systematic order,
referring to the more general categories first and the more specific ones later.

In addition to conveying information provided by the various cells of the display, the script should contain other
information. This script information includes anecdotes, information that is unusual or unexpected, and facts about
entries on the display. As a rule of thumb, about 25 percent of the total scripted information should consist of
information that is not directly shown on the display.

The criteria for constructing a script serves as a basis for testing whether the script is adequate. If the script is
adequately designed, the answer to each question below should be, “Yes.”

• Does the script contain statements about every cell and about “connections” shown on the display?

• Does the script present this information in a logical order, starting with the general information?

• Does the script contain information that is not contained on the chart?

• Does each bit of secondary information relate to specific cells or spatial relationship cells on the chart?

• Does the secondary information comprise about 25 percent of the total information presented in the script?

Analysis of Preskills

Once designed, the display serves as a task. We can analyze the task as we can any other task. Ideally, we would
use a transformed-task approach. Our objective would be to assure that the learner understands all the prerequisite
concepts that are needed before the chart is presented. However, the display is usually capable of teaching most new
names.

Here are the entries that might appear on a chart dealing with cedar tree identification:

stomata—white lines

Port Orford Cedar—X’s on the underside

For the first entry, we would not preteach the meaning of stomata. The entry itself provides a description that is

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adequate for the purpose of identifying leaves of cedar trees. Stomata are the white, powdery lines that appear on some
leaves. If preteaching were necessary, it would involve teaching the meaning of white lines. The probability, however,
is great that the learner already understands this discrimination.

The entry, Port Orford cedar—X’s on the underside, assumes that the learner understands what X’s and underside
mean. If the learner does, no further preteaching is required. The entry itself is capable of teaching the relationship
about the identifying feature of the Port Orford leaf. (It has X’s on the underside.) The script presented with the display
could amplify this point: “If you look on the underside of Port Orford cedar leaves, you’ll see a line of X’s on top of
each other. The X’s will be in white stomata. Remember, Port Orford cedar—X’s on underside.”

Often, the sentences that define words do not imply preteaching for these reasons:

1. The facts that are presented of ten have very little generalizability and therefore do not imply heavy
discrimination teaching.

2. The type of test that is provided in connection with the visual-spatial display is usually adequate to assure
that the learner is discriminating highly similar members (such as Port Orford with its X’s, incense cedar
with its stomata pattern of wine glasses, and western red with its stomata butterflies).

3. The phrases or sentences used to describe new words are usually familiar to the learner. (Describing
stomata in terms of plant physiology would result in an item that would require extensive preteaching. Not
only would the learner be unaware of the meaning of stomata, but also the learner would probably have to
learn components of the physiological explanation. Typically, however, sentences that explain new words
phrase the explanation in words familiar to the learner.)

4. The script can be used to amplify or clarify meanings that may not be obvious.

Some entries on a visual-spatial display imply preteaching. Although it is possible to explain the four strokes of an
internal combustion engine without referring to any- thing beyond the engine, the transformed-task analysis would
suggest that the learner should have a generalized understanding of compression, exhaust, and intake. To achieve this
teaching, we would teach compression (possibly teaching it as a comparative—more compressed versus not-more
compressed). (We might press against a spring.) For exhaust and intake, we could use breathing in and breathing
out, with the learner labeling each example as intake and exhaust.

In summary, those entries that are defined or described on the chart usually need no preteaching if the explanation
presents a familiar idea and familiar words. Some entries, however, assume an understanding not provided by the
display. Preteaching for such entries is desirable if not necessary.

Scheduling Different Displays

The rules for ordering the displays are the same as those for other members. Each display is a member. Displays
that are minimally different should be separated in the order of introduction. If an introduction is difficult (involving
many new words or minimum-difference discriminations), the next display should not be as difficult.

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 The juxtaposed displays should look as different from each other as possible and the topics should be as different
as possible. Often, great difference in topic is not possible if the various charts are designed to develop a particular
subject. For example, if we are developing the subject of vertebrates, each display will deal with the same general
topics; however, as much as possible, the juxtaposed displays should deal with different aspects of the topic. One
display might deal with characteristics of fish, while the next deals with mammals. The two types of vertebrates are not
easily confused. The next display might deal with reptiles, followed by birds, followed by amphibians. The highly
similar members—reptiles and amphibians, and possibly fish, are separated in this order of introduction. Greater
separation of these members is possible if we intersperse displays that deal with geological information between those
that deal with types of vertebrates.

The juxtaposed displays should not be similar in form. As we mentioned earlier, the displays should be ordered so
that each is unique—ideally not highly similar to any of the others in the group that is to be presented. The extent to
which each display is unique is the extent to which the learner will not confuse elements or parts of different displays.
If charts are similar in shape or structure, they should deal with different information. (Two circular charts should not
be used for the same topic.)

Add-ons should be designed both so that the add-on part is unique in structure and so the total display (original and
add-on) is not highly similar to any other display.

Summary

Programs that use visual-spatial displays are effective for teaching the structure of topics for a wide range of
content.

The goal of a display is to teach a system of facts that are relevant to a topic. The learner understands the topic if
the learner understands the individual facts and their relationship. The visual-spatial display provides a framework for
inducing this understanding.

Although the goal is to teach only one thing (one topic) through a display, the teaching is complex because the
topic is composed of facts. The facts and their relationships are the features of the topic. The teaching therefore firms
the learner’s understanding of the facts.

The visual-spatial display uses codes for indicating information in various cells and codes for showing
relationships between cells.

The display should therefore not be used with learners who do not understand these codes and who may assume
that what is being shown is a real thing of some sort (rather than a representation of a system).

For the more sophisticated learner, the visual-spatial display is an extremely efficient way to learn about systems.
The spatial arrangement of elements provides visual prompts for retrieving any particular entry from memory. (By
remembering what is next to the cell, what above the cell, or what the shape of the cell is, the learner may be prompted
to retrieve the information for a particular cell.) Also, the wording presented for each cell provides a logical basis for

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retrieving the information.

Accompanying the display is a script that deals with the wording of each cell, amplifies what is said in each cell,
and provides perhaps 25% additional information that is not provided by the visual display.

Following the presentation of the display with words in the cells, the learner is tested on a blank-cell version of the
display, then engages in expansion activities that usually take the form of a game.

There are three types of visual-spatial displays: the natural-part, natural-whole; the natural-part, created- whole;
and the created-part, created-whole. Often information that involves a created-whole or classification system can be
presented as either a natural-part, created-whole, or a created-part, created-whole presentation.

If the new teaching focuses primarily on the organization, the created-part display is preferable. New words may be
introduced, but they should be “defined” with familiar words.

If the learner is to learn to identify parts that cannot be described with familiar words (such as game birds), the
natural-part display is preferable.

The general rules for designing the visual-spatial display are:

• Limit the information to 16 cells with only one “idea” presented by each cell.

• Use coordinate wording for cells that are coordinate in structure (visually coordinate).

• Show subordination by using lines or cell design. If three things are subordinated under a particular cell,
all will have lines leading to that cell or all will be included in a larger cell.

• Show that members are coordinate by lines and form. If cells are coordinate, they should occupy the same
relative position and they should be visually the same.

• If one of the group of coordinate entries is different from the others in an important way, we can show the
difference by adding a visual clue to the cell of the different member.

• Avoid spurious cues. A cue is spurious if it is designed to prompt a name that is familiar to the learner.
The cell may be in the shape of a letter or the shape of the familiar object.

Apply a transformed-task analysis to the display and identify any preskills that are to be taught. As a rule,
sentences or phrases that appear on the display do not imply preskill teaching. The explanation of the new word is
usually expressed in terms the learner can understand. Some labels imply preteaching. These are labels that are not
adequately taught through the display and that are important to the understanding of the displayed information.

To present a series of visual-spatial displays, use the procedures specified for introducing members to a set. Each
display is a member. No display should be juxtaposed to a display that is highly similar in shape or highly similar in
content.

Add-ons may be used when a topic is extensive (more than 16 cells). Two types of add-ons are possible. One type

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adds individual cells as interpolated or extrapolated members. These add-ons are possible when the cells are arranged
according to some generic ordering rule (such as weight, size, number, or other feature that admits to a clear
presentation on a continuum). By teaching the original display, information points on the continuum are provided.
These become the basis for possible interpolation or extrapolation.

The other add-on is a unit or group of cells that is added to the appropriate entry of an earlier chart. The procedure
is first to teach the original display, then (after the learner has had an opportunity to work with this display and learn it
thoroughly) introduce the add-on. The presentation of the add-on should not be juxtaposed with the presentation of the
original display. The add-on should be unique in shape. The display that results when the add-on is included should be
unique in form.

If more than one display is introduced to the learner, each display is a member of a set. Following the rules for
members, juxtaposed displays should be as maximally dissimilar as possible.

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 Section V

Complete Teaching
The sequences presented in Section II and Section III are initial teaching sequences. They communicate with the
learner in the most straightforward manner possible, showing what controls the discrimination response and how
changes in the features of the examples relate to possible changes in the responses. The initial teaching sequences,
however, do not provide for complete teaching. They do not show the learner the range of application for the
discrimination. And, they do not provide practice in which the discrimination is juxtaposed with different
discriminations. Solving the problems of stipulation and juxtaposition prompting is the goal of complete teaching.

The Problem of Stipulation

The initial-teaching sequences are guilty of two types of stipulation: (1) they stipulate the range of examples, and
(2) they stipulate the responses that are called for. Following a presentation of the concept farther apart demonstrated
with the teacher’s hands, one learner may think that farther apart applies only to the hands; another may think that it
applies only to humans; another may think that it applies to any physical thing. All interpretations are consistent with
the presentation, which means that the presentation has done nothing to confirm or reject any of these speculations
about the range of application. The longer the teacher works on a limited range of examples or a particular response,
the greater the amount of stipulation that results. If a teacher repeated a demonstration of farther apart on ten
different occasions and never showed that farther apart applies to anything but hands, naive learners would tend to
learn the stipulation much more than they would if the teacher presented the sequence only once or twice.

To solve the problem of stipulation, we:

1. Work on the initial-teaching sequence no longer than necessary.

2. Immediately follow the sequence with applications of the discrimination that use new examples and that
require new responses.

The Problem of Juxtaposition Prompting

Juxtaposition prompting occurs in most initial teaching sequences because tasks in the sequence teach within-set
discriminations, which means that the learner attends to the same instructions and responds to juxtaposed tasks that
involve the same response dimension. This pattern of juxtapositions prompts attention to particular features of the
examples and is relatively easy for the learner because it involves less memory of what to attend to and which label
goes with an observed change or feature. Our goal, however, is to teach the learner to deal with situations in which the
discrimination is not prompted by being preceded by a task that involves the same discrimination.

To “shape” the learner’s memory for the discrimination’s details and its label, we shape the context or the pattern
of juxtapositions. To do this, we sandwich different tasks between tasks that involve the newly-taught discrimination.

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Think of interspersed activities as interference. In the initial teaching sequence, there is no interference. Immediately
after operating on one example in a specified way, the learner operates on another example in basically the same way.
No interference occurs between trials. By interspersing other activities between the trials with the target discrimination,
we increase the interference. We increase the demands on the learner’s memory (and demonstrate that we expect the
learner to remember the label and the details of the examples). We increase the difficulty of dealing with the new
discrimination in stages, not in one leap from the initial teaching situation to situations that contain no juxtaposition
prompting, but through stages that increase the amount of interference.

General Directions for Expanded Teaching

We will look at a number of specific techniques for expanded teaching. All are effective in overcoming the
problems of stipulation and juxtaposition prompting. However, expansion activities are very natural and easy to
construct if the initial teaching is effective. The most frequently encountered problem observed in teaching situations is
that the learner has not been taught a discrimination, but adequate expansion activities are provided.

The simplest strategy for dealing with the problem of expanding what has been taught is to teach it and
immediately apply it in a significant way. When we follow the simple strategy of teaching what is to be used and using
what is taught, we eliminate many problems of “keeping discriminations alive” through techniques such as the
expansion activities described in this section.

Chapter 15 presents various expansion tasks that do not involve worksheets. Chapter 16 explains worksheet
strategies.

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 Expanded Teaching
The activities that we will deal with in this chapter are presented after the initial-teaching sequence. Their purpose
is to show the learner that the newly-learned discrimination applies to examples and tasks beyond those shown in the
initial teaching setup.

The five primary types of expansion activities are:

1. Manipulative tasks for basic-form and correlated-feature concepts.

2. Implied conclusion tasks for all concepts.

3. Divergent tasks for single and double transformation concepts.

4. Fooler games for all concepts.

5. Event-centered task series for all concepts.

Manipulative Tasks

A manipulative task calls for a production response. The learner is told to produce an example of the concept.
Instead of calling for a verbal or symbolic response, however, the manipulative task calls for a non-verbal response.
Manipulative tasks may be presented immediately after the learner has successfully performed on the initial-teaching
sequence for any basic discrimination and for many correlated-feature relationships. Here are several tasks that could
follow the presentation of the positional concept over.

“Your turn. Take the block and hold it over this table . . .”

“Now put it so that it is not over the table . . .”

“Now hold it over the table again . . .”

Possibly, the teacher could introduce different objects for the child to hold the object over:

“I’ll name things. You hold the block over them.

Hold it over the table.

Hold it over this chair.

Hold it over the floor.”

Manipulative tasks may be presented to a group. For instance, the teacher puts his hand on his knee:

“Put your hand like this.” He prompts children to put their hand on their knee.

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 “Your hand is not over your knee. When I clap, put your hand over your knee.” Teacher claps. “Good.”

“When I clap again, hold your hand so that it is not-over your knee.” Teacher claps. “Good.”

“When I clap again, hold your hand so that it is way over your knee.” Teacher claps.

Many variations of manipulative tasks are possible. To construct them, follow these rules:

1. Design the tasks so they do not involve much that is new, particularly new instruction forms.

2. Use the same setup for more than one positive example.

The illustrations just presented introduce some new wording. The choice-response sequence teaches over—not
over without naming the second object. (“The eraser is over,” not, “The eraser is over the table.”) The use of
manipulative tasks makes it fairly easy for the learner to respond to these new instructions. If the learner understands
over, the first series of tasks is obvious because all tasks involve “this table,” pointed to by the teacher. The task
involving “the knees” is of the same form. The tasks involving different objects involve the same block, so mistakes are
improbable.

Manipulative tasks can be designed so they go beyond what had been taught in the initial-teaching sequence;
however, the gap between the manipulative task setup and the initial-teaching setup should not be too great initially. If
it is, the learner’s responses will probably provide us with feedback that we have made the task too difficult.

Manipulative tasks are easily designed to expand comparatives. Here is a possible sequence of manipulative tasks
that might follow the presentation of higher.

“Everybody, hold your hand out like this . . . When I clap, move it a little higher.” Teacher claps. “Good.”

“Let’s do it again. Hold your hand out like this . . . When I clap, move it a lot higher.” Teacher claps.

Variations of this kind of manipulation are appropriate for most comparatives. A similar variation is possible for
correlated-feature relationships that involve comparatives. The following are some manipulative tasks that could follow
the presentation of the relationship: The hotter it gets, the higher the line in a thermometer.

“Your turn. Hold your hand in front of you, like this . . . Your finger is the top of the line in a thermometer. I’ll tell
you what happens to the temperature. When I clap, you show me how the line moves.”

“The air gets a little hotter . . .” Teacher claps.

“The air gets a lot hotter . . .” Teacher claps.

“The air stays the same temperature . . .” Teacher claps.

“The air cools a little tiny bit . . .” Teacher claps.

“The air gets a little bit hotter . . .” Teacher claps.

“The air gets a little hotter . . .” Teacher claps.

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 The teacher could add a second question to each example:

“The air gets a little hotter . . . Why did you move up a little bit? . . . Yes, because the air got a little hotter, so the
line moved up a little bit.”

This type of expansion activity is well-designed for work with a group because it provides clear feedback on the
understanding of individual group members. Many trials can be juxtaposed and the teacher can readily observe
problems.

Manipulative tasks can be designed so they test the discrimination or so that they do not actually test it. If the
learner does not actually produce an example of the discrimination being taught, the task is not actually testing the
discrimination. Here are manipulative tasks that do not test the discrimination being taught:

“There’s a rabbit in this room. And that rabbit is going to hide under different things. I’ll tell you where the rabbit
is hiding. You point to the place the rabbit is . . .”

“The rabbit runs under the table . . .”

“Now the rabbit goes under the window . . .”

“Now the rabbit is under Janey’s desk . . .”

“Now the rabbit is under my desk . . .”

If this sequence is presented to a group of children, we could not actually tell whether each child was pointing
under the various objects. Possibly some children simply point at the objects named. We could change the response so
that the children clearly produce positive examples of under. “Touch the place the rabbit is . . .” Now, however, the
task becomes perfectly unmanageable with children tumbling about the room, making a game of the touching, and
making it impossible for the teacher to maintain any sort of reasonable pacing from one task to another.

The pointing task above may easily be justified on the ground that although it does not clearly require the learner to
produce positive examples, it does acquaint the learner with the idea that the concept under can be used in connection
with a variety of objects.

By changing the task somewhat, we could make it manageable.

“There are many rabbits in the room. One of them is on your chair. Touch the place that rabbit is.”

“Now that rabbit is under your chair. Touch the place that rabbit is . . .”

In summary, the manipulative task consists of instructions for the learner to produce a non-verbal response. Ideally,
the task should be designed so that the learner produces examples of the concept. The manipulative task can be used to
expand any basic concept and for many correlated-feature relationships. It may be introduced as soon as the learner
performs successfully on an initial teaching sequence, even as the final task in the series (following the test segment).

Manipulative tasks are most manageably presented as a series that involve a common setup. For example:

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 1. The teacher begins with general instructions, telling: (a) what the teacher will do, and (b) how the learner
is to respond.

2. The teacher then presents 3 to 6 examples.

3. Although some new information may be introduced through these examples, the tasks should not be
drastically different from those presented in the initial-teaching sequence.

Implied-Conclusion Tasks

The initial teaching sequences are direct. They show examples of the discrimination being taught and they ask
questions about that discrimination. They do not ask about other discriminations. For instance, the learner is asked
about bigger. The learner is not asked to name the object that gets bigger, to respond to the position of the object that
gets bigger, or to label anything other than the discrimination being taught. The learner’s responses change in a way
that parallels changes in the example, so the most direct relationship possible exists between the examples and the
response.

Once the learner has performed on an initial-teaching sequence, we can introduce tasks of a different nature:
implied-conclusion tasks. For these, the learner uses the discrimination that was taught to draw a conclusion. The
learner will not produce overt responses of “bigger” and “not-bigger” for the implied-conclusion tasks; however, the
learner will use the discrimination bigger—not-bigger in responding to these tasks.

Here is an example. The teacher presents three familiar objects: a dog, a horse, and a man. The teacher says, “One
of these objects is bigger than the others. Which object is bigger?”

The response: “The horse.”

Note that the learner is required to use the concept bigger. But instead of saying “bigger,” the learner was required
to name the thing that was bigger.

The teacher could have asked a variety of other questions, such as: “Is the bigger object sitting or standing?” “What
color is the bigger object?” “Where is the bigger object?” “What is the bigger object doing?” Note that the teacher
would present one of these tasks, not the series. For each of these tasks, the learner must: (1) first find the object that is
bigger, and (2) follow the response conventions called for by the questions.

Designing Implied-Conclusion Tasks

The easiest way to figure out how to construct implied-conclusion tasks is to make up a statement that names the
newly-taught discrimination in a fact. Then create a display consistent with the fact and that requires the learner to
discriminate. If the newly-taught discrimination is gymnosperm, the statement might be: “A girl sat under the
gymnosperm.” If the newly-taught discrimination is bigger, the statement might be: “The red object is bigger than the
other objects.” If the newly-taught discrimination is rhyming, the statement might be: “John rhymed with the word
am.”

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 Each fact tells about a possible implied-conclusion display. Consider the sentence, “The red object is bigger than
the other objects.” We can create a display that is consistent with this fact. The display would show a group of colored
objects. The red object would be bigger than the others.

For the fact, “A girl sat under the gymnosperm,” we would show a girl sitting under a gymnosperm tree.

We now adjust the display to make sure that the learner discriminates. To do this, we add other objects to each
display.

For the display that shows the red object bigger than the others, we could ask, “What color is the bigger object?”
Or we could ask, “Is the bigger object yellow? . . . Is the bigger object blue? . . . Is the bigger object red?” Or we ask,
“Where is the biggest object?”

To answer the question, “What color is the biggest object?” the learner must first find the object that is biggest,
then respond to the color of that object. We design the display so that the learner must find the bigger object to answer
the question. (Only one object is red.)

For the display of John rhyming with am, we could ask: “Who is rhyming with am?” or, “Is Terry rhyming with
am? . . . Is John rhyming with am? . . . Is Martha rhyming with am?”

The task does not require the learner to rhyme with am (which is what the initial teaching sequence teaches).
However, to respond correctly, the learner must first attend to what John is rhyming with, then respond to the task that
is presented.

If John is the only person rhyming, the learner could respond to the task, “Who is rhyming with am?” by simply
naming the only person who is rhyming. Therefore, the display would show not only John, but also the other people,
each of whom would rhyme with words other than am.

For the display that shows a girl sitting under the gymnosperm, we could ask the following questions: “Who is
sitting under the gymnosperm?” “What is the person under the gymnosperm doing?” “Is there a girl under the
gymnosperm?” “Is there a boy under the gymnosperm?”

To assure that the learner uses the discrimination gymnosperm in answering these questions, we would make sure
that there is more than one person in the picture (ideally someone under an angiosperm tree). This person would not be
sitting if we ask the question, “What is the person under the gymnosperm doing?” We would make sure that the person
under the angiosperm is not a girl, if the learner is asked, Is there a boy under the gymnosperm?”

The following is a summary of the steps for constructing implied-conclusion tasks:

1. Construct different facts that include the name or label of the discrimination that had been taught.
Underline the discrimination in each sentence. “Martha constructed a fraction equivalent to two-thirds.”

2. Make up a display that shows what the fact says. For the fact, “Martha constructed a fraction equivalent to
two-thirds,” we would show a girl (labeled Martha) holding a fraction that is equivalent to two-thirds

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 (e.g., 4/6). For the fact, “Five of the eight fractions were equivalent to two-thirds,” we would show eight
fractions, five of which are equivalent to two-thirds.

3. Present a task that does not call for the label or name of the discrimination. For the displays involving
fractions, the tasks could be: “Tell who constructed a fraction that is equivalent to two-thirds,” and “Name
all the fractions that are equivalent to two-thirds.

4. Make sure that the learner is not able to respond correctly without using the discrimination. Add negatives
to the display that share features with the positives. Show people other than Martha holding up fractions.

Divergent Activities for Transformation Concepts

Divergent tasks are most appropriate for single-transformation and double transformation concepts. It is usually
possible to construct manipulative tasks for the transformation concepts. We could require the learner to “Touch the
word that rhymes with am,” which would satisfy the requirement for manipulative tasks. The learner creates an
example of rhyming with am and the response is non-verbal. It is also usually possible to construct implied-conclusion
tasks for transformation concepts. (“Who is rhyming with am?”) Often, however, the best expansion tasks for
transformation concepts are divergent tasks. “Say some words that rhyme with am.”

The difference between divergent tasks and the type presented in the initial-teaching sequence is that there is a
range of correct responses for the divergent task.

If the learner has been taught to construct predicates through a single-transformation sequence, we can construct
divergent tasks that require the learner to make up predicates. If the learner has been taught to tell where something is
through a single-transformation sequence, we can design divergent tasks that require the learner to tell where different
things are. “Make up sentences that tell where the things in this room are.” Like other expanded activities, the divergent
tasks can be presented as soon as the learner has mastered the initial-teaching sequence.

Here is a possible sequence of divergent tasks that might follow the teaching of predicates:

Teacher: “My turn to make up predicates for sentences that start out with the subject, the girl. The girl” (pause)
“went to school. The girl” (pause) “liked ice cream.” Your turn. Make up predicates for sentences that start out
with the subject, the girl.”

After teaching rhyming, the teacher could present this divergent sequence:

“I’m going to rhyme with am. Ram . . . Sam . . . Your turn. Make up some new words that rhyme with am.”

After teaching simple addition facts, the teacher might present this sequence:

“I’m going to say addition statements with numbers that add up to four. Listen. Four plus zero equals four. Your
turn. See if you can make up more statements that add up to four.”

After teaching simple spelling of words that end in the letters a-n:

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 “I’ll spell some words that end with the letters a-n. Listen. C-a-n spells can. M-a-n spells man. Your turn. Make
up some words that end with the letters a-n.”

To design divergent tasks:

1. Select a task for which at least five answers are correct and available to the learner.

2. Model one or two responses.

3. Require the learner to make up at least two more examples.

4. Reinforce acceptable responses.

Reinforcing Divergent Responses

Although the illustrations above do not specify teacher reinforcement, reinforcement is very important for success
in divergent activities because these activities involve varying degrees of “risk taking.” Unless there is some potential
payoff for taking risks, the learner will be reluctant to produce more than very safe responses. Also, if there is a great
penalty for creating an unacceptable example, the learner will not be very adventurous.

If the teacher is working with more than one learner, the teacher can show the kind of responses that are worthy of
“ooos and aaahhs” by responding with obvious admiration to learners that produce examples somewhat different from
those modeled by the teacher. If no student produces these, the teacher can model a couple of them and brag about how
smart she is. If the teacher is a good actor, the students will usually work very hard to emulate her model. As a rule,
however, divergent tasks are far less manageable than other tasks, particularly within the group setting. Of ten this
setting gives the impression of great discovery and application because many of the students may be performing. If the
goal is to teach all, however, the teacher must have systematic procedures both for monitoring all the students and for
providing those who do not perform with prompting and reinforcement.

One possible solution to the problem of reinforcing divergent responses in a group is to use a “gamble” format.
Everybody in the group receives a desirable reward if the work of a randomly-selected learner meets specified criteria.
For instance, students write down four words that rhyme with at. The teacher secretly selects the work of one student.
If all of the student’s responses are correct, the teacher rewards everybody in the group. If the student’s responses are
not correct, the teacher does not disclose the student’s name, but does not give any member of the group the reward.

All other solutions to the problem of reinforcing divergent responses in a group setting are similar to the one above
in that they require the teacher to “sample” the work of each student frequently enough to provide feedback and
encouragement.

Fooler Games

Foolers are expansion activities that can be used after nearly any initial teaching sequence. The details of various
foolers differ; however, all are based on the idea that the teacher will make some mistakes and the learner will be able
to catch those mistakes if the learner uses knowledge of the newly-taught concept. If the learner has been taught

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bigger, the teacher may present different pairs of discs. One disc in each pair is bigger than the other.

Teacher: “I’m going to try to fool you. If you see me make a mistake, say: ‘You didn’t touch the bigger disc.’ What
are you going to say if I make a mistake?”

“Here I go.” Teacher touches bigger disc. “I touched the bigger disc.”

Teacher replaces one disc, touches bigger disc. “I touched the bigger disc.”

Teacher replaces one disc, touches smaller disc. “I touched the bigger disc . . . Show me what I should have done . .
. Oh, that’s the bigger disc.”

The fooler game has the potential of being challenging to the learner, because the learner has the opportunity to
correct the teacher by using knowledge of bigger.

Many variations of the fooler game can be created. Instead of using different pairs, the teacher could present a
group of five discs that vary in size. By adding a disc or taking one away, the teacher could create a series of examples
for the game. “One disc is bigger than any other disc. I’ll touch that disc.” After touching a disc, the teacher says: “I
touched the disc that is bigger.” The children respond to examples in which the teacher is wrong by saying, “You didn’t
touch the disc that is bigger.” The rest of the procedures are the same as those for the original fooler game.

The fooler game above calls for three different types of responses:

1. Responses to correct teacher assertions (by saying nothing).

2. Responses to incorrect teacher assertions (by using the label for the new discrimination, “You did not
touch the bigger disc”).

3. Manipulation responses. (In response to, “Show me what I should have done.”)

A similar fooler format can be used with transformation concepts. For example, after the learner has been taught to
rhyme, the teacher may present this type of fooler:

“See if I can fool you. I’ll say words that rhyme with am. When you hear me make a mistake, tell me to stop. am,
bam, jam, jack . . . Starting over: ram, ham, hat . . . Starting over: ran, cat . . .”

A powerful fooler might be used after the learner has been taught to read simple words.

“Follow along as I read. See if I can fool you. When I make a mistake, tell me to stop. That . . . dog . . . can . . . ran
. . . What should I say? Yes, run. Starting over: That . . . dog . . . can . . . run . . . with . . . me . . . What should I
say? Yes, us. Starting over: That . . . dog . . . can . . . run . . . with . . . us . . . We . . . have . . . What should I say?
Yes, had.”

Here is a fooler that might be presented after the learner has been taught to identify the subject of simple sentences:

Teacher shows a list of sentences. “Let’s see if I can fool you. I’ll say the subject of each sentence. If I say it
wrong, tell me to stop. The boy went to the lake. The subject is: the boy.
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 “The boy and a dog went to the lake. The subject is: the boy . . . What is the subject?

“My mother was sitting on the grass. The subject is: my mother was . . . What is the subject?”

Here is a fooler that might be presented following the initial teaching sequence for the relationship: The steeper
the grade, the faster the stream flows.

“I’ll tell you a story about a woman who is paddling down a river. See if I can fool you. When you hear me make a
mistake, say ‘stop.’ The woman started out going down a very steep grade. The water was moving very fast. Then
the stream leveled off a bit and the stream slowed down. Then the stream leveled off more and the stream moved
faster . . . What’s wrong? . . . What should have happened? . . . How do you know? . . . Then the grade became
really steep and she had to paddle to keep the canoe moving . . . What’s wrong? . . . How do you know? . . . What
should have happened? . . . How do you know? . . . Then the grade flattened out completely and she had to paddle
to keep the canoe moving . . .”

To construct the fooler game:

1. Design a setup or story line that will permit you to present at least six examples.

2. Make about one third of the examples foolers.

3. Precede the series with general instructions to indicate what you will tell the learner and what the learner is
to do if a mistake occurs.

4. If practical, ask the learner what you should have said instead of what you did say when you made a
mistake.

Event-Centered Tasks

Event-centered tasks can be used to expand any newly-taught discrimination. An event-centered series is created
when the teacher presents a single object or example and asks different questions about that object, including questions
that call for the newly-taught discrimination.

All questions asked about the object involve skills that have been taught to the learner.

Let’s say that the learner has been taught names of common objects, basic color names, position names: round,
slanted, and smooth. The most recently taught concept is the classification of vehicles.

Teacher presents a blue truck to the learner. “What color is this object? . . . Touch a part of the object that is round .
. . Is this object a vehicle? . . . What kind of vehicle is it? . . . What are trucks used for? . . . What kind of things
would a truck like this carry? . . . Do you know that a truck like this could carry all the furniture that you would
find in three houses?”

Ideally, the series of questions should be designed so that:

1. The newly-taught skill (in this case, vehicle and truck) is juxtaposed with the other discriminations
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 associated with a particular object. This juxtaposition assures that the learner becomes familiar with the
more common “associations” for a newly-taught discrimination.

2. A variety of tasks is presented. In the sequence above, there is a manipulative task, a yes-no task, a
divergent-response task, and a rhetorical question.

3. The target tasks (those dealing with vehicle and truck) do not appear at the beginning of the series.
Ideally, they should occur after two or three other tasks have been presented.

4. Not all tasks are necessarily pretaught. For instance, the task about what kinds of things the truck might
carry might not be pretaught.

5. Included in the series is some additional information. To make the presentation interesting to the learner,
the teacher should tell something about the object. In the series above, the telling takes the form of the last
question.

The event-centered series is perhaps the easiest for traditional teachers to construct because it involves the variety
of tasks that are typically presented when an object, event, or picture is analyzed (an activity historically referred to as
an object lesson). The teacher asks about specific details of the presentation. The teacher also asks questions of the
form, “What do you think? . . .” And the teacher presents some new interesting information about the object or part of
the object.

Event-centered tasks are very effective for removing what had been taught from the juxtaposition context of the
initial teaching series.

Using Expanded Activities in Programs

To expand a newly-taught discrimination, you may use any combination of the five expansion activities—
manipulative tasks, implied conclusion tasks, divergent tasks, fooler games, and event-centered series. Not all
expansion activities are perfectly appropriate or manageable for every discrimination, and some activities are generally
more useful than others. The most widely-applicable activities are: implied conclusion tasks, fooler games, and event-
centered series. The event-centered series may include manipulative tasks, so there is often no need to present
additional manipulative activities. Divergent tasks are very good; however, they are not manageable for many concepts.
Also, it is possible to incorporate divergent tasks in an event-centered sequence.

The schedule in Figure 15.1 shows how the various expanded activities might be used in connection with the
teaching of the concepts on and over and between.

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 Figure 15.1
An Expanded Activities Program
Components Lessons

1 2 3 4 5 6 7 8 9 10 11 12 13 14

1. on X X

2. over X X

3. between X X
F2 I E1 F1 I1 F2 E1,2,3 I2,3
Expansion
F1 I1 E1 E1,2 1,2 I2 E2 F2 E1,2 I1,2,3 F3
Cumulative 1,
Review 1,2 1,2 2,3
I = Implied conclusion F = Fooler E = Event-centered

The initial teaching of new discriminations is repeated (on two consecutive lessons in the schedule). After the
second member (over) has been introduced, a cumulative review is presented. Between lessons 6 and 11, the
discriminations on and over are reviewed and expanded. Another cumulative review occurs on the second initial
teaching lesson for between (lesson 12). The assignment of particular activities on different days is arbitrary. The
general pattern is for two activities to be presented as part of the expansion—one to deal with the most recently taught
discrimination, and the other to deal with the earlier-taught concepts.

Figure 15.2 shows the expansion activities for lesson 1.

Figure 15.2
Expansion Activities for Lesson 1

Example Teacher Wording

Teacher places objects on, I’m going to try to fool you.

under, and above a table. If you hear me make a
mistake say, “stop.”
Touches object on table. “This object is on the table.”
Touches object above table. “This object is on the table.
Show me an object I
should have touched.”
Touches object on table. “This object is on the table.”
Touches object on table. “This object is on the table.”
Touches object on table. “This object is on the table.”
Touches object above table. “This object is on the table.
Move it so that it is on the
table.”

Figure 15.3 shows the activity that is presented as part of lesson 13. Note that the single activity presents event-
centered tasks involving on, over, and between. It also presents implied conclusion tasks for on, over, and between.

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 Figure 15.3

Teacher touches the horse. “What kind of animal is this? What are those things in the sky called?
What kind of animal is over a tree? Why do you think that cat is climbing the tree?
Touch the animal that is climbing something. What do cats like to hunt?
What kind of animal is that? What object is right over the house with the chicken on it?
Where is the chicken? Which animal is between two trees?
What is between the two clouds? Where is the flying bird?
What is the horse on? Name the animals the cat is between.
Which animal is on the tree?

The series contains some tasks that have nothing to do with on, over, and between.

The schedule of expanded activities for the prepositions assumes no other applications of the concept are provided
for the learner, which means that all expansion must occur through the activities specified in the teaching schedule.
Hopefully, this situation does not occur frequently. In more typical situations, fewer expansion activities are needed
because the learner uses what has been taught. For instance, the learner engages in a “building project” that utilizes
prepositional concepts as soon as they are taught. When engaged in this activity, the learner receives instructions that
refer to prepositions, and the learner gives instructions to others. The building activity serves as the primary expansion.
It provides an ongoing, “cumulative review,” and it automatically provides for a variety of tasks.

With such an activity scheduled on a regular basis, the number and type of expansion activities (and review
activities) can be reduced (see Figure 15.4).
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 Figure 15.4
A Reduced Expanded Activities Program
Components Lessons

1 2 3 4 5 6 7 8 9 10

1. on X X

2. over X X

3. between X X

Expansion F1 E1,2 F1,2

Building 1 1 1,2 1,2 1,2 1,2,3 1,2,3 1,2,3

F = Fooler E = Event-centered

Note that the rate of introduction for prepositions is faster. The reason is that the building activity is a better vehicle
for consolidating these concepts. The learner works for longer periods of time and receives stronger associations for
using the newly-taught concepts.

Expansion for Advanced Applications

For the advanced learner, the problem of stipulation is not severe; however, expansion activities of some sort
reduce possible juxtaposition prompting and shape the learner’s memory of the discrimination. Part of the expansion
for the advanced learner may take the form of worksheet activities (Chapter 16). Therefore, only infrequent expansion
tasks may be needed for advanced applications. The activities that are generally most useful are the same as the most
useful ones for the fairly naive learners—foolers, implied conclusions, and event-centered series.

Figure 15.5 shows a variety of a fooler game that might be presented after the learner has learned to discriminate
between white oak, red oak, and sycamore leaves. The teacher presents a variety of leaves.

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 Figure 15.5
Expansion Activities for Advanced Applications
Example Teacher Wording

I’ll point to each leaf and tell you

what kind it is. Correct me if I
make a mistake.
Points to black oak This is a black oak . . .
Points to white oak This is a black oak . . . How do
you know it’s not a black oak
. . . What is it?
Points to sycamore This is a sycamore.
Points to black oak This is a sycamore . . . How do
you know it’s not a sycamore?
. . . What is it?
Points to white oak This is a white oak.
Points to sycamore This is a black oak . . . How do
you know it’s not a black oak?
. . . What is it?

As a rule, expansion activities are appropriate for the sophisticated learner if no worksheet activities are presented,
and if there is no ongoing application that requires the discrimination. The schedule of expansion activities, however, is
more sparse than that used for the naive learner. Whether we work with the naive learner or the sophisticated learner,
what is taught must be used and it must be applied to new contexts. If this requirement is not met through worksheets
and other ongoing activities, it must be met through expansion activities. Conversely, if the expansion functions are
effectively handled through activities for which the discriminations were needed in the first place, the formal expansion
activities can be reduced drastically or even eliminated.

Summary

Expanded activities are designed to amplify the learner’s understanding of the discrimination that is taught through
the initial-teaching sequences. The expanded activities follow immediately after the completion of the initial-teaching
sequence (either following the first appearance of the sequence or following the second appearance). Expanded
activities consist of a group of tasks, as few as three and possibly as many as twelve.

The five types of expanded activities are:

1. Manipulative tasks

2. Implied-conclusion tasks

3. Divergent tasks

4. Fooler games

5. Event-centered series

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 The most useful types are the implied-conclusion tasks, fooler games, and event-centered series. These types can
be used for nearly all discriminations.

Manipulative tasks are well-designed to amplify the teaching of basic-form concepts and correlated-feature
relationships. They permit the teacher to monitor responses of various individuals within a group.

Implied conclusion tasks are appropriate for all discriminations. When presented with an implied conclusion task,
the learner must use the newly-taught discriminations to figure out which example is referred to. The response that the
learner produces, however, is different from the one used in the initial-teaching sequence. (Instead of identifying the
color of a circle the learner might answer a question such as, “What shape is the orange object?”)

Divergent tasks are appropriate for single-transformation and double-transformation concepts. The teacher models
one or two examples; the learner makes up some examples of the same type.

Fooler games are appropriate for any discrimination. The teacher presents positive examples and negative
examples of the discrimination. The learner responds to the negative examples by indicating that they are wrong and by
telling either why they are wrong or what should be done to them to make them right.

Event-centered series present the newly-taught discrimination in the context of other tasks and skills that have been
taught. The sequence provides assurance that the new discrimination is recognized in this context. A variety of tasks
may be included in the sequence—implied-conclusion tasks, manipulative tasks, and tasks that are not related to the
new discrimination. The use of different types of expansion activities assures that the discrimination is processed in
different contexts.

Discriminations taught to naive learners generally need more careful expansion than those presented to advanced
learners. One reason is that the problems of stipulation may not be as great with advanced learners. Effective expansion
for both naive and sophisticated learners coordinates activities with cumulative review exercises to assure that the new
learning is thoroughly integrated with familiar concepts and skills.

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 Worksheet Items
The three primary advantages of worksheet items are:

1. They provide independent practice.

2. They are good vehicles for reducing juxtaposition prompting.

3. They are good for introducing a variety of new task forms.

The major disadvantage of worksheet items is that they do not provide corrective feedback.

Their use, therefore, should be limited to situations in which the probability of errors is low. The strategy that we
will use with worksheets involves:

1. First teaching the learner a discrimination through initial teaching sequences.

2. Possibly expanding the discrimination through expansion activities and oral review activities.

3. Presenting worksheet items as part of the expansion and review.

4. Providing a workcheck to assure that the learner’s worksheet performance is promptly checked and the
learner is provided with corrective feedback.

Worksheet Items for Different Jobs

Some worksheet items are better suited than others for particular jobs. Some require strong responses; others do
not. Some are more manageable for particular discriminations. Some can be used as an extension of the initial-teaching
sequence for hard-to-teach concepts that may require a very careful transition; others are not well-suited for this job.

A summary of the various types of worksheet items and an indication of their strength is given in Table 16.1. As
the classification system suggests, there is more to worksheet items than multiple-choice items, and even within the
domain of multiple-choice items, some are better-suited than others to achieve certain instructional objectives. This
classification does not suggest that we must use a particular type if we wish to achieve specific objectives. The
classification merely suggests options and alerts us to possible problems. There is no exact formula for using the types.

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 Table 16.1
Types of Worksheet Items
Choice items
1. Choice of examples. Items consists of single label and
different examples. Items are good for strengthening the
discrimination of features of examples–the discrimination
taught through the initial-teaching sequence.
2. Choice of label. Items consist of a single example and
different labels. Items are good for strengthening
discrimination of labels.
3. Matching labels and examples. Items consist of different
labels and different examples. These are space-efficient
items that reinforce discrimination of label and of example.
Production items
4. Production of label. Items consist of an example that
the learner labels (or answers questions about). Items
are good for strengthening production responses used in
transformation, noun sequences, or correlated-feature
sequence.
5. Production of example. Items consist of label. Learner
creates example to fit label. Items are good for
strengthening discriminations taught through any sequence.

Just as it is possible to create choice-response sequences for any concept, it is possible to create choice-response
worksheet items for any concept. They are generally easy to manage and they are usually unambiguous. Their weakness
is that they lead to fairly weak responses and of ten responses that are artificial. For example, the type of item that
presents a passage and requires the learner to choose the appropriate main idea is artificial with respect to real-life
applications. Rarely would one be in a position of listening to an argument or proposal and then receiving the
instructions, “What was the main idea of the argument? Indicate choice A, B, or C.” Obviously, the response is weak.

While the production-response items require stronger responses than choice-response items, they are often more
unmanageable and are mechanically hard to “grade.” They also introduce a variable that may not be present in choice
items. The learner who is able to read the item, understand it, and understand the discrimination, may fail the
production-response item if the learner cannot produce the response called for by the item. For example, the learner
may not be able to make up a reasonable main-idea sentence even though the learner understands what that sentence
should be and would be able to discrimination between a choice of sentences. Also, the learner may not draw
acceptable pictures of an oak tree, although the learner understands the item and can discriminate between oak trees
and others.

Illustration of Five Worksheet Item-Types

We will illustrate the five different items and then discuss how to design items for various purposes. The
illustration of the item types (Figure 16.1) provides an overview of the range of skill expansion that is possible through
worksheet applications.

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 Figure 16.1
Five Worksheet Item Types
1. Choice-of-Examples
The choice-of-examples form presents positive and
negative examples of the discrimination and a single Cup
label.

a.
Tree

Circle the CUPS.

1
b. 100 7 1 6 a+1 5 6 a—1 4 b. More than 1 5
101 5 1 6 a+0 6 5 a+1 3
5
Circle the fractions that are equal to more than 1. Less than 1 1
c. Cross out every adjective in this passage: 5
Equals 1 5
The oldest boy-wonder in the history of funambulistic
gyrations was Ron (“Lizard”) Torpy, who, at the ripe old c. Adjective Why should we go?
age of 14, was already a 45-year-old person.
Adverb That ship went under.
2. Choice-of-Labels
The choice-of-labels form presents one example and a Pronoun Where is she?
choice of labels.
Verb Why should we go?
a. Circle the correct label.
Tree Noun Running is tiring.

Bottle 4. Production of Labels

Cup This item form presents an example that is to be
Dish “described” by the learner.

b. Circle the correct label. a. What kind of animal is this?

More than 1
Less than 1 2
More than 2
Less than 2
2 b. How many people are illustrated?
How many legs are illustrated?
How many smiles are illustrated?
c. The underlined word is:
adjective noun pronoun adverb
c. What is happening in this picture sequence?
He turned around.

3. Match of Labels and Examples

This form presents more than one example and more
than one label.

a. Girl
2
d. Solve this problem: 3 R = 5
R =
Dog
e. Read the passage on page 112 and state the main idea.

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 Figure 16.1 (Continued)
5. Production of Examples
Instructions direct the learner to create or complete
the example.

a. Complete this sequence.

b. Draw the shadow of the girl.

c. Make leaf 1 a black oak, and leaf 2 a white oak.

1. 2.
7
d. Make up four fractions so that each is equivalent to 8 .

e. Make up a paragraph for this main idea: The railroads

opened the West.

Designing Worksheet Items

Here are the general strategies for item design:

1. Use production-response items if possible, to create items that are manageable and unambiguous.

2. If the discrimination cannot be adequately controlled as a production-response item, present it as a choice-

response item.

3. If the discrimination is difficult (involving either a complicated discrimination or difficult responses),

present it first as a choice-response (easier item) and then as a production-response (more difficult item).

Using Choice-of-Example Items

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 A series of these items may be patterned after initial-teaching sequences. The series can be used to help firm the
learner on attending to features of the examples. This item type is not particularly useful in reinforcing the use of the
label because the same response dimension is used with juxtaposed examples. Item order and form may be modeled
after those in the initial-teaching sequence.

If the learner has been recently taught slanted, not-slanted, we could introduce the following choice-of-examples
worksheet items.

Circle every slanted line.

The choice-of-example item is well designed for the non-reader because only one response dimension is used. The
teacher can therefore present the instructions verbally.

A variety of response conventions are available for a series. The instructions for the example above direct the
learner to circle the positive examples. A similar sequence could be presented with the instructions: “Cross out every
line that is not slanted,” or, “Make a box around every slanted line.”

Using Choice-of-Label Items

These items serve well as review items. Since they present a single example and a choice of labels, they help firm
the learner on the use of different labels. A single item serves as an event-centered series. Let’s say that the learner has
been taught the discriminations in front of, under, and in back of. Choice-of-labels items could be used to firm the
different labels.

The girl is on the chair.

The girl is under the chair.

The girl is in front of the chair.

The girl is in back of the chair.

Note that different labels are juxtaposed; therefore, the item type firms the discrimination of the labels. The choice-
of-label items are most appropriate for learners who can read. If the learner cannot read well, the reliability of these
items drops and the mechanical problems associated with presenting the items increase.

A variation of the choice-of-labels items can be created for basic discriminations. Each item presents two choices
as in the next illustration (positive label and negative label).

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 Circle the right words.
Slanted not-slanted

Slanted not-slanted

Slanted not-slanted

If the learner has been taught names that are similar or that describe members of the same class, use choice-of-label
items to firm the learner’s understanding of the names. Choice-of-label items are well-suited for reviewing nouns,
hierarchical classes, and other systems of names (color names, names of position or orientation, and names of parts of
speech).

Using Match-of-Label-and-Examples Items

These items are space-efficient and provide for firming of both the labels and the features of the examples. If the
learner has been taught the following concepts: slanted, vertical, and horizontal and has been firmed on these
concepts in simpler worksheets, we could use the matching format to review the concept:

Connect each word with the right line.

Horizontal

Vertical

Slanted

Variations of this item may be created so that more than one example of each type of line is presented.

The following is a possible item that could be presented after the learner has been taught the names of four
vehicles:

Connect the words with the pictures.

Boat

Truck

Car

Train

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 Figure 16.2 illustrates a match-of-labels-and-examples item that includes some “left-over” pictures. Variations in
which there are left-over labels or left-over pictures are better than those that call for one-to-one matching because they
tend to discourage the learner from using an “elimination” strategy. With one-to-one matching, the learner can skip
uncertain items. By eliminating the various items, the learner is often able to respond correctly to one or more uncertain
items.

Figure 16.2
Draw lines to the right picture.

The girl is
under the bed.

The plane is
over the tree.

The truck is
under a bridge.

The man is in
a box.

If we wished to teach the learner an elimination strategy, however, the one-to-one matching item is ideal. For
instance, we might present this item:

Draw a line from each word to the meaning.

Then write a sentence using the word indolent.
store Something that lets you talk to
people far away.

telephone Lazy

swim A place where you buy things.

happy What you do when you move in

water.

indolent How you feel when things go well.

By figuring out the known words, the learner discovers the meaning of indolent.
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 Match-of-labels-and-examples items can be used in situations for which choice-of-labels items are appropriate.
Generally, match-of-labels-and-examples items are more difficult than choice-of-label or choice-of-example items.

Using Production-of-Label Items

These items ask about the example. Either a single question or a series is asked. This item form can serve as a
worksheet version of an event-centered series of tasks.

What small class is this object in?

What larger class is it in?
How many stacks does it have?
What’s another name for its front?

It is possible to test comparatives by presenting a picture of more than one object and using a production-of-label
form.

Which person is taller?

Jane Jill

Here’s a more sophisticated item:

What element is the left atom?

What element is the right atom?
Which atom is heavier?
Which atom would not enter into a chemical reaction?
How do you know?

A variation of the production-of-label item can be used when no actual object is present. Possible questions are: “In
what year did Columbus discover America?” “How many trucks were in the school yard this morning?” These
questions ask about factual relationships.

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Using Production-of-Example Items

These items are useful in reviews. The items name things that had been taught; the learner creates or completes the
examples.

1. Draw a white oak leaf.

2. Make an acute angle.
3. Show particles in a rhombic formation.
4. Draw four more weights on the scale so that the scale
balances.

5. Draw a shape that is more elliptical than the one

shown.

6. Tank 1 Tank 2

Draw the water level in tank 2 so that the pressure at

the bottom of that tank is the same as at the bottom of
tank 1.
Place an O in tank 1 so that the O has more pressure
than A or B.
Place an X in tank 1 so that it has more pressure than
A but less than B.
Place a V in tank 2 so that it has as much pressure as
B.

Production-of-example items are very good for firming the learner on following instructions. The most efficient
instructions are “low-probability” items, which means that the response is not apparent unless the instructions are read.
Here are examples:

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 Make a box on the left end of the line.
Make a circle just below the right end of the line.
In the middle of the line, make a right-angled triangle.

In summary, choice-response items offer control. They permit testing a variety of skills. They are particularly
useful for giving the learner practice in discriminating different labels that are introduced (choice-of-labels, and match-
of-labels-and-examples). The choice items usually test the concept directly. The reason is that the items vary along a
single response dimension. Therefore, the choice items are best for immediate, direct testing of a newly-taught concept.

The production-response forms permit activities that parallel the verbal expansion activities presented in Chapter
15. Implied conclusion tasks can be created, as well as event-centered series that incorporate a variety of different
questions. The production forms are good for testing the learner on labels that have been taught, for providing practice
with transformations, for reinforcing the skill of following directions, for testing applications of rule-related
information, and for reducing the prompts associated with the initial presentation of the concept.

Using Worksheet Items in Programs

There are three basic uses of worksheet items: review, expansion, and integration.

For a given discrimination or set of discriminations, you may use worksheets to achieve one or more than one of
these functions.

Review Functions

The general strategy for using the worksheet items as a review is to substitute worksheet items for oral cumulative
review exercises. Note that such substitution is not possible for all concepts.

To design a review block, use item forms that provide a fairly direct test of the material that is being reviewed.
Include enough items to provide a reasonable test.

Here is a possible review item for the discriminations gymnosperm-angiosperm.

Circle the correct class name for each tree.

Dawns redwood Red cedar
gymnosperm gymnosperm
angiosperm angiosperm

Hemlock Live oak

gymnosperm gymnosperm
angiosperm angiosperm

Pacific yew Vine Maple

gymnosperm gymnosperm
angiosperm angiosperm

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The review block consists of six items. It provides a test of the higher-order class names. The items are choice-of-label
items.

The best forms for reviewing lower-order members is match-of-label-and-examples items or production-response
items that involve a non-symbolic referent.

Figure 16.3 shows two possible blocks.

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 Figure 16.3
Illustrations of Review Blocks
a. Connect each leaf with its correct name.

Elm

Vine maple

Black oak

Redwood

Hemlock

b. Write the common name for each leaf.

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 Many other review blocks are possible. Remember, review is necessary when different coordinate members have
been introduced into the set. The learner may confuse these members either because they are similar in appearance,
similar in label, or similar in both appearance and label. A review block should require the learner to use labels that
may be confused.

One way to present review blocks is to use a structured presentation with learners before having them work the
items independently. This precorrection approach assures that the students will first reach an acceptable criterion of
performance on the various tasks before working independently. It reduces the need for immediate feedback following
the independent work.

For the structured presentation, the teacher would simply go through each worksheet item:

“Don’t write anything. Look at the leaves in part 3 of your worksheet.

The first leaf: what kind? Next leaf: what kind?”

Ideally, the teacher would engage the students in several intervening activities before instructing them to do the written
items. To succeed, the students must remember the various items. The prompt of going over the items before the
independent work can be dropped after the students have performed successfully on a similar review block.

Expansion Functions

When items are used to expand a discrimination, they do not necessarily occur in a block of items of the same type.
These items would introduce new response forms or extend the range of examples that had been presented. Production-
response items are generally the best for expanding.

By using the same procedures used to create verbal implied-conclusion tasks, we can create good worksheet tasks
that expand a newly-taught concept. We start with a statement that uses the discrimination taught. For the concept rate
increase, we create an item by starting with a sentence that uses the discrimination, “The boy’s heart rate increased.”
We use this sentence to design an item:

Which activity caused an increase in the boy’s heart rate?

Other items are possible, such as, “If his heart rate is 72 when sitting, what do you know about the rate when he is
running?”

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 Another good expansion activity is one that presents an event-centered series of tasks.

Here is a statement:

The boy got up from his desk and began to walk briskly, then run.

What happened to the boy’s heart rate during the second activity and the third activity?

What is a “normal” rate of heartbeat?

If the boy is an athlete, how would his rate of heartbeat compare to that of a non-athlete?

Scheduling expansion activities. Figure 16.4 shows a schedule for worksheet activities that serve both review and
expansion functions.

Figure 16.4
Components Lessons

1 2 3 4 5 6 7 8 9 10

Discrimination A X X

Discrimination B X X

Discrimination C X X

Expansion Activities A A B C
1 2 3 4 4 2 3 4
Worksheet activities *
A A AB AB AB ABC ABC ABC
*1. Choice of example. 2. Choice of label. 3. Match of labels and examples. 4. Production-response responses.

Note the pattern that occurs with discrimination C. First it is verbally expanded. On the following day, choice-of-
label worksheet exercises are introduced. These require the learner to discriminate the newly-taught label (C) from the
familiar labels (A and B). On the third day of C, a matching task is introduced (involving A, B, and C). Finally, a
production-response expansion activity is presented on lesson 10. This activity requires the learner to apply knowledge
of labels and examples to production-response worksheet items.

Integration Functions

When we deal with single-transformation and double-transformation items, we can use blocks of items to firm and
even teach relationships. This type of integration is modeled after the double-transformation program.

• First the block of new items is separated from the block of familiar items.

• Then items from both blocks are mixed.

Figure 16.5 shows a single list of worksheet items that follows this general pattern. The new subtype being taught
is the algebra-addition form of 2 + [6] = 8; 3 + [5] = 8; 7 + [1] = 8. The familiar type is the regular addition form.

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 Figure 16.5
Problems

{
2 + = 8
New 7 + = 8
5 + = 8

Review { 5 + 3 =
2 + 6 =

{
2 + = 8
5 + 3 =
7 + = 8
Mixed
5 + = 8
2 + 6 =
7 + = 8

The transformation form may be used to firm weak discriminations. The three facts that are checked in Figure 16.6
have been missed four or more times. We can treat these facts as members of a new or unfamiliar set. The other facts
become the familiar facts in a shortened version of the standard transformation sequence.

Figure 16.6
Problem Errors
7 + 1 2
9 + 1 1
4 + 1 4
8 + 1 5
6 + 1 2
2 + 1 2
7 + 1 0
5 + 1 1
3 + 1 5
6 + 1 2
8 + 1 2
2 + 1 1

Figure 16.7 shows a sequence that integrates the unfamiliar and familiar facts.

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 Figure 16.7

{
{
4 + 1 2 + 1
New 8 + 1 8 + 1
3 + 1 1 + 1
3 + 1
Familiar { 27 + 1
+ 1 Mixed 6
4
+
+
1
1

{
8 + 1 7 + 1
3 + 1 9 + 1
Mixed
4 + 1 8 + 1
9 + 1

This worksheet can be used to firm any subtype that the learner has trouble with. A good procedure is to use
structured teaching with the new set before the learner works independently. The teacher tests the learner on the first
three items. The learner responds orally. Later the learner works all the items independently. (If the learner performed
quite poorly on the oral activity, the teacher could provide oral practice on all items in the sequence before permitting
the learner to work the items independently.)

The scheduling strategy for the integration parallels that of the double-transformation program. Figure 16.8 shows
the kind of schedule that would be appropriate for difficult discriminations. At first the worksheets present only the
blocks of familiar and blocks of transformed items. Later, the juxtapositions shift, so that the mixed part of the double-
transformation program is presented. The mixed items continue until the learner is firm on the various items within the
context of the mixed block. Finally, the individual items from the sequence are dispersed throughout the worksheet.
The steps in this program proceed from blocks that provide a great deal of juxtaposition prompting to contexts in which
there is no prompting for the individual items.

Figure 16.8
Components Lessons

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

A1B1

A Familiar set items (blocked).

B Transformed or subset items (blocked).
C Mixed set of familiar and transformed (blocked).
A1B1 Familiar and transformed items (distributed throughout parts of worksheet).

For integrations that require less careful scheduling, the same strategy would be followed (from pure blocks to
dispersed items); however, the procedure would be accelerated, requiring perhaps no more than three or four lessons
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for relatively easy integrations.

Transformation worksheets. Specific worksheet forms are well-designed for integrating transformation concepts.
More forms clearly show the across-set relationships. Let’s say that the learner has been taught the relationship between
words that end in consonant-y, and the same words with the ending iest. Here is a possible sequence:

Write the words in each blank.

funny funniest
happiest
lonely
hardy
clammiest
stingy

The layout makes the relationship obvious, a prompt that helps the learner who is beginning to work with the
transformation.

If the learner has been taught the relationship between percent, fractions with a denominator of 100, and decimals,
a three-column variation can be designed:

Fill in the blanks.

50
100 50% .50
27
100

4%

3.50

58%
475
100

.40
15
100

152%

.09

The transformation worksheet form shows the range of variation for each set (column) because the juxtaposed
examples differ greatly from each other. The across-set relationship (rows) make the transformation relationship clear
because the row items are minimally different.

The transformation worksheet form provides a great deal of prompting. Therefore, after items have appeared in the
transformation worksheet form, they should be moved into some other form. Remember, the transformation form

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prompts across-set understanding. Once this understanding has been demonstrated by the learner, more difficult, less
prompted worksheet item forms should be presented.

In summary, the integration procedures are modeled after the double-transformation program. The basic
procedures can be used for three purposes:

• To integrate a set of new members or facts with familiar facts.

• To teach a particular subtype.

• To firm particular items or facts that give the learner problems.

The transformation worksheet involves blocks or sets of items—a large block for the new set of discriminations, a
smaller block for the familiar discriminations, and a block that mixes the new with the familiar. As the learner masters
the transformation, the learner works on separate blocks. Each block contains only familiar or only transformed items.
Next, the mixed-item block is introduced. The mixed-item block continues until the learner is firm on items in this
context.

Summary

Worksheet items serve review, expansion, and integration functions. They are effectively used to reduce
juxtaposition prompting and stipulation caused by the initial teaching sequence.

Different worksheet item-types are better-suited for different jobs.

The choice-of-example worksheet firms a discrimination. It is a worksheet parallel of the initial-teaching sequence,
with the same response dimension (instructions) presented for a series of juxtaposed items.

The choice-of-label worksheet firms the label. This item form presents a choice of labels for a single example. The
learner must attend to the differences in labels to perform adequately. This worksheet form, therefore, serves as a good
review item.

The match-of-examples-and-labels form is a space-efficient means of providing practice both on the features of the
examples and on the features of different labels. The learner matches labels with names, a procedure that focuses
attention on both labels and names.

The production-of-label form is one that requires the learner to use an example as the basis for answering a
question. The example may be a picture, a passage, or a fact. These items do not provide the degree of control that
choice items do; however, they are very effective at firming labels. The reason is that they require the learner to
produce the label, not merely choose it from a group of candidates.

The production-of-example form requires the learner to create or complete an example. These items provide a good
test of the learner’s understanding of the features of the examples. However, they may call for responses that are
difficult or may create situations that are not manageable.

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 As a general rule, production-response items are better than choice-items for testing the learner’s understanding;
however, choice items are more manageable than production items.

When used as part of a program, worksheets augment other review and expansion activities. If there is a need for a
careful progression of item difficulty with the worksheet items, the schedule would first call for choice items, and later
for production items. The most efficient choice items are those that require matching. They are efficient because they
test a large number of labels and examples in a relatively small space.

A variation of the double transformation program may be used for integrating new subtypes or new transformations
that are taught. The strategy involves creating item blocks that correspond to the three parts of the double-
transformation program—one block for the familiar set, one of the transformed set (or new subtype), and one for the
mix of the two sets.

To integrate a transformation, first present the isolated blocks. When the learner performs acceptably, introduce the
block of mixed items, and drop the other blocks. Finally, disperse individual items throughout the worksheet. A special
worksheet form makes the across-set differences apparent for transformations. This form presents the familiar-set items
in one column and the transformed-set items in the next. The rows show how each item of the familiar set is modified
to create a corresponding member of the transformed set.

This form is space-efficient and instructive; however, it provides very strong prompts about how to create items.
Following the learner’s successful performance on it, it should be replaced by worksheet forms that do not provide as
much prompting.

We have many options when designing worksheet items. There is no exact formula for using them, except that they
should not be introduced unless there is reason to believe that the learner will perform well on them. Also, the
instructional effort should be designed to provide feedback soon after the learner has completed the worksheet. This
requirement is necessary when the learner first works with items that introduce a new discrimination. The need for
feedback diminishes as the learner’s performance improves.

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 Section VI

Constructing Cognitive Routines

Chapters 17, 18, and 19 present procedures for developing and sequencing cognitive routines. These routines are
designed to show the learner how to handle complex cognitive operations, such as solving fraction problems or
extracting the main idea from a passage. The philosophy of designing such routines or “algorithms” is the opposite of
“discovery-learning” philosophy. The structured-communication philosophy holds that using algorithms or guides for
solving cognitive problems will not stifle the learner, will not inhibit generalizations, and will not make the learner a
passive receiver of information rather than an active seeker of information.

A logical analysis of any discovery situation reveals why it is far inferior to a more structured format for
communicating a particular concept to the learner. The learner in a discovery situation is expected to discover some
sameness from exposure to concrete situations. The concrete situations have many features. Therefore, any observed
sameness across these situations: (1) may involve features that are irrelevant to the discrimination or concept; (2) may
be associated only with a particular subset of the examples, not with the full range for which the discrimination holds.

The first possibility above suggests that the presentation is capable of generating misrules. The second possibility
implies stipulation problems.

Consider the situation in which the child is using rods of different colors and lengths to solve “arithmetic”
problems. The teacher presents a green rod (10 units long). Next to it, the learner places two yellows (each 5 units
long). With the two yellow sticks positioned end to end, they are as long as the green rod. By repeating similar
“matching” exercises, the learner becomes facile at making strings of rods that are as long as the pattern presented by
the teacher.

A perfectly credulous interpretation of this situation is that the learner has learned the various numerical
equivalences. The learner has responded to the teacher’s instructions, “Show me how many fives it takes to equal
ten,” by placing two yellows next to the green.

There is no question about what the learner does. The question is whether it is possible for the learner to respond
in the manner observed without attending to numerical features of the examples. Stated differently, if the presentation
is consistent with more than one possible interpretation, the learner could learn misrules. For example, it is quite
possible for the learner to perform in the observed manner by attending to length—not number. (We do not have to
count to judge things the same length.) Once shown that the activity is a matching game, the learner may ignore the
instructions and make rows as long as the one presented by the teacher.

Even if the learner had the idea that the rods were associated with “number” (the idea that all fives are the same
color and length), the learner might suppose that the relationship holds only for measuring length, or that there is
some sort of special relationship between a color (yellow) and a number value (five). (Perhaps only yellow things can

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 have a value of five. Perhaps five refers to a comparative relationship—with the smaller object always yellow.)

To correct the logical communication problems of the discovery situation, we would have to make the learning
consistent with a single interpretation. For instance, we would have to show the learner how all examples of two-
fives-equal-ten are the same.

If we consider what relevant sameness obtains for all possible applications of two-fives-equal-ten, we discover
that it is the numerical operation, expressed as: 5+5=10, or 5x2=10. If this feature is the irreducible sameness
possessed by the various examples of the two-fives-equals-ten, this feature is the one we must teach—not color,
position, or “matching.”

The routine that we design to achieve this teaching must have the following properties:

1. It must permit the learner to handle any application of two-fives-equals-ten by responding in the same
way.

2. It must require responses that are unambiguously directed at the numerical features of the examples, not at
irrelevant aspects of the examples.

3. It must provide a behavioral sameness for every relevant sameness in the examples.

For efficiency in teaching an operation a well-designed routine will prove to be far superior to discovery because
of the inherent communication clarity provided by the routine.

If the goal of instruction is to teach the learner to discover a particular relationship, actual practice in discovery is
imperative. Such practice can be provided through a cognitive routine, however. The routine is demonstrated with
some examples. The learner then applies the routine to other examples. By encouraging the learner to make up
problems that may be solved by the routine and then testing them to see if the routine works, we provide the learner
with a framework for discovery. The idea is to find examples that do not work. If the learner is not able to find any,
the discovery validates the routine. If some are found, the limits of the routine are clarified.

Before leaving the question of discovery, we should emphasize one point. The learner becomes practiced in
discovery only through discovering. Therefore, any program that is designed to develop specific discovery skills must
have provisions for adequate practice. If the practice is reasonably well-designed, it will make the learner increasingly
proficient in discovering relationships.

As noted earlier, although the learner may benefit from unguided practice of physical skills, unguided practice of
cognitive skills cannot lead to the same types of benefits. The reason is that the physical environment provides
feedback on attempts to perform physical acts; however, the physical environment does not provide feedback about
cognitive acts.

The following are properties of any physical operation, such as throwing stones, opening a door, tying a shoe,
doing a backward somersault, performing a specific shot on a pool table, and any other physical operation.

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 1. All attempts that lead to the successful outcome of the operation have common, observable features which
are overt behaviors.

2. The overt behaviors completely account for the outcome of the operation.

3. The learner is prevented by the physical environment from achieving the desired outcome unless the
learner performs the necessary set of overt behaviors in the appropriate sequence.

4. Since the learner achieves the outcome only if an ad- equate sequence of behavior is produced, the
physical environment provides feedback on every trial—information that a set of overt behaviors are
inadequate for successful trials.

5. The learner may successfully practice the physical skill independently so long as the learner understands
the goal of the physical operation (for example, understanding that the ball is to go into the corner pocket,
that the pieces of the puzzle should fit together with none left over, etc.).

None of the five statements above apply to any cognitive skill. Consider reading or any arithmetic skill, such as
working a story problem.

1. For any cognitive operation, there is no common set of overt behaviors. One learner may read a word out
loud. Another may read the same word covertly. One learner may solve a story problem by writing many
numbers and signs on paper. Another learner may frown for a moment and say the answer to the question.
For all instances of a particular physical operation, a common set of overt behaviors is observed.

2. Since there is no necessary overt behavior associated with cognitive operations, the overt behaviors cannot
account for the successful outcome of the operation. When the learner throws a stone (a physical
operation), we can see that the outcome is completely explained in terms of the behaviors that we observe.
Furthermore, it is impossible for a learner to successfully throw the stone without producing observable
behaviors that completely account for the outcome. When we observe the learner who frowns before
giving the answer to a story problem, we would be ill-advised to use the observable behavior as the basis
for explaining the outcome. It is theoretically possible for any cognitive operation to be performed
covertly, with a minimum of overt behavior; therefore, the overt behavior cannot explain the outcome.

3. The physical environment does not prevent the learner from producing inappropriate responses for a
cognitive operation. The physical environment prevents the learner from achieving the outcome for a
physical operation unless the necessary set of steps is produced in the appropriate sequence. The learner
who does not produce the response of hitting the 5-ball in the correct spot will not achieve the goal of
sinking the 5-ball in the corner pocket. The physical environment exercises no such control over cognitive
operations. If the learner misreads a word, the physical environment does not prevent the learner from
saying the wrong word and does not give the learner the slightest clue that the response is wrong.
Similarly, the environment does not respond to the learner’s wrong answer to the story problem, or to any
other cognitive problem.

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 4. The physical environment cannot provide feedback on cognitive operations. The environment provides
feedback on every trial of a physical operation. If the ball did not go in the pocket, the behaviors that led to
this outcome were wrong. If the ball did go in the pocket, the learner received information that the
behaviors leading to the outcome were appropriate. No such corrections or reinforcement are provided for
cognitive operations. If the learner misreads a word, the physical environment does not “correct” the
response and may not even imply that it is wrong. Correctly reading the word does not lead to different
reinforcement.

5. Independent practice on new cognitive operations does not necessarily imply that the skills will improve.
The learner who understands the objective of a physical skill may safely practice the skill independently
because the learner is never really “alone.” The physical environment provides feedback on every trial and
this feedback has the potential for shaping the learner’s behaviors. The cognitive operation does not run a
parallel course. The learner may practice superstitious behavior, misrules, or inadequate strategies, but the
physical environment will not provide direct feedback and therefore cannot shape the learner’s
understanding.

Because cognitive operations do not parallel physical operations, we must create routines so that they have the
important properties of physical operations.

1. We make all necessary steps that lead to the desired outcome overt so that by producing the overt steps,
the learner would achieve the outcome.

2. We design these steps so that the set of steps works for every example that is to be processed by the
cognitive operation.

3. When the learner produces the behaviors that lead to the outcome, we provide feedback—corrections for
incorrect behaviors and reinforcement for appropriate behaviors.

By following these design requirements, we can create routines that communicate as effectively as physical
operations.

There are two tests for whether it is reasonable to design a cognitive routine for teaching a particular
discrimination:

1. Can the operation be expressed as a series of steps that holds for all instances of the operation?

2. Would less teaching be involved in teaching the various steps or in teaching the operation in some other
way—perhaps as a basic discrimination?

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 Cognitive Routines
Cognitive routines grow from a need for the presentation to be consistent with a single interpretation. For
most of the concepts we have dealt with, we can achieve a single interpretation if we simply show how
changes in the example lead to corresponding changes in the responses. The relationship between features of
the example and changes in the responses is not obvious for some concepts, which means that if we simply
show changes in the examples and corresponding changes in the response, our presentation may not be
consistent with a single interpretation. We can illustrate the problem with the following sequence.

Teacher Wording
59 is not a grommel.
49 is not a grommel.
48 is a grommel.
24 is a grommel.
80 is a grommel.
Is 81 a grommel? (No.)
Is 12 a grommel? (No.)
Is 8 a grommel? (Yes.)
Is 16 a grommel? (No.)
Is 15 a grommel? (Yes.)
Is 35 a grommel? (Yes.)
Etc.

Note that the sequence follows the basic format for choice-response discriminations. Minimum
differences are provided. So is a range of positives.

If we presented enough examples, we would certainly make it possible for the learner to discover what
grommels are. The presentation above, however, falls far short of making the meaning clear.

A correlated-feature sequence, such as the one that follows, makes the critical features of the
discrimination clear.

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 My turn:
Is 24 a grommel? Yes.
How do you know? Because there is a difference of 2
between the numbers multiplied
together.
Your turn:
Is 15 a grommel? (“Yes.”)
How do you know? (“There’s a difference of 2 between
the numbers multiplied together.”)
Is 16 a grommel? (“No.”)
How do you know? (“There is not a difference of 2
between the numbers multiplied
together.”
Is 35 a grommel? (“Yes.”)
How do you know? (“There’s a difference of 2 between
the numbers multiplied together.”)

The clarity of grommel comes from the answer to, “How do you know?” The answer discloses that: (1)
the arithmetic operation of multiplication is involved in determining whether the example is a grommel; and
(2) the relationship between the size of the numbers multiplied is relevant to whether the example is a
grommel.

Grommel is different from discriminations we have dealt with because the labeling of grommel depends
less on the immediately-observable features of the examples than on the procedure for creating grommels. A
particular number can be reached through an incredible variety of procedures. It is therefore necessary to
specify the procedure used and the steps that were taken.

The correlated-feature sequence does not fully satisfy our requirements for creating a presentation of
grommel that is consistent with only one interpretation. When the learner answers the questions, “How do
you know?” in sequences above, the learner describes the operation, but we do not know that the learner has
actually applied the operation and determined the numbers. The solution to the ambiguity problem is to add a
question to the pair of questions used in the correlated-feature items.

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 My turn
Is 24 a grommel? Yes.
How do you know? Because there is a difference
of 2 between the numbers
multiplied together.
What are those numbers? Six and four.
Your turn:
Is 15 a grommel?
How do you know?
What are those two numbers?
Is 16 a grommel?
How do you know?
What are those two numbers?
Is 35 a grommel?
How do you know?
What are those two numbers?

By satisfying any doubts about whether the learner is attending to the relevant details of grommel, we
have created a cognitive routine—a series of steps that solves any problem of a given type. Although the
cognitive routine for grommel above may not be the best designed one that we could create, it has the
properties of cognitive routines.

1. The routine consists of a series of more than two steps, which means that the learner is asked a
series of more than two questions or is given a series of more than two instructions each time the
routine is applied to a positive example of grommel.

2. The routine requires the learner to produce overt responses and these responses tell us whether the
learner is approaching each example appropriately.

These features of a cognitive routine imply its weakness and strength. The primary weakness is that the
juxtaposition pattern is not well-designed to teach component discriminations. The juxtaposition pattern of
component discriminations that is easiest for the learner is one that requires the learner to do the same thing
on juxtaposed examples. After successfully responding to question 1 in the routine, the learner must respond
to questions 2 and 3 before having another opportunity to respond to a new example of question 1. Questions
2 and 3 function as interference, pulling the learner’s attention from those variables associated with question
1. Because this juxtaposition pattern is not well-designed for initial teaching, the component discriminations
involved in a cognitive routine should be pretaught before the routine is introduced.

The routine’s principle strength is that the routine is a good vehicle for teaching the learner to solve a
problem by chaining a series of steps together. The cognitive routine is well-designed to teach the learner the
chain of steps because the routine stipulates the same series of steps for all positive examples.

As the grommel illustration points out, the cognitive routine assures that the learner attends to the
appropriate features of the examples. If all grommels are characterized by a multiplication relationship, the

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 learner must attend to this feature of the example and not to irrelevant features. We know that the learner is
attending to the appropriate features if the learner produces a series of responses. The series would be
virtually impossible to produce if the learner attended to irrelevant features.

Designing Cognitive Routines

Designing cognitive routines is difficult. The guidelines that we present for designing cognitive routines
provide the criteria for determining that one is well-designed. These general guidelines, however, do not lead
to the same degree of rigor that is possible with the component discriminations. The reason is that there are
many decisions that must be made in designing these routines. And at each decision point, there is more than
one acceptable solution.

Throughout the process of designing the routine, we must consider the routine in the perspective of other
routines. The routine we design is not an island. Rather, it will exist in the context of other routines and the
parts of the routine we design will often appear in other routines. The equal sign in addition problems has the
same function as the equal sign in subtraction and multiplication problems. We must honor this sameness. If
the parts are the same in related operations, we should treat these parts the same way in these operations,
thereby demonstrating that they are the same.

There are four steps that we follow to design a cognitive routine.

1. Specify the range of examples for which the operation will work.

2. Make up a descriptive rule that tells exactly what the learner must do to attack every example
within that range. (The same rule must hold for all examples.)

3. Design a task that tests each component discrimination mentioned in the descriptive rule.

4. Construct a chain composed of tasks that test the component skills or the “steps” in the operation.

Before presenting details of how to perform each of these design steps, we will illustrate the procedure
with two examples.

Illustration 1: Stop-Sound-First Words

The illustration deals with teaching the learner to read regularly-spelled words that begin with stop
sounds (can, bad, pin, top, prod, etc.).

Specify the range of examples. The range we will deal with is regularly-spelled words of not more than
four sounds (at least for the initial teaching of the routine).

Make up a descriptive rule that tells what the learner must do to attack every example. To read these

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 words, the learner first sounds out and identifies the part of the word that includes all but the first sound; then
the learner identifies the entire word by rhyming with the part that has been identified.

Design specific tasks for every component mentioned in the descriptive rule. We will illustrate the tasks
by referring to the teaching sequences in which the tasks would be used.

a. The descriptive rule refers to sounding out the word. Sounding out could be taught through a
series of verbal activities such as: “Sound out this word: op. Sound out this word: ip,” etc. The
same procedure could be used with written words (op, ip, etc.)

b. Identifying the word part could be initially taught as a verbal discrimination. “Listen: aaammm.
Say it fast. Listen: aaaat. Say it fast,” etc. Later, the learner could produce the same responses
with written words instead of oral words.

c. The descriptive rule refers to rhyming with the word part that had been identified. If the word is
pat, for instance, the learner would rhyme with at. The discrimination could initially be presented
as a task that presents written symbols.

f Rhyme with at (fat)

t Rhyme with at (tat)
p Rhyme with at (pat)

Note that all the discriminations named in the descriptive rule have now been translated into
discriminations that can be taught through specific tasks.

Construct a chain composed of tasks used to test the component skills. Here is a possible routine that is
composed of the component skills and that incorporates the same wording as that used with the component
skills.

Example: pat

Teacher covers p and point to at. “Sound out this part.”

Teacher touches under a and t as learer says: “aaat.”

Teacher: “Say it fast.”

Learner: “at.”

Teacher uncovers p. “This word rhymes with at.”

Teacher touches under p.

Learner: “pat.”

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 The routine provides overt behavior for every discrimination that is mentioned in the descriptive rule.
The routine, in other words, translates the descriptive rule into a chain of behaviors, which shows that the
learner is producing the discriminations named in the descriptive rule.

Illustration 2: Factoring Any Number From an Expression

Specify the range of examples. Any expression involving numerals or letters.

Make up a descriptive rule. To factor any value from an expression, create a fraction equal to one, with
the desired value in the numerator and denominator of the fraction. Rewrite the initial expression so that it is
multiplied by this fraction of 1. Then rewrite the fraction so the denominator of the fraction is distributed.

Design tasks for each component discrimination. The rule refers to creating a fraction equal to one with
the desired value in the numerator and denominator. Here is the discrimination:

Make up a fraction of one that has six in the denominator (six sixths).
Make up a fraction of one that has six j in the numerator (six j over six j).
Etc.

The rule also refers to rewriting an expression so that it is multiplied by a fraction of one. Here is the
beginning of a possible sequence:

Example Teacher Wording

3–4j My turn to show how to rewrite 3–4j
when we multiply by four-fourths:
4 (3–4j)
4
3–4j Your turn. Show how to rewrite 3–4j
when we multiply by 4N over 4N.
4N (3–4j)
4N

Finally, the descriptive rule specifies that the fraction of one is to be “distributed.”

Example Teacher Wording

4 (3+2–5+1) Here’s how we rewrite the four-fourths:
4
(
4 3 2 5 1
+ – +
4 4 4 4 )
6
6 (3+2–5+1)
Here’s how we rewrite six-sixths:
(
6 3+2–5+1
6 6 6 6 )
Construct a chain composed of tasks used to test component skills.

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 Example: 3+2–5+1
Teacher: We’re going to factor 7 from this expression.
The first thing we do is make up a fraction of
1 that has 7 in the numerator. What fraction?
(Signal.) 7
7
Teacher: 7 (3+2–5+1)
7
Now we rewrite the fraction. Do it.
Learner:
(
7 3+2–5 1
+
7 7 7 7 )
Teacher: You factored seven from the expression.

The routine above could be changed after one or two examples have been processed. The changes would
require the learner to do more of the operation. For instance:

We’re going to factor 7 from this expression. So what’s the first thing we do? (Signal.) “Make up a
fraction that has seven in the numerator.”
What fraction? (Signal.) “Seven-sevenths.”
Now what do we do? (Signal.) “Multiply the expression by seven-sevenths.”
Do it. (Learner multiplies.)
Now we rewrite the fraction. Do it.
(Learner writes.)

The two routines we have illustrated for rhyming differ in many respects from the fraction routine. The
rhyming routine is designed for a relatively small range of words. The factoring example applies to any
expression, not merely those that contain multiples of the number that is to be factored. The rhyming routine
directs the learner to produce sounds. The routine for factoring directs the learner to answer questions and to
write responses.

Despite their differences, the routines are the same from a construction standpoint. The starting point is a
descriptive rule that tells about the essential steps needed to attack all examples that are to be processed
through the routine. If the descriptive rule is accurate, the resulting routine should provide behavioral steps
for each discrimination named in the descriptive rule.

Identifying the Range of Examples and the Descriptive Rule

The simplest procedure for determining the range of examples to be covered by the routine and the
descriptive rule is to begin with the broadest possible range of concrete examples that could possibly be dealt
with by a single operation, and then use the examples to determine the descriptive rule. Following are the
steps involved in this procedure. (Note that the procedure is similar to the subtype analysis described in
Chapter 10. The difference between the two procedures is that the present one is designed to identify what is
the same about all the examples within the range.)

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 1. Arrange a series of examples in a way that would communicate sameness. Juxtapose examples
that differ greatly and that cover the full range of possible variation for an operation. Try to
include all subtypes of the operation.

2. Make a list of operational samenesses that are shared by all concrete examples in the list. An
operational sameness would refer to a common detail of all examples that must be responded to in
a specific way. For instance, “All problems have denominators.”

3. Evaluate the list of samenesses. If there is a statement of sameness for every detail of each
example that would logically have to be processed to solve the problem, the list of samenesses
provides a basis for an adequate descriptive rule. If important details for some examples are not
referred to by the statement of sameness, the basis for an adequate descriptive rule does not exist.

4. Adjust the set of examples if the statements of sameness do not include all examples. The simplest
adjustment is to eliminate the examples that are not sufficiently described by the statements of
sameness.

5. Make up a descriptive rule that is based on the statements of sameness.

Illustration: Sounding out simple words. If we follow the steps above for dealing with sounding-out
beginning reading words, we would start by juxtaposing beginning words in a way that implies sameness. For
our present purpose, we will limit the words to those having no more than four letters. This is a sample of
possible words we might start with: from, me, and, belt, if, meet, ate, any, then.

We now make up a list of operational samenesses:

• All have letters.

• The sounds for the letters roughly correspond to the left-to-right arrangement of letters.

• For each letter or pair of letters, some sound is called for.

This set of samenesses does not imply a descriptive rule that would deal with every critical feature of
each word. Therefore, we must either modify some of the examples or eliminate them from the set.

If we eliminate the two-letter combinations that make a single sound (ee and th), and eliminate those
words that do not have a standardized relationship between the sound that is pronounced for a given letter
(from, me, any, ate), we have a much smaller range of variation. The convention that we will arbitrarily
adopt is that of using only short-vowel words, e.g.: and, belt, if.

We can add more words of this general type to the set. However, even if we do not, we are now able to
make a more precise set of sameness statements.

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 • All words have a standard sound for each symbol.

• All sounds are produced in order, starting with the sound for the left letter.

These two statements of sameness imply a descriptive rule for sounding out that will account for all important
details of each word:

To sound out the word, start with the left letter and say the specified sound for each letter in order.

If we wish the learner to hold every sound of the word before going on to the next sound, we have to
modify the set further. The reason is that it is impossible to hold sounds like b. We would therefore have to
eliminate belt from the set (and any other word that begins with b, c, d, g, h, j, k, p, q, t).

The set would now consist of two-letter, three-letter, and four-letter words that begin with continuous
sounds only (a, e, f, i, l, m, n, o, r, s, u, v, w, y, z), and that do not have stop sounds except as the last letter of
the word. The examples in this set have an additional sameness: All sounds except possibly the last sound can
be held.

We can now modify the sounding-out operation:

To sound out the word, start with the left letter, hold the specified sound for that letter, and repeat the
procedure for each remaining letter in the word except possibly the last letter (which may not be held).

The rule now describes an operation that deals with all relevant aspects of each example. The operation
that would be used for the last set of words eliminated from the larger set (those that begin with stop sounds)
could be processed through the rhyming operation.

Illustration: Factoring. By following the same procedure for determining an operation for factoring, we
begin with a list of examples sequenced to show sameness:

1
a. / 4 + 1/ 2 + 3/ 4

b. A2 + 2AB + b2

c. 12 – 9 + 3 – 6

d. – 5 – 6J

12
e. /3B + 5 – 1/2J – 6R

Next we make up a list of statements that hold for all examples:

• Each expression can be written as the product of a fraction of one times the original expression.

• The denominator of the fraction can be distributed so that it appears in the denominator of each

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 term.

These statements of sameness imply an adequate descriptive rule: To factor a number value, first write
the expression as a product of a fraction of one and the original expression. Then distribute the denominator.

Since this operation will apply to all expressions from which a single value is to be factored, we do not
have to eliminate any example types from the original set.

The preceding illustrations point out some important facts about designing cognitive routines. A cognitive
routine works only for examples that share a common set of features. The routine may cover a very broad
range of examples or a narrow one. We can design a routine for rewriting fractions and decimals that involves
only positive numbers or that involves positives and negatives. We can design a routine for decoding words
that applies to all English words, one that applies to a specified set of words, one that applies to all simple
regularly-spelled words, or one that applies to some other set of words.

When initially designing cognitive routines, we should try to identify the broadest possible range of
application for the routine.

The rule should contain two sections. The first tells about the goal of the rule. The second tells about the
overt behavior that the learner must engage in to process every example. Start the rule by referring to the
goal:

To decode all regularly-spelled simple words . . .

To convert fractions into decimals . . .

To find the angle of reflection . . .

The second part of the rule should specify at least two overt things that the learner must do. Initially, you
may write the rule in a way that describes only one thing that is done, such as: To make a grommel, multiply
two numbers that have a difference of two. Certainly this rule implies that you must identify the numbers
before you multiply them. To make it easier to design a routine, recast the rule so that it expresses at least two
steps:

To make a grommel:

1. Select a pair of numbers that have a difference of two.

2. Multiply them together.

The requirement that the rule must specify at least two overt behaviors is a logical one. If you can specify
only one behavior that is needed for all examples (“To read a word, you say it”). The skill is most properly
taught as a discrimination, not as something processed through a cognitive routine.

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 The descriptive rule does not necessarily contain the words that the learner will encounter. The rule is
for the instructional designer. It provides a dear list of skills that we must teach. For example, a descriptive
rule for reading regularly-spelled words may refer to “beginning with the left letter of a word.” This phrase
does not imply that you teach labels, “begin,” “left,” “letter,” and “word,” to the learner. However, the
routine must account for overt responses that show whether the learner is beginning with the left letter of a
word. The wording in the routine may be, “Touch the starting point.” The learner touches the left letter. The
wording may be: “Touch the beginning of the word,” or “Name the letter that is at the beginning of the
word,” or “Tell me the sound of the left letter.” All are acceptable, because all satisfy the requirement of
providing the learner with a dear signal to respond to the left letter, and all lead to a response that leaves little
doubt about whether the learner understands the discrimination of “the left letter.”

Constructing a Routine for a Set of Examples

Perhaps the greatest mistake that designers make when trying to design routines is to start with some sort
of task that is to be analyzed. Although it is possible that somebody has worked out the best possible
procedure for solving problems, we should not begin with this assumption. We should recognize that
cognitive operations are not like physical ones, that the procedures people have developed for solving
different types of problems are not necessarily the only ones possible, and that even behaviors that may seem
perfectly essential may not be essential if we change the operation. For example, the process of “borrowing”
seems to be necessary for solving column subtraction problems like:

52013
– 31425

The following is a method for solving the problem that does not involve borrowing. The method is “the
mirror” method.

522 = 588
52013
– 31425
20
(Answer) = 20588
Solving the problem through the mirror method involves subtracting one value from another in each
column, writing the answer nearest (above or below) the smaller number in that column, adding 1 to the
bottom number of the next column if the answer for the preceding column is written above, and converting
the mirrors on top to the corresponding values on the bottom, Subtract 3 from 5 in the one’s column and write
it above the 3. Then add one to the bottom number of the tens column, making the value 3. Now do the same
thing in the tens column, starting with 3 and subtracting 1, and writing the answer closest to the 1. After all

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 numbers have been computed, convert all mirrored numbers on top to their real value. 9 is the mirror of 1; 8
is the mirror of 2; 7 is the mirror of 3; etc.

The operation requires carrying, not borrowing. Whether or not we would judge it superior to some other
operation depends on how well it fits in with other skills we teach. Possibly, this method would never be
recommended. The point, however, is that we cannot start out by accepting conventional borrowing as a
given operation. The various methods of borrowing are merely algorithms for converting a very difficult
discrimination (saying the answer to the problem) into a series of overt behavior steps that yield the answer.

Designing Tasks for Components Named in a Descriptive Rule

Once we have developed a descriptive rule that seems adequate, we must transform the rule into a series
of behaviors. The rule tells us what the learner is to do. In translating the rule into behavior, we can either try
to design the cognitive routine and then identify the components, or we can first determine how we might
teach the various components named in the rule. If we first specify the tasks for teaching the components, we
create specific wording that will then be used in the routine we create. Designing the final routine involves
simply “chaining” the component descriptions together. Although the approach of identifying components
first and then chaining them may be somewhat mechanical, it is reliable.

To use this approach, we translate every discrimination named in the descriptive rule into a task that tests
the discrimination. Each task calls for an overt response that leaves little doubt that the learner is
appropriately attending to relevant details of the example and operating on them in the prescribed manner.

Responses. When we design specific tasks for the components of the descriptive rule, we should try to
use strong responses. The strongest responses are those that actually require the learner to do something
rather than describe it. For verbal responses, production-responses are stronger than choice-responses.

Choice responses. These responses are the weakest verbal responses because:

1. The probability is higher that the learner will answer correctly by guessing, since the number of
choices is severely limited.

2. Spurious transformations are prompted since the same task leads to answers that may be
minimally different (e.g., reversals). (For example, the task, “Tell me when I touch the left of the
word,” may create reversals because the only difference in examples is where the teacher touches.
The task, “Touch the left of the word,” stipulates the correct behavior and reduces the possibility
of spurious transformations.)

The great single advantage of the choice-response item is that it is manageable and frequently
unambiguous. Here is a production-response item presented in working a long-division problem. “Where do I

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 write the 3?” The answers range from “right there,” to “under the 7.” When working with a group, this range
of response variation creates management problems. The problems can be reduced by several techniques,
such as a more precise production-response task. “Under which number do I write the 3?” A choice response
alternative—but not necessarily the most desirable—might be, “Tell me where to write the 3, under the 7 or
under the 0.”

Another possible advantage of choice-response tasks is that we can construct a yes-no choice for any
discrimination or relationship. Therefore, if we have trouble figuring out how to ask about a particular
discrimination, we can construct a yes-no task or a choice-task as a last resort.

In summary, the advantages of choice-response tasks are that the tasks are manageable, they can be
designed so they are clear, and they are possible options for any discrimination or relationship.

Production-responses. The advantages of production responses are that they do not promote spurious
transformations; they provide for very “strong” responses (because they require the learner to create either the
label that is being taught or an example for the label); and they are logically capable of testing more
information with fewer examples. If we were to test the learner’s understanding of rhyming through yes-no
tasks, we would first have to test on words that rhyme with one ending, then on words that rhyme with
another, and so forth. To test each ending would require at least three items. If we used production responses,
a single item for each ending would provide much information. (“Rhyme with am and start with the sound
ssss . . . Rhyme with at and start with sss . . .” etc.)

The most serious problem with production responses within the context of a cognitive routine is that the
learner may not be able to produce them quickly. Responses to “Write it,” “Draw it,” “Do it,” “Say it,” “Tell
where,” “Tell how,” and other production-response items may require five seconds or more. This pause
seriously affects the flow of the cognitive routine. During the time that the learner “Writes it,” the link of the
writing step with the rest of the routine may have been lost. If the learner is to learn a chain of behaviors,
these behaviors should occur in an obvious sequence. The sequence becomes less obvious if the individual
responses are long or require firming by the teacher. The problem of the long production response is less
serious if the routine has only three steps. If it is a relatively long routine, long responses become increasingly
troublesome.

Guidelines for using the names that appear in the descriptive rule. If we were to make up a cognitive
routine for determining whether a fraction is more than one whole, we might state the rule so that it refers to
numerator and denominator. If the learner is not required to use these words in other contexts, there is no
particular reason to teach them. The learner can learn the discrimination about the fractions by referring to
the top number and the bottom number—familiar terms. The introduction of new vocabulary simply
lengthens the amount of teaching that is required for the discrimination. It also introduces another possible

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 source of error. Unless the new words are thoroughly taught, the learner may begin to reverse them, a
problem that will not exist if the familiar terms top and bottom are used in the task.

Sometimes new vocabulary may be necessary. For example, if our rule had to do with teaching the
learner to express fractions such as:

A  x  A  x  A
= A!
A

we might find it necessary to refer to the base number and the exponent. If we try to refer to the base
number as the “big number” and the exponent as the “little number” we may create confusion when the base
number is smaller than the exponent (e.g., 45). If we try calling the base number the “lower number,” we may
have trouble communicating with the learner when we introduce problems that have bases above and below
the fraction line:

 2!
 3!

(The 3 is an upper and a lower). Granted, there may be solutions to these problems. However, if these
solutions seriously limit the range of examples presented, require new teaching, or lead to contrived wording
that will later have to be dropped, the conventional terms base and exponent may present the most
straightforward solution.

In summary, if it is possible to design discrimination teaching that uses only familiar names, do so. If
communication problems result or if the teaching become contrived, teach the words that are in your
descriptive rule. Note, however, that the determination about using words does not come from a reference
book or from an examination of how the operation is traditionally taught. It comes from trying out the
simplest wordings and noting the problems that result. The problems encountered suggest whether more
precise terms should be introduced.

Responses that Describe Behavior. Here is a task that calls for a description of behavior:

Teacher: What do you do first when reading a word?

Student: Start with the left letter of the word.

Although the response was correct, we have no compelling reason to believe that the learner carries out
this response when reading a word. The task provides only a verbal response. A more convincing
demonstration would be provided if the learner showed the starting point. Furthermore, if the learner
produced this response, there might be no reason for the learner first describing what to do.

For some situations, we may want the learner to produce descriptive responses like the one above. As a
general procedure, we would avoid descriptive responses unless they set the stage for steps the learner is to

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 carry out when working independently. If we decide to include descriptive responses, an additional step must
follow, in which the learner carries out the behavior described.

Teacher: What do you do first when reading a word?

Student: Start with the left letter of the word.
Teacher: Do it.
(Student touches left letter of word.)

Constructing tasks for component discriminations. The wording that will direct the learner in each step or
task determines how each component discrimination is to be pretaught. If we refer to the top of the fraction
and the learner is to respond to the top of the fraction, we can require the learner either to identify the top or
touch the top. “Tell me the part of the fraction I touch . . .” This task could imply a choice-response sequence.
If the descriptive rule says that the learner is to make fractions of one with specific values in the denominator,
we could present a single-transformation sequence. “I’ll show you numbers that are on the bottom of
fractions that equal one. You tell me the fraction of one for each bottom number.”

The descriptive rule for the mirror-method of borrowing suggests that the learner finds the final answer
by converting mirror numbers into “regular” numbers. We are not required to refer to “mirror numbers.” We
can call them anything we wish. A good name is one that prompts their relationship to regular members. We
may decide to call them wrong-way numbers, because this name suggests that they are not “right” in their
present form.

Once we decide on the wording for the task, we can create a sequence for preteaching the relationship
between wrong-way numbers and their counterparts. One procedure would be to make a “folding” number
line with 5 in the middle. When folded, the 9 would be under the 1, the 8 under the 2, the 7 under the 3, the 6
under the 4, and nothing under the 5. The dotted segment shows the number line when it is unfolded. The
arrow shows the direction this part moves when folding.

0 1 2 3 4 5 6 7 8 9 10
6 7 8 9 10

Teacher presents the folded number line.

Teacher: The wrong-way numbers are those that are under
the other numbers.
My turn: What’s the wrong-way number for 3? 7.
What’s the wrong-way number for 2? 8.
Your turn. What’s the wrong-way number for 9?
What’s the wrong-way number for 5?
Etc.

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 The sequence above presents the relationship through a single-transformation sequence. Another option
is a correlated-feature sequence. This sequence adds a question.

Teacher: My turn. What’s the wrong-way number for 3? 7.

How do I know? Because 3 is over 7.
My turn again. What’s the wrong-way number for
4? 6.
How do I know? Because 4 is over 6.
Your turn. What’s the wrong-way number for 6?
How do you know?
Etc.

Following the introduction, the learner would be required to perform when the number line is unfolded.
For corrections, the teacher would fold the line and show the answer.

There are different ways of presenting the wrong-way discrimination. A possible approach is this:

0 1 2 3 4 5 6 7 8 9 0

Teacher: (Touches under 5 with chalk and draws a line that

looks under 4 and ends at 3.)
1 2 3 4 5 6 7

My turn. What’s the wrong-way number for 3?

(Draws loop from 5 to 6 to 7.) 7 is the wrong-way
number.
1 2 4 5 6 7 8 9 0

(Draws one more loop to 2.) My turn. What’s the

wrong-way number for 2?
(Adds one loop to right.) 8.
Your turn. What’s the wrong-way number for 8?
(Signal.) 2.
(Loops from 5 to 4.) What’s the wrong-way num-
ber for 4? (Signal.)

The wording of the task could be changed to be more consistent with what the learner will do in the
subtraction problems, which is to convert wrong-way numbers into right-way numbers. The wording might
therefore be, “If 4 is a wrong-way number, what’s the right-way number?” Regardless of the specific
wording, the product of translating a discrimination that is named in a descriptive rule is a step for a cognitive
routine. That step will have the same wording as the task that is created to teach the dis- crimination named in
the descriptive rule.

Component skills that are modified. Sometimes, the descriptive rule specifies behaviors that are not well-
designed for an introductory cognitive routine. As noted above, the steps in a cognitive routine should be
chained without interruption, particularly if the routine is relatively long. Responses that interrupt the
chaining should be eliminated from initial routines.

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 The descriptive rule for mirror numbers indicates that the learner will convert wrong-way numbers by
writing their right-way counterparts. The writing is probably not desirable for the initial teaching; however,
we should try to make the component that deals with converting the wrong-way numbers as similar as
possible to the component that will appear in the final routine. Here’s a possible setup:

2 6 9 1 0 5 3

Teacher: The numbers above the lines are wrong-way

numbers. For each wrong-way number, tell me the
right-way number that goes below the line.

After the learner responds to each number, the teacher writes it below the bottom line. The pacing is
maintained. The learner receives models of where the numbers will be written. And the tasks are presented in
the same context that will be presented by the cognitive routine. When the routine for wrong-way numbers
first appears, the learner will tell and the teacher will write. Later, the learner will write the converted
numbers.

Chaining the Discriminations Together

Usually, all discriminations referred to in the descriptive rule should be pretaught, then chained together
in a routine.

1. The wording used in the cognitive routine should be as close as possible to the wording used in
the teaching of the component discriminations.

2. The set of steps designed from the descriptive rule should provide a good test:

a. Some items should test on the mechanical details that might interfere with the learner’s
performance.

b. Some items should test whether the learner attends to the appropriate features of example and
responds appropriately.

c. All items should provide “strong” responses (doing rather than choosing or describing) if
possible.

Wording. If the wording used to teach numerator is “top number,” the same wording would be used in
the cognitive routine. If mirrored numbers are referred to as “wrong-way numbers” they would retain this
designation in preteaching and in the routine. If the discrimination of beginning at the left has been labeled
“touching the starting point,” the same wording would be used in the cognitive routine.

If we discover that the wording is ambiguous, awkward, or inaccurate when it is used in the routine, we
change it both in the routine and in the teaching of the component discriminations.

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 Order of steps. After the component skills have been taught, the idea is to arrange steps into an order that
will permit the learner to move through various examples smoothly.

1. Ideally, the first steps should deal with mechanical details.

2. The remaining steps should reflect the order of events suggested by the descriptive rule.

When we put mechanical details first, we first require the learner to read the problem or identify symbols
that are involved in the routine. By dealing with mechanical details first, we reduce the possible causes of
failure on later steps in the routine. We therefore increase the potential for diagnosing the learner’s specific
problems if the learner does not perform in parts of the routine. When we know that the learner is accurately
decoding the problem, we know that misidentifying the symbols cannot contribute to failure in working the
problem. If we are not sure whether the learner is accurately decoding the problem, we cannot tell whether
failure on later steps results from misidentification, from faulty understanding of an operation, or from a
combination of problems. If we are reasonably sure that the learner is firm on mechanical details, we should
not begin the routine with such detail.

Ideally, we should design the remaining steps (those that deal with the procedure) so they form a kind of
acting-out of the descriptive rule.

Let’s say that we have designed the descriptive rule: “To find out if the fraction reduces to a whole
number, count by the bottom number. If you say the top number, the fraction reduces.”

Assume we have taught counting by different numbers, identifying the bottom number and identifying
the top number of fractions. Now we must chain the components together to achieve the operation called for
by the rule. We start with mechanical details.

Here is a possible beginning of the cognitive routine:

Example Teacher Wording

24 What’s the bottom number?
6 What’s the top number?

Now the operation follows. According to the descriptive rule, we want the learner to count by the bottom
number and determine whether the top number is said in the process. The top number is critical, because the
learner must compare it with the numbers said when counting by 6. Therefore, we could name the top number
in our instructions to prompt the learner. “Count by the bottom number and see if you say 24. Get ready . . .”

The rest of the routine will involve drawing a conclusion about whether the fraction reduces.

“What do you know about the fraction?” (It reduces to a whole number.)
“How do you know it reduces to a whole number?” (Because I said the top number.)

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 A variation of this routine can be used with various fractions. Here is the entire routine with italicized
parts indicating words that change when different examples are introduced.

1. What’s the bottom number?

2. What’s the top number?

3. Count by the bottom number and see if you say 24. Get ready.

4. What do you know about the fraction?

5. How do you know it reduces to a whole number?

The 24 is italicized because any number could be used as the top number of the fraction. Also, the
reduces in step 5 is italicized because this step might say, “How do you know it does not reduce to a whole
number?”

Although the routine could be used with any number, it is not as smooth as it might be. The question,
“What do you know about the fraction?” is possibly vague. Furthermore, the routine does not contain very
good provisions for teaching the correlated-feature relationship between the top number of the fraction and
whether the fraction reduces. To correct these problems, we may add a step to the beginning of the routine.

1. If you say the top number when counting by the bottom number, what do you know about the
fraction? (It reduces to a whole number.)

2. What’s the bottom number of this fraction? (Six.)

3. What’s the top number of this fraction? (Twenty-four.)

4. Tell me the number you’re counting by.

5. Count by that number and see if you say 24. Get ready . . . (6, 12, 18, 24)

6. What do you know about the fraction? (It reduces to a whole number.)

7. How do you know it reduces? (Because I said 24.)

Step 6 would be easier if it were changed to a choice-response task: “So does this fraction reduce or not
reduce?”

Comparing the Routine with the Components

Now that the routine has been designed, we have a final check on the various component skills that
should be pretaught. We see that step 1 is a question about correlated features. Step 2 and step 3 involve
numeral identification (which skill would be taught through noun sequences that introduce the various

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 numerals). Step 4 is a correlated-feature relationship that is not stated. “If the number is on the bottom, you
count by it.” Step 5 calls for another correlated-feature response. Steps 6 and 7 present a correlated-feature
relationship.

The general order of events in this routine divides into two parts. The first is the mechanical (steps 1
through 3). The second part is the application (steps 4 through 7). In the first part, the learner describes the
criterion used for determining whether a fraction reduces and identifies the numerals involved in the
particular problem. In the second part, the learner applies the descriptions and the identification to determine
whether the fraction reduces.

The relationship that we added as step 1 of the routine implies possible preteaching. The relationship
might be taught through a series of sequences, each showing how the counting operation applies to fractions
with a particular denominator.

Example Teacher
19 I’ll tell you about fractions that have the bot-
6 tom number of 6.
My turn: When you count by the bottom
number 6, you don’t say 19.
So what do you know about the fraction 19
over 6? It doesn’t reduce to a whole number.
How do I know? Because you don’t say 19
when counting by 6.
18 My turn: When you count by the bottom
6 number 6, you say 18.
So what do you know about the fraction 18
over 6? It reduced to a whole number. How
do I know? Because you say 18 when counting
by 6.
20 Your turn. When you count by the bottom
6 number 6, you don’t say 20.
What do you know about the fraction 20 over
6?
How do you know?
Etc.

The same type of correlated-feature sequence would be repeated with fractions having denominators of 4,
9, 5, and other numbers the learner can count by.

Testing the Learners Understanding

If the cognitive routine is adequately designed, it provides overt responses for the key discriminations
named in the descriptive rule. By observing how the learner performs these overt steps, there would be little
doubt about whether the learner was attending to relevant details of that example. If we observed the learner
performing on an example of a fraction that reduces to a whole number, for example, we might be tempted to
conclude that the learner understands the operation; however, the learner’s performance does not suggest how

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 the learner would perform when presented with a fraction that does not reduce. We need a test of more than
one application.

The series of examples that we use to test a routine should parallel the series of examples used to test
simpler discriminations.

We test the learner’s understanding of simple discriminations by requiring the learner to respond to a
variety of examples, arranged so that there is no predictable relationship between examples. (These examples
are designed to test sameness across a range of possible variation.) If the learner performs appropriately on
the set, we judge that the learner understands the discrimination.

In following the same procedure for the cognitive routine, we test the learner by requiring the learner to
perform on a variety of examples. These are ordered so that there is no predictable relationship from example
to example. The learner carries out the same steps of the routine on different examples. Successful
performance implies that the learner understands the concept.

Summary

A cognitive routine is a creation designed to make problem-solving easier for the learner.

Cognitive routines deal with discriminations; therefore, anything presented through a cognitive routine
could be presented as a simple discrimination. The communication will be better, however, if the problem-
solving is designed as a series of overt steps that lead to the solution.

Our guide is physical operations. We construct the routines so they incorporate the three primary features
of these operations—the functional nature of the various steps, the essential overtness of steps, and the
feedback on every application. We try to design the steps so that each leads us closer to the outcome, and so
that each is overt.

The steps for designing a cognitive routine are:

1. Specify the range of application for the routine.

2. Construct a descriptive rule that indicates what the learner is to do.

3. Indicate a task that tests each discrimination named in the descriptive rule.

4. Chain the tasks designed into a routine.

Start the chain with mechanical details. Then present the events that parallel those described by the rule.

The easiest way to evaluate the overtness of the routine is to play devil’s advocate. Ask, “Would the
learner’s successful performance on examples of that routine convince me that the learner is attending to the

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 appropriate details of the example and is operating on them in a way that is called for by all examples of the
routine?” If the answer is “Yes,” the routine is adequate. If the answer is “No,” the routine must be
redesigned to show which details the learner is processing and to show what the learner does to create a
solution to the problem.

If we cannot specify tasks for parts of the rule that we have constructed, either we do not understand the
discrimination, or there is no discrimination. The intent of specifying the tasks for everything named in the
descriptive rule is not so much to articulate the tasks involved in the preteaching as it is to force us to specify
each discrimination as a task. That task must either be a basic form or a joining form.

The great strength of the cognitive routine is that it stipulates a series of steps that, if followed, will
appropriately process all examples of a given type.

The routine’s greatest weakness is that it is not well-designed for initial teaching, because the pattern is
juxtaposed if the routine consists of different steps. This weakness suggests that the component skills that are
to appear in a cognitive routine should be pretaught, according to the general rules for introducing
discriminations into a set.

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 Illustrations of Cognitive Routines
Chapter 17 presented a general procedure for designing cognitive routines: start by specifying the range
of examples; specify a descriptive rule that accommodates any application within this range; design tasks for
the component skills named in the rule; and construct a routine composed of the component tasks. Chapter 17
demonstrated the procedure with several routines, but it did not show the range of variation in routines. The
purpose of this chapter is to demonstrate the range. We will look at some routines that involve the prediction
of physical outcomes and the verification of these outcomes. We will deal with routines that are designed for
the sophisticated learner who has reasonable verbal understanding and the motor skills that are involved in
the routine. The focus of these routines is on providing a list of things the learner is to do, not on providing
actual teaching for the component behaviors. The words that appear in the descriptive rule often appear in the
routine that we design for the sophisticated learner. At the other end of the continuum is the unsophisticated
learner, who has neither a great deal of verbal knowledge nor the motor skills called for by the routine. The
emphasis for this learner is on communicating the actual behavior. The routine probably will not contain the
language that appears in the descriptive rule.

This chapter also illustrates that although some descriptive rules are very complicated, the resulting
routine may be fairly simple. In contrast, other descriptive rules may be relatively simple, but the routine that
is called for may be quite elaborate—so elaborate that it may have to be presented a part at a time.

Routines Based on Rules that Predict Physical Outcomes

The rule, Objects fall at the same rate, predicts that two objects quite different in size and weight will fall
at the same rate. Other rules of science and of arithmetic are the same in that they predict particular outcomes.
When we work with these rules, we should try to design them according to an ideal formula.

1. The learner takes the steps necessary to arrive at the appropriate answer.

2. A confirmation occurs after these overt steps have been taken.

3. The confirmation takes the form of an observation of something familiar to the learner.

For some routines, the confirmation may not take the form of a physical outcome, but of verbal
confirmation by the teacher. This confirmation is usually not as desirable, but is often far more practical than
observation of a physical outcome. (Confirmation for the rule about falling objects would provide the learner
with a demonstration that two dissimilar objects, such as a feather and a lead ball, fall at the same rate. A
verbal confirmation would simply be a verbal statement that the objects fall at the same rate.)

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 Illustration 1: Angle of Incidence and Angle of Reflection

The setup for the routine is very important. It is a format or structure that permits us to process a range of
examples by conducting the same series of steps. The setup is a device that makes the sameness of the various
examples obvious, because it shows that each example is processed in the same way.

We want to teach this idea: The angle of reflection equals the angle of incidence. The rule applies to the
reflection of light, ricocheting of balls, etc. Here is a possible descriptive rule based on the idea: To figure the
angle of reflection, you first figure the angle of incidence and then mark off the same angle for the reflection.

Identifying components. How do we show that two angles are the same? The temptation is to deal with
numbers, perhaps constructing tasks such as, “What’s the same angle as forty-five degrees?” The answer
would not convince us that the learner would be able to recognize angles that are the same. A far more
convincing task would require the learner to construct angles that are the same.

Figure 18.1 shows the beginning of a possible task series for teaching same angle. These tasks are
presented as worksheet items. This preskill actually teaches the same basic discrimination that is involved in
the angle of reflection.

Figure 18.1
Examples Teacher Wording

My turn: The dotted lines have equal angles because both

4 4 dotted lines go to the number 2.

3 3
2 2
1 1

What number does this dotted line go to? Make another

4 4 dotted line with the same angle.

3 3
2 2
1 1

The setup for the routine. Figure 18.2 shows a diagram of a physical setup that permits verification that
the correct angle has been constructed. Note that it is quite similar to the worksheet items the learner works
for figuring equal angles. This setup involves actual objects. A small mirror is attached to the wall. A center
line marks the 0 angle. A semi-circular chalk line on the floor shows the path that is used for figuring the
“same angle.” The basic game is for a person to figure out where to stand to see an object. Confirmation is

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 possible by standing in the specified place and looking into the mirror.

Figure 18.2
Wall
Mirror

The new descriptive rule. Now that we have designed the setup, we can construct a rule that is more
precise than the original descriptive rule.

To see a target in the mirror (to predict the angle of reflection), first see how many marks the target is from
the center line, then mark off the same number of places on the other side.

This new rule suggests the preskills of judging how many places from the center line a target is, marking
off the same number of places on the other side. If the learner did not have these preskills, we would design
teaching for them.

The routine. The following routine begins with the mirror covered.

1. Teacher: If you want to see me in the mirror, you

have to make the same angle on the other
side. I start in the center and then I move to
the third mark. Which mark did I move to?
Learner: The third mark.
2. Teacher: (Moves to the third mark on left.) If you
want to see me in the mirror, which mark
do you have to move to on the other side?
Learner: The third mark.
3. Teacher: Do it.
Learner: (Moves three marks from the center line.)
4. Teacher: You made the same angle on the other side.
So will you be able to see me in the mirror?
Learner: Yes.
5. Teacher: How do you know?
Learner: Because I made the same angle on the other
side.
6. Teacher: (Uncovers mirror.)

Note that the verification comes after the learner has carried out the steps. This point is important. The
verification is a contingency, so that the verification functions in the same way that a successful outcome
functions when the learner is engaged in a physical operation, such as throwing a ball at a target. Unless the

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 routine places emphasis on the steps that lead to the verification, the routine will be weak. If we had designed
the routine so the learner merely walked around the semi-circle until the teacher was visible in the mirror,
there would be no compelling reason for the learner to learn the relationship between the angles on either side
of the center line. The learner could simply attend to seeing the teacher in the mirror, not to first figuring out
where to stand to see the teacher. The teacher’s explanations of why she is visible in the mirror would be
after-the-fact explanations that do not require the learner to attend to the angles.

If the routine is designed so the learner must take certain steps and figure out the answer before receiving
verification of the answer, the routine works like a physical operation. The outcome depends on the
successful performance of certain steps.

Illustration 2: Finding the Number You Multiply By

The idea that we start with is: You can change any number into any other number by multiplying. The
number you multiply by is simply that value that is needed to make the sides of the equation equal. The
number is always expressed in terms of the other two numbers.

In a general way, we can express the rule as:

a
b (bcad) = c
d

The value that is in parentheses is what we multiply a/b by to change it into c/d. We can show that the value
in parentheses is correct by analyzing how many are on each side of the equation. The first two fractions on
the left side:

a x b
b a
equals one (abab)
because the top and bottom numbers are the same. What remains on the left side of the equation is c/d. The
same value is on the right side of the equal sign. Since c/d=c/d, the value inside the parentheses must be
correct.

The objective of our routine is to teach the learner how to figure out what goes into the parentheses for
any problem such as:

4 ( ) = 6
a ( ) 1
=
3 b
9 ( ) 4R
=
3 7

The rule and the setup. As with the routine for the angle of reflection, the learner should go through the

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 steps that will lead to the correct answer. Then the learner should receive some form of verification that the
answer is correct (possibly a compelling observation). The routine in Figure 18.3 assumes that the learner can
reduce fractions to whole numbers and that the learner understands whether fractions equal one whole.

Figure 18.3
Example: 3 ( ) = 6
3( )=6
1. Teacher: You know the answer to this problem. But
we’re going to work it a new way and see
if we get the right answer. Three times
some number equals six. What number?
Learner: Two.
Teacher writes 2 in top problem: 3(2)=6.
2. Teacher: Now let’s work the same problem another
way. I’ll pretend that I don’t know what to
multiply three by to change it into six, but
I do know what to multiply one by. So
first I multiply to change three into one.
What do I multiply by?
Learner: One-third.
Teacher writes 3( 1 )=6
3
3. Teacher: Are the sides of the equation equal now?
Learner: No.
4. Teacher
points How many are on this side?
left:
Learner: One.

5. Teacher
points How many are on this side?
right:
Learner: Six.
6. Teacher: So we multiply by six inside the
parentheses.
Teacher writes: 3( 1 x 6 )=6
3 1
7. Teacher: Multiply and tell me the fraction that is in
the parentheses.
Learner: Six-thirds.
8. Teacher: What does six-thirds reduce to?
Learner: Two.
9. Teacher: How do you know?
Learner: Because the top is two times bigger than the
bottom.
10. Teacher: We ended with the right number when we
worked the problem this way.

There are two important points about this routine.

1. The attack is logical. First we make the left side equal to one. Then we multiply by whatever is

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 needed to make the left side equal to the right. This basic procedure is logically compelling.

2. The validation comes from applying a familiar reducing procedure that is assumed to be valid.

Instead of designing the initial routine so that it deals with examples that are not readily verifiable, such
as 4/3( )=2/8, we present an example that reduces to a whole number. This procedure validates the routine. If
the routine works with different values that can be verified through reduction, the routine should work for
other values. Once the validity of the routine has been established, a variation that involves no validation is
applied to a full range of examples. The routine in Figure 18.4 is changed slightly so that it provides less
prompting. The same routine will work for any value.

Figure 18.4
Example: 43 ( )= 78

1. Teacher: I don’t know what to multiply four-thirds by

to change it into seven-eights, but I do
know what to multiply some number by.
What number?
Learner: One.
2. Teacher: So what do I multiply by first?
Learner: Three-fourths.
Teacher writes: 4 ( 3 )= 7
3 4 8
3. Teacher: How many are on the side now?
Learner: One.
4. Teacher: So what do I do to make the sides equal?
Learner: Multiply by seven-eights.
5. Teacher: What fraction is inside the parenthesis?
Learner: Twenty-one thirty-seconds.
Teacher writes: 4 ( 3 x 7 )= 7
3 4 8 8
6. Teacher: That is the answer.

Illustration 3: Rewriting Numbers That Have Exponents

This illustration involves a descriptive rule that is quite complicated; however, the resulting routine is not
very complicated. This relationship is observed in many skills.

The goal of the teaching is to show the learner how to write exponents such as N4 or N-4 as N x N x N x
!
N and . For this teaching, we do not encounter the problem that we encountered in the preceding
!  !  !  !  !  !  !

illustrations. For these problems, we had to design a setup. For the exponent routine, we are going to use the
conventional setup or notation system and simply teach the learner how it works. Our major concern is that of
dealing with a complicated descriptive rule:

To write numbers that have exponents as fractions that show repeated multiplication, multiply the base the
number of times the exponent indicates; use the sign of the exponent to determine where the multiplication

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 occurs—with the multiplication occurring where the exponent is shown if the exponent is positive and the
multiplication occurring in the opposite part of the fraction if the exponent is negative.

!
The exponent is written on either the top of the fraction or on the bottom: N2 or . If the exponent is
!!

positive (+), the multiplication occurs in the same place the exponent appears (top or bottom). If the exponent
is negative (N-2 or 1/N-2), the multiplication occurs in the opposite part of the fraction (on the bottom for N-2
and on the top for 1/N-2).

We next translate each component discrimination into a task that tests the discrimination. The
discrimination named first in the descriptive rule is: “The base number is multiplied.” The task could be
“Which number is multiplied?” We could use a single-transformation sequence to teach this skill as in Figure
18.5.

Figure 18.5

Example Teacher Wording

46 My turn. Which number is multiplied? Four.

64 Your turn. Which number is multiplied?

65 Which number is multiplied?

56 Which number is multiplied?

N4 Which number is multiplied?

21-7 Which number is multiplied?

To teach the discrimination that the exponent tells how many times the base is multiplied, we could use a
pair of correlated-features tasks (illustrated in Figure 18.6).

Figure 18.6

Example Teacher Wording

N5 My turn. How many times is the base

number multiplied? (Five.) How do you
know? (Because the exponent is five.)

N15 My turn again. How many times is the

base multiplied? (Fifteen.) How do you
know? (Because the exponent is fifteen.)

5N15 Your turn. How many times is the base

multiplied? (Fifteen.) How do you
know? (Because the exponent is fifteen.)

Etc.

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 We may have some problems with the next idea expressed by our descriptive rule. The idea is that the
sign of the exponent tells whether the multiplication occurs on the same part of the fraction as the exponent.
We should probably use correlated-feature tasks for this discrimination.

Figure 18.7 shows the test wording of four examples that teaches the relationship.

Figure 18.7

Example Teacher Wording

N-5 Where is the base multiplied? (Away

1 from the exponent.) How do you know?
(Because the exponent is negative.)

N5 Where is the base multiplied? (Where

1 the exponent is.) How do you know?
(Because the exponent is positive.)

1 Where is the base multiplied? (Where

N5 the exponent is.) How do you know?
(Because the exponent is positive.)

1 Where is the base multiplied? (Away

N-5 from the exponent.) How do you know?
(Because the exponent is negative.)

The final routine requires the learner to tell us how to rewrite different numbers that have exponents. The
routine is created by chaining the various tasks that teach the component skills. We simply add steps that
require the learner to carry out the multiplication. See Figure 18.8.

Figure 18.8

Example Teacher Wording

1 1. What’s the base? (N)

N5
2. What’s the exponent? (Five.)

3. How many times is the base multiplied?

(Five times.)

4. How do you know? (Because the expo-

nent is five.)

5. Where is the base multiplied? (Where

the exponent is.)

6. How do you know? (Because the expo-

nent is positive.)

7. What’s another way of saying: “One

over N to the five?” (One over
NxNxNxNxN.)

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 Although the descriptive rule from which we started is quite complicated, the resulting routine is simple.
The learner first responds to two mechanical questions. The bulk of the routine consists of two correlated-
feature relationships.

To firm the concept of multiplying away from where the base is and multiplying where the base is, we
would present minimally different applications of the routine as illustrated in Figure 18.9.

Figure 18.9

Example Teacher Wording

1 1. What’s the base? (N)

N5
2. What’s the exponent? (Five.)

3. How many times is the base multi-

plied? (Five times.)

4. How do you know? (Because the expo-

nent is five.)

5. Where is the base multiplied? (Where

the exponent is.)

6. How do you know? (Because the expo-

nent is positive.)

7. What’s another way of saying: “One

over N to the five?” (One over
NxNxNxNxN.)

Illustration 4: Column Addition

Some complex descriptive rules imply a series of routines or a variation of the basic routine. Column
addition illustrates this type.

Let’s say that the idea is to teach procedures for column addition problems that do not involve carrying.
The problems would include different types:

203 41
55 57 34
+ 320 + 92 + 2

Before we can design a descriptive rule, we must determine a procedure that we will use to achieve the
addition. (No single procedure holds for all column-addition problems because some later problems will
require carrying.) If we used a conventional approach, the rule for non-carrying problems could be expressed:

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 To solve the problem, first add the numbers in the ones column and write the answer. Then add the
numbers in the tens column and write the answer.

To design a non-carrying routine, we first specify tasks for the various parts of the descriptive rule.

The rule requires the learner to discriminate ones column and tens column.

We should teach the learner to operate first in the ones column, but we should try to avoid teaching a
discrimination (ones column vs. tens column) that would prompt possible reversals (spurious
transformations). (See Figure 18.10.)

Figure 18.10
Example Teacher Wording
We always start adding in the ones column.
That’s the last column.
A. 24
+ 31 Touch the ones column for problem A.

B. 340
+ 246 Touch the ones column for problem B.

C. 798
101
3
+ 52 Touch the ones column for problem C.

By touching only the correct column, the learner is not required to discriminate between the two labels
and the two columns, merely to learn which column is the one you start with. The task, “Touch the ones
column,” becomes a component or step in the routine.

Now we teach the routine for solving the problem in the ones column. The routine stipulates the first
behaviors involved in solving all problems. (See Figure 18.11.)

Figure 18.11
Example Teacher Wording

20 Read problem A. (Learner reads.)

15 Touch the column you start adding
+ 23 in. (Learner touches ones column.)
Which column did you touch? (Ones
column.)

Write the answer for the ones column.

(Learner writes.)

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 By presenting this routine with juxtaposed examples, we show the learner that the same steps apply to all
problems.

To assure that the learner writes the answer in the appropriate place (directly under the ones column), we
could make answer boxes under the ones column and under the tens column. (Note that if we placed an
answer box only under the ones column, the box would serve as a spurious prompt about where to add first.)

After the routine of starting problems in the ones column has been applied to perhaps a dozen problems,
we could introduce an entire routine for dealing with the ones column and the tens column. (See Figure
18.12.)

Figure 18.12

Example Teacher Wording

30 Read problem A. (Learner reads.)

15 Touch the column you start adding in.
+ 23 (Learner touches ones column.)

Read the problems in the ones column.

(Learner reads.)

Write the answer for the ones column.

(Learner writes 8.)

Now read the problem in the tens column.

(Learner reads.)

Add the tens and write the answer below

the tens column. (Learner writes 6.)

How many are in the tens column? (“Six.”)

When you add 30 and 15 and 23, how

many do you end up with? (“Sixty-
eight.”)

The questions that the teacher presents require the learner to discriminate between the ones column and
the tens column, the total for the ones column, the total for the tens column, and the total for the whole
problem. The questions are presented so that they minimize reversal problems. The reason is that the routine
presents a chain of behaviors that always occur in the same order.

An alternative routine that would provide better juxtaposition of examples for the tens column could be
introduced after the learner has practiced working with the ones column only. This intermediate routine
would be introduced before the total routine is presented. (See Figure 18.13.)

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 Figure 18.13

Example Teacher Wording

20 Touch the columns you start adding in.

56 (Learner responds.)
+ 13
The answer is nine. Write the answer in
the ones column. (Learner writes.)

Read everything in the tens column.

(Learner reads.)

How many tens is two tens, five tens

and one ten?

Write the number of tens in the tens

column.

The juxtaposition of examples is improved because the learner does not have to work the ones column,
merely touch the column and write the answer. After the learner has worked 5 to 10 problems with this
routine, the total routine could be introduced.

In summary, the descriptive rule for these addition problems implied a program, a sequence of more than
one routine. Certainly we could have started with the final routine; however, the juxtapositions would have
been very poor. The central thrust of the program is to stipulate the first behavior’ that the learner is to
perform when working the column-addition problems. This starting behavior is critical, because after it is
established, the remaining steps tend to follow naturally. Perhaps the best way to demonstrate this feature is
to compare the final routine with one that has weak beginning steps and non-functional discrimination tasks
(Figure 18.14). Note the choice.

Figure 18.14

Teacher Wording

When you start adding, do you begin in the tens column

or the ones column?
Read the problem in that column.
Do you write the answer to that column under the tens
column or under the ones column?
Do it. (Check.)
Which column do you add next?
Read the problem in that column.
Do you write the answer to that column under the tens
column or under the ones column?
Do it. (Check.)

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 Although this routine is not a perfect disaster (and would certainly work with most learners), it is
relatively weak because it does not strongly prompt the starting steps and it does not stipulate these steps in a
way that makes them different from other steps. The routine presents two questions about choosing the ones
column or the tens column. Although the learner may respond correctly to step 1 of the routine, the learner
may start to read the problem incorrectly in step 2 because the routine does not provide us with a strong
indication that the learner knows which column is the ones column. The second question about where the
answer is written is not functional. If the learner has already written an answer under the ones column, where
could the answer for the tens column go, except under the tens column?

Illustration 5: Routines that Call for Descriptive Responses

Earlier, we cautioned against the use of descriptive responses. We noted that if a descriptive response is
used, the learner should be required to actually perform the step described. In some situations, descriptive
steps are both reasonable and efficient, even if they are not followed by behavioral demonstrations. These
steps are used extensively in procedures that direct adults and sophisticated learners to perform physical
manipulations—such as achieving a high-brace turn in a canoe, tying a square knot, hypnotizing yourself,
preparing egg salad, etc.

These routines are justified if they clearly specify the intended behaviors and if the learner is capable of
translating the descriptive steps into appropriate behavior. For these situations, the descriptive steps of the
routine are not intended to teach actual behavior, but to teach discriminations or facts. Consider operations
such as fixing an omelet, repairing a tire, dressing to look slim, or bathing a dog. If we were to teach the
behaviors, we would begin with the component skills. We would then introduce the routine and teach it. In
many situations, however, we cannot take responsibility for the entire teaching. Further- more, we assume
that it has been taught. We therefore may decide to design routines that concentrate only on the learner’s
symbolic understanding of the situation, not on the actual behavior. These routines provide information
necessary to chain known behaviors together—however, they do not teach the actual behaviors.

Let’s say that we wanted to teach the learner a procedure for remaking a garment to get rid of wrinkles in
the sleeves.

The descriptive rule. To get rid of wrinkles in a sleeve of a garment that is being made, increase
the sleeve seam and taper the seam from under the arm to the cuff.

The descriptive rule not only describes the discrimination that we must teach; it probably also contains
the language and the order of events.

Increase the seam. We will say that the seam is increased if no part of it is smaller than it had been. We
use a variation of a non-comparative sequence (see Figure 18.15).

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 Figure 18.15

Example Teacher Wording

The solid lines show the original

seam. The dotted lines show the
new seam.
Is this seam increased? Yes.

Your Turn: Is this seam increased?

Yes.

Is this seam increased? Yes.

Etc.

Tapering from the shoulder to the neck. This discrimination is best taught as a non-comparative. Figure
18.16 shows the first part of a possible sequence.

Figure 18.16

Example Teacher Wording

UA My turn. How does this seam

taper? It doesn’t.

C
How does this seam taper?
UA From the cuff to under the
arm.

C
How does this seam taper?
UA From under the arm to the
cuff.

C
Your turn. How does this seam
taper?
Etc.

The final routine is simply a correlated-feature sequence that expresses the two things that are done to get
rid of wrinkles (Figure 18.17). Because the final routine is verbal and does not involve actually doing the
operation, it begins by expressing the two things that must be done to get rid of the wrinkles. Following the
mechanical test on this information are applications. The applications are simply correlated-feature examples.

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 Figure 18.17

Example Teacher Wording

To get rid of the wrinkle, you do

two things. You have to in-
crease the sleeve seam, and
taper it from under the arm
to the cuff. What are the two
things?
Shows increased seam My turn. Could this get rid of the
tapered from under wrinkle? Yes. How do I know?
the arm to the cuff. The seam increased and is
tapered from under the arm to
the cuff.
Show seams not Your turn. Could this get rid
increased. of the wrinkle? (No.) How do
you know? (The seam is not
increased.)
Show seams not Could this get rid of the wrinkle?
tapered. (No.) How do you know? (The
seam does not taper from
under the arm to the cuff.)

Note that the routine begins with a positive to assure that both criteria are presented to the learner as
quickly as possible.

Following the routine above could be applications that require the learner to perform the operation. The
routine for the application is quite simple.

Teacher: What are the two things you do to get rid of a

wrinkle in the sleeve?
(Learner responds.) Do it.

Although the final routine requires the learner to “do it,” the teacher emphasis is on the description of
what is to be done.

Illustration 6: The Concept of Specific Gravity

This illustration points out that the procedures specified for designing cognitive routines work for any
concept that can be treated as a problem-solving routine. The concept of specific gravity is one that is
sometimes used as a developmental marker and one that is not frequently taught in a direct manner. We can
create a descriptive rule that satisfies the requirements for describing specific gravity; therefore, we can
design a routine for teaching the concept.

The descriptive rule. To determine whether an object sinks or floats in a medium, the learner

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 receives information about the weight of the object as it compares to the weight of the medium in
which the object is to be placed. The learner then concludes that: (a) the object will float (if lighter
than an equal volume of the medium); or (b) the object will sink (if heavier than an equal volume of
the medium).

Before we can design the routine, we have to make a decision about how the learner receives the
information about the density of the object, and of the medium. The information could be provided through
verbal statements of fact or through some sort of physical demonstration that involves weighing the object
and the medium. Unless we make some decisions about how the learner will receive this information, we do
not know what kind of setup to create for the routine.

Let’s say that the learner receives density information by weighing part of the medium and an equal-size
part of the object. One possible setup consists of an object that is composed of blocks. To determine the
relative weight of the medium and object, the learner selects part of the object and fills a container the same
size as the block with the medium. Both are weighed on a balance scale. The learner then predicts whether
the object will float or sink in the medium. (See Figure 18.18.)

Figure 18.18

Because of the large number of component discriminations, we will present the routine first (Figure
18.19), then show the component skills. Note, however, that if we were designing the routine, we would start
with identifying the various discriminations referred to in the descriptive rule.

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 Figure 18.19

Teacher Wording Learner’s Response

(Presents object.) What do Weigh a block and weigh

you do to figure out wheth- a piece of the medium the
er the blocks will float or same size.
sink in the medium?
Do it. (Detaches one block from
object, fills second block
with water from the tank,
and places one object on
either side of the balance
scale. The block from the
object goes up.)
What did the object do on Went up.
the scale?
So what will the object do Float.
in the medium?
How do you know? Because it’s lighter than
a piece of medium the
same size.
Let’s see if you’re right. Floating.
(Places object on the medi-
um.) What’s it doing?

The learner first predicts what will happen, then receives confirmation in the form of an observation. The
confirmation serves as reinforcement for working the problem and predicting correctly.

Let’s assume the learner has to be pretaught float and sink.

Teacher: Listen. Another way of saying that an object

goes does in air is this: An object sinks in air.
What’s another way of saying an object goes
down in water?
What’s another way of saying an object goes
down to the bottom of a lake?
What’s another way of saying an object does
not go down in gasoline?
Etc.

The purpose of this wording is to show what is the same about floating and sinking in any medium. By
making the wording the same for different media, we prompt the sameness.

To teach the discrimination of medium, we could do another transformation sequence (Figure 18.20).

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 Figure 18.20

Example Teacher Wording

An object is always in a medium.

(Holds egg in air.) What medium is the egg in now?
(Holds egg in sand.) What medium is the egg in now?
(Holds egg in water.) What medium is the egg in now?
Etc.

To teach the same size, we would use a correlated-feature relationship. For this series, the teacher pours
sand from the one container into containers of different shapes and sizes (Figure 18.21).

Figure 18.21

Example Teacher Wording

Teacher presents sand in Here’s a rule. If things hold
flat container. the same amount, they are
the same size.
Teacher pours sand into Are the containers the same
round container of same size? (Yes.) How do you
size. know? (They hold the same
amount.)
Teacher pours sand into Are the containers the same
rectangular container size? How do you know?
same size.
Teacher pours sand into Are they the same size? How
rectangular container do you know?
not the same size.
Etc.

In addition to the single-transformation sequence for floating, the learner would be tested on concrete
applications that require the learner to tell whether the object sinks or floats, and to identify the medium
(Figure 18.22).

Figure 18.22

Example Teacher Wording

Object sinks in water. What did the object do? What
did it sink in?
Object floats in water. What did the object do? What
did it float in?
Object floats in air. What did the object do? What
did it float in?

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 Another skill would be to determine which object is lighter. To teach this skill, the balance scale would
be used. The concept is a correlated-features relationship.

Teacher: Which is lighter, the block or the egg?

Learner: The block.
Teacher: How do you know the block is lighter?
Learner: The block went up on the scale.

We could also teach the relationship between the behavior of objects on the scale to their floating and
sinking behavior (Figure 18.23).

Figure 18.23

Example Teacher Wording

Teacher presents two (Points to A.) This is the object.
objects, A and B. (Points to B.) This is part of the
medium we will put the object
in. The object and the medium
are the same size. How do I
know that? (They hold the same
amount.)
Teacher places object What did the object do on the
and medium on a scale? That’s what it will do in
scale. the medium.
Object goes up on What will this object do in the
scale. medium? How do you know?
Object goes up on What will this object do in the
scale. medium? How do you know?
Object goes down on What will this object do in the
scale. medium? How do you know?
Etc.

This sequence is possibly weak because the learner should understand that the objects must be the same
“size” or volume. This discrimination is not required by the tasks, however. We would correct this problem
by requiring the learner to produce “stronger” responses (Figure 18.24).

Figure 18.24

Example Teacher Wording

Object goes up on What will the object do in the

scale. medium? (Float.) How do you
know? (It’s lighter than a piece of
medium the same size.)
Object goes down What will the object do in the
on scale. medium? (Sink.) How do you
know? (It’s heavier than a piece of
medium the same size.)

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 The routine that appears on page 211 is a chaining of the various skills that would have been pretaught.
Like the other cognitive routines, it serves as a model that prompts a complex discrimination. In this case, the
routine must prompt the idea that for homogeneous objects the whole object sinks if any part of the object
sinks. After demonstrating with the routine that the whole object floats, the teacher might say, “Listen. If each
part of the object is lighter than a piece of the medium the same size, each part will float.” This step expresses
the idea that there is a constant ratio of object-to-medium of the same size. If the routine prompts this notion,
it communicates the essence of the concept of specific gravity.

Although the concept of specific gravity is seen by some cognitive psychologists as something that
operates within the learner’s mind and that evolves through unspecified development, it can be taught if we
formulate a descriptive rule that expresses what behaviors would be accepted as evidence that the learner has
the concept. The routine that would succeed in teaching this concept to learners has the same properties as the
routine for teaching angle of incidence, beginning decoding, or any other cognitive skill. Overt behaviors are
designed for the various component discriminations named in the descriptive rule. These discriminations are
then chained together in a routine that can be used for all examples of the cognitive skill (in any medium and
with any object.)

Summary

There is no right way to construct cognitive routines. Rather, there are strategies that are acceptable and
some that are better than others.

When we deal with rules about the physical world, we want the learner to understand that the routine is
actually a vehicle for predicting outcomes. We therefore design the routine so that the observation of the
predicted outcome comes only after the learner has taken the appropriate steps. The observation is treated as
“reinforcement” for solving the problem. It serves as an analogue to a successful trial with a physical
operation.

Some routines simply list the various steps the learner is to perform. The emphasis is placed not so much
on the actual skills, but on the cognitive discriminations. The idea is to prepare the learner with
understanding of how the operation should work, not necessarily with the motor skills of doing it. For such
routines, conventional wording is usually acceptable (the wording that appears in the descriptive rule).

The opposite situation occurs if new, difficult responses are called for by the routine. The wording that is
used by the teacher becomes arbitrary. The focus of the routine is on doing. This routine is designed to
prompt the actual behavior—not merely to serve as a checklist.

Some routines deal with skills that are not conventionally taught (such as specific gravity). The
procedure is basically the same for these routines so long as we can express a rule that describes the things

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 that a learner must do to solve the problem.

When routines do not involve a conventional setup, we must design one. For many problems, we accept
the conventional setup (such as the conventional symbol system for numbers). However, we may decide to
reject the conventional setup, or we may have to create one if it does not exist.

Some descriptive rules need more than one routine. These are usually rules that involve long strings of
behavior. If we introduce one routine that contains all the behavior, the juxtaposition of steps is not well-
designed to stipulate a chain of behaviors, particularly the very important first steps. Therefore, we create one
routine that deals with the starting behavior, then another that presents the starting behavior and additional
behavior.

Although there is an incredible variation of routines, all are designed to provide a series of overt steps
that lead to solutions to a class of problems. And all are composed entirely of components that can be
identified and pretaught.

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 Scheduling Routines and Their Examples
This chapter deals with three issues of coordinating routines:

1. How do we schedule examples for the different components of a routine?

2. How do we schedule the examples that are to be processed by a complete routine?

3. How do we schedule conditional parts of routines?

Scheduling Components

Once the routine has been tentatively designed, the components have been identified. The routine
functions as a “format.” By substituting different instances, we create a range of applications (examples) for
the routine. The various components of the routine are members. The procedures for introducing the members
are those that we follow for all members. We separate highly similar members. We avoid juxtaposing
introductions that involve minimum-differences, and we provide for the firming of each component skill.
Each skill must be:

1. Introduced through an initial teaching sequence.

2. Expanded in activities that require responses and contexts different from those in the initial-
teaching sequence.

3. Reviewed until the component is used in the routine (or in some other application).

The order in which the components are introduced for a particular routine does not necessarily reflect the
order in which the skill appears in the routine. Decisions about the order of introduction are based on: (1)
features of the components; and (2) the complexity of the components (with more practice time allotted to
components that require difficult discriminations and difficult responses).

Minimum-Time Schedules

In its purest form, the schedule for the components of a particular routine would be specified in much the
same way as those for other programs that derive from tasks. In practice, some components would have been
pretaught to achieve objectives that are not related to the routine. The schedule for the components of a
particular routine would therefore show some components being taught months, perhaps years, before the
introduction routine. If we assume that none of the components involved in the routine have been pretaught,

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 we design a minimum-time schedule. This schedule provides a careful introduction and shows the best
estimate of the minimum time that would be required for the introduction.

Illustration: Beginning word reading. To teach the reading of regular words, we must teach some letters
that can be identified as sounds and we must teach oral blending. For the schedule in Figure 19.1, sounds are
introduced at the average rate of one new sound every three days. On the first few days, however, the rate is
somewhat higher because the discriminations are relatively easier. (Note that the letter e is pronounced “ee,”
not the short sound.) Oral blending exercises begin on lesson 1 and continue through lesson 22. The sound
review begins on lesson 3 after three sounds have been introduced. The word-reading routine starts on lesson
9. At this time, the learner has been introduced to the sounds for r, a, s, m, and e. The routine continues on
every lesson.

Figure 19.1

Components Lessons
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Sounds a,r a,r s s m m e e d d u u l l o o g g t t
Review X X X X X X X X X X X X X X X X X X X X X X X
Oral
Blending X X X X X X X X X X X X X X X X X X X X X X X

Word-
Reading X X X X X X X X X X X X X X X X X
Routine

The routine processes only those sounds that are firm, which means that the learner has worked with
them for at least two lessons. The routine can safely be introduced nine lessons after the program begins. This
is the minimum schedule that is required for a careful introduction. Information about specific problems the
learner encounters or that the teachers experience in presenting the tasks may lead to modification of what we
consider a minimum schedule. Also, if learners are sophisticated (which means that they have learned some
of the specific skills demanded by the program and a general “learning strategy”), we may accelerate the rate
of introduction. Possibly, the learner has already learned sounds for some of the letters. Possibly, the learner
already knows the names of the letters and therefore requires less instruction to master the sounds. For this
learner, the introduction of the cognitive routine would appear on lesson 3 or 4; however, the 9-lesson
presentation would be safe for most learners who have received no appreciable preteaching.

Illustration: Getting rid of wrinkles in a sleeve. This routine involves two discrimination components (“Is
the seam increased?” and “How does this seam taper?”), a correlated-feature sequence, and a routine that
combines the components. Figure 19.2 shows a possible schedule.

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 Figure 19.2

Components Lessons
1 2 3
Discrimination:
Is the seam increased? X X
Discrimination:
How does this seam taper? X X

Correlated feature relationship X X

Routine X X

Note that this sequence assumes that the learner is fairly sophisticated and that the skills are introduced
for the first time. Each skill appears on two consecutive lessons, with the preskills for the routine preceding
the introduction of the routine by one lesson.

In most situations, the information about increasing the seam and tapering a seam would not be delayed
until we teach about removing wrinkles. These skills would play a role in many more elementary tasks;
therefore, they would be introduced long before the correlated-feature relationship and the routine. A more
reasonable program for removing the wrinkles would assume that the only skills directly implied by the
routine are the correlated-features relationship and the routine (Figure 19.3).

Figure 19.3

Components Lessons
1 2

Correlated feature relationship X X

Routine X X

This schedule presents both the correlated-feature relationship and the routine on the same lessons. (The
reason is that this program introduces less that is new to the learner.) The correlated-feature relationship deals
with discriminations familiar to the learner; therefore, the application (the routine) may immediately follow
the verbal training (the correlated-feature sequence).

Scheduling Clusters

Some routines have many steps. Routines that involve long division or similar long arithmetic operations
may have as many as 30 steps. If we follow the basic procedure of teaching each component individually and
then combining them into the routine, we create a rather rough transition. We observed earlier that the pattern
of juxtaposition for long routines makes it difficult for the learner to remember the sequence of steps. To

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 make the transition easier, we introduce an intermediate step. Instead of proceeding directly from individual
components to the cognitive routine, we proceed from the individual components to small groups or
“clusters” of components. The schedule in Figure 19.4 shows the strategy. Note that a cluster is composed of
juxtaposed steps; however, the steps are not necessarily scheduled according to their occurrence in the routine
(with AB first, BC next, etc.). In the schedule, AB and CH are scheduled at the same time.

Figure 19.4

Components Lessons
1 2 3 4 5 6 7 8 9 10
Skills: A

B AB AB AB

C
D CD CD

E
F EF EF

G
H GH GH GH
Routine A-H A-H A-H

The pattern shown in Figure 19.4 groups the component skills into pairs. These pairs would be two
consecutive steps in the final routine. Not all skills can be grouped gracefully into pairs. For some routines, a
more logical grouping might involve one cluster composed of four steps and perhaps another composed of
two. The goal, however, is to present subparts of the routine as intact clusters. When the learner practices a
cluster, the transition from clusters to routine is easier because less new learning is involved. Instead of being
faced with a series of familiar independent steps that are brought together in an unfamiliar chain, the learner
simply chains two or three smaller chains or clusters together.

We can illustrate the cluster strategy with the 7-step routine in Figure 19.5. This routine is designed to
teach the procedure for determining whether a fraction reduces to a whole number. Steps 1, 2 and 3 deal with
the mechanical details of the routine. Steps 4, 5, 6 and 7 analyze the fraction. We could create various clusters
from this routine. For instance, steps 2 and 3 should go together as a cluster. Possibly, 2, 3 and 4 could go
together as a cluster. Possibly 1, 6, and 7 (or a slight variation of these steps) could go together in a
correlated-feature sequence.

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 Figure 19.5

Example Teacher Wording

24 1. If you say the top number when

6 counting by the bottom number, what
do you know about the fraction? (It
reduces to a whole number.)
2. What’s the bottom number of this
fraction? (Six.)
3. What’s the top number of this fraction?
(Twenty-four.)
4. Touch the number you’re going to
count by. (Learner touches 6.)
5. Count by that number and see if you
say twenty-four. Get ready. . . (6, 12,
18, 24.)
6. What do you know about the fraction?
(It reduces to a whole number.)
7. How do you know it reduces? (Be-
cause I said the top number.)

The following is a brief description of how we might achieve major clusters, one composed of steps 2, 3,
4; another composed of steps 1, 6, 7; and a third made up of steps 4 and 5.

We would introduce steps 2 and 3 initially as a test. We would present various fractions and tell the
learner, “Touch the bottom number . . . Touch the top number.” We would present the tasks in the reverse
order for some fractions.

As soon as the learner is firm on steps 2 and 3, we would add step 4, creating the first cluster. Step 4
could be introduced as a separate step before being incorporated into the cluster of 2, 3, and 4.

Teacher: I’ll show you fractions you’re going to reduce later. When you reduce fractions, you count by the
bottom number.

a. What’s the bottom number of this fraction?

b. What’s the top number of this fraction?

c. Touch the number you’re going to count by.

Another cluster is formed from steps 1, 6, and 7. Although step 1 is not juxtaposed with step 6 in the
complete routine, it is needed to make 6 and 7 clear. Step 1 is a rule that precedes a correlated-feature
sequence:

Teacher: Listen. If you can say the top number when counting by the bottom number, the fraction

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 reduces to a whole number.

Listen again: If you say the top number when counting by the bottom number, what do you know
about the fraction?

After the learner is firm on the rule, the teacher presents a series of correlated-feature examples (Figure
19.6). The sequence contains the exact wording of steps 6 and 7. And the learner’s responses are the same as
those called for by steps 6 and 7.

Figure 19.6

Example Teacher Wording

I’ll tell you about each of these fractions.
You tell me if the fractions reduce to
whole numbers.
24 You don’t say the top number of this
6 fraction when counting by the bottom
numbers. So what do you know about
this fraction? . . . How do you know it
doesn’t reduce?
24 You say the top number of this fraction
6 when counting by the bottom number.
So what do you know about this frac-
tion? . . . How do you know it reduc-
es?
Etc.

To make sure that the learner counts by different numbers, we may introduce a cluster that involves step
4 and 5. (“Touch the number you’re going to count by . . . Count by that number and see if you say twenty-
four.”) We assume that the learner already has the counting skills and the skill of telling whether or not a
particular number was said when counting.

Figure 19.7 gives a possible schedule that shows the different clusters.

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 Figure 19.7

Components Lessons
1 2 3 4 5 6

Steps 1 X
2
3
4 X

5
6
7
Cluster 2,3 2,3 2,3,4
Cluster 1,6,7 1,6,7
Cluster 5 4,5
Routine X X X

For this routine, only two steps are taught as single-step tasks—steps 1 and 4. The teaching for each
occurs on only one lesson. Following that lesson, the skill is incorporated into a cluster.

This pattern of introduction is only one of many possible patterns. However, it illustrates the common
strategy, which is to create clusters of juxtaposed steps that are the same, (or nearly the same) as the steps that
will appear in the final routine. We should create clusters for long routines, usually those with more than six
steps. In some cases, we may not be able to use the exact wording that appears in the final routine. In other
cases, we may decide to create a cluster that consists of steps that will not be juxtaposed in the final routine
(as we did with 1, 6 and 7). If we make the latter decision, we should have strong logical grounds for creating
the cluster. Also, the cluster should contain at least two steps that will be juxtaposed in the routine.

Scheduling Examples In The Routine

The routine, or a cluster that presents part of the routine, is an initial-teaching tool. Like other initial-
teaching tools, it processes examples.

1. The set of examples must show the range of variation that the learner is expected to understand,
and show how all applications of the routine within this range are the same.

2. The set of examples must show how differences in the examples cause differences in the outcome
or responses.

3. The set of examples must provide an adequate test of the learner’s mastery of the operation that is

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 being taught.

Another parallel exists between the cognitive routine and other initial-teaching sequences. Since routines
are designed to establish initial teaching, they present a context that is very controlled—one that provides
minimum noise and maximum direction. Routines are therefore susceptible to the same problems we observe
in other initial teaching procedures. If the routine is retained too long, stipulation misrules may result. The
learner may become too dependent on the teacher’s direction for initiating the steps. The routine, therefore,
must be dropped or changed as soon as it has done its job. The routine should not be dropped, however, until
an appropriate range of examples has been processed.

Below are the five procedural rules for ordering and designing the examples of a routine:

1. We generally construct the same sequence of examples that would be presented through basic
teaching sequences.

2. We present examples that could be failed because of difficult responses near the end of the
sequence of examples.

3. We show subtypes (the range of variation within the positive examples) as early in the sequence
as possible.

4. We process minor “exceptions” through the routine if such processing violates conventions or
idioms, but does not affect the operation or discrimination.

5. If there are major subtypes that cannot be processed through the routine, we introduce examples
of the subtype early, possibly as negative examples.

Samenesses and Differences

Cognitive routines show differences and samenesses. To show differences, we follow the same rule of
juxtaposition that we use when dealing with discriminations: juxtapose examples that are minimally different
and respond to them differently. Similarly, the routine must convey information about sameness. To show
this sameness, we follow the rule of juxtaposing examples that differ greatly. By using the same steps to
approach each example, we treat examples in the same way. When greatly different examples are juxtaposed,
we show that we treat them in the same way, thereby showing that the examples are the same.

Since we have the same basic concerns with the examples processed through a routine or cluster that we
have for examples processed through a discrimination sequence, we pattern the sequence of examples for a
routine after the sequences for more basic forms. Most routines are single-transformation discriminations, in
that the learner produces different “answers” for different examples. (The learner reads different words, or
figures out different answers, in response to the same basic set of steps.) Therefore, the most frequent patterns

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 of ordering examples for a routine require starting with mid-range examples and introducing progressively
arranged minimum differences at the beginning of the sequence, followed by larger differences near the end
of the sequence.

Some routines have conditional parts. The specific steps the learner takes depends on a choice-response
discrimination. If one outcome occurs, the learner proceeds one way. If another outcome occurs, a different
set of steps follows. For these routines, the choice-response pattern of juxtaposition is generally most
appropriate. One contingency is treated as the “positive” outcome; the other becomes the “negative.” The
sequence of examples follows the familiar form used for comparatives and non-comparatives: NNPPPN or
PPPNNP with minimum difference negatives and a range of positive variation.

Following the examples that show sameness and difference is a set of examples that test the learner. The
test juxtaposition used for the basic discrimination sequences is appropriate for routines. Examples are
juxtaposed so that no relationship is obvious from one example to the next. The examples require the learner
to use the information provided through the earlier examples. A set of examples for a routine provides an
adequate test if the set requires the learner to work juxtaposed examples that differ greatly from each other.

As a general rule, 12 to 14 examples of the routine are needed to teach the procedure and to provide an
adequate test. For more sophisticated learners and for routines that have relatively few steps, as few as 5 or 6
examples may be adequate. For routines that require a large number of steps or responses that are elaborate,
as many as 30 or 40 examples may be needed. Also, a large number of examples is needed if only a few
applications of a routine are presented during a session. Generally, however, the learner should be considered
firm after performing successfully on 3 or 4 consecutive applications. The test segment should, therefore, be
5 to 7 examples long.

Examples with Difficult Responses

In a production-response sequence, examples that call for difficult responses are presented near the end
of the sequence. The sequence first permits the learner to concentrate on the features that control the
discrimination in a relatively easy context. After the learner has performed in this context, the context is made
more difficult by introducing examples that demand more difficult responses. If the learner fails these
examples after performing successfully on examples that present only modest response difficulty, we know
that the learner’s problem has to do with the response production, and we work on response production—not
on the discrimination. If the responses that call for difficult responses are placed earlier in the sequence,
however (before we have received a demonstration that the learner can handle relatively easy examples), we
do not know how to interpret the failure of a learner. Perhaps the learner has trouble producing the response.
Perhaps the learner does not understand the discrimination being taught.

Cognitive routines are governed by the same considerations as the sequence. By placing the difficult-

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 response examples near the end of the sequence, after the learner has performed on a series of examples that
involve relatively simple responses, we limit what the learner has to learn when working on these difficult-
responses and we increase the information that we receive about the learner’s performance.

A routine for figuring out how many minutes after the hour the clock shows requires the learner to start at
zero and count by fives for each number until reaching the minute hand. The response for figuring out 55
minutes after the hour is far more elaborate than the response for determining 20 minutes after the hour. We
would therefore sequence the examples so that the learner dealt with a range of examples not much more
difficult than 20 minutes after (15 minutes, 25 minutes, 10) before we introduced examples that were as
difficult as 55 minutes after. Similarly, if we were teaching single-syllable words that ended in consonant-
vowel-consonant, we would not include splat or strip at the beginning of the sequence, because the initial
consonant combinations in these words make the decoding far more difficult than that for words like lat and
rip. The less elaborate words would therefore occur early in the sequence and splat and strip would occur
near the end, after the learner had demonstrated ability to handle words that involved easier responses.

The Range

We want to show the range of variation as early as possible in the sequence of examples; however, this
procedure must be considered in connection with placing the difficult responses at the end of the sequence.
The examples that do not involve difficult responses should be designed so that they show a fair range of
variation early in the sequence. If the routine deals with fractions, the early examples should include those
that involve single-digit numbers, letters, and combinations of letters and numbers. If simple, regular words
are introduced, early examples should include vowel-consonant words, consonant-vowel-consonant words,
consonant-vowel words, and possibly consonant-vowel-consonant-consonant words.

It is not always necessary to show every subtype of example that is to be taught. If the examples include
a fair variety of types that can be processed through the routine, the set of examples implies that types not
shown may also be processed through the same routine. Let’s say that this set of examples is presented to
teach carrying:

23 24 67
4 4 23 1
+ 17 + 17 + 17 + 4

781 5234 28
236 9 432 34
+ 918 + 505 + 306 + 44

All problems involve carrying to the tens column only. Therefore, the routine used to process these
problems should not run too long. However, if we were to specify every type of problem in which carrying

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 occurs in the ones column, we would find that the set shows only a few of the possible types. The set does not
show problems that involve adding four numbers, five numbers, six numbers, twenty numbers, and so forth.
However, the range of variation is sufficient to imply a generalization to types not shown because they are
not generically different from the types shown. We can state this sort of extrapolation as a general rule: If the
same operation is contingent on a feature of the examples, and if the feature is demonstrated across a set of
examples that shows some variation, the operation is implied for the full range of examples.

This rule is a restatement of the double-transformation principle. We do not have to teach the entire
transformed set. The carrying operation is the standard transformation. The familiar set is the non-carrying
problems the learner has worked. If we show the “transformation” with a fair range of examples, we assure
generalization. The procedure that we should follow is to show as much variation in examples as we can
reasonably process through the routine without violating the constraints imposed by showing samenesses and
differences and by withholding examples that call for difficult responses.

Scheduling Limited Generalizations

We have referred to the teaching of irregular words and irregular notations for decoding numerals.
Irregulars occur in other areas that involve sets of symbols. From the standpoint of teaching, irregulars imply
that the generalizations the learner is taught must be limited to some arbitrarily designated members. For an
arbitrary reason, 14 is pronounced “fourteen,” but 11 is not pronounced “oneteen.” The word that is spelled
chef is pronounced as if it is spelled shef. Despite these limiting examples, a generalization is possible for
most of the teen numbers and a different generalization is possible for most words spelled with the letters ch.

Three instructional design problems are associated with limited generalizations:

1. Teaching the generalization for the large set (the regularly-spelled words or the regularly-
pronounced numerals).

2. Teaching the negative examples for this large-set generalization (irregulars).

3. Teaching negatives early enough to alert the learner to the fact that the generalization for the
larger set does not apply to all examples.

If we introduce the negatives of the generalizations early, they serve as advance organizers that alert the
learner to the fact that the generalization is not universal. For the decoding of irregularly-spelled words like
was, the strategy would be to apply the routine for 15 to 40 regularly-spelled words and then introduce a new
routine for irregularly-spelled words.

For other limited generalizations, however, we can introduce the negatives for the limited generalizations
earlier, even before the routine for the limited generalization is introduced. Our guide for introducing the

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 negatives is the similarity of the routines for processing the positives and negatives. If the routine for the
negatives is not highly similar to that for the positives, the negatives can be introduced even before the
positives. If the routine for handling negatives is highly similar to that for positives, a range of positives
should be introduced through the routine before the negatives are introduced through its routine. The latter
situation occurs with decoding, which is why the routine for regular words is presented before the routine for
irregulars. The routine for irregulars involves many of the same steps involved in decoding regularly-spelled
words. The learner sounds out the word and says something fast. The difference is that what the learner says
fast is not the sum of the regular letter sounds. The sounds are transformed somewhat (which is the only
difference between regularly-spelled words and irregularly-spelled words).

It might seem that decoding irregular words could be introduced before regulars if we make the routines
different. The regulars would be “sounded out” while the irregulars might be introduced as “sight words.”
The problem with this analysis is that the learner has no direct way of knowing whether a word is a sight
word. Every word is a group of letters. If the learner knows how a particular word is pronounced, the learner
knows whether or not it is irregular. But the learner is not given information about how to pronounce the
word. The learner is simply presented with a group of letters. The decision about how to pronounce the word
must therefore be based on the arrangement of these letters. And it is impossible for the learner to identify
whether the word is irregular until after the learner attends to the arrangement of the letters. (The only
difference between said, which is irregular, and sad, which is regular, is a difference in a single letter.) Unless
the routine for irregulars points out the arrangement of letters, the routine will not convey information about
the features that distinguish a particular irregular from other words. However, if the routine for irregulars
deals with the individual letters in a word and their order, the routine will be highly similar to the routine for
regular words. This routine should be delayed until the learner has mastered the routines for regulars.

Spelling words with e-a. Spelling words presents a situation quite different from that of decoding words.
Negatives for limited generalizations can frequently be introduced very early because they can be processed
through a routine not highly similar to that for the positives. To avoid confusing spelling problems with
decoding problems, we must remain sensitive to the context of spelling. The learner is presented with a
verbal word and indicates the symbols for that particular sound pattern. Generalizations are often possible
because a particular sound (such as the f sound) may reliably signal a particular spelling. The generalization
that is possible for some other sounds is more limited. For instance, the /ĕ/ sound that occurs in words like
head is spelled with the letters e-a. The /ĕ/ sound in other words, however, is spelled with a single e (red).
The two groups of words present the same /ĕ/ sound but call for different behavior.

A second problem exists because the response of spelling with the letter e or with the letters ea is not
limited to words that have the sound /ĕ/. Words like meal and me are spelled with the same letters that occur
in the spellings for /ĕ/ words (head and met).

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 Here is a summary of the major subtypes associated with the short-e sound:

• For some words, the sound /ĕ/ leads to spelling with e.

• For some words, the sound /ĕ/ leads to spelling with ea.

• For some words, the sound /not-ĕ/leads to spelling with e.

• For some words, the sound /not-ĕ/ leads to spelling with ea.

The generalization problem is complicated, but is basically the same as that for decoding examples. If we
wait too long before alerting the learner that the four possibilities exist, we may induce serious
misgeneralizations. The learner may come to believe that any word with the sound /ĕ/ spelled ea, and that no
word with a /not-ĕ/ sound is spelled ea.

We can introduce some negatives before we introduce the generalization for spelling /ĕ/ words with ea
because we can present a unique context for the negatives. Let’s say that we wish to teach the spelling of
words like head, instead, bread, tread, etc. Before we present the routine for spelling these words, we
introduce several words that contain the sound /ĕ/, but that are spelled differently (bed, met, send, etc.) We
also introduce words spelled with ea that do not make the sound /ĕ/ (meat, seal). We teach these token
negatives in a sentence such as: Her letter was ten years late. By memorizing the spelling of words in this
sentence, the learner learns that the sound /ĕ/ can be spelled with the single letter (in the words letter and
ten); that the letter e does not always produce the sound /ĕ/ (late and her); and that the letters ea may be
pronounced as /ē/ (years). After the sentence has been memorized, we can introduce the routine for
processing words that make the sound /ĕ/ spelled with ea.

Although we do not introduce many negatives, we introduce enough to assure that the learner will
approach the words bread, instead, etc., with a perspective about their sounds and spelling. Furthermore, the
sentence presents a context that is different enough to militate against confusion. The sentence is treated as an
event. The words occur only in the context of this event until the spelling has been highly stipulated. Then the
words are integrated with other words that have been taught.

In summary, it is possible to inform the learner about the limits of generalizations before serious misrules
have been induced. If a generalization is limited, some negatives (perhaps only a few “token” negatives),
should be introduced as early as possible. If the routine for processing the negatives is highly similar to that
for positives, the positives should be firmed first. If there is a reasonable way of presenting the negatives
through a routine that is quite different from the routine for the positives, the negatives can be introduced
quite early, perhaps before the positives are presented.

The test for sets that involve limited generalizations involves two questions:

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 1. Does the same symbol or feature call for one type of response in some examples and another type
of response in other examples?

2. Does a different symbol or feature call for the response associated with the symbol or feature
under consideration?

If the answer to either question is “Yes,” negatives for the generalization are implied. We should try to
design the routines for these negatives at the same time we design the routine for the positives. If there is
doubt about the extent to which the very early introduction of negatives may interfere with the teaching for
the positives, we can delay the introduction of the negatives until at least some positives are firm. Another
possibility, that of regularizing the examples through prompts, is presented in Chapter 20.

Scheduling Routines with Conditional Parts

Special identification and sequencing problems occur when routines have conditional parts. As the name
suggests, a conditional part is a part that occurs only under certain conditions. The part of a routine that deals
with the carrying operation occurs only under certain conditions—the total of a column adding up to more
than 9. Similarly, steps that follow when the trial answer in a division problem is too large occur only under
certain conditions (the answer being tried is too large).

Applications that involve conditional parts are a subtype of all the applications for the routine. The
conditional-part routine is not totally different from the non-conditional routine. In fact, the routines are
identical until they reach the point where the conditional outcome occurs.

If we tried to introduce both the conditional-part routine and the non-conditional-part routine at the same
time, the schedule might not be well-designed to teach either the non-conditional or the conditional operation.
Therefore, we firm the non-conditional routine first, then introduce the conditional part. We try to schedule
examples so that we do not induce stipulation by working too long on the non-conditional routine and so that
the learner is reasonably firm on the non-conditional routine before the conditional routine is introduced.

Structure of Conditional Parts

All conditional parts work in the same way. A key discrimination determines whether or not the
conditional steps are to be taken. If a specific condition occurs, the conditional operation is performed. If it
does not occur, the non-conditional operation is followed.

In dealing with the conditional parts, we answer two questions:

1. What detail or feature of the problem tells us that the learner must take the conditional steps?

2. What are these conditional steps?

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 The first question involves a discrimination. Often, the discrimination can be communicated to the
learner as a correlated-feature relationship: If X happens, you do Y. The second question implies a
descriptive rule, derived by the same procedure used for other cognitive operations.

The most efficient strategy for introducing contingent parts is to: (1) work on the discrimination that keys
the conditional part, and (2) chain a shortened version of the original routine with the discrimination and the
contingent part.

Illustration. Let’s say that we have already taught the learner column-addition problems involving no-
carrying. We now introduce the discrimination for carrying (if the answer in a column is more than 9, you
carry the tens). The discrimination is expressed as a correlated-feature relationship. Figure 19.8 illustrates a
possible sequence.

Figure 19.8

Example Teacher Wording

35 What’s the answer in the ones

+ 24 column? (9)
My turn. Do we carry? No.
How do I know? Because the
answer is not more than 9.
35 What’s the answer in the ones
+ 25 column? (10)
My turn. Do we carry? Yes.
How do I know? Because the
answer is more than 9.
35 What’s the answer in the ones
+ 29 column? (14)
Your turn. Do you carry? (Yes.)
How do you know? (Because
the answer is more than 9.)
39 What’s the answer in the ones
+ 29 column? (18)
Do you carry? (Yes.)
How do you know? (Because
the answer is more than 9.)
39 What’s the answer in the ones
+ 20 column? (9)
Do you carry? (No.)
How do you know? (Because
the answer is not more than 9.)

The question, “What’s the answer?” is added to the two correlated-feature questions to assure that the
learner produces the correct answer. The examples are designed after the negative-first choice-response
sequence.

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 The sequence in Figure 19.9 deals with the mechanical problems of writing an answer to the ones column
that is more than nine. The steps that take place before the carrying operation are minimized.

Figure 19.9

Example Teacher Wording

39 The answer in the ones column

46 is 23. How many tens in 23? So
+ 29 we write 2 in the tens column
and 3 in the ones column
(teacher demonstrates).
39 The answer in the ones column
46 is 24. How many tens in 24?
+ 29 So what do you write in the
tens column? And what do you
write in the ones column?. . .
Do it.
39 The answer in the ones column
46 is 15. How many tens in 15?
+ 20 So what do you write in the
tens column? And what do you
write in the ones column?. . .
Do it.
9 The answer in the ones column
39 is 36. How many tens in 36? So
49 what do you write in the tens
+ 29 column?. . . And what do you
write in the ones column?. . .
Do it.

Additional examples involving teen answers would probably be needed because the name of these
numbers does not indicate how many tens they contain.

Also, examples that involve no-carrying could be presented through the routine shown in Figure 19.10.

Figure 19.10

Example Teacher Wording

37 The answer in the ones column

41 is 8. How many tens in 8?
+ 24 (None.)
So what do you write in the
tens column? (Nothing.)
And what do you write in the
ones column? (Eight.)

Following the teaching of the discrimination and of the routine for the conditional part, we could

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 introduce a routine that involves application of the discrimination and the new step. For this routine, the
learner performs the discrimination in all problems, but works only the problems that involve carrying
(Figure 19.11).

Figure 19.11

Example Teacher Wording

29 You’ll work only the problems

+ 48 that involve carrying.
What’s the answer in the ones
column?
Do you carry?
How do you know?
Write the answer in the tens
column and the ones column.
69 What’s the answer in the ones
+ 99 column?
Do you carry?
How do you know? Write the
answer in the tens column and
the ones column.

Once the learner has successfully performed on the discrimination and on the procedures for writing an
answer that is more than ten, very little additional teaching remains for the contingent part. The learner must
be reminded to “Add everything in the tens column, starting with the one ten that you carried.” However,
beyond this prompt the remaining steps are the same as those for the regular addition problem.

The schedule in Figure 19.12 shows the different activities for carrying.

Figure 19.12

Components Lessons
1 2 3 4 5 6 7 8
1. No carrying
routine
2. Discrimination
(Carrying, no
carrying) X X

3. Steps for
carrying X X
4. Combination
discriminations
and carrying X X

Worksheet appli- 1 1 1 3 4 4 1-4 1-4

cations for 2 2 2

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 This schedule does not show the teaching of the no-carrying routine (1). This routine had been taught
earlier and is reviewed through worksheet applications on lessons 1, 2 and 3. Note that the discrimination (2)
appears on worksheets starting with lesson 2 and continues through lesson 4. The routine for determining
whether carrying is involved and for carrying out the operation is presented on lessons 4 and 5.

The rate of introduction for the conditional part could be accelerated. However, the same progression
would be followed. This discrimination would come first, followed by the operation. The worksheet exercise
would be introduced early and would continue until the learner was quite proficient at using the routine.

A schedule similar to the one for carrying is appropriate for different conditional-part routines. First, the
original routine is firmed. Then the conditional part is taught (discrimination and procedures) and the non-
conditional operation is applied to worksheet exercises or to review items. When the learner is firm on the
conditional operation, the types are integrated in a way that is modeled after a double transformation program
(see Chapter 13). The final integration is a mix of problems with non-conditional parts and problems with
conditional parts. (In the schedule above, this integration is referred to as worksheet applications 1-4 on
lessons 7 and 8.)

Illustration: Over-estimating answers in long division. One conditional part in a long division routine
deals with a trial number in the answer that is too large. The discrimination is based on details of the
partially-worked problem.

7
33 211
231

The number in the answer is too large because 231 is larger than the number above it (211).

The discrimination can be expressed as a correlated-feature relationship: If the number subtracted is

bigger than the number above it, the number in the answer is too large. This rule is involved, but manageable.
(See Figure 19.13.)

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 Figure 19.13

Example Teacher Wording

7 The 7 in the answer is not too big.

30 211 How do I know? Because 7x30 is not
– 210 more than 211.

7 The 7 in the answer is too big. How

31 211 do I know? Because 7x30 is more
– 217 than 211.

4 The 4 in the answer is too big. How

57 211 do I know?
– 228

Etc.
Additional steps may be included in the routine to make
sure that the learner knows how much 4x57 or 7x31 equal.
4 What number is the answer? (4)
57 211 What does 4x57 equal? (228)
– 228 Is the 4 in the answer too big? (Yes.)
How do you know? (228 is bigger
than 211.)

The steps for the conditional part involve making the number in the answer smaller and multiplying it to
see if the smaller number works. Figure 19.14 shows a routine for the conditional part.

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 Figure 19.14

Example Teacher Wording

7 The number in the answer is too

31 216 big, so we erase the number we are
– 217 subtracting and the number in the
answer. We write a number that is
smaller in the answer, then we multi-
ply. (Teacher demonstrates):
6
31 216
– 186
30

7 (Teacher returns to original problem.)

31 216 Your turn. The number in the answer
– 217 is too big. So what do you erase?
And what number do you write in the
answer?
Do it.
Now what do you do?
(Multiply.)
Do it.

The routine in Figure 19.14 does not deal with the discrimination of whether the answer is too big—
merely with the steps involved in the conditional part. We would next present a routine that involves the
discrimination and the steps for carrying out the contingent part if the answer is too large. A possible strategy
would require the learner to work a series of problems in which the answers would follow the positives-first
pattern: P, P, P, N, N, P. The first five problems might be:

2 6 9 9 6
29 568 31 188 56 505 56 503 31 180
– 58 – 186 – 504 – 504 – 186

Following the variation above, a similar sequence of problems that give only the answer could be
presented:

Example Teaching Wording

9 Is the number in the answer too big?

56 505 How do you know? (If it is too big:)
What do you do? Do it.

We could use a variation that requires the learner to, “First circle all the problems with answers that are
too big.” After the learner has circled these problems, we ask, “How do you know the answer is too big in
this problem? . . . So what do you do? . . .” The learner then works these problems. When this procedure is
used, the spatial juxtaposition of the problems is not as important as it is for the other procedure. The reason

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 is that the learner “groups” the positive examples of over-estimated numbers by circling all examples.
Regardless of their positioning on the page, these problems are juxtaposed when they are worked.

Figure 19.15 shows a schedule for the activities involved in teaching the conditional part.

Figure 19.15

Components Lessons
1 2 3 4 5
A. Problems that do not
involve over-estimation
B. Discrimination X X
C. Contingent part X X

D. Combination X X
E. Worksheet A A,B C A,D D

The answer is written for each worksheet problem. When working independently, the learner marks or
indicates problems in which the number in the answer is too large. The learner works on the conditional part
by crossing out the number in the answer and the number subtracted. The learner then replaces the number in
the answer, multiplies by the new number, and subtracts to verify that the new number is not too big.

The development of conditional parts presents nothing that is generically new. Some particular feature or
detail of the problem keys the contingent behavior. We can identify that detail and we can design a
communication that teaches it. Furthermore, we can design the communication so that we achieve relatively
efficient juxtaposition of events. Juxtapositions are more efficient if we can shorten the chain of behavior and
give the learner repeated practice on “doing the same thing.” This axiom translates into abbreviated routines.
First, we work with the discrimination that keys the conditional part. Then we chain the discrimination to the
routine for the conditional part, requiring the learner to identify whether the conditional routine is called for
and to work problems that require the conditional routine. When the learner is facile with this abbreviated
routine, we can introduce the entire routine and require the learner to work all problems. We integrate the
types (the conditional-part routine and the non-conditional part routine). First we block examples; then we
disperse examples of the two types. Now the discrimination is both firm and integrated within the context of
the problem types that call for the discrimination.

Summary

Strategies for scheduling of component skills, for presenting examples of cognitive routines, and for
dealing with conditional parts of routines can be largely explained with reference to basic instructional-design
procedures.

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 Component skills that are brought together in a routine can be viewed as members of a set and scheduled
accordingly. They are introduced through an initial teaching sequence, reviewed, and integrated with other
skills that have been taught.

The routine is an initial teaching tool, designed to acquaint the learner with a procedure that is to be
applied to a range of examples. Sequencing examples of a routine initial teaching sequence for basic-form
discriminations.

Examples should show the range of variation, should show how differences in the examples lead to
specific differences in the response, and should provide an adequate test. Examples that require difficult
responses should either be eliminated from the initial presentations or should be included at the end of the
example sequence—appearing after the learner has demonstrated mastery of the routine with simpler
examples.

To avoid stipulation for routines that deal with “limited generalizations,” the negative examples should
be introduced early. (Irregulars should be presented early enough to assure that the learner does not
overgeneralize that all examples will be regular.)

The procedures for dealing with conditional parts of routines derive primarily from concerns over
juxtaposition of steps. We encounter the same problem with long routines. If the routine is very long, it is not
well-designed to stipulate the various steps or to provide practice for those steps that may be troublesome.
We first teach the non-conditional routine. Teaching the conditional part involves introducing: (1) a
discrimination, and (2) a conditional routine.

The schedule of events for the conditional routine first introduces the discrimination that triggers the
contingent steps, then introduces the steps for dealing with the conditional part.

Conditional parts present one of the most elaborate programming forms that we use in communicating
with the learner. The conditional part must be coordinated with the non-conditional routine.

Despite the complexity of the routine with conditional parts, this design problem presents very little that
is new. The types of discriminations processed through these routines admit to the same analysis that serves
us in other design situations. The procedures for introducing members to a set, providing initial teaching,
reviewing discriminations, and expanding them, are common to the procedures used for many more basic
communication forms.

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 Prompting Examples
A prompt of an example is a detail that is added to an example so that relevant features of the example
are more obvious than they are in the unprompted example. A prompt is used only for initial teaching and is
then removed. (Verbal instructions are not prompts of the examples as we use the term.)

The form of a routine depends on whether the examples are prompted. If the word-reading examples are
prompted with an arrow that runs under each word, showing the direction in which the word is to be decoded,
the routine can refer to the arrow. If no arrow exists, the routine must change. The routine could still refer to
the “starting point” of the word, because the word still possesses the feature of having a “left letter” or the
one that produces the first sound. The arrow simply adds details that make the starting point more obvious.

All prompted examples work in basically the same way as the arrow. They provide details that increase
differences. Both ends of a word may look the same to the naive learner. With the addition of the arrow,
however, one end looks discriminably different from the other:

if
The letters b and d are highly similar because they are the same object with different orientations. A
prompt can create additional discriminable differences between the letters.

bd bd
unprompted prompted
one-feature difference four-feature difference

The two basic assumptions underlying prompting are:

1. Prompted examples are easier because they make the relevant features of the prompted example
more obvious (increasing the probability that the learner will attend to some relevant detail of the
prompted example.)

2. After prompts have been removed, the learner will perform with the unprompted example.

Difference Prompts and Sameness Prompts

There are two generically different types of situations in which prompting is appropriate. One deals with

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 difference between examples of a particular type and examples that are minimally different. (The b-d
discrimination is an illustration of this type.) The other type deals with sameness. The purpose of the
sameness prompt is to show the learner something that is the same (but possibly not obvious) about all the
examples that will be dealt with. The arrow under the word illustrates this type of prompt. All words are read
from left to right, and the arrow prompts this common feature. Different procedures are used for analyzing,
designing, and fading the two types of prompts.

Difference Prompts

These prompts are used to create a greater difference between minimally different examples. As the
description of the b-d examples indicates, the prompt increases the number of features that are different.
Therefore, the prompt increases the probability that the learner will attend to some difference between the
examples. If there is only one difference between the positives and negatives, the learner must attend to that
difference. If four differences are present, the learner may reliably discriminate between the examples by
attending to any of the four differences or any combination of the differences.

For difference prompting, we prompt only those features of the example that are relevant to the
discrimination. The features that are relevant to the discrimination are those that are different between the
positive examples and the negative examples. To determine which features are different, we juxtapose a
positive example (for which a prompt is being considered) with a negative that is minimally different. The
difference between the two examples is the only feature that should be prompted.

If we were interested in designing a prompt for adding fractions that would alert the learner not to add the
bottom numbers, we would create an example of a fraction-addition problem and juxtapose it with a fraction
multiplication problem that has the same numbers. These examples are minimally different.

4 6
+ =
3 3

4 6
x =
3 3

The features that are different across these two examples are the features that are relevant to the
discrimination of addition and multiplication of fractions. The only difference is the sign; therefore, the only
element that should be prompted is the sign.

If we modify the signs so they are more greatly different from each other, we create no misrule problem.
If we prompt any other elements of the problem, however, we create a situation in which the learner can
solve the problem by attending to the irrelevant details of the problem, not the relevant ones. We could create
an inappropriate prompt for the discrimination of addition and multiplication by using two types of numerals
as prompts:

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 4 6
+ =
3 3

4 6
x =
3 3

Only the top numbers in the addition problem are italic. Therefore, the learner operates only on the top
(by adding).

Both top and bottom numbers in the multiplication problem are italic. Therefore, the learner operates on
both the top and the bottom of this problem (by multiplying across the top and multiplying across the
bottom).

The spurious nature of the prompt can be determined by simply answering the question, “Would it be
possible for the learner to discriminate between prompted multiplication and addition problems without ever
attending to the detail of the example that is relevant to the dis- crimination (the signs)?” The answer is
“Yes.” The learner could develop the strategy of simply finding the bold numerals and operating on these
numerals. In this case, the prompt is a success if we look only at the learner’s performance on the prompted
examples. When we remove the prompts, however, we will certainly find that some learners will be unable to
perform, because they were never required to attend to the relevant difference between the two types of
problems. A number of appropriate prompts are possible with the signs of the problem. Below is a pair of
prompted signs.

4 3
=
3 3

4 3
=
3 3

The addition sign shows that you operate only on the top. The multiplication sign points to the top and to
the bottom, indicating that you must multiply on both top and bottom. These signs are simple modifications
of the already-existing signs. Some features have been added to direct the learner. The learner, however, must
look at the sign to find the prompt. The learner is therefore attending to the element that is relevant to the
discrimination. After the prompts have been removed, the unprompted element still provides hints that were
made more salient by the prompted signs. The unprompted addition sign has only one bar that points to the
following fraction, while the times sign has bars that point to top and bottom. This prompt, while not
essential to teach the discrimination of multiplication and addition of fractions, is not as potentially dangerous
as a prompt that makes the irrelevant features of the example more salient or discriminable.

Spurious prompts may be created for many types of cognitive knowledge. We could prompt the naive,
non-reading learner to read color words by printing each word in the color that the word names (presenting
the word red in red, the word yellow in yellow, etc.). A comparison of the word yellow and any word that is

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 minimally different discloses that the relevant difference between the word has to do with the letters:

yellow
yeller

The color prompt, therefore, permits the learner to identify the word by attending to features that are
irrelevant to the discrimination of yellow and any other word. Just as the learner who is provided with the
irrelevant prompt for adding fractions might work every problem correctly without attending to the part of the
example that is relevant to the discrimination, the naive reader is provided with the option of “reading” every
word correctly without attending to the details of the word that are relevant to the discrimination. (Even if we
“fade” the prompt out, so that a word like yellow is primarily black with only tiny flecks of yellow, the
learner is still provided with an irrelevant prompt.) When we remove all prompts, the prediction would be
that a fair percentage of naive readers would call every color word, “black.”

The procedure for analyzing discriminated features that may be prompted is to juxtapose examples of the
type that is to be discriminated with minimally-different negative examples. The difference between the
positives and negatives indicates the features that may be prompted.

The assumption of prompting or exaggerating the difference is that once appropriate behavior has been
established with the prompted example, the learner will attend to other features, including the features that
are relevant to the unprompted examples. Therefore:

1. The relevant features of the unprompted examples must be present in the prompted counterpart.

2. The teaching of the prompted examples must be designed so that the learner attends to the
features that are relevant to the unprompted example.

3. The transition from prompted to unprompted examples must be executed to assure that the learner
attends to relevant features.

Difference prompts must take the form of additions or simple transformations, not substitutions. We
cannot replace d with a symbol such as ∞. While the learner may have little difficulty discriminating between
b and ∞, the new symbol is not a variation of d and does not have the features of d; therefore, the symbol
does not serve well as a prompt. It requires initial learning (identifying the symbol) which proves to be dead-
end learning when the transition occurs and the symbol d is introduced. The prompted example ɗ has all the
parts of the unprompted example. The shape has simply been transformed somewhat to create difference.

Teaching with prompted examples. If the learner does not attend to the features of the example that will
be present when the example is no longer prompted, the learner will not recognize the unprompted example.
We must assure that this does not happen. One teaching procedure that requires attention to the features of

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 examples that will remain relevant involves pairing juxtaposed prompted and unprompted examples. For
instance, the learner may be required to trace sentences that do not use the prompted letter d.

his dad dug

his dad dug
Note that the dotted letters show the conventional d. The learner first reads the sentence above (with
prompted d’s), then traces the sentence and reads the traced sentence. When performing this last step, the
learner is actually reading unprompted orthography.

Other discrimination tasks could include:

ddddddd
Teacher: “Look at the d’s. Two of them look more like b’s. Circle those d’s.”

This task alerts the learner to the idea that although the objects are still labeled d, they differ somewhat
from those with which the learner is familiar. The task can be dangerous for low performers because it calls
attention to b and possibly prompts a spurious transformation. This problem could be avoided if the teacher
circled one of the conventional d’s and told the learner, “Circle all the d’s that look like this d.”

Transition to unprompted examples. The transition for a difference prompt should occur: (1) after the
learner has developed some fluency using the prompted examples, but (2) before the prompted examples
become so heavily stipulated that the learner has trouble with the unprompted examples.

As a rule, the difference prompts should be retained until the learner has become reasonably facile with
the prompted examples, but has not attained a “high” criterion of performance. As soon as the behavior has
been reliably established, we should provide a transition to unprompted examples.

Sameness Prompts

The prompting procedures that we have outlined apply to types of examples that are to be discriminated
from other types. In some cases, however, we are interested in prompting common features, not those that are
different. Both the analysis of sameness features and the procedures for prompting them are different from
those used for discriminated features.

If the feature is the same across all examples the learner will encounter, the feature may be prompted by

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 creating a detail that has no counterpart in the unprompted examples. As noted earlier, a common feature of
all examples of reading words is the left-to-right progression. There are no examples in which the learner
reads from right to left. Since this left-to-right orientation is common to all examples the learner will work
with, we may create a prompt that has no counterpart in the unprompted examples. For instance, we may
create an arrow under the word:

m a t
This prompt provides the learner with a basis for approaching any word. Because the learner’s behavior is the
same for every word with respect to starting left and proceeding right, the prompt creates no serious problem.
After the behavior has been established and is stipulated through many practice trials, the prompt is removed
and the behavior will remain (because of the stipulation).

This situation is greatly different from the one in which we prompted the signs in the fraction problems.
In creating prompts for the fraction problems, we had to modify each sign so that it was a clear variation of
the sign that would occur in the unprompted examples. We were permitted to prompt only the sign because
only the sign discriminates the addition and multiplication operations. The word-reading example, on the
other hand, involves a feature of the examples that is common to all examples. The function of the prompt is
not to call attention to a detail that may be confused with some other detail. Rather, the function is to stipulate
the common behavior.

Consider working column problems of addition, subtraction, or multiplication. The learner always begins
with the ones column. Therefore, it would be perfectly reasonable to create a prompt that makes the ones
column different from the other columns.

23
+ 16

Although an analogous prompt (bold digits) would be spurious for prompting the difference between adding
fractions and multiplying fractions, it is not spurious for the problem above because it deals with an
operational feature common to all column problems. We could therefore introduce the bold-digit prompt
without running the risk that the learner will develop a misrule. It is not possible for the learner to attend to
the prompt without attending to a feature of the example that is relevant to the operation.

Because we are prompting something that is the same for all examples, we are permitted to introduce
details that will not appear in the unprompted examples. Therefore, we could use this prompt for the column
problems:

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 23

+ 16

No serious misrule problem is created.

Teaching with sameness prompts. A general procedure for teaching with sameness prompts involves
pointing out the prompt and interpreting it to the learner. When introducing the column problems, the teacher
would point out the bold numbers and present this rule: “You always start with these heavy numbers. Then
you add the other numbers.” The fact that the learner performs the operation provides sufficient evidence that
the learner is attending to some relevant details of the problems.

Transition to unprompted examples. If the goal of the prompt is to direct the learner to perform some
operational step that is the same for all problems, the problem of stipulation is not serious. Therefore, the
prompt may be retained longer than a prompt for demonstrating difference between minimally different
examples. As a rule, we can retain the sameness prompts until the learner becomes quite facile; however, we
may remove them as soon as the learner becomes reliable in performing the operation.

Summary of Prompting Procedures

Prompts may be used to show differences between minimally different examples or used to show
operational sameness. The simplest procedure for determining the type of prompt that is appropriate is to
juxtapose minimally-different examples. The features that are different between these examples are features
that may be prompted through difference prompts. The features that are common to the examples are features
that may be prompted with sameness prompts. If the equal sign is a common feature that is relevant to both
examples, the equal sign may be prompted in both examples (a sameness prompt). If the two problems have
different signs and if this difference is relevant to the operation, the signs may be prompted differentially (a
difference prompt).

For difference prompts, we may only modify the unprompted example.

For sameness prompts, we may modify the unprompted example or create a new addition or detail to the
example.

For difference prompts, the transition to the unprompted examples must occur as soon as we can
practically engineer it.

For sameness prompts, the transition may occur early or may be delayed. The delay will not create a
serious stipulation problem.

Prompts and Cognitive Routines

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 The context in which prompts become most important is the cognitive routine. There are two reasons:

1. Prompts may be used to facilitate the creation of unambiguous overt responses.

2. Prompts may be used to reduce irregular examples.

A cognitive routine works only for examples that have a fixed value and that work in the same way as the
other examples processed by the routine. For instance, we can use this routine: “Say the sounds . . . Now say
the sounds fast,” for decoding words such as met and top, but not for words such as me, and so. The letters e
and o make different sounds in the different words.

The solution to this problem that we offered in Chapter 17 involved adjusting the range of application
and the descriptive rule until:

1. There was a sufficient amount of overt behavior for the routine.

2. The same behavior worked for every application in the specified range.

If we applied this solution, we would make up a routine that did not apply to words like go and me. A variant
routine would be needed for these examples. The original routine would be retained for words like met, got,
and other short-vowel words.

Another possible solution to the problem of irregular examples is to use prompts to regularize the
symbols. This solution involves creating new symbols that are prompted. According to the basic procedure
for creating prompted elements for differences, we would design them so they are obvious variations of their
unprompted counterparts. The letter e is the unprompted symbol that will appear in words like me. Therefore,
the prompted counterpart that we create must be an obvious variation of the unprompted symbol. At the same
time, the prompted variations must be clearly different from the unprompted counterparts. For instance, we
could create these symbols: ē and ō. Each would signal the long vowel sound (and only this sound). By
adding these symbols to the set, we can use one routine (“Say the sounds . . . Say the sounds fast . . .”) for all
examples that have long vowels and all examples that have short vowels. Here is the orthography for some
words: gō, sō, mē, bē, met, bell, got, mom.

We refer to this solution as regularizing the examples by creating new prompted examples that will later
be dropped.

A problem similar to the decoding problem occurs with numerals. The symbol 1 does not always stand
for a value of one. In 103, it stands for one hundred, while in 310, it has a value of ten. The value is not
obvious from inspection of the symbol. Just as we can assign prompts to make perceptible differences
between the various e’s we can assign prompts to make perceptible differences between the various 1’s. By
writing 10 in this manner: 10, and 100 in this manner: 100; we can use these symbols when the fixed value 10

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 or 100 is called for in a problem. We would write 18 as 108 and 108 as 1008. The regularized set includes the
symbol 1 and these other symbol variations. All prompted 1’s are highly similar to 1. Each numeral in the
prompted set has a fixed value and always retains this value. The symbol 10 always has the value of ten in the
regularized set, never any deviant value.

There are other solutions to the regularizing problem. We could introduce a routine that requires the
learner to indicate the number of places and then identify whether the unprompted 1 in the numeral tells
about one, one ten, one hundred, etc. For some situations, this solution would be a better choice. For others
(particularly those that are designed for relatively naive learners), the prompted variations would present the
most efficient solution. In any case, the prompt serves as an alternative approach to dealing with symbolic
matter that is not perfectly regular. We can create new examples that are prompted and that may be processed
through the same routine that processes the unprompted examples.

In summary, the prompts used with symbol systems:

1. Permit the use of a particular cognitive routine across a much broader range of applications.

2. Eliminate or reduce the number of irregular examples.

3. Create a perceptible variation in an element or a symbol for each response.

The process of making up regularized (or partially regularized sets) follows these three rules:

1. All elements (symbols or objects) that appear in the conventional set must appear in the
regularized set.

2. All elements in the regularized set must have a fixed value from application to application (always
calling for the same response).

3. All new elements created for the regularized set must be obvious variations of the corresponding
elements in the conventional set.

Stipulation

Like all techniques that make the communication more articulate, the completely regularized set involves
trade-offs, the greatest of which is the possible stipulation that occurs if the learner is exposed to only
“regular” examples. This teaching stipulates that there are no exceptions and does not adequately prepare
the learner for the strategies required when the prompts are removed. Also, the transition from the
regularized set of symbols to the conventional set may be difficult. The more completely the set has been
regularized, the more involved the transition becomes. The longer the learner has worked with the regularized
examples, the greater the probability that the learner will have trouble with the transition.

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 Regularized Symbols for Decoding

According to the three rules for regularizing a set, we would include all conventional letters in the
regularized words (rule 1). Only one response would be produced for any element in the regularized set (rule
2). Any new element we created must be an obvious variation of the corresponding element in the
conventional set (rule 3).

If we apply the prompting procedures to letters like t, we must create a prompted variation that applies to
the th sound. It must look like a t, but should be linked with the h and should look different from the t that
signals the regular t sound. Here are three possible conventions:

math math math

If we use the underlining convention, the underlined letters have one fixed value. Letters not underlined
have another. Although the t and the h are quite visible in the underlined variation, the underlining provides a
cue that a different response is called for by the underlined t. The other prompts function in the same way as
the underlined th functions. To transition from the regularized set to the conventional set, we simply
eliminate the prompt. In the meantime, the learner reads the regularized elements, a skill that should easily
transfer to the conventional orthography.

A similar regularizing problem exists with respect to silent letters. In the word have, the final e does not
make a sound. According to the three criteria for creating a regularized set, we must somehow retain the letter
for the silent e, create a discriminable difference between a silent letter and one that is not silent, and design a
silent letter so that it is a recognizable variation of e.

Below are three conventions for creating regularized silent e.

have hav e hav e

Each convention satisfies the requirements for regularized elements. If we use the convention of making
silent e smaller, every smaller e has a fixed value (it is silent or not read). The learner can easily discriminate
between a normal-sized e and a smaller e. The conventions adopted for regularizing silent e’s could be used
for other silent letters, such as the a in breath, or the i in paint.

The orthographic illustrations of prompting are not intended to be exhaustive. They merely point out
some options that are available. There is no prompt that is “correct.” A prompt serves a need. If it is
constructed according to the appropriate procedures, it will serve the need.

Illustration: Telling Time

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 Just as the set of symbols used in beginning reading and those used in beginning arithmetic may benefit
from regularization, the set of examples used in initial time-telling may benefit from regularization.

The hands on a clock are minimally different in features; however, each hand signals a different
operation. The hour-hand signals the learner to read a number on the clock face. The minute hand signals the
learner to multiply the number on the clock face by five. Although it is possible to teach the differences
between the hands and teach which hand signals us to count by five (rather than to read), the elements can be
regularized to show when counting is called for.

On the clock face in Figure 20.1, the long hand (minute hand) terminates in a large stylized 5. It points to
the number that is to be reached through counting by fives. The shorter hand has an arrow. It points to the
number to be read.

Figure 20.1

12
11 1

10 2

9 3

8 4

7 5
6
Teacher: Touch the hand that counts by five.
Learner: (Touches tip of hand.)
Teacher: Start at the top of the clock and count to the
5 hand.
Learner: (Counts 0, 5, 10, 15, then says:) Fifteen after.
Teacher: After what?
Learner: (Touches tip of hand.)

Semi-Specific Prompts

The prompts that we have been using are specific. One prompt is used for one subtype of example. All
subtypes of long-vowel sound receive the same prompt (i.e., a line). All subtypes of silent letters receive the

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 same prompt (a smaller letter). And so forth. When we use a semi-specific prompt, we use the same prompt
for a group of subtypes. For instance, we might use a line over any pair of letters in a word that makes an
irregular sound in that word:

mail bite part soil ship

Because the same prompt is used for various subtypes, it cannot signal a specific response. It merely
points out that the prompted element is “unusual” in some way. To determine how it is unusual, the learner
must attend to the features of the prompted element and apply knowledge of the response that the prompted
letter combination calls for.

The semi-specific prompt may be used as an initial prompt or as a transitional prompt for groups of
previously prompted elements. For example, after th, sh, ai, and i-e have been taught with individual
prompts, this type of transitional exercise might be introduced:

Example Teaching Wording

that First tell me the sound of the part

of each word that has a line
over it. Then tell me the word.

wish (Point to the overlined part of

each word and ask:) What
sound?

mail (Point to the beginning of each

word and ask:) What word?

bite

Figure 20.2 is a flow diagram indicating how the semi-specific prompts can be used in a transition.

Figure 20.2
Original Prompt Semi-Specific Prompt Unprompted
Prompt I

Prompt II Prompt N (no prompts in exercise)

Prompt III Replaces Prompts I-III

The three original prompts are replaced by a single semi-specific prompt (N) after the learner has met
acceptable criteria of performance with the original prompts. The semi-specific prompt requires the learner to
attend to the non-prompted features of the examples. Finally, the unprompted examples are introduced. No

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 prompt signals that the element requires special processing.

For most situations, the use of semi-specific prompts presents too much new teaching to be efficient as
transitional devices. They may be used as initial prompts for sophisticated learners. If the set of examples for
a system is only partially regularized, some symbols may have specific prompts. Others may have semi-
specific prompts. For instance, the same specific prompt may be used for a group of sound combinations (ar,
al, oi, er, ou). In addition to the semi-specific prompts, specific prompts may be used for “joined letters” such
as th, sh, and ch. The specific prompts direct the learner to make a unique response for each pair of joined
letters. The semi-specific prompt merely alerts the learner that the letters make a variant sound.

Prompting Through Juxtaposing or Grouping Examples

A final type of prompt is achieved by grouping examples together and indicating that all examples in the
group are the same in some way. The grouping strategy is appropriate when the learner has been taught some
examples of a particular subtype and we want the learner to: (1) learn new examples of that subtype, or (2)
learn an unprompted form of the prompted examples.

Both situations involve some form of standard transformation. Therefore, a variation of the double-
transformation program can be used to teach the desired behaviors.

If we have taught a prompted th and we want the learner to respond to unprompted th, we could present
the sequence shown in Figure 20.3. To introduce the sequence, the teacher might say:

“You’re going to read some words that have the sound ‘ththth.’ Then you’re going to read the same
words when the letters for ‘ththth’ look different.”

Figure 20.3

Prompted Unprompted Integration

1. that 5. with 13. top

2. math 6. math 14. bat
3. than 7. then 15. math
4. with 8. that 16. hat
9. the 17. than
10. these 18. that
11. bath 19. mat
12. those 20. hen
21. tan
22. bath
23. then

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 The sequence is different from the double-transformation sequence in two ways: (1) additional examples
appear at the end of the unprompted set; and (2) the integration of types consists of the unprompted examples
and examples that have t or h, but that do not have the sound combination th.

Figure 20.4 shows a possible sequence for dealing with long-e words. The sequence ends with an
integration set consisting of the unprompted examples and other words that are not minimally different from
the new type.

Figure 20.4

Prompted Unprompted Integration

1. she 5. with 12. at 19. reed

2. seem 6. math 13. peep 20. she
3. weed 7. then 14. ship 21. sand
4. me 8. that 15. red 22. met
9. the 16. sat 23. sit
10. these 17. me 24. seem
11. bath 18. mad

The juxtaposition prompting strategy is particularly appropriate for teaching families of irregulars. If the
learner has been taught the words other, come, and some, we could use the sequence below to introduce new
words.

come brother
some done
other won
mother

To introduce the sequence, the teacher could say: “The o in all these words makes the same sound.”

For such skills as spelling, the juxtaposition strategy is probably the most important prompt we can use.
The learner must learn that the e sound in some words is represented with the letters ee, in other words with
ea, and in still other words with e. No before-the-fact rules are adequate for describing which spelling is
appropriate for a new word that makes the sound. By grouping the words, we can establish the
“generalization” of using a particular spelling, and we can limit the generalization to the examples we show.

Let’s say that the learner can already spell bean and seal. The following is a possible juxtaposition
sequence:

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 Teacher Wording
You know how to spell some words that are spelled
with the letters e-a. You’re going to spell some new
words that have the letters e-a.
Seal. What word?
How is the sound ee in seal spelled?
Spell seal.
Bean. What word?
How is the sound ee in bean spelled?
Spell bean.
Eat. What word?
How is the sound ee in eat spelled?
Spell eat.
Seat. What word?
How is the sound ee in seat spelled?
Spell seat.
Meat. What word?
How is the sound ee in meat spelled?
Spell meat.
Leap. What word?
How is the sound ee in leap spelled?
Spell leap.

Note that this routine is quite powerful in prompting the desired behavior because it establishes a three-
step attack with familiar examples. The same steps are then presented with unfamiliar examples, starting with
seat. This sameness suggests to the learner that the new examples are the same.

Although grouping is a powerful strategy for showing how members of a family are the same, the
grouping must be followed by practice with non-grouped examples. The grouping strategy involves creating
“blocks” or groups of examples, all of which are the same with respect to a particular operation. We saw
variations of the strategy in designing worksheet items (Chapter 16). After items appear in a block, the items
are dispersed. By breaking up the groups, we create contexts of increasing difficulty. The transition from
grouped to dispersed examples does not necessarily occur as a single all-or-none change. Rather, the larger
blocks might first be broken into smaller blocks, each of which is dispersed. After the learner performs well
on the smaller blocks, the final dispersion occurs.

As a further transition from the grouped examples, we can mix examples of the prompted type with other
examples in a way that still prompts the desired responses. If we retain a high percentage of the previously
prompted examples in a particular block, we prompt the desired response. For instance, if we mix 50 spelling
examples, half of which involve the letters ea, the mixed group continues to prompt the ea spelling, simply
because the examples of this spelling occur very frequently, although examples are not juxtaposed. This
unprompting strategy shapes the context in which the response occurs, starting with the most highly-
prompted context (the juxtaposed examples) and proceeding to the least-prompted context (the unpredictable

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 mixture of the previously prompted examples with other familiar examples). We will encounter this context-
shaping strategy again when we deal with non-cognitive operations (Section VII).

Complete Regularizing Versus Partial Regularizing

Regularized elements are inventions. We should not become enamored with inventing, without
considering ultimate effectiveness. The ultimate effectiveness of a regularized set of elements depends
greatly on the future curricula slated for the learner. If we could design the learner’s future so that everything
the learner read followed orthographic conventions that we specified, we could design a perfectly regularized
code. We could redesign letters, spell phonetically, rename letters as sounds, and generally design the set of
elements so that there are no highly similar members, and so that learning to read is a simple variation of
learning to spell. For this code, spelling words would involve nothing more than saying them slowly.

Unfortunately, the learner is expected to read conventional orthography. Spelling English words is not
perfectly congruent with reading them. Therefore, we must consider the number and type of misrules
associated with different regularizing conventions. If we make everything in the reading code regular, we are
inducing a possibly serious misrule, which is that written words have a perfect correspondence to spoken
words. To teach the learner a basic tenet of our conventional orthographic code, we must introduce some
early words that are irregular. These words teach the learner that there is a type of word that is not
pronounced as the spelling would suggest. This code is only partly regularized.

Questions about the future skills that are scheduled for the learner determine the extent to which
arithmetic may be regularized. Arithmetic notions are perhaps as irregular as those associated with
conventional orthography used in reading. To convert a fraction notation into a notation of division, rotate the
fraction ninety degrees in the clockwise direction:

1 4 1
4

To convert a fraction into a fraction that use s this “divide by” sign: ÷, the fraction is rotated in the opposite
direction:

1 1÷4
4

If the learner is expected to deal with this horrible inconsistency, we must design the teaching so that
both notations are taught. However, the job would be much easier if the notation 1+4 were completely
discarded.

If we are assured of working with the learner for the first four or five years of arithmetic instruction, we
can create notations far superior to those used in traditional arithmetic, and we will have ample time to

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 transition the learner from these to the traditional notations. Furthermore, we will be able to teach skills in an
order that is far more efficient than that of traditional sequences. One thing we might do is regularize the
notation system so that any problem can be read (and worked) by approaching them from the left or from the
right. The traditional notation does not permit the learner to approach this problem from the right:

2J + 5 = 7 – 15

If we read the problem from the right, the problem says: “Fifteen minus seven.” The minus sign,
however, belongs to the fifteen. We could change the notation system so that the act of “combining” is
separated from the sign of the number. Here is such a notation:

+ + + –
2J + 5 = 7 – 15

Multiplication problems would have similar notations, but use a different sign between the numbers to
indicate that the values are combined through multiplication.

+ + – + –
2xR=5x4x 3
7

All numbers have signs. The sign between numbers tells how the groups are to be combined. The learner
can work the problems from the right or from the left.

Perhaps the biggest advantage of this notation is the conceptual one. It shows that there is a difference
between the sign of the number and the sign used to combine the numbers. One sign stands for an absolute
value, while the other signals an operation-information that is poorly conveyed by conventional notations.

No matter how attractive a system may be, our decision to use it must be based on the learner’s future. If
the learner will soon be required to engage in conventional arithmetic, it may be more efficient to work with
the conventional notation setup.

The beginning Distar Arithmetic program (Distar Arithmetic I) uses little zeros as place-value indicators.
These are faded or dropped after the learner has been taught to respond to the digits. A great advantage of the
little zeros is that it makes the operation of carrying very easy and regular. What is carried to the tens column
is never a one, but a ten. Figure 20.5 shows a routine that introduces carrying.

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 Figure 20.5
Teacher: Read the problem.
30 7
Learner: 37 plus 28.
+ 20 8
Teacher: Where do we add first?
Learner: In the ones column.
Teacher: Listen. If you ever see a ten in the ones
column, say: “Ten go over to the other side.”
This dotted line shows the sides. What are
you going to say if you see a ten in the ones
column?
Learner: Ten go over to the other side.
Teacher: How many ones are we adding?
Learner: Seven plus eight.
Teacher: What does seven plus eight equal?
Learner: Fifteen.
Teacher: (Writes 105 and says:) Oh-oh.
Learner: Ten go over to the other side.
Teacher: (Erases 10 and writes it above the 10
30.) 30 7
Now what do we do? + 20 8
Learner: Add the tens. 5
Teacher: How many tens are we adding?
Learner: One plus three plus two.
Teacher: How many is one plus three?
Learner: Four.
Teacher: How many is four plus two?
Learner: Six.
Teacher: What numeral do I write for six tens?
Learner: Sixty.
Teacher: (Writes 60.)

The operation of carrying is much easier, because the learner can see that the carried digit belongs in the
tens column. It has sameness features that are shared by the other digits in this column (the little O’s).
Reversals in carrying (carrying the wrong digit) are minimized. Also, the concept of what is happening when
carrying occurs is made obvious.

Regularizing the operation of carrying and similar operations can be achieved within a relatively short
period of time (one or two school years). The regularization, therefore, is reasonable. If the regularization
requires a greater amount of time, however, its appeal of efficiency is seriously tempered by practical
problems.

Transitions to Unprompted Examples

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 Two important transition issues are:

1. How quickly should the prompt be faded?

2. How are new examples introduced after the prompts have been removed?

Two strategies are used in transitioning to unprompted examples: removing prompts and modifying
prompts.

Here is an illustration of prompt removal. The learner has been taught to decode words that have joined

letters. After the learner has become proficient at decoding joined-letter words, such as that , the joined
letters are removed. The learner now reads the word that. Removal of prompts is an all-or-none procedure.

When prompts are modified, they are changed in the direction of the unprompted example. If the learner

has been taught to read with small, silent letters, the size of the silent letters are increased (from to

). The prompt is still present; however, since the “small letter” is now less discriminable from regular-
sized letters, the learner is provided with less of a prompt. If the prompt is still present, but the prompted
element is less discriminable from its unprompted counterpart, the prompt has been modified in the direction
of an unprompted element.

Here is a test to determine whether prompt modification is advisable: If a particular detail that appears
in prompted examples also appears in the unprompted example, it can be faded through prompt modification.
The word shave is an unprompted example that contains the detail e. Since the e appears in the unprompted
example, we may use a series of fades to modify a small e in the prompted example. If our prompt is a box
around the e, however, we should not modify the box through a series of fades. Instead we should simply
remove the box. The unit sh is a discriminable element that is present in unprompted examples of the word
shave, which means that we can fade a prompted sh through modification. If our initial prompt involves
italicized letters, sh, we can reduce the degree of italics. If our initial prompt is a joined sh, we can unjoin the
letters through a series of fades.

If the prompted variation of the word is:

shave
details have been added to the a that have no counterparts in the unprompted word shave. Also, the arrow
does not appear in the unprompted word. Therefore, prompt modification is not appropriate for the a or the
arrow. The reason is that if we modified the prompts, the final fade before the removal of the prompts might
be:

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 shave
This example provides the learner with as much information as the original prompted example. There is a
marker to distinguish the a from short-vowel counterparts. Another marker shows the beginning of the word.
The learner may rely as completely on these marks as on longer marks. So long as any detail of a prompt that
is not in the unprompted example is present, the learner may rely on the part. (Note that it would be possible
to introduce a prompt for the long-a that could be modified rather than be removed, perhaps an italicized a.)

Illustration 1. The following are prompts that have been introduced in a reading program. For some
prompts, modification is possible. For others, removal is implied.

1. moon 3. sister
2. fARm 4. sinq
1. A series of prompt modifications is possible for moon. The reason is that the elements oo appear
in the unprompted words moon, soon, loot, etc.

2. Prompt-removal is the most appropriate for sister because the unprompted counterpart, sister,
does not have an element that corresponds to the wiggly line.

3. Prompt removal is required for farm because the unprompted word farm has no shapes like those
in the capital letters AR. Therefore, it is probably not practical to deal with prompt modification.
Although it would be possible to superimpose the lower-case letters over the originals and fade
out the originals, the learner could respond to the presence of the capital letters until the removal
of the upper-case letters had been achieved.

4. Prompt modification for

sinq is possible, because the prompted detail (the tail) appears in the
unprompted word sing.

Illustration 2. Figure 20.6 shows prompts that are used on clock faces.

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 Figure 20.6
1. 12 2.
11 1 12
11 1
10 2
10 2
9 3 9 3
8 4
8 4
7 5
7 5 6
6

3. 4.

1. Prompt removal is called for on clock face 1. There is no line down the middle of an unprompted
clock face and no shading on one side.

2. Prompt removal is called for with clock face 2. The unprompted clock has no arrows outside the
face.

3. A prompt modification series could be used with face 3. The minute hand is present on the
unprompted clock, and the minute hand is “longer” than the other hand. Therefore, it would be
possible to create a series of fades from the prompted hand to the unprompted counterpart.

4. Although the 5 is not present on the unprompted clock face, prompt modification could be used
for face 4. If we treat the 5 as an arrowhead, we can achieve the following type of modification:

It may be argued that so long as there is difference between the points of the long hand and the short
hand, it is possible for the learner to discriminate between the hands by attending to this difference, not to
length. This shape difference is therefore “spurious” because it does not appear on the unprompted clock
face. According to this argument, prompt modification is not advised. Since modification is questionable, we

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 should probably not use it.

Remember, if it is possible for the learner to discriminate successfully by attending to some feature other
than the one that will be present when the prompts have been removed, a misrule is possible. The misrule is
possible as long as the feature exists. The feature, therefore, should be eliminated through prompt-removal.

Illustration 3. We indicated earlier that the double-transformation strategy provides the simplest
procedure for prompt removal. First the learner goes through a series of prompted examples, then the learner
goes through the same examples with the prompts removed. The learner is also introduced to new
unprompted examples. The double transformation juxtapositions show the differences between prompted
examples and their unprompted counterparts.

Figure 20.7 shows a possible series of examples involving a prompted clock face. The first two examples
are prompted. The third example is the same as the preceding example except that the prompts have been
removed. Example 4 is the same as example 1. It is followed by a new example.

Figure 20.7
prompted unprompted
0 0
55 12 5 55 12 5 12 12 12
11 1 11 1 11 1 11 1 11 1
50 10 50 10
10 2 10 2 10 2 10 2 10 2
45 9 3 15 45 9 3 15 9 3 9 3 9 3
8 4 8 4 8 4 8 4 8 4
40 20 40 20
7 5 7 5 7 5 7 5 7 5
6 6 6 6 6
35 25 35 25
30 30

1 2 minimum 3 4 5
difference
The overtized routine for the prompted example 1:

1. Teacher: Touch the hour hand.

(Learner touches short hand.)
2. Teacher: Figure out the hour.
(Learner touches 5 and says, “Five.”)
3. Teacher: Figure out the minutes.
(Learner touches 0, 5, 10, and says: “Ten minutes.”)
4. Teacher: Tell me what time it is.
(Learner says, “Five ten.”)

Before the first unprompted example is presented, the teacher explains: “You’re going to figure out the same times you
just worked, but the clock doesn’t show the numbers that you count by. Pretend that they are there. Start at the top of the
clock with zero and touch where each numeral should be.

The same basic double-transformation strategy is effective for nearly all types of prompt removal. Note
that the prompted and unprompted sets are followed by the integration set.

Prompt modification versus prompt removal. It is possible to modify the prompt if the detail or feature
that is prompted occurs in the unprompted example. Merely because prompt modification is possible does not

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 mean that it is either desirable or efficient in a particular situation. The guiding rule should be to fade
prompts as quickly as possible and with the least elaborate procedure. In some situations, prompt
modification is a reasonable technique. For instance, if we are trying to maintain rate behavior (such as
reading rate), prompt modification through a series of small changes in the program would be desirable
because it would not disrupt the learner’s rate performance. For most situations, however, we should try to
remove the prompt. If the removal program leads to an obvious regression in the learner’s performance, a
series of prompt modifications might be tried.

Correcting mistakes on unprompted examples. If the learner can respond appropriately to fully-prompted
examples, but is making mistakes on unprompted examples, the correction should present a juxtaposition
prompt or a semi-specific prompt.

Let’s say that the learner makes a mistake on the unprompted word tooth by calling the word “Toot.”
The word occurs in the sentence, “She had a sore tooth.” The correction using a semi-specific prompt is as
follows:

Teacher: (Points to the last two letters.) These two letters go

together. What sound do they make?
Learner: Thththth.
Teacher: So what word?
Learner: Tooth.
Teacher: Let’s go back to the beginning of the sentence
and read it again.

The semi-specific prompt merely pointed out that the two letters go together. It did not tell what sound
they made. Note that the last step in the correction involves rereading the sentence—the context in which the
mistake occurred.

The following shows the same correction with the addition of a juxtaposition prompt:

Teacher: (Points to the last two letters.) These two letters go

together. What sound do they make?
Learner: Thththth.
Teacher: So what word?
Learner: Tooth.
Teacher: (Writes on board: booth math them.)
See if you can read each of these words.
Learner: (Reads words.)
Teacher: Now let’s go back to the sentence that begins,
‘She had . . .’

The addition of the grouping step would be called for only if the learner tended to make the mistake
frequently or failed to read the sentence correctly as the last step of the semi-specific-prompt correction.

The following shows the correction with no semi-specific-prompt, merely a juxtaposition prompt:

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 Teacher: The word is tooth. What word?
(Writes on board: booth math them mother
this.)
Read these words.
Learner: (Reads.)
Teacher: Now let’s go back to the sentence that begins,
‘She had . . .’

A pre-correction that provides a juxtaposition prompt consists of a number of examples that share the
same features. For instance, the teacher presents the words math, tooth, thing, and other. The instructions:
“Read these words.” This prompt reduces the probability of a mistake on the examples that follow.

Summary

Prompts are devices that increase differences in examples. Prompts simplify the use of cognitive routines
and increase the range of a routine’s application.

Prompts permit the guidance of behavior that is more overtized, more precise.

Prompts permit the set of examples to be regularized, which means that fewer cognitive routines are
needed to process all the examples in the set.

If a prompt is designed to show the difference between two elements that are minimally different, the
prompt must involve the features that are different between these two elements. The prompt consists of a
modification of the unprompted elements. If the prompt is designed to show an operational sameness, the
prompt may involve the creation of some new element or detail that is not found in the prompted example.

If the set of examples is completely regularized, every element or symbol used has a fixed value (signals
only one response). Since there are no irregular examples (variant values) for the elements in a fully-
regularized set, the learner does not need rules, routines, or reminders for processing irregular examples.
Elements are always processed in the same manner.

Regularization establishes the behavior across a range of types that will later become “irregular.” A
problem occurs because the same behavior will not be used for all examples when the prompts are removed.

The prompted, regularized set will have more symbols or types than the unprompted set, because it will
contain all elements of the original set and as many additional elements as are needed to accommodate all
examples that are created.

The transition from prompted to unprompted examples may be achieved through prompt modification or
prompt removal.

Prompt modification may be used if the detail that is being modified is retained in the unprompted

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 example. If the detail is not present in the unprompted example, however, prompt removal is most
appropriate.

Two types of prompts have extensive application in the transition process—the semi-specific prompt and
the juxtaposition or grouping prompt.

The semi-specific prompt provides the learner with the location of the prompted element and an
indication that the part of the example contains an irregular element. This prompt, however, does not suggest
a fixed value for the prompted elements. The same semi-specific prompt may be used to process examples of
various subtypes.

The juxtaposition prompt is based on the assumption that if new examples in the group have the same set
of features as a familiar unprompted example, the juxtaposition makes common features more obvious. The
grouping prompt may be used as part of the prompt-removal procedure. This prompt also serves to introduce
new, unprompted examples of a particular type.

The juxtaposition prompting procedure is modeled after the double-transformation program. By grouping
the prompted examples together and by following the set with the same examples in their unprompted form,
we show the learner the standard transformation that converts prompted members into their unprompted
counterpart.

The best prompts to use for corrections are the juxtaposition prompt and the semi-specific prompt. Both
prompts require the learner to attend to the features of the unprompted example. At the same time, they
provide the learner with information that part of the example is to be treated in a special or irregular way. For
chronic errors, the semi-specific prompt or the juxtaposition prompt can be used to pre-correct possible
errors.

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 Covertization
Instruction that involves prompts shifts progressively from highly-prompted examples to unprompted
examples. Stated differently, the instruction shapes the context in which the responses occur. First, the
responses are established in the simplest context. Then, the context is changed (as the prompts are either
modified or removed) as the responses are maintained.

Cognitive routines run a course parallel to that of prompts. The routines that are created for initial
instruction are like prompted examples in that they make the context for solving the problem as easy as
possible. The initial routines are designed so that the learner produces overt responses to specific teacher
directions; however, the ultimate goal is for the learner to internalize a problem-solving operation. The
program must therefore provide a transition from a highly-structured routine to one that contains very few
directions. This process shapes the context in which the responses occur. It assures that some form of the
original behavior is maintained as the steps in the routine are removed or modified. In the end, the responses
are to be produced covertly by the learner, with no direction from the teacher.

Covertized Routines

The highly overtized routine is an initial-teaching communication. It is followed by a series of routines

that provide for fewer overt responses or fewer responses that are signaled by specific directions from the
teacher. Each routine in this series is similar enough to the preceding one to prompt the appropriate behavior.

The process of replacing the highly overtized routine with less structured routines is covertization. The
routines are referred to as covertized routines.

As a rule:

1. Each covertization calls for one-half to three-fourths the overt responses as the preceding routine.

2. The learner should usually work on a covertized routine for at least two days before the next
covertization is presented.

3. The covertization process should be no more elaborate than is necessary to assure that the
transition from highly overtized routines to independent work is relatively errorless.

An illustration of an initial routine and four covertizations (A, B, C, D) follows. Note that these
covertizations show the different covertization strategies. The illustration is not necessarily good at
demonstrating how many or what kind of covertizations should be used. It simply shows the types of

 
 covertization strategies.

Original routine: =////

1. Teacher: Touch the equal.
Learner: (Touches.)
2. Teacher: What’s the rule about the equal sign?
Learner: We must end up with the same number
on this side and on the other side.
3. Teacher: Touch the side you can start counting on.
Learner: (Touches side with the lines.)
4. Teacher: Count how many are on that side.
Learner: (Touches each line as he counts: “One,
two, three, four.”)
5. Teacher: How many did you end up with on this
side?
Learner: Four.
6. Teacher: So how many must you end up with on
the other side?
Learner: Four.
7. Teacher: Write 4 in the box.
Learner: (Writes.)

The initial routine contains overt steps to assure that the learner can: (1) say the rule about the equal sign;
(2) identify the side on which he is to count; and (3) apply the equality rule by “ending up with the same
number on the other side.”

Four covertization sequences follow after the learner has responded successfully to about eight
applications of the original routine.

 
 Covertization A
1. Teacher: What’s the rule about the equal sign?
Learner: We must end up with the same number
on this side and on the other side.
2. Teacher: Touch the side you can start counting on.
Learner: (Touches side with vertical lines.)
3. Teacher: Count how many are on that side.
Learner: (Touches each line as he counts: “One,
two, three, four.”)
4. Teacher: Make the other side equal.
Learner: (Writes 4 in box.)
Covertization B
1. Teacher: What’s the rule about the equal sign?
Learner: We must end up with the same number
this side and on the other side.
2. Teacher: Touch the side you start counting on.
Learner: (Touches side with lines.)
3. Teacher: Count the lines and make the other side
Learner: equal.
(Counts lines and writes 4 in box on the
other side.)
Covertization C
Teacher: You’re going to make the sides equal in
this problem. First you’re going to touch
the side you start counting on. Then you’re
going to count everything on that side. Then
you’re going to make the other side equal.
Do it.
Learner: (Counts lines and writes 4 in box.)
Covertization D
Teacher: Make the sides equal in this problem.
Learner: (Completes problem.)

The following strategies were used in the A, B, C, and D covertizations:

1. Dropping a step or steps. In covertization A, step 1 was dropped from the original routine
(“Touch the equal sign”).

2. Regrouping a chain of instructions and responses, so the interaction between teacher and learner
changes from T1-L1, T2-L2 to T1-T2, L1-L2. (T stands for a teacher instruction; L stands for a
learner response.) Covertization B contains the teacher instruction: “Count the lines and make the
other side equal.” This instruction involves two instructions that had been presented separately in
Covertization A (“Count how many are on that side” and “Make the other side equal.”) This
interaction pattern is T1-L1, T2-L2. The interaction in Covertization B followed the pattern:

 
 Teacher instruction (T1-T2) followed by learner responses (L1-L2).

3. Replacing a series of specific instructions with one inclusive instruction. In covertization A, steps
5, 6 and 7 of the original routines were replaced by a single step (step 4). The wording of this
replacement step, “Make the other side equal,” was not identical to the original wording in steps
5, 6, or 7. The replacement step simply functioned as 5, 6, or 7 had functioned in the original
routine.

4. Creating pairs of equivalent instructions. In Covertization C, the teacher begins by saying:

“You’re going to make the sides equal in this problem.” The teacher then provides instructions
that are equivalent to the first statement. “First you’re going to touch the side you start counting
on; then you’re going to count everything on that side; then you’re going to make the other side
equal.” These directions are simply another way of saying, “You’re going to make the sides equal
in this problem.” In Covertization D, only part of the equivalent instructions appear. (“You’re
going to make the sides equal in this problem.”)

Uses of the Covertization Techniques

Dropping Steps

Steps that are not critical once the learner has developed some facility with the routine may be dropped
from the subsequent covertization. Also, parts of steps may be dropped when they make the routine awkward.
Here’s the first-step of a routine: Teacher says, “Read the problem.” Possibly, this step could be dropped for
the next covertization. The decision to drop depends on the learner’s reading performance. If the learner only
rarely makes mistakes in identifying numerals or signs, the reading could be dropped.

Parts of steps may also be dropped if they are part of a chain that is always produced in the same order.

Teacher: You’re going to multiply across the top and write the answer.
Then you’re going to multiply across the bottom and write the answer.
Do it.

After the learner has worked many problems in the same way, this chain needs less verbal prompting.
Therefore, in a covertization we can drop reference to writing the answers.

The best guideline for whether steps or parts of steps can be dropped (without some form of replacement)
is the learner’s behavior. In the absence of information about learner behavior, we can follow this rule of
thumb: About one-fourth of all steps in a routine can be dropped after the routine has been presented on
three consecutive sessions with a total of eight or more examples.

Dropping steps, however, is not always the most appropriate covertization. The semi-overtized routine

 
 that follows illustrates this point. The routine is designed to prompt the learner to attend first to the bottom
number and then to the top number of the fraction.

12
Example:
4
1. Teacher: What’s the bottom number of this fraction?
Learner: Four.
2. Teacher: How many times bigger than four is the
top number?
Learner: Three.
3. Teacher: So what does the fraction equal?
Learner: Three wholes.
4. Teacher: How do you know that it equals three wholes?
Learner: Because the top is three times bigger than
the bottom.

If step 1 were dropped, the first step of the routine (step 2) would tell the learner the bottom number (4),
but would not require the learner to attend to it in the fraction.

If we drop step 2 instead of step 1, the resulting routine is difficult.

1. Teacher: What’s the bottom number of this fraction?

Learner: Four.
2. Teacher: What does this fraction equal?
Learner: Three wholes.

The juxtaposition of questions is particularly misleading in this covertization. Both questions ask “what.”
The answer to the first question is based on inspection. The answer to the second involves a “figuring out.”
The juxtaposition may prompt a hasty and possibly wrong answer to the second question.

A slightly better routine would be:

1. Teacher: What’s the bottom number of this fraction?

Learner: Four.
2. Teacher: Figure out what the fraction equals.
Learner: Three wholes.

Step 3 is necessary to make sure that the learner is discriminating between “numbers” that are parts of
fractions (such as the denominator 4) and the “whole” number that the fraction equals.

Step 4 should not be dropped because it is the only step in which the learner responds with the statement
about the relationship between top and bottom numbers. (In step 2, the teacher asked about the relationship,
but the learner did not express the relationship.)

This illustration demonstrates the test for dropping steps. If we can identify possible problems created by

 
 dropping steps, we should consider alternative covertization techniques.

Regrouping a Chain of Instructions and Responses

When steps are regrouped, a single step provides the instructions that had been conveyed through two or
more separate steps of the earlier routine. For this covertization strategy, the teacher tells the learner to do
two or more things. The learner then does them.

The general rule for using the regrouping technique is: Regroup separate steps that lead to the same goal
or that are part of the same fixed series of steps. When the learner figures out minutes on the clock the
learner goes through a fixed series of steps that lead to this goal. Possibly, some of these steps can be grouped
together. When carrying a number of a multiplication problem, the learner carries out a series of steps
(writing one of the digits, carrying the other, multiplying and then adding the amount carried.) These steps
always occur in a fixed order; therefore, it may be possible to regroup some of them.

Let’s say that the learner has successfully responded to an overtized routine for multiplying factions and
to a covertized variation. Here’s the latter routine:

3 2 =
Example:
4 3
1. Teacher: Read the problem.
Learner: Three-fourths times two-thirds.
2. Teacher: Multiply across the top and write the
answer to the top.
Learner: (Multiplies three times two and writes six.)
3. Teacher: Now multiply across the bottom and write
the answer on the bottom.
Learner: (Multiplies four times three and writes
twelve.)

The pattern of responses in the format is: T1-L1, T2-L2, T3-L3.

We could create the next covertization by regrouping the parts of the series.

1. Teacher: Read the problem.

Learner: Three-fourths times two-thirds.
2. Teacher: You’re going to multiply across the top
and write the answer. Then you’re going to
multiply across the bottom and write the
answer. Do it.
Learner: (Multiplies across the top and writes six.
Multiplies across the bottom and writes
twelve.)

The instructions in step 2 tell the learner two things to do.

 
 Earlier we observed that dropping steps might not be the most effective approach for covertizing the
fraction-reducing operation. The regrouping strategy could be used more effectively. Here is a possible
covertization of the fraction reduction routine:

1. Teacher: First look at the bottom number. When I

signal, tell me how many times bigger the
top number is.
Learner: Three.
2. Teacher: So what does this fraction equal?
Learner: Three wholes.
3. Teacher: How do you know it equals three wholes?
Learner: Because the top is three times bigger than
the bottom.

Note that two ideas conveyed by the original instructions are chained together in step 1; however, the
wording is not identical to that of the original steps. The learner is told to attend to the bottom number, not to
respond to it. The learner is also told to tell how many times bigger the top number is; however, the learner is
not told to tell how many times bigger than four the top number is. These variations are permissible if it
becomes either awkward or too highly prompted to regroup steps in a way that retains the precise wording of
the original steps.

The regrouping strategy should be used judiciously, with awareness of the responses the learner is to
make. We should avoid creating illogical responses or those that are difficult to correct. If we regrouped the
first two steps of the more highly structured routine by retaining the first question of the original routine, the
step would be: “Tell me the bottom number and then how many times bigger the top number is.” The
learner’s response “four . . . three,” is not very logical. It treats the numbers as if they are coordinate with no
qualification about how they are related. If we regrouped the last two steps of the original routine, the
resulting step would ask the learner about the number of wholes and about the evidence of this conclusion:
“Tell me what the fraction equals and how you know how many wholes it equals.” The response the learner
is to produce is elaborate: “Three wholes because the top is three times bigger than the bottom.” With this
change, we would have to spend time teaching the learner the conventions for responding.

The following is an overtized routine for telling time. Steps 3 and 4 could be regrouped when the routine
is covertized.

 
 Example: 12
11 1
10 2
9 3
8 4
7 5
6
1. Teacher: Touch the minute hand.
Learner: (Touches.)
2. Teacher: How do you figure out minutes?
Learner: Start at the top of the clock and count by
fives to the hand.
3. Teacher: Touch the top of the clock and say zero.
Learner: (Touches 12 and says: “Zero.”)
4. Teacher: Now count by fives to the hand.
Learner: Five . . . ten . . . fifteen.
5. Teacher: Tell me what the hand shows.
Learner: Fifteen minutes.

We could regroup steps 3 and 4 in this way: “Touch the top of the clock, say zero, and count by fives to
the hand.” This is not a particularly difficult step because it deals with a chain of behaviors that the learner
has performed.

Not all juxtaposed steps can be regrouped. For instance, we would not provide combined instructions for
steps 2 and 3. (“Tell me how you figure out minutes” and “Touch the top of the clock and say zero.”)

Steps 4 and 5 could possibly be combined, and there might be some justification in such chaining.
However, if we had desired this behavior, we should have taught it initially, not as a part of a covertization.
The teacher could have structured the step in the overtized routine as follows: “When you count minutes, you
stop at the hand then say the minutes. My turn: Zero, five, ten, fifteen. Fifteen minutes. Your turn . . .”

Inclusive Instruction

Inclusive instructions involve new wording. The wording presents less detail than the original
instructions but results in the same set of behaviors. Inclusive instructions cover behaviors that had been
treated as two or more steps in the original routine.

The original routine for the multiplying fractions can be covertized many different ways through the use
of inclusive steps. Here is one possibility:

 
 3 2 =
Example:
4 3
1. Teacher: Read the problem.
Learner: Three-fourths times two-thirds.
2. Teacher: Figure out the answer.
Learner: (Multiplies across the top and writes the
answer. Multiplies across the bottom and
writes the answer.)

The instructions in step 2 do not provide the information about multiplying, or multiplying across the top
and then across the bottom, but they lead to the same behavioral outcome as instructions in the original
routine. Note the difference between inclusive steps and regrouped steps. When steps are regrouped, the
resulting instructions tend to contain the same words as the original steps. For inclusive steps, more general
wording is introduced (such as “figure out”).

Although the inclusive step does not involve the exact wording of earlier routines, it should use
vocabulary already familiar to the learner. For long routines, there are usually many possible uses of inclusive
instructions. The following is an overtized routine for an addition problem.

 
 Example: 7 3=

1. Teacher: Read the problem.

Learner: Seven plus three equals how many.
2. Teacher: (Points to 7 + 3 side.) Can we start
counting on this side?
Learner: Yes.
3. Teacher: (Points to box side). Can we start counting
on this side?
Learner: No.
4. Teacher: Why not?
Learner: Because the box doesn’t tell how many.
5. Teacher: Circle everything on the side you’re going
to count on.
Learner: (Circles 7 + 3 side.)
6. Teacher: (Points to 7.) How many in this group?
Learner: Seven.
7. Teacher: And what does the problem tell you to do
next?
Learner: Plus three.
8. Teacher: Do it.
Learner: (Makes three lines under the three.)
9. Teacher: Now count everything on the 7 + 3 side.
Learner: (Touches under seven and says, “Seven.”
Then touches each line under the 3 and
counts “Eight, nine, ten.”)
10. Teacher: So how many go on the side with the box?
Learner: Ten.

We could provide an inclusive step for figuring out the side the learner starts counting on. This step
would replace steps 2 and 3 with one that required a strong response: “Show me the side you start counting
on,” or “Read everything on the side you start counting on.”

Steps 7 and 8 of the original routine involve making lines for 3. These steps could be replaced with the
step: “Make the lines.” Although this instruction may seem more specific than any in the original routine, the
learner must figure out where the lines are to be made and how many are needed. (The learner makes lines
only under the 3.)

This covertized routine utilizes inclusive steps for steps 2-3 and for steps 6, 7, and 8.

 
 1. Teacher: Read the problem.
Learner: Seven plus three equals how many.
2. Teacher: Read everything on the side you start
counting on.
Learner: Seven plus three.
3. Teacher: (Points to box side). Why can’t you start
counting on this side?
Learner: Because the box doesn’t tell how many.
4. Teacher: Circle everything on the side you’re going
to count on.
Learner: (Circles 7 + 3 side.)
5. Teacher: Make the lines.
Learner: (Makes three lines under the 3.)
6. Teacher: Now count everything on the 7 + 3 side.
(Touches under seven and says: “Seven.”
Learner: Then touches each line and counts to
10.)
7. Teacher: So how many go on the side with the box?
Learner: Ten.

The resulting routine is shortened to 7 steps.

Equivalent Pairs of Instructions

The use of equivalent pairs is more complicated than the use of the other covertization techniques
because the equivalent pairs strategy generally requires three covertizations. Equivalent pairing is necessary
when the instructions that are to appear in a later covertization do not appear in the original routine. For
instance, the later covertization introduces the instructions, “Find the numerator.” If this instruction does not
appear in the original routine, equivalent pairing is needed. Equivalent pairs consist of two instructions—one
that the learner has mastered (A) and another that calls for the same behavior, but does so with unfamiliar
instructions (B). The equivalent pairing technique assures that the learner will respond to B.

The general procedure for introducing equivalent instructions is first to present them in the A-B order,
then in the B-A order, and finally to present only B.

A-B pairing. The first pairing (in the first of the three covertized routines) is one in which A appears first
and B immediately follows. The learner responds to the familiar instructions, after which the teacher
introduces new wording. (“You’re going to tell me the opposite of big. That is, an antonym for big.”) The
learner does not have to attend to the meaning of the word antonym. It comes after the fact—after the learner
has already listened to the instructions that convey the necessary information.

A variation of the A-B pairing is one in which the new term (B) is presented after the learner has
responded to the A instructions: “Tell me the opposite of big . . . Good, you told me the opposite of big. You

 
 told me an antonym for big.”

The basic task may be embellished by adding tests of the new word. “What kind of word did you give me
for big?” In either case, this basic pairing (A followed by B) has the least potential to teach the learner
because the learner receives all the necessary information from the A instructions and therefore is not
required to attend to B. (The learner could respond appropriately by attending only to the familiar
instruction.)

B-A Pairing. The pairing for the next covertization is a B-A pairing in which the new instruction (B)
comes first. “You’re going to tell me the antonym for a word. To do that, you tell me the opposite.” This
pairing has a greater potential to teach the meaning of antonym because the learner is required to attend to it
to a greater degree.

B Alone. Following two or three sessions of the B-A pairing would be the final covertization, in which
the A is dropped completely and the learner is required to operate only from B. The step in the new operation
would be, “Tell me the antonym for big.” If the learner failed to respond or responded with the synonym, the
basic correction would be to use the B-A pairing, “You’re going to tell me the antonym for big. To do that,
tell me the opposite.”

Illustration: Multiplying fractions. Let’s say that the learner performed well on the fraction-
multiplication operation and we wanted the learner to operate from the instructions: “Find the product.” The
instructions provided in the original operation are treated as the familiar instruction A (multiply across the top
and the bottom). We could introduce the following progression of covertization.

The A-B Pairing

1. Teacher: Read the problem.
Learner: Three-fourths times two-thirds.
2. Teacher: Multiply across the top and across the
bottom.
Learner: (Performs operation.)
3. Teacher: You multiplied across the top and across
the bottom. So you found the product.
What did you find by multiplying?
Learner: The product.

The A-B pairing occurs in step 3 above.

 
 The B-A Pairing
1. Teacher: Read the problem.
Learner: Three-fourths times two-thirds.
2. Teacher: You’re going to find the product. You do
that by multiplying across the top and
across the bottom. How do you find
the product?
Learner: Multiply across the top and across the
bottom.
3. Teacher: Do it. Find the product.
Learner: (Multiplies and writes: 6 )
12

The pairing is introduced in step 2. Note that the step is designed so the learner responds to the question,
“How do you find the product?” The learner, however, has been told what to do (multiply across the top and
the bottom). Although step 3 tells the learner to find the product, the learner is not actually required to operate
from a knowledge of the meaning of product because of the pairing of instructions that occurs in step 2.

B Alone
1. Teacher: Read the problem.
Learner: (Reads.)
2. Teacher: Find the product.
Learner: (Multiplies and writes: 6 )
12

The A instructions are dropped and the learner must operate from the B instructions only.

Illustration: Expanded notation. The learner has been taught this operation for expressing two-digit
numerals as addition facts:

Example: 37
Teacher: Tell how many tens and how many ones
are in this numeral.
Learner: Three tens plus seven ones.

We want the learner to indicate the addition fact in response to the direction: “Expand this numeral.”

 
 A-B Pairing
Teacher: Tell how many tens and how many ones
are in this numeral.
Learner: Three tens plus seven ones.
Teacher: You told me how many tens and how
many ones are in the numerals so you
expanded this numeral.
B-A Pairing
Teacher: You’re going to expand this numeral. To
do that, you tell how many tens and how
many ones are in this numeral.
How do you expand this numeral?
Learner: Tell how many tens and how many ones
are in this numeral.
Teacher: Do it. Expand the numeral.
Learner: Three tens plus seven ones.
B Alone
Teacher: Expand the numeral.
Learner: Three tens plus seven ones.

Illustration: Beginning word reading. The learner can respond to this routine:

 
 Example: mat
1. Teacher: You’re going to sound out this word and
say it fast. What are you going to do?
Learner: Sound out this word and say it fast.
2. Teacher: (Touches ball of arrow.) Sound it out.
(Teacher moves to each sound as
learner says: “mmmmaaaat.”)
3. Teacher: Say it fast.
Learner: Mat.

A-B Pairing
1. Teacher: You’re going to sound out this word and
say it fast. What are you going to
do?
Learner: Sound out this word and say it fast.
2. Teacher: (Touches ball of arrow.) Sound it out.
(Teacher moves to each sound as
learner says: “mmmmaaaat.”)
3. Teacher: Say it fast.
Learner: Mat.
4. Teacher: You sounded out this word and said it
fast. You showed you could read this
word.

B-A Pairing
1. Teacher: You’re going to read this word. To do
that you sound it out and say it fast.
How do you read a word?
Learner: Sound it out and say it fast.
2. Teacher: (Touches ball of arrow.) Sound it out.
(Teacher moves to each sound as
learner says: “mmmmaaaat.”)
3. Teacher: Say it fast.
Learner: Mat.

B Alone
1. Teacher: You’re going to read this word. How do
you do that?
Learner: Sound it out and say it fast.
2. Teacher: (Touches ball of arrow.) Sound it out.
(Teacher moves to each sound as
learner says: “mmmmaaaat.”)
3. Teacher: Say it fast.
Learner: Mat.

Choice of Covertization Techniques

Here are general rules about using various techniques:

 
 1. If the terminal operation involves instructions that are in the initial overtized routine, no
equivalent pairing is required. The strategies will be limited to dropping steps, regrouping, and
creating inclusive steps.

2. If different wording is called for by the terminal operation, equivalent pairing is needed in the
covertizations.

3. If the original routine involves steps that relate to mechanical details that are assumed to be
firmed through the practice with the original routine, drop the steps or parts of steps that refer to
the mechanical detail.

4. If the original routine involves a large number of steps, first try to group these steps to form
“units,” or a series of logically related behaviors. Use regrouping and inclusive instructions.

Let us say that the learner has mastered the following overtized operation:

5
Example: ( )=( )=1
8
1. Teacher: To change any fraction into one by multiply-
ing, you turn the fraction upside-down and
multiply. How do you change any fraction
into one by multiplying?
Learner: Turn the fraction upside-down and multiply.
2. Teacher: What fraction are we starting with?
Learner: Five-eighths.
3. Teacher: What is five-eighths turned upside-down?
Learner: Eight-fifths.

4. Teacher: Multiply eight-fifths times five-eighths.

Learner: (Writes five-eighths times eight-fifths and
multiplies and writes forty-fortieths.)
5. Teacher: What’s the answer for the top?
Learner: Forty.
6. Teacher: What’s the answer for the bottom?
Learner: Forty.
7. Teacher: What does forty over forty equal?
Learner: One.
8. Teacher: We changed five eighths into one by multi-
plying.

Here’s the terminal covertization that will be presented:

5
Example: ( )=( )=1
8
Teacher: Change the fraction into one by multiplying.

 
 By comparing the terminal covertization with the fully overtized routine, we see that the wording used in
the terminal task appears in the highly overtized routine. Therefore, we will not use equivalent-pair strategies
in the covertizations.

Our first covertization lumps component steps together and makes the learner responsible for producing
the rule about multiplying.

1. Teacher: How do you change any fraction into one by

multiplying?
Learner: Turn the fraction upside-down and multiply.
2. Teacher: What fraction are you starting with?
Learner: Five-eighths.
3. Teacher: Change five-eighths into one by multiplying.
Learner: Writes eight-fifths and forty-fortieths.

4. Teacher: What does forty over forty equal?

Learner: One.
5. Teacher: We changed five-eighths into one by multi-
plying.

The most critical aspects of the covertization have to do with dropping the instructions from step 1, and
the inclusive instructions for the behaviors of the multiplication process.

Step 1 tests the learner on the rule. Step 2 (which could possibly be dropped with no great loss) remains
overt. Step 3 is an inclusive instruction including behaviors presented in steps 3 to 6 of the original operation.
This lumping is permissible because the learner has already practiced the individual behaviors as a chain that
occurs in a fixed order.

Following the covertizations above would be the terminal task: “Change this number into one by
multiplying.” It would be possible to interpolate other covertizations before presenting the terminal task;
however, the reduction of steps would be somewhat trivial. We assume that the learner can read the fraction
and can perform the step of turning the number upside-down and multiplying. The learner can therefore
perform all the steps covertly.

Timetable

We cannot provide a precise specification of how many covertizations should be introduced or how long
the learner should be exposed to each. However, we can provide rule-of-thumb guidelines based on logical
considerations.

1. To be safe, retain each routine for two days or lessons. The two-day exposure guarantees that the
learner works the sequence of steps frequently enough to make the subsequent covertization safe.

 
 2. Process at least two examples through the routine each time it appears. For some routines, four or
five examples should be processed each lesson.

3. For routines of 8 steps or more, do not change the preceding routine more than 25-30% for
relatively naive learners and no more than 50% for experienced learners. To be safe, the 30% rule
can be followed for all learners. For routines shorter than 8 steps, as much as 50% of the routine
can be changed.

As the rules-of-thumb indicate, the longer routines require a larger number of covertizations than shorter
routines. An 8-step routine can be reduced first to 5 steps, then possibly to 3, and finally to 1—the goal
operation. A 20-step routine, on the other hand, would be reduced first to around 15 steps, then to 10, then to
7, then to 5, then to 1. If each variation of the routine is retained for 2 lessons (and if there is no overlapping
of the routines presented on each lesson), the covertization process would require 12 lessons.

Overlapping Schedule

The problem associated with longer routines is stipulation. Since teacher directs the activities in an
atomic manner, the learner may understandably have trouble working independently if the process of
covertization takes too long. To hasten it, routines at different stages of covertization may be introduced in
the same lesson. (This procedure is analogous to the response-shaping procedure that is described in Section
VII.)

The basic overlapping procedure is to introduce two routines during the same lesson, the first of which is
more highly structured than the second. The first-presented routine functions as the “familiar” routine. It
prompts the series of behaviors that is to be followed in the subsequent routine. By juxtaposing the routines
so that the less-highly structured one immediately follows the more highly-structured one, we prompt the
learner to perform on the less-highly structured routine. (The learner does “the same thing” on juxtaposed
examples.) This procedure helps to avoid stipulation because it shows the learner the “direction” in which
instruction is progressing.

Table 21.1 gives a timetable for overlapping covertization of a long routine.

 
 Figure 21.1

Components Lessons
1 2 3 4 5 6 7 8 9
Routine A 4 2 1
Routine B 2 4 4
Routine C 2 4 2
Routine D 2 5 8 8 8

The numbers in each cell indicate how many examples are processed through the routine. Each routine is
presented on three successive lessons; however, in nine lessons, the learner is taken through four routines. By
the seventh lesson, the learner is performing on independent work exclusively. On nearly every lesson before
7 the learner works on two routines. On lesson 6, for example, the learner is presented with two examples of
the C covertization, followed by five examples of the D covertization (which is independent work). The work
on the C covertization prompts the skills that are called for by the independent work.

Correcting Mistakes in Covertized Routines

The correction for mistakes in covertized routines always involves overtizing the part of routine on which
the mistake occurred using overt steps presented previously in the covertization series.

The correction involves:

1. Presenting those steps from an earlier covertization that relate to the learner’s mistake.

2. Repeating the entire covertized problem on which the learner made an error.

Step 1 of this correction procedure is designed to “prompt” the behaviors called for. Step 2 is designed to
test the learner in the context in which the mistake originally occurred.

Note that the correction takes the form similar to that of an A-B pairing. The A is the familiar routine; the
B is the covertized routine. When the learner makes a mistake on B, A is introduced to prompt the
appropriate behavior. B is then presented to assure that the learner can perform on the covertized routine.

Let us say that the learner is performing on this covertization:

 
 5
Example: ( )=( )=1
8

1. Teacher: How do you change any fraction into one by

multiplying?
Learner: Turn the fraction upside-down and multiply.

2. Teacher: What fraction are you starting with?

Learner: Five-eighths.
3. Teacher: Change five-eights into one by multiplying.
Learner: (Writes: 5 ( 5 ) = ( ) = 1 )
8 8

The teacher stops the learner at this point and provides a correction:

1. Teacher: What fraction are we starting with?

Learner: Five-eighths.
2. Teacher: What’s five-eighths turned upside-down?
Learner: Eight-fifths.
3. Teacher: Multiply five-eighths time eight-fifths.
Learner: (Writes: 8 and multiplies.)
5
4. Teacher: (Praises, erases work.)
5. Teacher: What fraction are you starting with?
Learner: Five-eighths.
6. Teacher: Change five-eighths into one by multiplying.
Learner: (Writes: 5 ( 8 ) = 40 = 1 )
8 5 40
7. Teacher: What does forty over forty equal?
Learner: One.
8. Teacher: We changed five-eights into one by multiply-
ing.

Summary

To transport the learner from the highly overtized routine to one that requires independent work, a series
of covertizations or routines of decreasing structure are introduced. Each routine in a covertized series is
similar enough to a preceding one to prompt the appropriate series of steps, but different enough to represent
a significant movement toward making the routine more independent.

The four covertization techniques are: dropping steps, regrouping interactions, providing inclusive
instructions, and providing equivalent instructions.

Step-dropping, regrouping, and inclusive instructions are appropriate for virtually any routine. The
equivalent-pair strategy is more elaborate than the others and is appropriate when the original overtized
routine does not contain the words that will appear in the final routine.

 
 Typically, more than one technique is used in designing a covertization for a routine.

The steps may be regrouped so the learner performs a logically related set of behaviors in response to the
instructions for the set.

Steps or parts of steps that are needed for any but the introductory routine are dropped.

Inclusive instructions may be presented to alert the learner to a group of related steps that are to be
performed.

When appropriate, the equivalent pairing prepares the learner for new vocabulary that will signal
responses in later routines.

Both the inclusive instructions and the equivalent pairing introduce new words into the covertizations.
The instructions processed through the inclusive-instruction technique are “familiar” to the learner in other
contexts. The instructions introduced as equivalent pairs present new vocabulary.

As a general rule, covertizations of longer routines should involve an initial shortening of 30%. When
inclusive instructions or equivalent pairs of instructions are introduced, the part of the routine in which they
occur is lengthened. Other parts, however, are dropped, regrouped, or changed. Routines of 8 steps or fewer
may be shortened initially by 50%.

If the learner works on highly structured, teacher-directed routines for a long period of time, the learner
may understandably expect to be led through the operation, without initiating the various steps. To buttress
against this type of stipulation, the covertizations of initially long routines are overlapped, so that on many
lessons, the learner works problems of more than one covertization. The more-highly-structured routine is
presented first. It serves as a prompt. After the learner has successfully worked one or more problems using
this routine, the less-structured routine is introduced. This juxtaposition of a more highly-structured routine
and a less-structured one implies to the learner that the same operation is used by the less-structured routine.

As a rule of thumb, each routine should be presented for at least two days, and each routine should
process a total of at least ten examples.

The basic correction procedure involves making the steps involved in the operation overt. If the learner
misses a step in a later covertization that had earlier been processed through four steps, the teacher takes the
learner through the four steps, then presents the routine that requires only one step. By making the steps
overt, the teacher can provide a precise diagnosis of the learner’s problem. By returning to the less-structured
routine, the learner is tested on applying the steps to the context in which the mistake occurred.
 

 
 Section VII

Response-Locus Analysis
So far, we have dealt with sequences and presentations that are based primarily on the analysis of stimuli.
Although the pure stimulus-locus assumption has been modulated somewhat, it remains paramount in these
communication forms. The design of communication forms has been dominated by the notion that the learner
is a perfect receiver and that if we present a communication that is consistent with a single interpretation, the
learner will learn. Evidence of the learning will be the pattern of responses the learner produces and
generalization to examples for which generalization is logically implied.

As noted earlier, the response-locus analysis is an analysis of the learner, not of the stimulus. The analysis
is based on the lawful tendencies of the learner’s behavior to change in response to practice. It is used when the
learner does not respond to a logically faultless communication. For example, we present a faultless
communication that shows the learner exactly how to perform a back somersault in the air. When we test the
learner on the understanding of what is to be done, we observe that the learner is able to describe the motor
response, discriminate it, and label it. However, when asked to perform, the learner fails to produce the
response. Repeatedly and in response to various reinforcers or punishers, the learner fails.

The response-locus analysis now comes into play because we must teach the learner how to produce the
response. This process is slow, requiring many, many trials. The focus of this teaching is on the production of
the motor response (doing a backward somersault in the air), not on verbal discriminations.

Parallels Between Stimulus-Locus and Response-Locus Communications

Although the emphasis of the response-locus communication is different from the ones we have dealt with
in the preceding sections, there are parallels.

1. The procedures for inducing generalization of a new response are basically the same as those used
to induce generalization for discriminations. To show how a particular motor response applies to a
range of variation, we juxtapose applications that are greatly different. If the learner responds “in
the same way” to these applications, the learner should be able to respond to any new application
that falls within the range of variation that had been practiced.

2. The problem of stipulation has a counterpart in new-response teaching. Just as extended practice
on a range of examples that presents a very limited range of variation induces stipulation, practice
on a range of new-response applications that presents a very limited range of variation tends to
produce responses that are distorted. These responses do not readily generalize to new

 
 applications.

3. Juxtaposed practice examples present the simplest context for new responses. Just as juxtaposed
examples of the same discrimination present the least demanding context for showing what is the
same and what is different about the various examples of a discrimination, juxtaposed practice
with a new response prompts what is the same about the various examples of the new response.

4. The various applications that the learner is to respond to admit to a subtype analysis. This analysis
suggests subtypes that are relatively simple (involving applications that require fewer responses,
less coordination of responses, or shorter chains of responses), and subtypes that are relatively
difficult (requiring a greater number of responses, more coordination of responses, or longer
chains of responses). This analysis suggests details of a program for teaching the new response.

Differences Between Stimulus-Locus and Response-Locus Communications

There are three major differences between stimulus-locus communication and response-locus
communications:

1. The goal of response-locus teaching is to teach specific responses. The stimulus-locus analysis is
concerned with teaching a “pattern” that would be manifested with a variety of responses
(pointing, verbal, manipulative). Through any of these responses, the learner could demonstrate
knowledge of what is the same about the various examples. The analysis, therefore, is one of the
stimulus quality that is the same across all examples, and the teaching is designed to induce
knowledge of this sameness. The new-response teaching, on the other hand, has a completely
different objective: to teach specific responses the learner cannot produce. The analysis therefore
begins with the specific task that the learner cannot produce. The response that is called for may
not be substituted for other responses. The teaching focuses on inducing the response. Much of
the information that is relevant to the efficient induction of the new response comes from an
analysis of the learner, what type of fatigue patterns are typically associated with different types
of practice, what rate of learning can be expected, what type of reinforcers are relatively efficient,
and so forth. Certainly, not all of the information that is relevant to new-response teaching comes
from an analysis of the learner. The learner who is trying to learn a back somersault in the air is
the same learner who learns qualitative samenesses and who generalizes on the basis of these
samenesses. The learner can therefore be expected to apply this same generalization strategy to
responses and to learn “misrules” if they are implied by instruction. However, the instructional
procedures are basically different because the task is mandated as something the learner must do.
The focus of instruction is on modifying the learner and inducing the response to the task.

2. The end product of response-teaching is a specific set of overt behaviors. When we teach the

 
 learner to write, the learner continues to produce the overt writing response after the program has
terminated. These responses are not faded, covertized, or changed. This situation is quite different
from instruction associated with concepts. For concept learning, no particular behavior is implied,
and the goal of the program is usually to covertize the skill as much as possible. The responses
that the learner produces are not the same responses the learner will produce after the program is
completed. The routine that had been constructed to guide the learner and to permit feedback is
replaced by some kind of internalized counterpart that may bear little resemblance to the overtized
series of steps in the routine.

3. The juxtaposition pattern of examples cannot be controlled for response teaching the way it can be
controlled for concept teaching. This difference is the most serious and carries the most extensive
implications for the specific techniques used to teach responses.

Because the learner does not produce a new response appropriately during initial teaching, we are faced
with a double problem: the first is to induce appropriate response to some examples. The second is to induce
generalization to a range of examples. When we deal with concepts, we do not face a problem of first inducing
the response with some examples. The learner is usually able to produce the responses on the first trial. When
the learner produces the responses for different examples on the first trial, there is a close correspondence
between the juxtaposition pattern of the examples and the learner’s pattern of correct responses.

When we deal with response-teaching, we find a great discrepancy between the juxtaposition of the
examples and the pattern of correct trials the learner produces. The reason is that the learner requires many
trials on each example before producing an appropriate response to that example. Since we do not know how
many trials will be required for the learner to master the early trials, we have very little control over the precise
juxtaposition of correct-trials.

Figure VII.1 illustrates the discrepancy between the juxtaposition of the learner’s correct trials with the
juxtaposition of examples. The circled X’s are successful trials. The learner is presented with trials on test
example 1 until an acceptable response is produced. Then the learner practices on example 2, and so forth. The
examples (1-8) are sequenced to show sameness and difference.

 
 The pattern of juxtaposition for the successful trials is completely different from the juxtaposition of the
examples. The successful performance of examples 1, 2 and 3 are not juxtaposed. Following the twelfth trial
with example 1, the first trial for example 2 is presented. This trial is not followed by example 3, but by seven
more trials on example 2. Near the end of the sequence, a parallel evolves between the juxtaposed examples
and juxtaposed trials. We would therefore predict that whatever sameness between examples exists at the end
of the sequence (examples 5 through 8) would be conveyed to the learner. During the first part of the sequence,
however, the discrepancy between the juxtaposition of examples and juxtaposition of successful trials creates a
presentation that is not well designed to demonstrate sameness or difference. The practice trials with the first
examples in the sequence are designed to induce stipulation in some form. (The learner repeatedly practices
the same example.)

Avoiding Traps

Problems of teaching new responses are well designed to lead us into four traps.

1. The first trap is that of displacing the problem from a new response to some sort of
discrimination. Instead of teaching the learner how to operate a handsaw, we require the learner to

 
 work on labeling parts of the handsaw or on other trials that have nothing to do with the central
behavior of operating the machine. The goal of response teaching is to establish words about the
behavior.

2. The next trap is that of identifying the entire event in which the new response is embedded as the
objective of instruction. Instead of teaching the child how to tie a shoelace, the program begins
with a long chain of behaviors—finding the shoes, putting them on, then tying them. Although we
ultimately want the learner to perform the shoe-tying behavior in the context of a chain, we do not
begin with this chain to teach shoelace-tying. We first teach the behavior of tying a shoe, and after
it is taught we incorporate it into a longer chain. The distinction is not whether the chain of
behaviors will be taught, but rather how we begin instruction. The shoe-tying should be removed
from the chain because massed practice on shoe-tying is not provided easily within the longer
chain.

3. The third trap is that of sequencing examples from easy to hard. It seems reasonable to assume
that if we were teaching the learner to operate with screwdrivers, we would bring the learner to a
high criterion of performance on the screwdrivers that involve the simplest behavior (those like
hexagonal wrenches that fit tightly onto the nut that is being turned). The strict easy-to-hard
progression is usually spurious. It fails to show the learner what is the same about the range of
applications. The learner will learn inappropriate behaviors that work with simple examples.
Later, the learner will have to relearn the behaviors. By following the principle of demonstrating
sameness, we avoid the teaching of inappropriate responses. To the extent that it is possible, we
sequence examples so that the learner works on juxtaposed examples that are greatly different.
This work requires the learner to perform the same operation with “easier” examples and with
juxtaposed harder examples. Knowledge of sameness is therefore induced.

Obviously, the demonstration of sameness must be modulated by practical considerations. If

the learner cannot produce even an approximation of the response called for by the more difficult
applications, work on these applications will not demonstrate sameness. We must therefore limit
the sameness description to the range of applications to which the learner can respond with
approximations of the desired response.

4. The final trap is that of practice. Most programs for teaching responses like cursive writing do not
provide for the amount or type of practice needed to induce the desired behaviors. Without
practice, the behaviors will not develop. With practice they will, even if the practice is not
provided as elegantly as possible. Just as there is a tendency to provide inadequate practice, there
is the opposite tendency to recognize practice as the only variable. On-task time and trials are not
the only variables affecting performance. Good programs make substantial differences in the

 
 learner’s performance; however, these programs may show no particular advantage at first.

A related practice issue concerns the criterion that we use for judging that the learner is
“successful” on a particular task that involves new-response learning. A single successful
performance does not imply that the learner has achieved an acceptable criterion of performance.
The learner must demonstrate consistency in the appropriate response. The fact that the learner is
able to produce the desired response in a highly prompted context (following the teacher’s
demonstration and help in producing the response) does not imply that the learner can produce the
response consistently or in various contexts. A much more convincing performance is needed
before we judge the learner to be successful.

Perspective

Chapters 22, 23, and 24 deal with different facets of teaching new responses. Chapter 22 presents the basic
response-induction technique. Chapters 23 and 24 deal with complex physical operations like throwing a ball,
tying a shoe, or performing the chain of behaviors required to get on a bus.

The range of new-response-teaching applications is very broad, but all applications are the same in an
important way. The immediate goal of instruction is to change the learner so that the learner can perform
appropriately on the task. A clearly understood assumption is that we will proceed as rapidly as possible. We
will not attempt to see how many “steps” we can introduce in instruction, how many discriminations we can
teach, or how much practice we can present. Our goal is scrupulous efficiency, which means that we will be
very careful about the starting point for each program, the order and type of examples, and the various
techniques that we may be able to use to achieve the desired behavior as quickly as possible.
 

 
 New Response Teaching Procedures

Identifying New-Response Problems

To identify a new-response problem, we first determine what the learner is to do. We then provide
reasonable evidence that the learner cannot do it. As part of this process, we “reduce” or simplify the task so
that we strip the unessential details from it.

Selecting a Single Objective

Initial teaching for responses is like that for discriminations in that it teaches only one thing at a time. The
teaching may involve a complex response such as shoe-tying, or it may deal with a simple response, such as
holding something between the thumb and forefinger.

We can appreciate the problem of specifying objectives by observing a naive learner eating. We do not
like the way the learner is sitting, the inappropriate use of the fork, the slobbering, the spilling from the cup,
the eating with fingers, etc. We may change all this behavior, but we do not start with “eating properly” as our
objective. We must be far more precise. What aspect of eating properly do we wish to select for instruction?
We are provided with many options, and our ultimate selection will be arbitrary, but we must identify one
specific objective from the array of possibilities.

The good way to identify objectives is to group behavior by goals. What is the learner trying to do with the
fork? (Transport green beans from plate to mouth.) What is the learner trying to do with the glass? (Transport
milk to the mouth.) Each behavioral goal implies activities that are associated with particular goals.

By viewing objectives according to what the learner is trying to do, we automatically identify the
functional behaviors. Furthermore, we avoid the peripheral behaviors.

Demonstrating Behavioral Deficiency

After we identify units of behavior related to a specific goal, we test to make sure that the learner cannot
produce the behavior. For instance, if we identify the behavior of drinking from the glass without spilling, we
must now demonstrate that the learner cannot do it. The demonstration serves as a placement test or pretest for
the program.

Our strategy is to rule out all possible variables that could account for the observed behavior. Perhaps the

 
learner is not trying. Perhaps we observed a non-representative sample of behavior. Perhaps the learner can
perform on the particular behavior when it is removed from the context of the activity in which we observed it.
Perhaps the learner does not understand what to do. To provide evidence that the observed performance is not
controlled by these possibilities, we should:

1. Remove the behavior from the chain in which it occurred. Remove the drinking from eating a
meal.

2. Remove all unnecessary discriminations from the task, and model the exact behavior that is
expected of the learner. Do not use questionable verbal explanations. Simply show the learner
what is expected, possibly with verbal explanations after the fact—as statements that describe
what you did.

3. Arrange the testing setup so that trials of the task can be juxtaposed and presented quickly. If
consecutive trials require more than 10 seconds, something is probably wrong with the skill that
had been identified or with the setup. If you can identify the part of the response that prevents the
learner from completing the trial, discard the original test and test the learner on the troublesome
part.

4. Control reinforcement or attention variables if there is a question about whether the learner is
trying. A number of techniques are available, including presenting a series of familiar tasks,
which is immediately followed by the target task. If the learner performs appropriately on
consecutive trials of the familiar task, and if the learner fails the new task, the probability is great
that the learner cannot produce the response. For the tasks that precede the target tasks, the teacher
directs and the learner complies. Therefore, the learner will probably try to “do the same thing”
(comply) with the target task.

The learner’s performance on the four-step task above not only provides strong evidence about the
behavior the learner cannot perform; it also helps us modify the task so that:

1. Juxtaposed trials can be presented quickly to the learner.

2. Trials do not require great understanding of verbal instructions or discriminations that are
irrelevant to the new response.

3. There are reinforcing contingencies for trials that are judged reinforceable.

These requirements are essential for response teaching.

Reducing Tasks to a Simpler Form

When we design tasks, we try to find a simpler form. Obviously, we cannot simplify tasks indefinitely. We

 
reach a point at which the simplified task no longer requires the intended response. A simpler form is one that
requires the intended response and that has these features:

1. It shares all the essential response features with all other examples of the response.

2. It contains as few component behaviors as any example of the response we can identify.

The response that we design shares all essential response features if it possesses those features observed in
all positive instances of the response, but in no negative instances. Judgment about whether the instance
contains no more components than any other instance of the response depends on our ability to produce
“creative” examples.

The point of applying the tests is to avoid infinite reduction of the operation or response. At some point,
the operation of shoe-tying ceases to be shoe-tying. We want to identify a simple example of shoe-tying, but it
must be an example of shoe-tying, not of making an Indian knot, of holding the teacher’s hands as she ties, or
of dealing with a component of shoe-tying, such as making a loop or pulling on the ends of the laces.

If the objective of the response is to transport food on a fork to the learner’s mouth, the simplest form of
the response is one in which the learner is not required to spear the food, to pick up the fork, or to do any
behaviors except transport the forkful of food from the plate to the mouth. All other behaviors are removed. If
we carry the removal of behaviors to an extreme, we could eliminate some of those associated with holding the
fork. Perhaps the fork is attached to a glove the learner wears. The learner is still responsible for transferring
the material, but the response requires coordination of fewer behaviors.

If the goal of an operation is to throw a ball, the learner should be responsible for the throwing, not for
picking up or any other ancillary behaviors.

Continuous Parts and Non-Continuous Parts

Some responses involve non-continuous behaviors that are strung together. Others consist of continuous
behaviors that cannot be stopped. Tying a shoe involves various non-continuous parts. We can cross the laces
and then stop. We can next loop them and then stop, etc. Throwing a ball, saying the word me, spinning a top,
and doing a somersault, require continuous behavior. We cannot stop (or start) in the middle. The idea is not to
reduce all responses to a continuous behavior, merely to reduce them to a simple form. The ultimate form may
have non-continuous responses chained together; or may consist of continuous behaviors. In either case, the
simple form will have no unnecessary components.

Response-Teaching Strategies

There are three response-teaching problems. Each suggests a different strategy.

 
 1. The learner is capable of producing the response that is called for, but cannot produce the
response in the context of the task that we present. Although the response has been observed in
other contexts, the learner apparently does not understand the new context, cannot remember the
response that is called for, or is not attending to the appropriate details of the example. In any
case, the learner does not produce the response in the new context.

2. The learner apparently understands the directions of the task but is incapable of producing the
response and has never produced it in any context.

3. The learner apparently does not understand the directions and has never produced the response
called for by the task in any context.

The implied remedy for type 1 is to first establish the behavior in a simpler context, then to modify the
context so that it progressively approximates the desired context. This program works on teaching the learner
when to produce a particular response. It shapes the context.

Type 2 suggests shaping the response. The learner gives indications of knowing what to do; however, the
learner cannot produce the response under any reinforcing condition or in any context. Therefore, the
instruction must help the learner form the response. The learner does not need information about when to
produce the response, but needs information about how to respond and needs practice designed to change the
learner’s ability to respond. The program shapes the response through successive approximations.

Type 3 presents a double problem. The learner does not give any indication of knowing what kind of
response is called for and the learner has never demonstrated the response in question. The strategy must make
the learner aware of the type of response called for and must shape the response.

Type-1 Strategies:
Teaching When to Produce Responses

Type-1 strategies show the learner when to produce the response. The learner has a problem in
“associating” a particular response with the signal or cue. For whatever reason, the learner does not decode this
signal as a prompt or does not recall it, and therefore does not produce the response that is called for.

Teaching highly unfamiliar discriminations calls for type-1 strategies. Let us say that we were interested in
teaching the learner to identify notes that we play on the piano. The correct responses are “A,” or “C,” or “G.”
The learner has no trouble producing these responses (saying “A” or “G”). However, after repeated trials, the
learner may not produce the appropriate response when presented with an “A” note or a “G” note. The same
problem is observed with mentally retarded and autistic children, who often require many trials before
successfully responding to a task such as, “Come here.” Yet the behaviors of coming here are not difficult for

 
the learners to produce.

The strategy for remedying type-1 deficiencies is to provide systematic interruptions of trials of the task
being taught. The strategy is based on a fact of juxtaposition, which is that if the learner is required to do the
same thing on juxtaposed examples, the series requires less attention and less memory. Conversely, if the
learner is required to do different things on juxtaposed examples, the series is more difficult—requiring more
attention and more memory. Accordingly, we can make an item or task relatively easy or relatively difficult
according to how the task to be taught is juxtaposed with other events.

Let’s say that we wish to teach task A. Here’s the type of juxtaposition that would make performance on
task A relatively easy:

The tasks are presented in rapid sequence. The circled A is the target task. As the sequence shows, the circled
A is preceded by repetitions of the same task. The probability of appropriate performance of the circled A is
great if the learner has performed successfully on the preceding A’s.

A more difficult pattern of juxtaposition is one that increases the amount and type of interference between
the presentation of A’s. Here is a pattern of juxtaposition that requires the learner to attend to another task
before returning to A.

Assume that the learner performed successfully on the first A and assume that B is a familiar task (one the
learner can perform consistently) that is greatly different from A. The probability that the learner will perform
correctly on the circled A is reduced because the circled A is more difficult by virtue of the juxtapositions. The
reason is that the learner’s attention is pulled from A by the presentation of B. The learner must now remember
more details of how to discriminate A or about how to produce the response that is called for.

We can make the task even more difficult by creating greater separations of the A’s. The more interference
we introduce between the presentations of A, the more difficult the circled A. To create additional interference,
we can pause a longer period of time, introduce a greater number of interpolated tasks, or make the
interpolated tasks more similar to A or more difficult with respect to attention and response demands.

Here is a sequence that is more difficult than those illustrated above.

 
 The separations of A are created by presenting familiar tasks B, C, and D.

The most difficult sequence, with respect to the juxtaposition of examples, would be one in which the
initial A is removed by perhaps hours from the presentation of the circled A. All the interpolated activities that
occur between occurrences of A serve as potential interference for A.

The relative difficulty based on the patterns of juxtaposition provides the basis for designing teaching that
proceeds from relatively easy tasks to those that are more difficult. Below are three levels of difficulty:

The three levels of difficulty imply a three-level strategy. Since level 1 is the easiest level, it is the starting
level. The pattern shown for level 1 is repeated until the learner performs successfully on three or four
consecutive trials. Note that they must be consecutive. The fact that the learner performs once in a while is not
an indication that the learner has worked out whatever operation is required to respond consistently to that
task.

The number of trials presented for a given learner varies according to the learner’s performance, which
means that one learner may require 18 trials and another may require 5.

Here is the presentation of level 1 to a learner:

 
 The learner performed correctly on four consecutive trials. Immediately following this performance, the
teacher presents level 2 juxtapositions. This level is characterized by the interpolation of B tasks. These are
familiar tasks that are greatly different from the A tasks.

Immediately following Henry’s level-1 performance, the teacher presents these tasks. (Note that the
sequence begins with Henry’s last successful performance on level 1.)

The learner performed correctly on four consecutive trials, so the teacher may immediately proceed to the

 
next level of difficulty—level 3 juxtapositions. (Note that the sequence begins with Henry’s last successful
performance on level 2.)

Tasks introduced as B, C, and D in the level-3 pattern are tasks that have been pretaught.

The learner performed successfully, so the basic program has been completed. The teacher now introduces
integration activities.

Features of the Strategy

The three-level context-shaping strategy has the following advantages:

1. It does not make assumptions about how much practice will be required for the learner to perform
on the task when it is presented in the juxtaposition pattern of level 3. The learner’s performance
is the sole guide to whether the teacher stays on a particular level or moves to the next level.

2. Its design assures that if the teacher follows the rules, the learner will receive reinforcement on at
least 70 percent of the trials once the learner masters level 1. Initially, the learner may not receive
positive reinforcement at this rate; however, in most cases, the learner very quickly performs on
level 1 with enough consistency to earn reinforcement on at least 70 percent of the trials.

3. A variation of the strategy can be used in any situation that involves teaching when to produce a
response. Whether we are working with a traumatic stroke victim, an autistic child, or a person

 
 learning a new and difficult discrimination, we can design the practice according to the three-level
strategy. We simply begin with level 1 and work on it until the learner performs on three or four
consecutive trials, then immediately proceed to level 2 using the same criterion of performance,
then to level 3. Also, the procedure can be individualized to any rate of learning. If the learner
requires 2,000 trials to complete the program, the learner receives the necessary trials.

4. The three-level strategy permits the teacher to predict when mistakes will occur and why they will
occur. When the learner proceeds from one level of difficulty to the next, the probability is
increased that a mistake will occur. It will occur because the task is more difficult than the
preceding task.

5. Given the relative difficulty of the levels, the appropriate corrections for mistakes is implied.
Return to an easier level. More specifically, if the learner makes a mistake on level 3, return to
level 1 until the learner performs on two consecutive trials, move to level 2 until the learner
performs on two consecutive trials, then present level 3 tasks. Through this type of correction, the
learner still receives reinforcement on most trials.

6. The procedure permits the massing of practice. One problem with discriminations that are highly
unfamiliar to the learner is that the learner will require many practice trials before reaching an
acceptable level of performance. If we mass trials the practice is less efficient than it is if we
distribute the trials. However, if we mass the trials, we can reach our objective in far less clock
time than we could by distributing practice. Let us say it requires 500 practice trials if we
distribute practice and 800 if we mass the practice. Let us say that when we distribute the practice,
we present 20 trials a day. With the massed procedure we reach procedure criterion in four days.
The distributed practice requires 25 days.

The three-level sequence is different from any that we have dealt with in that it does not specify a number
of examples. Instead it specifies a procedure and a criterion of performance. This feature follows from the
nature of new-response teaching. We do not have perfect control over the juxtaposition of successful trials the
learner receives. Therefore, we design the juxtapositions that are relatively more difficult and less difficult.
The pattern that is presented is governed by the learner, not by a specified order of different examples.

Integration

The integration steps that we will describe involve two separate types of activities. The first is a careful
context integration. The newly-taught item is first integrated with instructions that are highly unlike those of
the target task and that call for responses that are greatly different from those signaled by the target task. Later,
responses that are increasingly similar or that are signaled by instructions highly similar to those for the new
response are juxtaposed with the new response.

 
 As this integration goes on, event-centered applications are introduced. These applications associate the
newly- taught response with a situation or event that calls for the response. They also associate the newly-
taught response with other familiar responses typically involved in the situation.

Context Integration. Context integration consists of test sequences composed of the new item and other
items familiar to the learner. The tests are ordered so that later tests are more difficult than earlier tests.

The progression of initial teaching events and context integration is outlined in Figure 22.1. Each letter
represents a different task. Tasks X and Y are highly similar to A. The others are not highly similar. Each
integration sequence is repeated until firm. The first integration sequence may be presented during three or
four sessions before the learner performs on the sequence without a mistake. If the sequence must be repeated
many times, however, the order of items in the sequence should be changed to prevent the learner from
memorizing the sequence of events.

As soon as the learner finishes one integration sequence, the next is presented. The last sequence contains
the two tasks that are most similar to A (X and Y). The non-similar task, R, is included to prevent the series
from being entirely composed of highly similar discriminations or responses.

Before the final integration, the learner deals with X and Y in one context and with A in another. The
learner is not alerted to the fact that all may occur in the same context or activity and that there are specific
features of A that distinguish it from X and Y. Although we have used a variation of this strategy for
introducing coordinate members into a set, a strict logical analysis suggests that it is not efficient. From a
logical standpoint, X and Y should be introduced very early. The reason is that if the learner can discriminate
between A and those discriminations that are most highly similar to A (X and Y), the learner will
automatically be able to discriminate between A and any discrimination that is less similar to A than X and Y.

We do not follow a strict logical course when introducing coordinate members, and we do not follow it
here because of the problems associated with firming the learner on an early set consisting of highly similar
discriminations (A, X, Y). The set would be extremely difficult, particularly for the naive learner. (Facts about

 
the learner support this contention.) We therefore shape the context that occurs in the integration, following the
same procedures used for introducing coordinate members into a set. We first place A in the simplest context.
We then modify the context by introducing discriminations that are more highly similar to A. Although this
strategy may not always result in a savings in terms of the number of trials needed to achieve consistent
performance on the set of A, X, Y, the ratio of successful trials to unsuccessful ones will be much higher,
suggesting that the instruction will be more reinforcing. Although we remain aware of the problem created by
this easy-to-hard progression, the learner’s performance gives us little choice. We must demur logic to the
facts about the learner.

Event-centered applications. The initial teaching provided by the three-level sequence prompts the new
response through careful manipulation of the context in which the response occurs. The integration sequences
remove many of these prompts. The purpose of the event-centered applications is to provide the learner with a
different set of prompts for producing the response. A given prompt, such as “What’s your name?” predictably
occurs in situations that involve somebody for the first time. The question, “What’s your name?” is not the
only question that will probably occur in this situation. Other probable questions include, “Where do you live?
. . What school do you go to? . .” and similar questions.

To present these event-centered applications, we create a mock situation that calls for the newly-taught
response and other responses that are called for by that situation. For example, after teaching Henry his name,
we include the question “What’s your name?” in the context of questions such as, “How old are you?” and
“Where do you go to school?” These questions would not always be presented in the same order; however, all
would be presented. The teacher would frame the event as a “pretend” situation. “I’m going to walk into the
room. Pretend that I don’t know you . . . Hello. What’s your name? . . Where do you go to school? . .”

These event-centered applications are effective for two reasons:

1. They prompt the various responses by virtue of their association with the same event.

2. They often help to solve problems of highly similar items because these items may not occur in
the same event-centered application.

If the event does call for highly similar items (such as “What’s your name?” and “What’s your teacher’s
name?”) we can make the discrimination easier by creating a response difference for one of them. For instance,
we could get Henry to initiate the response, “Hello, my name is Henry.” This initiation creates a difference
between the response for “your name” and for “your teacher’s name.” The response to “What’s your teacher’s
name?” is produced only as an answer to the question. Therefore, there are both differences between the tasks
that call for the response, and differences between the responses.

The procedures for using event-centered applications are fairly simple. As soon as the learner completes
the three-level sequence, these applications may be introduced (as the other integration activities are

 
proceeding). We create a mock situation similar to those in which the response would frequently occur. We
add to this situation responses to the other items that could occur in this situation. These responses would be
familiar to the learner. We then present the situation frequently. A more detailed discussion of these
“rehearsals” and their final application to real life situations is provided in Chapter 24.

Corrections

If the learner makes a mistake on the newly-taught task in any integration context, we use a shortened
variation of the three-level strategy to firm the learner. We then return to the integration activity in which the
mistake occurred. For example, if the learner failed to remember the sound for the letter l in the integration
exercise, the teacher would first tell the learner the sound and would then present the level 1 juxtapositions.
“Yes, what sound? . . . Good, what sound? . . .” followed by a level 2 and 3 juxtaposition (“What’s your name?
. . . What’s my name? . . . What sound? . . . Good”).

The steps in executing this correction are:

1. We tell the learner the correct response.

2. We present one level 1 task.

3. If the learner makes no mistakes, we follow with one level 2 task and one level 3 task. If the
learner makes a mistake, we present a larger number of tasks for each level.

4. We repeat the activity in which the mistake occurred.

A shortened version of the three-level sequence is usually sufficient for the correction because the learner
has already demonstrated successful performance on level 3. The correction therefore simply prompts the
correct response through a few items and then returns to the context in which the mistake occurred.

Type-2 Strategies:
Teaching How to Produce Responses

Type-2 strategies cover a large range of responses, from simple grasping responses to executing a giant
swing on the high bar. They apply to various situations in which:

1. The learner can produce the response in some form but cannot produce it at the desired rate.

2. The learner cannot produce a simple response.

3. The learner cannot produce a complex response.

Different strategies are appropriate for prompting the learner and for simplifying complex operations.
However, the common feature of these teaching situations is that the learner cannot produce the response or

 
cannot produce the response at a desired rate-accuracy criterion. The teaching strategies parallel those for
teaching the learner when to produce the response. To teach the learner when to respond, we manipulate when
(or in which juxtaposition context) we present the task. To teach the learner how to provide the response, we
manipulate the criteria for reinforcing the learner’s performance.

This procedure shapes how the response is produced and is appropriate only when we train the learner to
perform at a higher rate, with greater accuracy, or with a different response configuration. It is not appropriate
for teaching concepts or discriminations.

When we shape responses, we begin with a criterion of performance that the learner is capable of meeting
on at least some of the trials. We reinforce trials that meet the criterion. As the learner practices the response,
the responses tend to improve. We change the criterion of the performances that are closer approximations of
the ultimate criterion of performance.

If we wanted a learner to sit on a chair and if the learner is perfectly capable of producing the response,
we would not shape. (We would not reinforce the learner every time the learner got closer to the chair.) The
reason is that shaping induces misrules. The learner is reinforced for some behavior. The reinforcement signals
that the responses are appropriate and therefore should be retained and repeated. Every time we reinforce an
approximation, we run the potential risk of suggesting to the learner that the behavior we reinforced is the
behavior we desire. Therefore, we use shaping only when the learner is not able to produce the desired
response under any reinforcing conditions, which means that we have no alternative other than working from
approximations.

If the learner cannot say, “I am thirsty,” at an acceptable rate, shaping is implied. Let’s say that the learner
can say “I,” can say “am,” and can say “thirsty,” when these words are presented in isolation. Furthermore, the
learner can say “I am” and “Am thirsty.” The learner, however, cannot say “I am thirsty,” without stumbling,
pausing, or distorting one of the words. This situation meets the criteria for shaping. Although the learner can
produce components, the learner cannot produce the response. This point is important. The fact that the learner
can produce the components does not imply that the learner can produce the whole response.

Shaping the Response

Here are workable procedures for shaping simple responses.

1. Take baseline data on the learner’s performance. Present the signal that you will use throughout
the program. Possibly have another person model the response for the learner by producing the
appropriate behavior in response to the signal. Present nine trials to the learner. Provide some sort
of encouragement or reinforcement to insure that the learner is trying to produce the response.
Carefully record the learner’s performance, either by noting the details of the responses or by

 
 using some sort of recording that you can study later (such as video tape). The responses will
vary, with some approximations “closer” to being appropriate and some not as good. (Some
responses will be faster than others. Some will be less distorted.)

2. Use the baseline sample to assign three groups of responses.

a. Non-reinforceable responses. These are the worst three responses produced. During the initial
shaping, you will not reinforce responses of this type.

b. Single-reinforceable responses. These are the middle three responses. Responses of this type
will receive single-reinforcement during initial shaping.

c. Double-reinforcement responses. These are the best three responses. Responses of this type
receive double reinforcement.

3. Present trials and reinforce responses according to the criteria above. The resulting reinforcement
schedule is designed to show the learner the direction in which the shaping will move. Some
responses lead to no reinforcement. Each type of response has different features. By creating a
correspondence between the features of the response and the amount of reinforcement, we
communicate precise information about how reinforcement changes as the responses change.

Furthermore, the reinforcement procedure provides the learner with a sufficient amount of
reinforcement to assure that the practice will not become unduly punishing. If the learner
performs no better than the baseline performance, the learner will receive reinforcement on about
66% of the trials. With only 10% improvement (the elimination of some non-reinforceable
responses), the learner will receive reinforcement on over 70% of the trials.

4. Change the criterion for awarding the reinforcement when the learner receives reinforcement on
about 90% of the trials. The simplest procedure would be to look at the learner’s last nine trials
and assign the trials to three groups, the lowest group exemplifying the new standard for no
reinforcement, and the highest group providing the criterion for double-reinforcement.

5. Repeat the cycle of reassigning criteria for reinforcement each time the learner consistently
receives reinforcement on about 90% of the trials. Each cycle brings the learner closer to the
desired response. After three or four cycles, the learner is typically responding in the acceptable
range.

The procedure refers to double-reinforcers. A double-reinforcer is not necessarily two times a single, but
may be quite different from the single. The double-reinforcer is something that is more desirable to the learner
than the single. The test of which reinforcers are more desirable is to give the learner a choice of various
reinforcers. The one that is consistently picked over the others is the strongest or most desirable reinforcer. In

 
most situations, verbal statements and praise work perfectly well as reinforcers.

For singles, the teacher says something like, “Not Bad.”

For doubles, “I can’t believe that,” or “Fantastic.”

Points can also be used. For singles, the teacher might award one point or two points (somewhat
randomly). For doubles, the teacher might award 5 points. If points are used, the teacher should also use verbal
praise: “Five points again. Terrific.” If necessary, the points can be exchanged for tangibles.

Reinforcers should not satiate the learner. A wedge of cake is a very poor reinforcer. A sip of juice may be
a good one. Verbal praise is probably the best and may be used in connection with other reinforcers.

Illustrations. If the learner cannot color within spaces, we take baseline on 9 simple examples (coloring
inside 9 small boxes). We then identify the top three, middle three, and bottom three responses.

We then present trials of coloring inside outlines (perhaps starting with circles and then changing to
something else). We model how to do it and help the learner with a box or two. The examples should be
designed so the learner is able to complete one in 15 seconds or less. (For repetitive activities, such as making
the back-and-forth strokes used in coloring, we can waive the 10-second limitation.) After each reinforceable
trial, the learner gets points, praise, privileges, candy, or whatever the reinforcers are. After trials that do not
earn reinforcers, we simply acknowledge it. “Okay, let’s try another one.” We do not punish the learner for
non-reinforceable responses.

As the learner improves (achieving reinforcement on 90% of the coloring trials) we change the criterion of
performance based on the learner’s latest performance. The procedure is repeated until the learner’s
performance is judged acceptable.

We would use the same procedure to shape the learner’s typing rate to 100 words per minute. We would
first establish baseline performance by requiring the learner to type for nine 20-second samples. We would
then examine each, identifying the rate-accuracy characteristics for the slowest three passages (the non-
reinforcement samples), assigning the rate-accuracy range for the middle three as single-reinforcement status,
and identifying the range for the top three as double-reinforcement.

We would now proceed with the program by giving the learner one-minute typing tests or longer tests that
are rated on the word-per-minute basis. When the learner receives reinforcement on about 90% of the trials, the
schedule changes based on the latest 9 trials. The procedure continues until the learner has achieved the goal.

Applying the Skill That is Taught

Just as applications follow the teaching of when to respond, applications should follow the shaping of the
response. The purpose of shaping is to induce capabilities the learner did not previously possess. Once a

 
particular skill has been shaped to the desired criterion of performance the skill should be used or applied. The
situations in which it is applied should provide for reinforcing the learner. Of ten reinforcing applications for
shaped responses are achieved with less effort than for the type-1 skills (learning when to respond). The reason
is that the skill that is shaped often becomes a component of a physical operation. The physical operation
provides feedback to the learner on all trials so long as the learner knows what the objective of the operation is.
Therefore, the learner can practice the physical operation “independently” without direction or reinforcement
from the teacher. When performing the operation, the learner uses the component skill. Therefore, the
component skill is reinforced. For instance, if the learner is taught to press against the edge of a button with a
thumb, the learner is taught a component skill that may later be used in the operation of buttoning. When
pushing the button through the hole, the learner uses the component response. The learner receives
reinforcement from the physical environment on each successful trial of buttoning. (Additional reinforcers may
be added during the initial practice applications.)

Type-3 Strategies:
Teaching When and How to Produce Responses

In some situations, the learner is incapable of producing the response and is not firm on when to produce
the response. For example, a deaf child who is learning responses to questions may not acceptably say
“Henry,” and may not remember how to produce even an approximation of the response to “What’s your
name?” Obviously, a great deal of teaching is involved in bringing the learner to an adequate criterion of
performance.

The guiding principle for type-3 training situations is that the learner must receive reinforcement on at
least two-thirds of the trials. Unless this principle is followed, the program will probably be ineffective. To
assure this level of performance, we may consider three strategies:

1. Teach when to respond first and then shape the response.

2. Shape the response first (using cues or prompts so that the learner does not have to remember
when to produce the response) and then teach when to respond.

3. Initially use separate teaching for when and for shaping the response, then put them together.

Choice 3 is preferable. It probably achieves the objective with the fewest misrules because it shows the
learner very early in instruction both what type of response is called for and when the response is to be
produced. In addition, the procedure is manageable. When the teacher works on when to produce the response,
the criteria for reinforcing and correcting the learner are clear. A different set of reinforcement criteria is
introduced when the teacher works on shaping the response. Although the teachings are separate, instruction in
both how and when to respond can be presented during the same day. First the teacher works on one aspect of

 
the task (when). At a later time, the teacher works on how. When the learner reaches an acceptable criterion of
performance, the teacher presents applications. If the learner masters when to produce the response before
learning to produce it correctly, the teacher introduces applications for when while continuing to work on the
form of the response.

The type-3 training does not introduce anything new with respect to techniques for teaching how or
teaching when. These skills are independent of each other and will probably be learned at different rates.

Firming Responses that Require Modest Practice

The basic strategies that we have presented for establishing responses are appropriate for responses that
require hundreds of trials. While it is important to understand the procedures for establishing these responses,
the response-teaching situations most frequently encountered by the teacher are not severe enough to warrant
taking baseline data and carefully establishing the program. Most frequently, the learner has some trouble
producing the response and remembering the response that is to be produced; however, the remedy requires
fewer than 100 trials. For these situations, the teacher needs efficient techniques that teach the learner both
when to produce the response and how to produce it.

There are two common situations that call for the abbreviated response-teaching procedures:

1. Statement and rule-saying.

2. Producing all the steps in a routine without prompting from the teacher.

Rules and statements. One of the most frequent new-response teaching applications involves rule-saying
or statement saying. The learner may be required to say: “When things get hotter, they expand,” before
working on a series of correlated-feature examples. The learner may be required to point to the bottom number
of the fraction ¾ and say: “Four parts in each group,” then point to the top number, saying, “And we have
three parts.” The learner may be required to produce the statement, “The ball is under the table.” The rule
saying or statement saying may occur as part of a complex routine, or it may be a simple task. In any case, it
often requires new-response teaching.

The steps for establishing verbal responses are:

• Model

• Lead

• Test

For the model step, the teacher says the rule or statement exactly as the learner is to produce it—at the
same rate, and with the same inflection. Possibly the teacher presents a multiple model by presenting the model

 
more than one time before the learner performs. The presentation of multiple models is a powerful technique
because it impresses on the learner the pattern that is to be followed: “My turn. Four parts in each group . . .
and we have three parts. My turn again: Four parts in each group . . . and we have three parts.” The model
should have: (1) unique inflection, and (2) unique pausing patterns.

The teacher might stress the words group and have in the sentences above. Also, the teacher might pause
before and after three. The pauses and the unique inflections make various parts of the sentences different
from each other and therefore easier to produce. If all parts of the sentence sound the same, the learner will
have more trouble differentiating the parts. So we try to say the sentence in a way that conveys the information
and that anticipates problems the learner will have in repeating it. (Expect the learner to reverse the words
groups and parts and forget to say “. . . and we have . . .” Design the model to compensate for these
problems.)

The lead step immediately follows the model. For the lead step, the teacher says the response with the
learner at the same rate that the model had been presented. “Say it with me: Four parts in each group . . . and
we have three parts. Again . . .” The lead step is repeated and alternated with the model step. If the learner
simply tries to imitate the teacher’s response, the teacher presents the model step several times before returning
to the lead step. Usually, the model step helps the learner produce the response. Lead steps are weak if they are
presented too frequently. As a rule, model and test steps should each be presented twice as frequently as lead
steps.

The test step immediately follows presentations of the lead step. The test step is presented when the learner
seems to be performing well on the lead step. There is no assumption that the learner will pass the “test” step
the first time it is presented. “Your turn . . .” The teacher points to the bottom number and then to the top as the
learner says, “Four parts in each group and we have three parts.”

If the learner fails to perform on the test step, the teacher models the correct answer and repeats the test
step. If the learner fails again, the teacher goes back to the model step followed by the lead step.

Chaining Parts of Statements

If the statement is too long for the learner to repeat, we present only part of the statement at a time. Some
practitioners suggest the use of backward chaining—a procedure that first teaches the learner to respond to the
last part of the sentence and then increases the length of the response by adding parts that precede the last

parts.∗ We do not recommend the use of backward chaining for statement-saying because the procedure of ten

                                                                                                                         
Backward chaining was developed for non-verbal organisms to insure adequate reinforcement (the end of the
∗

chain). With verbal prompting of the chain elements, this procedure is not necessary and, as noted in the text, often
creates additional problems.

 
prompts mislearning. For example, if we backward chain on the fraction-analysis sentence, the learner would
respond to the top number first. The instruction is carefully designed to assure that the learner is strongly
prompted to refer to the bottom number first. In this way, the learner finds out about the properties of the
groups before referring to the top number and finding out how many parts have been used. Not only does the
backward chain weaken the procedure of the bottom first then top; it also muddles what the learner will say.
The first thing the learner is saying in the backward chain is: “And we have three parts.” When the learner later
tries to work the whole chain starting with the bottom, the learner may have serious reversal problems, saying
something like, “We have four parts . . . and we have three groups,” or “We have four parts and we have four
groups.”

The simplest and least complicated procedure for teaching long statements is forward chaining. To present
a forward chain, simply have the learner say the first part of the sentence. Present the model, lead, and test for
this part:

“My turn: Four parts in each group . . . Again, four parts in each group . . . Say it with me: Four parts in
each group . . . Pretty good. Your turn: Four parts in each group. Say it. Once more. Four parts in each
group. Say it . . . Sounding good. Listen: Four parts in each group. Say it . . . Almost perfect. Listen: Four
parts in each group. Your turn . . . That’s it. Once more . . .”

The teacher does not stop as soon as the learner produces a perfect trial. If the learner requires quite a few
repetitions to meet criterion on saying the part, the learner should not be considered firm until the learner is
able to say the part on three or four consecutive trials. Use the same criterion that you would for level 1 of the
three-level strategy for firming responses.

When the learner is firm on the first part, the teacher introduces the entire sentence, modeling, leading, and
testing:

“My turn. Four parts in each group . . . and we have, three parts.

Listen again: Four parts in each group . . . and we have, three parts.

Say it with me: Four parts in each group . . . and we have, three parts.

Good. Now listen to the whole thing:

Four parts in each group . . . and we have three parts.

Say it with me:

Four parts in each group . . . and we have three parts . . . You’re getting it.

                                                                                                                                                                                                                                                                                                                                                                                                       
 

 
 Listen: Four parts in each group . . . and we have three parts. Say it . . .

Yes, and we have three parts . . . Say the whole thing . . .”

Note that the pause (. . .) is quite long between parts. The purpose of this pause is to prevent the parts from
being amalgamated or jumbled. The pause shows the learner that you first say the familiar part, then you say
the new part.

If the learner has serious problems saying the second part of the statement, the teacher could introduce a
variation of a backward chain. For this variation, the teacher says the first part very quickly, then pauses, and
presents the second part at the regular rate and with the regular inflection. “My turn: (four parts in each group)
and we have, three parts. Say the last part with me. (Four parts in each group) and we have, three parts.”

To indicate that the learner is not to respond to the first part of the sentence, the teacher simply points to
herself while saying the first part. To signal for the learner to respond, the teacher points to the learner just
before presenting the second part of the sentence.

The same three steps—model, lead, test—are used with the last part of the sentence. As soon as the learner
is firm on the last part of the sentence, the teacher models, leads, and tests on the entire sentence. If the learner
seems to get into a mistake pattern, we model the correct sentence and do something else for a few minutes.
When you return to the sentence, the learner may perform better. As a rule, present ten trials, then take a small
break (perhaps no more than half a minute) before presenting another block of ten trials. Always try to end a
block of trials with the appropriate response. If the learner’s performance on the last trial is poor, model the
correct response before terminating work on the sentence.

Applying the Three-Level Strategy

If the learner requires more than seven trials to say the sentence, we treat the sentence as a task in the
three-level series.

We repeatedly juxtapose the same task until the learner performs on three or four consecutive trials.
Because the learner had trouble achieving this criterion of performance, the learner will probably “forget” how
to produce the response if we present the task ten minutes later. To provide the learner with the kind of practice
needed to make the response firm at a later time, we design a transition that shapes the context. First create
simple interruptions:

 
Then create more elaborate ones.

We work on each level until the learner performs on three or four consecutive trials. We can illustrate the
procedure with the fraction-statement examples. After the learner has said the statement about the fraction, the
teacher provides an interruption. “Add these numbers in your head. Thirty, fifty, and ten. What’s the answer? .
. . Good job. Now say the whole statement about this fraction again. Get ready . . .”

The teacher presents similar single-task interruptions until the learner performs on three or four
consecutive trials. The teacher then introduces more lengthy interruptions: “Do all the addition problems on
part three of your worksheet. Then we’re going to come back and say the statement about the fraction again . .
.” Following the intervening activity, the teacher presents the fraction-statement task. The teacher continues to
present level-three activities until the learner is firm.

Chaining Parts of a Routine

The most typical problem with rule-saying or statement saying is that the learner is capable of saying the
individual words or individual phrases of the rule or statement, but is unable to produce the entire response. A
similar situation occurs when the learner is working on cognitive routines. This problem is perhaps most
noticeable when the routine is being covertized. Before covertization, the learner is not required to produce the
entire chain or routine in response to a simple instruction. When covertization occurs, however, the directions
for the different parts are removed and the learner may fail to perform, even though the learner is firm on the
component steps.

The solution is to apply the three-level strategy. First, the teacher models exactly what the learner is to do.
“Here’s how I figure out if this fraction is more than one whole group. First, I touch the bottom number and
say, ‘seven parts in each group,’ and then I touch the top number and say, ‘and we have eight parts.’ Then, I
ask if we have more than there are in each group. Yes. So this fraction is more than one group. Watch how I do
it.” Teacher models the behavior, talking to herself and pointing to the parts of the fraction before writing
“more than one group.” “Your turn. Let’s see you do it just like that. Start with the bottom number and talk to
yourself . . . Good. Next problem . . .” This is level 1 of the three-level strategy.

After the learner has performed on three or four applications, the teacher provides an interruption to create
a level-two difficulty of juxtaposition. “Time out, Henry. I notice that you’ve been doing a good job in your

 
reading. How many points did you earn this morning? . . . Wow, that’s really good. Are you going to do that
well in your arithmetic? . . . Let’s see if you remember to talk to yourself. Do problem three . . .” Note that the
interruption is nearly as long as a task. Follow this procedure for longer tasks.

The teacher repeats level-two juxtapositions until the learner performs adequately. For the level-three
presentation, the teacher presents a variety of problems with the fraction problems consisting of about one-
fourth of the total problems. The fraction problems are distributed among the others.

Teaching the learner to perform independently should be considered as both an accuracy and rate problem.
The strategy involves first making sure that the learner performs adequately, then shaping the learner’s rate. In
the illustration above, the learner may become quite accurate but may continue to say the steps aloud and
process each problem relatively slowly. So long as the performance tends to be accurate, we can begin to shape
the rate. Note, however, that we would not generally shape rate until we secure accuracy. The procedure for
rate-shaping is the same as that outlined earlier. The teacher takes baseline data on the amount of time required
to complete a worksheet. Criteria are established for no reinforcement (or points), singles and doubles. Note
that the criteria may involve a combination of rate and accuracy performance. For doubles, the learner may be
required to complete the worksheet in no more than five minutes, making no more than two mistakes. Possibly,
the teacher could design a point schedule that involves different possible combinations of rate and accuracy
outcomes. This schedule provides for more than single and double reinforcement. Perhaps four to six different
point totals could be earned for a given worksheet.

Although the schedule might be a little more elaborate than other response shaping routines we have
discussed, the shaping procedure follows the same general guideline outlined earlier. As the learner’s rate-
accuracy performance improves (manifested by higher average point totals), the schedule changes so that the
learner must work faster to receive reinforcements.

Summary

When the learner does not behave like a perfect receiver, we use a response-locus analysis to identify the
learner’s deficiency. Three possible problems emerge:

1. The learner is capable of producing the response, but cannot produce the response in the context
that is called for by the task the teacher specifies.

2. The learner is not capable of producing the response (under any reinforcing conditions), but does
produce some approximation of the response in the context of the task.

3. The learner is not capable of producing the response (under any reinforcing conditions) and
cannot perform in the context of the task.

 
 Each deficiency suggests an instructional solution. We assume that the learner’s behavior will be modified
if we provide the learner with adequate practice. Many trials may be necessary before the learner meets an
adequate criterion of performance.

Any techniques used to support the learner during these trials must be designed so the learner receives
reinforcement on at least two thirds of the trials. If reinforcement drops below this level, the schedule may
induce delayed responding, superstitious behavior, and a “dislike” of the training sessions.

The strategy for teaching when to respond (1 above) involves manipulation of the juxtapositions in which
the task occurs. We identify three levels of juxtaposition difficulty:

When the learner performs at a specified criterion of performance for level 1, the teacher immediately
proceeds to level 2. The same procedure is repeated as the teacher moves from level 2 to level 3. When the
learner has performed acceptably on level 3, the skill is integrated with other skills that have been taught. An
integration series presents the new task (A) and other tasks. At first, integration series contain the new task and
highly dissimilar tasks. Later, tasks that are similar to the new task are included in the integration.

In addition to the integration series, event-centered applications are presented. These provide a new set of
cues that increase the difference between the new task and all other tasks. They also provide a format that
permits the learner to use the skills in situations that are reinforcing.

Teaching the learner how to produce the response involves shaping the response by changing the criterion
for reinforcing performance. We first take baseline and identify the poorest responses, middle responses, and
best responses. The middle and best responses receive reinforcement during initial training, with the best
responses receiving “double” reinforcement (reinforcement obviously more desirable to the learner). When
reinforcement is provided on 90% of the trials, the teacher shifts the criterion of performance. The shift is
based on the learner’s most recent set of trials. Shifting the criterion is repeated until the learner’s responses
are consistently acceptable.

Teaching the learner both when to produce the response and how to produce the response involves double
teaching. The most efficient and manageable way is to teach when at specified times of the day and how at
other times. As the learner meets an acceptable criterion of performance on either how or when, the teacher
presents applications.

 
 If shaping the response does not require a great many trials, but more than seven, a workable procedure is
to shape the response first so the learner performs appropriately on the task, then to process the task through
the three-level strategy to assure that the learner is able to perform on the task when it occurs in an unprompted
context. To establish the response, the teacher typically uses three steps: model, lead, and test. The model step
shows how the response is produced. The lead step provides a prompt for the response. The test step gives the
learner unassisted practice in producing the response.
 

 
 Strategies for Teaching New Complex Responses
Chapter 22 introduced the basic procedures for inducing new responses. These procedures play a role in
the teaching of complex responses, such as buttoning, swimming, throwing a ball, turning a somersault, and
similar complex responses. Complex responses are those for which we can identify parts of component
responses. Because complex responses consist of component responses that are to be coordinated, we have a
number of teaching options. Just as it is possible to treat skills taught through a cognitive routine as
discriminations, it is possible to use basic response-teaching techniques to teach complex responses. Instead of
using a program of events for teaching swimming, we could simply treat swimming as a response that is to be
shaped. We could model the response, reinforce approximations, and in the end the learner would probably
learn to swim.

The advantages of an analysis and accompanying techniques that go beyond simple shaping parallel the
advantages of the cognitive routine over a discrimination sequence for teaching the same skills. By using a
program for complex responses, we gain greater control over what the learner does. We make the learning
easier and faster. Concepts that should be processed through a cognitive routine are not as obvious as simple
discriminations. In a parallel way, complex responses should be processed through procedures more
complicated than shaping because they are involved responses that are not easy to produce.

Misrules and Distortions

One of the great concerns in teaching new responses has to do with generalizations. We want the learner to
generalize to a full range of applications, and we want these generalizations to come about as quickly and
efficiently as possible.

The problems of establishing generalizations of new responses parallel those of establishing

generalizations for discriminations or concepts. When teaching concepts we must avoid misrules. The two
major types of misrules result from:

1. Presenting a set of examples that is consistent with more than one interpretation.

2. Stipulating that the discrimination is limited to a very narrow range.

The presentation that is consistent with more than one interpretation does not “guarantee” that a particular
learner will pick up a misrule. It simply provides the learner with the option of attending to features of the
presentation that are irrelevant to the desired generalization. The stipulation that can occur with initial training
examples is that all features of these examples are necessary. The learner tends not to generalize to examples

 
that do not have all the features of the training examples.

Both types of misrule have parallels in the teaching of response generalizations.

1. A presentation that is consistent with more than one set of responses leads to distortion. When
teaching the new response, we present examples or applications. If the series of examples on
which the learner initially practices are easy examples, they provide the learner with behavioral
options (just as the easy examples in a discrimination context provide many options of features
the learner may attend to). When presented with the easy-response example, the learner may
produce the response in a way that will permit generalization to all other instances of the
response; however, the learner may learn to perform on the easy examples in a way that will not
generalize. In this case, the learner’s responses are distorted. If the communication presents
examples that permit many response options, the communication may induce distortion.

2. The communication that presents a very narrow range of differences in the initial-teaching
applications for the response is guilty of stipulation. Following mastery on this set, the learner
may not respond to applications that differ from those in the initial-teaching set. The initial
instruction stipulates that a certain set of features must be present before the response is to be
produced. Absence of some features implies that the response is not to be produced. This situation
parallels that of inducing stipulation in concept teaching.

Remedies

The problem of stipulation results because the learner has not been shown what is the same about the
range of applications that call for the response. The remedy is to show the sameness. To show how
applications of a response are the same, we juxtapose applications that are greatly different and we treat each
application in the same way. When instruction violates this principle, stipulation occurs. We can use the
sameness principle to diagnose programs that are guilty of stipulation. If the learner is capable of producing
responses, but tends not to produce them for new applications, the program probably provided stipulated
practice. Let us say that following programs designed to teach “creative responses” the learner produces
unique, creative responses with hammers, but tends both to use the hammering response when producing
“creative responses” with other objects and tends not to produce many creative responses with these objects.
We can describe the program that induced this behavior. The use of the hammer was strongly stipulated, which
means that the learner worked on hammering initially and for a relatively long period of time. The learner
received a great deal of reinforcement for this work. The result was stipulation. A program that avoids
stipulation would have required the learner to work with juxtaposed applications that are quite different from
each other (hammers, lamps, pillows, books) and to treat them in the same way (by producing creative
responses).

 
 The problem of distortion is quite different. Unlike stipulation, the learner is not capable of producing the
responses that are called for by some subtypes of examples, although the learner can produce the response for
other subtypes. The problem is not one of “understanding,” but of being able to produce the response.

All applications of the response are the same in some ways; applications for subtypes also differ in critical
ways. And these critical differences preempt the learner from responding to some subtypes of examples.

Consider the generalization of buttoning buttons. Some buttoning tasks are easier than others. The smaller
the button, the more difficult the response. Figure 23.1 presents a diagram of responses for three buttons—
large, medium, and small. The height of each column represents the behavioral options that are available to the
learner. The column for the large button is the highest, signaling that this application is the least demanding
with regard to specific behaviors. The application permits many response options, all of which will lead to
successful completion of the task. The height of the column for the small button is the shortest, indicating that
this application provides the learner with the fewest behavioral options. Unless the learner produces certain,
precise responses, successful buttoning is not possible.

The unshaded area of each column shows that there are possible common responses for the three subtypes.
Whether the button is large or small, it is possible for the learner to perform in a way that involves the same
responses. By pressing against the edge of the button with the thumb and aligning the button with the button
hole, the learner could button all three types.

The shaded area in the first two columns indicates the extent to which other possible behaviors may be

 
employed to achieve the outcome. The height of the shaded area gives an indication of how “easy” the
example is. The large button is easier than the medium and the small buttons because it provides the largest
range of behavioral options that will lead to a successful outcome. Possibly, the learner grasps the flat sides of
the large buttons between thumb and forefinger, an option that is not readily available with the small button.
Possibly, the learner holds the large buttons by the edge (thumb and forefinger holding opposite edges of the
button). Other options are also possible.

These options are unfortunate because they are not options for all subtypes of the response. They therefore
imply non-generalizable learning. If the learner learns an option that permits successful outcomes only with the
large buttons, the learner must later learn new options for the medium-sized buttons, and ultimately new
options for the small buttons. The amount of learning would be greatly reduced if the instruction initially
taught the learner to produce the response options that are common to all subtypes.

Analysis of Examples Versus Analysis of Learner

The button diagram suggests the universal dilemma for sequencing applications that involve subtypes of
behavior. The smaller button is more difficult, simply because it provides the learner with no behavioral
options. Unless the learner masters the set of behaviors that will lead to the desired outcome for all buttons, the
learner will fail to perform with this button. Once the smallest button has been mastered, the common
behaviors for all applications have been learned, which means that it is relatively easy to achieve
generalization to new examples.

The analysis of the examples suggests, therefore, that the most efficient approach would be to present the
application that has the least “margin of error,” or that provides the fewest behavioral options. These
applications, therefore, function a great deal like a discrimination sequence that is consistent with one and only
one interpretation. The learner who masters the application will have learned the essential features that apply to
all applications. Theoretically, no misrules are possible except that stipulation may occur if we work too long
on the application.

The analysis of the learner, however, leads to quite different conclusions. The learner may have to work
on the example for hundreds of trials before producing a successful response (that of achieving the desired
buttoning outcome). Instruction is judged appropriate from the standpoint of the learner when that instruction
permits successful trials on at least two-thirds of the total trials. Working on an application that does not admit
to behavior options does not meet this standard. The use of shaping does not entirely solve this problem,
because the idea behind working on the hard application is that successful completion of this application would
assure generalization to “easier” examples (all of which can be processed with the same set of behaviors as
those used for the more difficult application). When we shape the response on the more difficult application,
the learner cannot benefit from more difficult examples because these benefits are possible only if the learner

 
successfully processes the examples.

Another option is to start with easy examples. Possibly, the learner can perform the buttoning operation in
15 trials with an easy example. If so, the learner will be performing the operation in some form after only
modest practice. Practicing some form of a successful application is usually far better than practicing an
approximation of the application. The approximation is usually capable of generating more possible misrules
and greater possible distortion.

Here is a summary of the issues in teaching complex physical operations:

1. If we work initially with difficult applications (those that permit no behavioral options):

a. We provide a presentation that is consistent with a single interpretation, and therefore

generalization to new and easier applications will be achieved very easily.

b. But we make the task very difficult for the learner because there is no latitude in the
behaviors permitted by the application.

2. If we try to reinforce the learner on two-thirds of the trials:

a. The learner does not produce the responses that would lead to the generalization.

b. The work on approximations of a difficult example probably does not teach the learner as
much as producing successful responses with a simpler example.

3. If we initially introduce an easier example instead of a difficult one:

a. The learner will probably produce successful responses with fewer trials because the example
permits a wider range of behavioral options.

b. The probability is great that the learner will learn a response that is distorted and that will not
generalize to new and more difficult applications.

c. However, this application provides successful practice, which usually teaches more, and more
rapidly than approximations.

Programs for Complex New Responses

The program for the complex new response (like buttoning, shoe-tying, and swimming) should be
designed to bring the learner to applications with the fewest behavioral options as quickly as possible. Note
that although the program may begin with easy examples, the program does not proceed from easy applications
to hard ones in progressive steps. The easy-to-hard progression systematically teaches possible misrules of
diminishing proportions. The hard applications are the ones that are capable of assuring the generalization to

 
all other examples. The longer it takes to present these applications, the more work it requires to teach the
generalization. Therefore, the goal is to reach the more difficult application quickly. The problem inherent in
the easy-to-hard sequence can be illustrated by referring to a similar problem in teaching a simple
discrimination.

If the positive examples are circles, and the negative examples are ovals that are very similar to circles, we

would not shape the context by first having the learner discriminate between these examples:

then these examples:

and finally these examples:

The context shaping would probably induce serious misunderstanding. The reason is that any difference
between the first pair of examples would permit the learner to discriminate reliably between the examples. The
learner could attend to the waviness of the lines, the open form, the “branches,” or any other difference. The
second pair of examples rules out some of the possible misrules because this example is closed and has no
“branches.” However, there are many differences that the learner may attend to and reliably discriminate. Not
until the learner is required to discriminate between the third set of examples is the learner provided with
information about which features of the positive example are relevant to the discrimination.

The problem with the easy-to-hard sequence for physical operations provides a parallel. The easy example
permits a wide range of “options.” The intermediate example reduces some of the options. However, not until
the learner encounters the most difficult example is the learner provided with information about the constraints
of the operation. In the meantime, the learner is reinforced for learning behaviors that work for the easier
examples, but will not generalize to more difficult examples.

Analyzing Complex Responses

The mechanics of engineering rapid progression to the difficult examples begins with an analysis of
complex responses. Responses such as shoe-tying, using a screwdriver, soldering, riding a bicycle, and other
operations are the same as cognitive routines in two ways:

1. All behaviors that account for the successful achievement of the outcome are overt.

2. The operation provides for feedback on every trial (given that the learner understands the goal of
the operation).

Complex physical operations differ from cognitive routines in the following ways:

 
 1. The overt behaviors will never be covertized.

2. The feedback is provided by the physical environment.

These unique features of physical operations suggest that the learner can benefit from practicing the
routine independently. Furthermore, this approach should be designed so that the learner engages in
independent practice as quickly as possible.

The job of designing instruction for complex responses is less complicated than designing instruction for
cognitive routines. To design instructions for cognitive problems, we must first design the routine and make
sure that it applies to the full range of examples; then we identify the various preskills implied by the routine.
The behaviors for complex routines already exist. Therefore, we simply observe the full range of applications
and then design the instruction that leads to the complex responses. Note that we do not have to design the
routine—merely analyze it.

The analysis of complex responses involves identification of temporal parts and simultaneous
components.

A temporal part is everything that goes on during one identifiable segment of the response. For instance,
everything that goes on when you pull the shoe laces before starting to tie the shoe is a temporal part.
Everything you do to loop one lace around the other is a temporal part. A temporal part should have a clearly-
marked beginning and a clearly-marked end.

It is easy to identify temporal parts for responses that have discontinuous parts. A part is discontinuous if it
can be executed outside the context of the operation as an isolated task without distortion of any sort. Although
the parts may be run together, they may also be separated in time. (The learner can perform some parts of the
shoe-tying operation and then stop before starting the next part. The learner can stop after printing part of the
letter b.)

Some operations are composed primarily of continuous parts. Parts are continuous if they can be produced
only within the context of the operation or within a similar operation. The follow through of a golf swing may
be identified as a part if it has distinguishable behaviors not observed in the other parts of the swing. However,
the follow through cannot be removed from the context of a swing. It cannot be practiced in isolation. It shares
the movement of the preceding part, and differs only in some behavioral details that overlay this movement.

The implication of discontinuous parts is that they can be practiced as units or steps that may later be
chained together. The implication of continuous parts is that they must be practiced within the context of the
operation in which they occur (or in a simplified variation of the context, which has the same continuous part).

In addition to identifying temporal parts, we also identify simultaneous components. A component is a

response detail that can be created in contexts other than the context of the operation. For example, a

 
component of shoe-tying is grasping the ends of the shoe lace. This particular component occurs in various
temporal parts of the response. This component, however, can be completely removed from the context of
shoes, shoe laces, and any temporal part of shoe-tying. We could present the learner with a hanging string that
is to be grasped between the thumb and index finger and that is to be held against a certain amount of
resistance. The behavior is the same one that is called for in various temporal parts of the shoe-tying operation;
however, when we remove the component from the operation, we try to present it in a context that is
simplified.

The context of a component is relatively simplified if:

1. There are fewer simultaneous components that occur with the target component.

2. Trials of the component may be juxtaposed faster.

In the context of the shoe-tying operation, the learner grasps the ends of the two loops and pulls them in
opposite directions. If we removed the component of grasping ends of looped strings, the learner could practice
trials that involve only one hand. The requirements of coordinating this component with others is therefore
simplified. Also, we could juxtapose trials of loop-grasping because the learner would not have to perform the
preceding parts of the shoe-tying operation before practicing the grasping component. A trial of grasping could
be followed by another trial of grasping.

Simultaneous components occur in temporal parts that are continuous or discontinuous. It is usually
possible to identify ways of removing components from discontinuous parts. We simply identify the behavior
that occurs within that part. We then design a simpler context for the target behavior. When we deal with parts
that are continuous, simultaneous components present a far more challenging problem when they are not
common to the preceding temporal part of the operation. Because they are not common to the preceding part
and are produced within a continuous-part context, we must retain the continuous-part context. We can
illustrate the problem with the response of saying the word mess. This response has parts—saying three
sounds. Each part has simultaneous components. For the sound m, the simultaneous components are:

1. Making a voiced sound.

2. Keeping the lips closed.

3. Allowing air to escape from nose.

Each of these behaviors can be performed independently of the others. For instance, we can perform
behaviors 2 and 3 without performing 1. (Simply breathe through the nose.) We can do 1 and 2 without doing
3. (With your nose held and lips pursed, make a voice sound.) We can do 1 and 3 without doing 2. (Make the
sound for n.)

 
 The three components are not produced within a continuous context; therefore, they can be practiced in
isolation, removed from the word mess. For instance, if the learner was not closing the lips when saying the
word mess, it would be quite easy to give the learner practice in imitating lip closing. The learner would
simply imitate the teacher on trials that required separating the lips and closing them. This component can be
removed from the context of mess because it is a component of m that does not have anything to do with the
fact that the m sound is a part that is joined continuously to the next sound. Many components cannot be
removed from the continuous context. For instance, the learner may be able to produce the s sound in isolation.
When saying the word mess, however, the learner may voice the s sound—saying the sound for z instead of s.
Stated differently, the learner has trouble stopping the voicing component for the s when it occurs within the
continuous context of the word mess.

We cannot work on this particular behavior apart from the continuous context of a word. We may be able
to construct simpler examples or examples which have fewer temporal parts or fewer additional changes. For
example, if the learner said, “es,” there would be a shorter chain:

Possibly, we can find a context for the unvoicing of the s sound that is “easier” than the one above. For
instance, we could require the learner to whisper the word es. This example requires the learner to produce the
unvoiced component for the s sound; however, it does not require the learner to produce it in a context that
goes from a voiced sound to an unvoiced sound, because the e sound is unvoiced also. The parts in this
example vary only in other components, not in voicing.

The opposite approach is also possible, that of presenting continuous parts that vary only in the voice
component. The word zs fulfills this requirement:

The only difference between the z sound and the s sound is the voicing. Work on this example would certainly
show the learner how to control the voice component. It might not be a very easy example, however, because
there is only one difference between the responses for the z part and the s part. The learner may have trouble

 
manipulating this difference.

In summary, there are three options for finding a simpler example of components that must be joined to
preceding continuous parts. They are:

1. Try to shorten the chain of events that precedes the part with the component in question.

2. Create an example in which the preceding part shares the component in question (all parts in the
whispered word share the unvoiced feature).

3. Create an example in which the preceding part shares all details except the component in question
(all components of the s sound except the unvoiced features are shared by z).

The type-2 strategy is the easiest for the learner, but is least capable of teaching the behavior, because it
actually eliminates the problem. The type-3 strategy is the most capable of communicating the behavior;
however, it may be difficult for the learner because it involves the most precise manipulation of the
component. The introduction of type-3 continuous part should be experimental. If the learner performs on it
with modest practice, a great deal is gained. If not, the example may be dropped and not a great deal is lost.

If we carefully analyze the complex response, we can often discover a way to apply the different
strategies. For instance, if a learner is having trouble with a golf swing and if the problem has to do with the
learner’s left arm, the option of making the chain shorter is not a significant one. By requiring the learner to
wear a brace on the left arm so that it could not bend, we would create an example in which the preceding part
of the swing shares the component in question with the problem part of the swing (strategy 2). If we require
the learner to swing a light object using only the left hand, we create an example in which the left arm must be
totally responsible for any changes in the latter part of the swing. This example therefore functions as an
example of strategy 3, with the parts the same except for a single response component. (Although we might
have some trouble categorizing particular applications as either a type-2 strategy or a type-3 strategy, we can
either eliminate the critical response through the choice of example or we can make the critical response the
most prominent aspect of the application.)

Three Program Types

There are three basic programs for teaching new, complex responses. The three types are distinguished on
the basis of how they start—the nature of the first step. Two programs begin with the learner being taught
skills within the context of the operation that is being taught. One begins with the learner performing on
components or parts that have been removed from the context of the operation.

The three programs are:

1. Essential-response-features program. At the beginning of this program, the learner works within

 
 the context of the operation, performing those features of the response that are essential to the
outcome; the non-essential features that are involved in the operation are not performed by the
learner.

2. Enabling-response-features program. The learner begins by working within the context of the
operation; however, the learner does not produce the essential features. The learner produces the
non-essential features while the essential features are supported.

3. Removed-component-behavior program. The first step in this program presents a context different
from that of the operation. The learner first works on some part of the response or some
component of the operation in this removed context.

Essential-response-features. To test whether the learner is producing the essential-response-features, we

ask two questions:

1. What is the objective of the operation?

2. Who actually performs the behaviors that achieve that objective?

If the answer to 2 is “the learner,” the learner produces the essential response features. Let’s say that the
objective of an operation is to hammer nails into a board. Who actually performs the act of hammering? If the
learner performs the actual hammering—that behavior that accounts for the nail being driven into the board—
the program is an essential-response-feature program. Possibly, the learner wears a glove and the handle of the
hammer is affixed to this glove. Possibly, the learner’s arm is raised by the teacher. These are not essential
details or features. The essential features of the behavior are those that bring the hammer into contact with the
nail head.

Let’s say that the learner is screwing screws with a screwdriver. If the learner turns the screwdriver, the
program is an essential-response-feature program. Again, the operation may be designed so that other
behaviors are removed or supported. The blade of the screwdriver may be modified so that it fits easily into the
slot. The handle of the screwdriver may be modified. But if the learner produces that part of the operation that
turns the screw in the appropriate direction, the learner produces the essential response features.

Deciding on the essential features of a response is sometimes difficult and sometimes arbitrary. What are
the essential features of riding a bicycle? The essential features are those aspects that distinguish the operation
from all others. For bike-riding, we would judge that they have to do with maintaining the balance. The other
behaviors—pedaling, steering, etc.—can be done with other vehicles that do not require the kind of balance
called for by a bicycle.

The first part of the essential-response-feature program requires the learner to produce the essential
response features in the context of the operation that is being taught. If the operation is riding a bike, the

 
learner actually performs the balancing—perhaps on a coasting vehicle—while some of the non-essential
behaviors (pedaling) are eliminated. The learner begins, in other words, with a simplified version of riding. The
program then adds additional response features as the learner becomes proficient with the essential-response
features in a context that is simplified.

If the operation involves hammering, the learner strikes the nail with the hammer. This essential-response
feature is produced by the learner. Other behaviors, such as gripping the hammer, are removed or supported.
Again, the strategy is to begin with a simplified example of hammering and then to add in details.

Enabling-response features. These behaviors are not essential to achieving the objective of the operation,
but occur in every instance of the operation. They are enabling responses that permit the essential responses to
be performed. For example, putting toothpaste on a brush is a feature that may be involved in every instance of
tooth-brushing; however, it is not essential to the act of brushing teeth. Pedaling is not essential to “riding a
bike.” Holding a hammer is not essential to hammering. Placing a screw in position is not essential to driving
the screw with a screwdriver.

The enabling-response-feature program is the opposite of the essential-response-feature program. Initially,

the enabling-response-feature program requires the learner to perform some non-essential aspects of the
operation. If the operation is tooth-brushing, the learner does not produce the behaviors of brushing teeth.
Someone (or something) performs this part of the operation, while the learner performs enabling behaviors,
such as putting toothpaste on the brush, holding the brush as it is manipulated, orienting the brush to different
positions. The learner’s role is passive with respect to the essential behavior. The learner does not actually
brush the teeth.

In bike-riding, the learner who is engaged in an enabling-response-feature program would perform the
ancillary behaviors, such as pedaling, steering, etc., while some other agent performed the essential part of the
operation—balancing the bike. For swimming, the learner would swing arms or kick legs; however, something
else would support the learner’s body as it moved through the water. If the response involved making the letter
A, the learner might passively hold the crayon as the teacher forms the letter. If the response involved walking,
the learner would simply move one leg and then another, perhaps pushing backward, as the stroller or similar
device achieves the essential-response features.

The enabling-response feature program slowly shifts responsibility for producing the essential part of the
response. Note, however, that the entire process takes place within the context of the operation that is being
taught. From the beginning of the program, the learner works on examples that involve shoe-tying, swimming,
bike-riding, or hammering. At the beginning, the learner does not produce the behaviors essential to the
outcome. Later, the learner assumes responsibility for these behaviors.

Removed component behaviors. The removed component program first presents a behavior that is

 
removed from the context of the operation that is being taught. Those components or parts that are identified
for such removal are those that prevent the learner from performing correctly on applications of the operation.
For instance, we may identify that the learner fails to tie shoes because of a poor grasp on the ends of the
strings when pressing the doubled-over strands between thumb and forefinger. This component can be
removed from the context of shoe-tying. We can present relatively simple examples of the skill that involve
wide strips of paper. (Hold a strip on edge. Touch the ends together. Instruct the learner to “fold” the paper by
grasping it between thumb and index finger.) We can proceed from this example to those that involve smaller
targets, more pliable material, and different orientations in space.

Similarly, if we note that the learner is not able to screw properly because the wrist turns inappropriately,
creating torque that flips the blade from the slot in the screw head, we can remove the component of turning
the hand appropriately. One variation would require the learner to hold a penlight and flash it on a target. The
learner is then required to rotate her wrist without moving the beam from the target. The target may be placed
in different positions so the learner must point ahead, down, and up.

Perhaps we notice that the learner fails to “crimp” smaller wires, but does an adequate job on larger ones.
Further investigation discloses that the learner does not press the crimper tight against the base of the wires
before squeezing the handle. We can remove part of the operation from the total operation. We might design a
device that “beeps” if something presses against it with enough pressure. The learner takes the device, pushes
it against each of the targets with enough force to make the “beep.” When the learner has become proficient at
“beeping” targets of different sizes in various positions, we introduce crimping. Although the setup for
crimping may be quite similar to that for the preskill, the preskill does not involve crimping. It involves only a
component behavior and that behavior is removed from the context of crimping.

The removed-component-behavior program introduces the actual operation after the learner has reached
mastery on the removed component. Perhaps more than one component or part is removed from the operation
and is pretaught. Following the completion of the removed-behavior work, the learner practices the operation.
Possibly, it is simplified (essential-response-feature strategy).

The removed-component strategy may make it possible for us to accelerate the introduction of difficult
subtypes of an operation. We noted earlier that we may induce distortion if we work too long on easy examples
of operations like buttoning. If we identify a component-behavior problem—possibly that the learner does not
have precise control over where to press on the smaller buttons—we may be able to design removed
component exercises that strengthen this skill. An exercise for pressing might be to grasp a suspended disk
between thumb and index finger. The disc hangs from a string and may be moved by pressing the thumb
against the disc’s edge. The index finger is on the other side of a “barrier.” (See Figure 23.2.) The goal is to
move the button to the index finger, then lift it over the top of the barrier.

 
 A variation of this activity is achieved by changing the barrier so that it is a fabric with a slit in it. The
learner must now swing the disc through the slit in the fabric and press it against the index finger.

Typically, if we identify why the learner is failing particular subtypes of examples, we discover some
precise behavioral problem. We can frequently design preskill activities that require only that behavior.
Following completion of the preskill we can then present a full range of applications and solve the
generalization problem. Instead of working only with relatively large buttons and then progressively working
toward small ones, we may now introduce smaller buttons relatively early in the sequence (perhaps not as early
as the first applications, but quite soon).

Illustrations. We can usually design all three types of programs for teaching operations such as shoe-tying,
ball throwing, swimming, etc. To teach overhand-ball throwing, we could begin with a number of essential-
response-feature programs. Possibly, the instruction begins with the learner standing on marks that position the
feet properly. The “mark” for the front foot may be a sponge that is to be stepped on lightly (not flattened). A
string hangs above and behind the learner. The learner moves the hand that holds the ball back until it touches

 
the string.

At this time the feet are properly positioned; the weight is distributed properly. The elbow is positioned
correctly. Now the learner swings the arm forward and “throws” the ball. Note that most of the non-essential
features have been controlled, so they require minimum distraction from the central operation, throwing.
Another possible essential-response-feature approach might begin by requiring the learner to sit at a table and
throw by moving only the lower arm. The elbow would be held in place on the table. The arm would be bent,
the hand extended back over the top of the shoulder. An object is placed on the fingertips, and the learner
brings the arm forward, propelling the object. The elbow stays in place. Not only is this a simplified version of
throwing; it is one that works on the most crucial feature of throwing—use of the wrist. If the learner is
reinforced for throwing the object further, the learner will quickly learn that more velocity is achieved by
“snapping” the wrist as the arm comes forward. The most difficult part of throwing is taught.

An enabling-response-feature approach might involve passive throwing. The learner is instructed to

position his weight so that the foot opposite the throwing hand is off the floor (which means that all weight is
on the back foot) and the throwing arm is brought back all the way. The teacher then brings the learner’s arm
forward to achieve the throwing. To achieve the appropriate position for the throwing, the teacher could use
the same marks on the floor used for the essential-response-feature program. The learner stands on the marks,
lifts the front foot off the ground, brings his arm back, and then the teacher effects the throwing.

A removed-component approach is one that deals with one of the more critical components of the
response. One such component is the swing. To teach a transferable version of the swing, we could require the
learner to stand in a certain position and then strike a target (such as a small punching bag) with a certain
amount of force. The device could be rigged so that if it swings up, with a certain force, a bell rings. We could
use markers on the floor to assure that the learner stands in the right position. These marks would place the
learner far enough from the target to discourage all swings but a roundhouse. They would also assure that the
learner starts with the weight on the appropriate foot and with the appropriate foot forward. (The marks would
be arranged so that the foot opposite the throwing arm is forward.) The learner would be permitted to leave the
mark for the back foot while swinging. This convention assures that the learner is “following through” on the
swing. The program teaches the learner to swing a roundhouse blow, using the body for additional leverage—
important components in achieving a hard throwing motion.

Teaching the learner to ride a bike could be taught as an essential-response-feature program, an enabling
response feature program, or a removed component program. The enabling-response program would begin
with training wheels. The learner performs the non-essential features of pedaling and steering, while the
essential-response features of maintaining balance have been removed. The first step of the essential-response-
features program would begin with the learner being propelled. The learner would steer and maintain
balance—the essential features of the operation. The removed-component program would begin with some

 
response components removed from the context of riding a bike. Possibly, the learner would balance on a pogo
stick, or practice standing in different positions while only one foot is on the floor. These activities would
teach the balancing component, removed from the context of bike riding.

With all programs, there would be a change in the examples after the learner mastered the first
applications. After performing adequately on the first application in essential-response-features programs, the
learner would work on examples that require increasing numbers and types of enabling-response features. The
second stage of the enabling-response-features program would integrate more of the essential-response
features. Following mastery of the removed-component, this program would proceed as either an essential-
response-features program or an enabling-response-features program.

Distortion

Programs for complex physical operations involve stages of training. We must look at the early stages in a
program very carefully with respect to the amount of distortion that is implied by work on the early
applications. As noted earlier, we can determine the potential amount of distortion by analyzing the range of
response options that lead to the goal and comparing this range of options with the range that will be permitted
by the more difficult examples. The greater the discrepancy in response options, the greater the possibility that
the learner will learn to produce a response that is successful for the easier examples, but that will not transfer
to the more difficult ones.

This analysis of potential distortion must be referenced to the examples that are to be introduced, not to the
learner. This point is extremely important. If we simply analyze the learner’s performance while proceeding
through a program, we may be completely deceived about the problems the program is creating. The learner
may perform very well on the early examples. It is not until later examples are presented that we discover that
the learner has a great deal of trouble. If the later examples are not mastered in fewer trials than the earlier
examples, the earlier examples apparently did not facilitate generalization to the later ones. The reason that the
generalization was not facilitated, however, is completely revealed only by analyzing the early examples and
the distortion they imply. Without this information, we observe the learning progressing well through the
program and then suddenly “regressing” when faced with more difficult examples.

The beginning steps for all three programs that we have described induce a certain amount of distortion
that will be evident with more difficult examples. However, the most serious distortions result from the
enabling-response features program. The reason is that at the beginning of this program, the learner produces
the non-essential features in a context that supports the essential features. Therefore, these features may be
produced in a way quite different from the way they must be produced when they are coordinated with or
subsumed by the essential response features. When the essential responses are added, the learner is required to:
(1) learn to produce the essential features, and (2) relearn the non-essential features that had been produced in

 
the context of the earlier examples. This double learning may be as great as the learning that would be required
if we merely started with the more difficult example. (The double learning would be particularly great if the
learner practiced the earlier examples until achieving a high criterion of performance.) Consider the operation
of walking with a “walker” or device that supports the learner’s weight and relieves the learner of the central
responsibility of maintaining balance. The learner can produce the response of moving one leg forward and
pushing it back in a context that requires no balance. Therefore, the learner will probably learn to produce the
enabling responses in a way that will have very little transference to walking without support.

The distortion problem is not as great for the essential-response-feature approach because:

1. The learner first masters a legitimate example of the complex operation.

2. The addition of enabling features requires the learner to modify the essential-response features
only to the extent that is required to accommodate the non-essential features. (In other words, the
learner learns a variation of the operation, not a totally new operation.)

3. The introduction of enabling features may be staged so that they provide a gradual introduction of
new-learning demands.

Although the distortion is theoretically less for the essential-response-feature program, distortion may be
induced if the learner works too long with the easy examples.

The amount of distortion implied by the removed-component program is not as great as that implied by the
enabling-response-features program because the initial practice is not produced within the context of the
operation. The variation of the component taught in isolation implies some distortion when it is placed in the
context of the operation; however, the learner is required to learn a variation of this component, not to learn the
entire component. Therefore, the preteaching with the isolated component should facilitate the learner’s
performance with the first examples that involve the operation. Again, however, if the component is practiced
too long in isolation (or practiced to a very high criterion of performance), the transition to the operational
context will probably be more difficult for the learner.

Combining Programs

The guiding principle for teaching complex responses or any physical skill is to achieve the terminal
behavior as quickly as possible. It is a mistake to think of physical operations as if the total operation is a sum
of the various parts. It is not. The sum is unique and not a simple addition of steps or parts. A learner may not
be able to perform on a removed component or part, but will perform the complex operation. (A learner may
not be able to say some component sounds for a word, but may be able to say the word.)

If our program proceeds atomically, it will not be totally effective. We may have to work on atomic

 
details—parts, component behaviors—because the work on parts provides us with a more manageable
instructional unit. The decision to work on a part, however, should come about only after we have determined
that the learner cannot perform on the whole. In other words, we may develop a program for working on
saying isolated sounds when teaching the speech-language delayed child. We may work on chaining sounds
together, once they have been mastered in isolation; however, we must treat this approach as one track of a
total program. The work on parts will teach the learner how the various sound parts retain their identity, how
they are transformed when joined to other sounds, and how they affect the total word. These learnings are
important. But other learnings that cannot be derived from work on the isolated components is equally
important. The atomic approach does little to teach how to match inflection, match stress, and match tones.
Initial work on the total word would not require the learner to produce the component sounds correctly. It
would reinforce the learner for producing approximations, for attempting to match patterns of inflection and
stress. When working on the production of isolated sounds and chaining them together, the program would
reinforce the learner for saying the sounds properly and would not be concerned with stress and inflection.

A single-track program that teaches only the atomic chaining or only imitating the gross prosodic features
of the word will have serious weaknesses. The atomic approach will produce slow, mechanical behavior
because the learner will be so strongly prompted to chain the parts carefully. The gross-feature approach will
be ineffective in teaching the learner to produce the parts or sounds that occur in different words.

Our first attempt should be to teach the total operation without teaching parts. We should try to design
some essential-feature applications that would permit the learner to practice the operation quickly. However, if
we see that the learner is having great difficulty with a particular part of the operation or with a particular
behavioral component, we should introduce a removed-component program for that part. (At one time during
the lesson, we would present practice trials on the removed behavior. At another time, we would practice the
easy form of the total operation.)

If the operation is particularly complex, a different combination of approaches is possible. We may

coordinate a non-essential features program and an essential-features program (the learner working on both
tracks during each lesson). For some trials, the learner produces only the non-essential responses called for by
the operation, and for other trials, the learner initiates the essential features. This approach is useful in
operations such as bike-riding that require the learner to coordinate many details.

Just as the juxtaposition of greatly different examples that are treated in the same way demonstrates
sameness, the juxtaposition of greatly different applications of operations that are treated in the same way
suggests a response sameness.

Here is the simplest pattern for prompting this sameness:

1. Sequence practice so the learner spends about two-thirds of the time on easy applications and one-

 
 third on more difficult applications.

2. Use a different criterion of performance for easy and for difficult examples (higher criterion for
the easy and lower for the more difficult).

3. Juxtapose practice blocks of examples—possibly six easy applications followed by three difficult
ones—so the learner goes quickly from one block to the next.

4. Use the same instructions for all blocks to show that the blocks involve the same response.

Immediately after practicing on a block of trials involving the easy application, the learner would work on
a smaller block that involves a more difficult application. The teacher would indicate that the responses are the
same. “Here are some more buttons to button. These are small. I’ll bet you can’t even get them started.” Note
that the teacher uses a different criterion of performance. For the easier example, the learner buttons them. For
the difficult application, the learner is reinforced for “getting them started,” (or lining them up, or doing some
of the behaviors associated with buttoning).

Note also that it is not important that the learner succeeds on the more difficult examples. It is important
that the learner produces some behaviors that are involved in the buttoning responses. If the learner attempts to
treat the examples from the two blocks in the same way, the juxtaposition of examples will have achieved its
objective—that of prompting the learner to treat the different examples in the same way.

In summary, we present the learner with the terminal, difficult operations as early as we can. We do not
teach parts or components unless we have to, because we recognize that we cannot readily generate a whole if
we “stack” these parts (particularly when we deal with responses that have continuous parts). The whole has
features that do not derive from the parts. And the simplest way to work on these features is to work on the
whole. If the response is complicated, like swimming the crawl stroke, we may superimpose different
programs on each other. We may use an essential-features program that presents operations of increasing
complexity (based on the amount of coordination that is required). The simplest application may require the
learner to float face-down, arms extended. A more complicated application may involve starting in the same
position, pulling one arm down, and simultaneously turning the head away from the arm. All these variations
build around the basic “balance” and orientation that is involved in the crawl stroke (legs basically straight,
face-down floating position, arms extending full-length as part of the stroke).

In addition, components may be identified—such as kicking while holding a kick board. The program for
each component represents a track.

Within each track, there is not a rigid step-by-step progression in difficulty. Rather, there is a systematic
skipping around, with most of the initial practice provided with easy applications, but with more difficult
examples interspersed. The criterion of performance is adjusted so that the learner receives reinforcement on

 
most trials of the difficult applications.

Note that empirical evidence will provide us with information about which applications are better than
others. We will certainly modify the program on the basis of this information. However, the twin problems of
distortion and of the learner’s inability to produce the response acceptably after only a few trials dictates the
general direction that our program should follow. We may discover that requiring the learner to try to move
arms when floating face down does not work as well as requiring the learner to move one arm—down, then
straight out again. We may find that we can introduce an application that requires even more coordination of
responses, such as kicking first, then stopping the kick and moving the arms—all while floating face down.
The specifics are not given by the analysis, merely the direction that should lead to the most effective
transmission of the skill.

Summary

The principles for teaching new and complex responses are complicated because we must somehow wed
techniques for showing sameness of examples with techniques that will permit us to introduce approximations
of operations. The guiding principle is to design the program so the learner performs on fully detailed
applications as quickly as possible. The program should contain the fewest possible steps—the sequence that is
least elaborate.

Some examples or applications are easier than others. These should be introduced early. Their advantage is
that they provide the learner with practice in the operation that is targeted for instruction. Their disadvantage is
that they permit a wider range of acceptable behavior than more difficult examples. Therefore, extended work
on them will:

1. Create distortion in the response called for by more difficult examples.

2. Stipulate that the operation involves only applications of the type practiced.

To discover the options that are available for sequencing examples, we analyze examples of the operation.
We identify those applications that are more difficult (those involving the more precise behaviors and
coordination of a larger number of behaviors) and those applications that are easier. We analyze the responses
according to their temporal parts and their simultaneous components. The analysis shows us whether parts are
joined continuously or discontinuously. If they are joined discontinuously, they are easy to work on in
isolation. If they are chained continuously, they must be worked on within the context similar to that of the
operation being taught.

To work on a behavior that occurs within in a continuous context, we can:

• Shorten the chain of events that precedes the component being taught.

 
 • Design the preceding event so that it also shares the behavior being taught.

• Design the preceding event so that it differs from the next part only in the behavior that is being
taught.

For many physical operations, not all options are readily available.

The three basic programs for teaching complex responses or operations are:

1. Essential-response-features program.

2. Enabling-response-features program.

3. Removed-component-behavior program.

Each program has a different first step. The essential-response-feature program requires the learner to
perform the behavior that accounts for the outcome of an operation while the non-essential features of the
application are eliminated from the example or are performed by someone else. Think of the essential-
response-feature program as one that begins with an easy (perhaps contrived) example of the operation being
taught.

The enabling-response-features program also begins with an example of the operation; however, the
learner does not produce those responses that lead to the outcome. These responses are supported or performed
by somebody else while the learner performs the ancillary or component behaviors associated with the goal-
achieving behaviors.

The first step of the removed-behavior program is not presented within the context of the operation that is
being taught. A totally different setup is introduced to practice the particular component or part that is a
potential stumbling block to making the learner proficient.

All programs proceed from the first step to the terminal or goal operation.

To reduce the problems of response distortion that results if we work too long on a limited range of
examples, we use a combination of programs.

Each program becomes a track in a total program. We juxtapose practice blocks within a lesson. This
juxtaposition shows that the various activities involve the same operational steps.

A different criterion is used for the hard examples. About two-thirds of the practice is devoted to work on
the easier examples, and about one-third to the harder applications.

If we provide a very logically-designed program that buttresses against distortion and stipulation, the
program may require fewer trials and the learning will go more smoothly than it will if we simply require

 
practice. However, the difference will be seen in the manageability of the sequence, the reinforcing quality of
the practice blocks, and the relative smoothness of the progression from activity to activity. There should also
be a savings in trials and an improvement in the generalizations.
 

 
 Expanded Chains
An expanded chain is a program designed to teach a central physical operation and the discriminations
associated with this operation. The basic program for tooth-brushing deals with those behaviors that are most
immediately associated with brushing teeth—the manipulations that account for the teeth being effectively
brushed. Similarly, the basic program for operating the radial arm saw deals with the manipulations directly
associated with sawing the wood. Associated with each basic operation are other discriminations and behaviors
that are typically produced in the same context as the central operation. An expanded chain for tooth-brushing
may involve everything from taking the brush from the rack to replacing the cap on the tube of toothpaste. The
expanded chain for operating the radial arm saw may include putting on safety glasses, checking the
equipment, selecting stock, turning on and turning off the machine.

An expanded chain has two features that set it apart from the other forms we have worked with:

1. The expanded chain consists of a chain of behaviors that does not become completely covertized.
Often, the central behavior remains overt throughout all applications of the chain.

2. The chain consists of both discriminations and new motor responses.

The description of expanded chains suggests that expanded chains are not limited to motor operations like
sawing or brushing teeth. The line between a cognitive operation and an expanded chain is fuzzy. The clear
examples of expanded chains are those that deal with physical operations. Putting money in the soft-drink
machine is the physical operation for an expanded chain that involves “counting out” money. Typing a
business letter is an expanded chain, built around the physical operation of typing. Sweeping the floor is an
expanded chain. In all cases of expanded chains, there is an integration of discriminations and motor responses.
You do not sweep the floor where you have already swept. You do not type the heading on the letter three
times. And you do not stick pennies in the soft-drink machine. These discriminations feed the central
operation. You discriminate whether the floor has been swept. Then you perform the appropriate response.
You determine which coins are appropriate for the machine; then you place them in the machine. You
determine which part of the letter you have typed; then you type the next part.

Expanded Chain Programs

Think of an expanded chain program as a series of separate tracks, each dealing with a particular skill.
Ultimately, the tracks converge. The tracks that need the most work should begin earlier. These tracks typically
involve the new motor responses. Motor skills will require more time to induce than the discriminations.
Therefore, by starting them early, we enable the other components of the terminal expanded chain to be

 
introduced sooner.

First, we teach each skill within a juxtaposition pattern that permits rapid repetition of the skill. If the skill
is screwing the cap on the toothpaste tube, we work on repeated examples of this skill. We do not require the
learner to complete the entire chain of brushing the teeth before practicing this behavior.

Following the initial juxtaposition, we introduce the skill into a small chain that is composed of the new
skill and several skills that occur with the new skill. When the learner becomes increasingly proficient with
this chain, we introduce a longer chain and so forth until the learner is performing the entire operation. The
strategy is not new. In one respect it is a variation of the 3-level strategy. First, the easiest juxtaposition is
presented; then, more interruptions are presented between examples of the newly-taught skill. For the
expanded chain, the interruptions are other parts of the chain. They will always tend to occur in the same order
and therefore the teaching is somewhat different from earlier forms. However, the procedure is one of going
from repeated juxtapositions of a single task or part to a chain consisting of many parts.

Integration Strategies

Figure 24.1 shows two different patterns for initial teaching and integration of expanded chains. The first
pattern is a unit integration. Individual skills (A, TO, B, T1) are taught and are then chained into intermediate
units (ATO and BT1) and are finally combined to form the terminal chain (ATOBT1). The second pattern is a
trunk integration. A trunk is established (ATO). New elements are added to the trunk until the terminal chain
(ATOBT1) is formed.

At each stage in the trunk integration, the learner is working on the chain that contains A and another

 
response (TO, B, and then T1). When the learner becomes firm on each skill in isolation, the skill is added to
the trunk. In the trunk integration shown in Figure 24.1, the newly-taught skill is always added to the end of
the chain being taught. The new skill, however, could be added to the beginning of the chain as well.

The schedule does not show all the teaching that is possible for each time period. A general rule is that the
learner may practice the chain that has already been developed or any part of that chain. For instance, when the
chain ATO has been established as a trunk, the learner may work on A in isolation, TO in isolation, and ATO
as a chain. This convention follows from the general correction procedures. If the learner has serious trouble
with A when it occurs in the context of ATO, the teacher removes A from the chain, practices it in isolation,
and then reintroduces the chain ATO.

Scheduling Parts of the Expanded Chain

The principles for scheduling the skills that appear in an expanded chain are:

1. Do not attempt to introduce the skills in the order in which they will occur in the terminal chain.

2. Begin instruction on the central operation as early as possible.

3. Schedule the teaching of new responses (either as removed components or as part of the
operation) as early as possible.

4. Make the program as simple as possible.

There are three reasons for teaching the central operation early, regardless of its position in the terminal
chain. The first reason is that the central operation provides the motivation or justification for the other
behaviors. The justification for checking traffic is much more easily communicated if it is linked to the actual
behavior of riding the bike.

The second reason is that the central operation requires new responses. Because new responses imply a
great deal of practice, the sooner we start on the central operation, the sooner we will achieve the amount of
practice needed to establish independent behavior for the central operation.

The third reason is that the central operation can be practiced independently once it is mastered. (The
physical environment provides feedback.) The scheduling of later skills is easier if the learner continues to
work independently on that central skill.

Principle 4 suggests that the program should be as simple as possible. As we observed earlier, we should
not see how elaborate the program can be, but how simple we can make it. The programs that we will illustrate
may seem to violate this principle. These programs are designed for the learner who would be relatively hard
to teach. Therefore, the programs provide for the careful teaching of each skill. More sophisticated learners
will not need all the steps that the programs specify.

 
Illustration: Bike Riding

Bike riding is not a particularly difficult skill to teach; however, the skill serves to illustrate the process of
integrating various components. Ultimately, we want the learner to exhibit the following behaviors:

• Check apparel.

• Check traffic to assure that riding is safe.

• Mount bicycle.

• Stay in marked lanes.

• Follow traffic rules, including giving signals.

• Turn and brake appropriately.

• Start and stop on level surfaces and hills.

We could add to the list. The more we add, however, the more we may obscure that central thrust of
instruction. To find that central thrust, we answer two questions:

1. Which skill (or skills) do the other skills tend to assume?

2. What is it about this skill that the learner cannot do?

The answer to the first question is “ride the bike.” All the information about following traffic rules,
stopping and starting, etc., assume the basic skill of riding a bicycle.

The second question asks, “What is it about this skill that the learner cannot do?” Let’s mentally put the
learner on the bicycle, make the machine move, and think about what the learner would do if the learner does
not have to worry about clothing, checking traffic, and other details of the chain that may complicate the
operation. The learner would probably wobble from side to side, take one foot from the pedal to maintain
balance, place it on the ground, thereby tipping the bike in the other direction, overcompensate with the handle
bars, and fall over.

The purpose of mentally constructing the simplest possible example, removed from the events that precede
the critical part of the operation, is to determine the setup for the central operation. If the learner probably
cannot perform the operation described above, a simplified variation of this activity would be the central new
response of the program. Although it would not be expedient to start all skills at the same time, we can
simultaneously introduce some skills that do not interfere with the actual riding operation or that depend on
the operation. Our choices include discriminations about safety, checking traffic, possibly even mounting the
bicycle or pedaling, if these operations can be designed in a way that they do not assume the riding ability.

 
Figure 24.2 shows a possible schedule that shows the strategy for the first five lessons.

As we observed earlier, the central skill is maintaining balance on a moving bicycle. We will refer to this
skill as “riding.”

The schedule does not show all the activities that occur on a given lesson, merely the most complicated
chain that is presented on that lesson. The arrows show the progression of the different activities. First, each
skill is taught in isolation and then added to the chain. The program involves a trunk integration of skills.
Traffic checking is taught as a discrimination. The teaching may involve a correlated-features sequence.

Mounting is taught as an independent operation during the first two days. Practice in this operation could
be made independent of riding by designing a device that would not fall over, but would move forward in the
way a bike does when the pedal is pushed down. The learner could practice pushing the pedal down and
swinging into a sitting position. This program would be a non-essential response program because the essential
balance features of the response have been removed from the operation. Therefore, we should not retain the
practice too long in this form. If we teach mounting through an essential-response program, we would have to
present it in a context that required the learner to balance the bike. The learner might learn to scoot the bicycle
(like a scooter) while keeping one foot on the pedal. When this operation is firm, the learner could learn to
mount. Or the mount could be delayed until the riding had been firmed and then added as a part at the
beginning of the riding operation. The schedule in Figure 24.2 assumes that we are presenting mounting
through a non-essential program. The mounting is integrated with the central operation (on the third lesson);
however, not all trials of riding would involve the three steps of mounting, pedaling, and riding.

The schedule introduces pedaling through a non-essential-response-features program. Part of the pedaling
operation could involve standing and pedaling (with the pressure adjusted so that the learner would have to
exert quite a bit of pressure on the pedals). The pedaling operation is taught as a non-essential-response
features skill so that it may be introduced early. On the second lesson, it is integrated with some practice in
riding. The quick integration of pedaling and riding assures that the pedaling response is not seriously

 
distorted.

The schedule for riding in Figure 24.3 shows the various practice blocks and indicates how the activities
from the other tracks may be juxtaposed with riding. This schedule specifies activities not presented on the
general schedule for this program.

Within the riding track are skills that are taught within the context of riding (turning and stopping).

 
 The juxtaposition of the early practice blocks is designed to permit rapid practice. On lessons in which
skills are to be integrated, the skill that is to be added to the trunk is first practiced in isolation. Immediately
following the successful practice of the skill in isolation, the skill is integrated into a chain. For instance, on
lesson 2 the learner first practices pedaling in isolation, then practices pedaling within the chain of straight
riding and pedaling. The work on the pedaling in isolation serves as a precorrection. The idea is to make sure
that the learner is firm on the component to be integrated and is prompted on producing it. Note however, that
the precorrection becomes counterproductive if the learner becomes fatigued while working on the component
or begins to produce poor responses. In these cases, the introduction of the chain would be delayed. A break
would be scheduled, followed by more work on pedaling in isolation.

Illustration: Driving Screws with a Screwdriver

The expanded chain that we wish to teach involves the following behaviors:

1. Picking up screws and inserting them into predrilled holes on a surface that may be oriented so
that the screw head is up, down, parallel to the floor, or in some intermediate orientation.

2. Turning the screw in with a screwdriver until the screw head is flush with the surface of the
material in which the screw is embedded.

3. Meeting a rate-accuracy criterion for a minimum number of screws per minute.

Only two major behaviors are involved in the operation; however, each of these behaviors has many
component behaviors. The act of screwing the screw with a screwdriver involves aligning the screwdriver,
applying pressure, turning in the appropriate direction, and continuing to turn until the screw is set. The
implication is that the learner knows the difference between a screw that has not been completely set and one
that is completely set.

As the first step in designing the program, we answer the questions: “Which skills do the other skills tend
to assume?” and “What is it about this skill that the learner cannot do?”

The essential part of the chain is operating a slot (or blade) screwdriver appropriately. The operation that
the learner cannot do consists of aligning the instrument, pushing, and rotating the handle in a clockwise
direction.

The sameness that we wish to teach is the basic operation of manipulating the screwdriver (aligning,
pushing, and turning). The range of variation is accounted for by the orientation of the screw, the type of head,
the type of screwdriver. Although these features vary, the basic operation (aligning, pushing, and turning) is
not affected.

Figure 24.4 shows the relationship between the sameness (align, push, turn) and the range of variation for

 
the operation of manipulating the screwdriver.

The diagram implies that we must design the operation so it is obviously the same operation across all
possible variations of orientation, head type, and size.

Adding behaviors to make the operation unique. The orientation of screws presents a double problem. If
we teach the learner to perform the operation with screws that are oriented head-up, and if the learner works on
these examples for a fairly extended period of time, the learner will probably reverse the turn when we present
screws that are head-down. An apparent solution to this problem is to introduce variations of head-up and
head-down screws early. The problem with this solution is that it involves a very difficult discrimination,
turning clockwise (in relationship to the ground) when the head is up, and counter-clockwise (in relationship to
the ground) when the head is down.

We can solve this problem by adding stipulated behaviors to the operation. The purpose of these is to
permit the learner to do the same thing, regardless of the screw’s orientation. Possibly, the learner is taught to
aim the screwdriver at the screw head (sighting down the shaft in the same way that you would aim a rifle).
The learner holds the screwdriver so that the thumb is facing the screw head (not the learner’s face). The
learner inserts the screwdriver and turns it in a clockwise direction.

By following these steps, the learner will always turn the instrument in the appropriate direction, whether
the screw is oriented head-up, head-down, or head-horizontal. No reversals are possible, because the screw
head (not the ground) determines which direction is clockwise.

Note that this strategy involves adding behaviors to the operation. These stipulated behaviors permit us to
introduce variations in spatial orientation very early. Instead of working only on head-up examples, we can
work on examples in a variety of orientations.

The potential problem associated with the other variables (type of head and size of screw) is response

 
distortion. We looked at this problem in Chapter 23. There are easy and hard examples of head types. The easy
examples have the head of a bolt. (The screwdriver handle is attached to a socket that fits over the head.) This
example is easy because the learner can succeed while performing a wide range of turning behaviors. The
learner may create great lateral torque while turning the handle. If this behavior is performed with a blade
screwdriver, the blade would disengage from the screw head and slip to the side. With the socket driver,
however, the distorted lateral-pressure response does not affect the outcome, although the turning response is
inappropriate for the blade screwdriver. If we introduce the socket variation at the beginning of the program,
the program actually becomes a non-essential-response features program, because the learner is not responsible
for alignment. (The learner initially aligns the screwdriver; however, continuous alignment is not necessary
during the turning operation. Therefore, great distortion in the turning response is prompted by this example.)

The solution to the problem of a distorted-turning response is to juxtapose examples that are greatly
different from each other, but that require the same basic operation. Here are examples that we will use. They
are arranged in order of increasing difficulty.

1. socket driver

2. phillips

3. big blade

4. small blade

We begin by presenting examples of the easy type, and the most difficult type for which the learner can
produce approximate responses. We present some examples of the socket driver and juxtapose these with
examples of the big blade screwdriver. We set a different criterion of performance for both applications. (We
might reinforce the learner for each complete rotation of the handle when working with the big blade and
reinforce the learner only after setting the bolt with the socket driver.) As the learner improves, we change both
criteria of performance, moving in the direction of increasingly difficult examples. This juxtaposition should
prompt the learner to treat the diverse examples in the same way (particularly if the same unique operational
steps are taken with each example).

Figure 24.5 is a possible schedule for high-criterion and low-criterion activities that would be juxtaposed
during different time units (which could be weeks or days).

 
 Note that the low criterion is used for more difficult examples and the high criterion for easy examples. At
time 1, the high criterion problems involve a socket driver, and the low-criterion problems involve large-blade
screwdrivers. As the learner improves, the context for both high and low criteria shift. At time 2, the most
difficult examples the learner works on are small-blade examples. During the same lesson range, the learner
also engages in high criterion practice with a phillips screwdriver (an example that is harder than a socket
driver, but not as hard as a blade) and a large blade. At time 3, the high criterion track presents only large
blades. At time 4, the low criterion track drops, because the learner is now performing at a high criterion on the
most difficult examples (small blades).

The review track picks up the various skills that had been taught earlier to a high criterion. After high-
criterion performance has been achieved with socket drivers, they move to the review track (time block 2)
followed by phillips, large-blade, and small-blade applications. For all review items, the high criterion is
retained.

The setup would be designed to accentuate the operational sameness. The various screws and bolts would
already be started for the learner. All would be mounted on a board that rotates or changes position after each
screw or bolt has been set. The learner first screws two or three bolts with the socket driver. Immediately
following the last socket application, the learner screws one large screw with the large-blade screwdriver. As
noted earlier, the criterion for reinforcing responses is different for the two examples. Perhaps a light goes on if
the learner performs an acceptable trial (screwing the bolt all the way in a specified time period, or completing
one rotation with the large-blade screwdriver).

The rules for the variation in the orientation of the board might follow a pattern of changing within the
range of 45° during time phase 1. After each example has been completed, the board would change from 0° to
45°. During time phase 2, the range of variation might increase to 90°, during phase 3 to 135°, and for the
remainder of the program to 180°. It may be expedient to introduce this range of variation only on some blocks
of practice trials during the different time periods. (Just as the learner does not have to engage in every aspect
of bike riding for every practice trial, the learner does not have to work on “difficult examples” on every trial.)

 
 In addition to directing applications involving screwdrivers, the program should account for other aspects
of the chain—placing the screws in by hand, selecting the appropriate screwdriver for the screw, aligning the
screwdriver (a preskill to the operation of manipulating the screwdriver) and pushing against the screwdriver
(another preskill).

The introduction of these skills follows the same pattern as the various secondary skills that were added to
the chain for bike riding. For instance, to teach the behavior of turning, we could use the same board on which
the screws are mounted, a flashlight, and a polarized lens.

After each trial, the board changes orientation. The learner must turn on the beam, and aim the flashlight at
a polarized lens that is mounted on the board. Then the learner rotates the flashlight a half turn so that the light
continues to strike the lens throughout the turning process. This movement is the same that is used in turning
the screwdriver. The flashlight is therefore held in the same manner, with the thumb pointing to the target. A
simple electric device could indicate whether each trial is successful. Rate criteria could be introduced. (The
learner would have to perform “so many” trials during a specified time period.)

We might add component skills when we observe particular problems that the learner has. One such skill
could be pressing the blade of the screwdriver into a slot and maintaining a certain amount of pressure. To
teach this important component, we could design a screwdriver that must be aligned with the screw and
pressed with a certain amount of pressure to sound a buzzer, which indicates a successful trial. The slots on the
screw heads could be arranged so that the learner would have to first rotate the screwdriver and align the blade
with the slot, then press with a certain amount of pressure, maintaining the screwdriver so that the shaft is
parallel to the shaft of the screw.

The work on these operations would be superimposed on the work with the combined operations that
occur in time periods 1 through 5. Figure 24.6 shows the various skills feeding into the main trunk of operating
the screwdriver.

 
 The preskills of placing, aligning, and turning are started in the first time unit. Placing continues through
the third time unit. The combined operation of placing, aligning, and turning starts in the second time unit and
continues to the end of the program. When performing this operation, the learner would place the screw with
fingers, then set it using a screwdriver. This work would be performed in addition to the massed practice of
working with screws that have already been placed.

This program calls for various automated devices—flashlights that give feedback, possibly pressure
devices that “beep,” boards with screws that change position from time to time. The program could be devised
so that these devices are not needed. If the teacher is working with a group of learners, however, the devices
become very important, because they permit the learners to work without continuous supervision from the
teacher. The devices simply perform the feedback role for operations that involve discriminations the learner
may not know. The learner may not know, for instance, how hard to press against the screw when turning. A
device can be designed to provide feedback. Similarly, the learner may not know what the limits are for
turning the handle while maintaining a true orientation. Again, a device that provides feedback can be
designed. These devices have the potential of changing a program from one that requires constant teacher
monitoring to one that provides more practice with far less monitoring.

Poorly Designed Expanded Chains

An expanded chain is an elaborate communication. If it inadequately conveys information to the learner,

the learner will have trouble, which may not become obvious until later in the program.

The major communication flaws that are most typically observed in poorly designed programs are:

1. Inappropriate juxtaposition of examples for the initial teaching of a component or skill.

2. Introduction of events in a way that implies a false operation.

3. An order of introduction that arranges subtypes in a progressive, easy-to-hard series.

Juxtaposition Flaws

When a new skill is being taught, it should be possible to present repeated trials of that skill at least every
10 seconds. If much more time is required, something is wrong with the presentation. The problem is probably
that the skill is embedded in too large a chain, or that the logistics and setup are not well-designed for initial
teaching. The prediction based on poor juxtapositions is that some learners will experience chronic or
persistent regressions.

Juxtaposition flaws typically occur when the only initial teaching of a skill occurs as the skill is added to
the chain of behaviors already taught. Typically, this type of program introduces the skills either in the order in
which they occur in the terminal chain or in the reverse order (backward chaining procedures). If the temporal

 
parts are A, B, C, D (in that order), an inappropriate procedure would be the one diagrammed below.

Note that B, C, and D are not first taught in isolation and then integrated into the chain. The introduction and
integration occur at the same time. The only time that C is taught is when it occurs in the context of A + B +
C. The juxtapositions become very inappropriate for teaching skills that occur late in the chain. If the skill E is
introduced only in the context of the chain A + B + C + D + E, the practice is poorly designed to teach E. We
can illustrate the problem with a chain for teaching the learner to cut a series of boards the same length, using a
radial arm saw.

Assume that the teaching for this operation occurs in the order of the events described in the routine above.
First, the learner learns to recite the rule. The learner then learns to recite the rule and square the end of the
board. The learner then learns to recite the rule, square the board, and operate the clamp. And so forth. Not
only is the juxtaposition pattern inappropriate for performing the actual operation of making repeated cuts; it is
inappropriate for the teaching of the other skills in the chain. To teach the learner to select stock appropriate
for the sawing operation, a simple discrimination series is implied. The rule recitation should be dropped
completely from the chain. The learner should be provided with massed practice in squaring the end of boards.
The learner should receive massed practice in setting the stop at different distances along the fence. (“Set the
stop at four inches” . . . “Set the stop at nine inches” . . . Etc.) Furthermore, the operation of making repeated
cuts (moving the board to the stop and cutting, removing the piece that had been cut and again moving the
                                                                                                                         
4  The
fence is a guide that orients the board. The stop is a device positioned at the end of the board so that there
is a fixed distance from the saw blade.  

 
board to the stop and cutting) should be introduced early. It is the central operation, and it should not be
delayed until the learner learns the rule, stop-setting, selecting stock, and squaring.

The two major flaws in this program are very common. The program introduces skills in the same order
they occur when the chain is completed. The program adds skills to the chain. These two flaws mean that the
initial practice for many of the skills violate the ten-second rule. The simplest way to identify the flaws is to
determine whether the learner would be capable of producing repeated responses to the entire chain used for
initial teaching in no more than ten seconds. The remedy for the problem is to eliminate as much of the chain
as necessary to create an appropriate pattern of juxtapositions for initial teaching. After the initial teaching has
occurred, the newly-taught skill may be integrated into a larger chain.

If we follow the procedure of identifying the central operation and teaching it as quickly and simply as
possible, the program for repeated cuts involves one sequence that deals with the stop and then an operation
that meets the juxtaposition criterion. The following illustrates a possible program sequence:

Teacher Wording
To make repeated cuts that are 17 inches long, I set the
stop 17 inches along the fence.
Where do you set the stop along the fence for repeated
cuts that are 18 inches long?
Where do you set the stop along the fence for repeated
cuts that are 28 inches long?
Where do you set the stop along the fence for repeated
cuts that are 13 inches long?
Let’s say you are going to make repeated cuts that are 20
inches long.
So where do you set the stop along the fence?
Do it.
Let’s say you are going to make repeated cuts that are 9
inches long.
So where do you set the stop?
Do it.
Let’s say you are going to make repeated cuts that are 30
inches long.
So where do you set the stop?
Do it.

This skill in integrated into a chain that involves only the new behavior and the sawing (C and D).

 
 Teacher Wording
You’re going to make repeated cuts that are 15 inches
long.
Where do you set the stop?
Do it.
Now make 4 repeated cuts.

Implying False Operations

Another type of problem occurs if the program attempts to teach both a new-response skill and a
discrimination in the same context or chain. This treatment may not effectively communicate to the learner
whether the learner is expected to learn a discrimination or a motor response. Two types of false operations are
possible. For the first, we are trying to teach a new motor skill; however, we emphasize discriminations or
concepts to a degree that implies that we are actually trying to teach discriminations or concepts.

The second type of false operation involves the opposite situation. We are trying to teach a discrimination
or concept; however, our emphasis on motor responses implies that the learner is supposed to learn a motor
skill.

The radial-arm saw operation presented a false operation of the first type. The emphasis on verbal
behavior (recitation of the rule) implied that the operation of making repeated cuts requires these verbal
discriminations. The emphasis is displaced to talking about sawing.

We can illustrate the second type (teaching a discrimination, but implying that we are teaching a motor
response) with a program designed to teach basic language skills to naive learners. The teacher presents
objects. For each, the teacher says, “What is this?” The learner responds by naming the object within a
statement form: “That is a _____.” The learner may not know the name of some objects. The learner also may
have trouble producing the complete statement (even for objects that are familiar to the learner). The typical
correction is for the teacher to model the correct response: “That is a lamp,” and require the learner to repeat
the statement until firm. The reinforcement is provided with the words “Good talking.”

The inappropriate strategy that the learner may develop from this presentation (particularly if there are
many unfamiliar objects) is to talk (produce the motor responses) without paying attention to the features of
the object. To remedy this flaw, we separate the motor skill from the discrimination. We first present the
discriminations. When the learner is firm on these, we present the motor skills. Let us say the learner is to
identify five objects—lamp, dog, cup, chair, and phone. The teacher first presents pictures of these, asking the
question that requires a relatively simple response: “What is this?” Response: “Lamp.” Reinforcement:
“Good.” The reinforcement now is clearly linked to identifying the object. The type of behavior the learner is
performing is the same as that performed in other situations that call for looking at things and identifying them.

 
 After the learner has performed on juxtaposed examples of identifying the objects with single-word
responses, the teacher can work on the responses. The teacher points to an object. “What is this? . . . Yes, a
lamp. Here’s how to say the whole thing about the lamp: That is a lamp. Say the whole thing about the lamp . .
. Again . . .” Reinforcement: “Good saying the whole thing.”

Now the separation is clear. The reinforcement for “saying the whole thing” shows that the teacher is
dealing with a new skill. If the teacher corrects a statement-saying response, the chances are greatly reduced
that the learner will be confused about what sort of behavior the correction deals with. The correction does not
imply that the learner should stop discriminating and simply produce “talking” responses.

Easy To Hard Examples

If the program sequences examples from easy to hard in a linear progression, we would predict serious
response distortion. In fact, the learner would probably learn better if the sequence were random. In an easy-
to-hard series of examples for screwdriving, the learner works first on bolt heads followed by phillips, large-
slot heads, and finally small-slot heads. As we observed earlier, the prediction implied by the sequence is that
the learner will have serious difficulty when the large-slot screws are introduced. The reason is that the
preceding examples permit the learner to exert inappropriate lateral force while turning the handle. The large-
slot screw does not permit this behavior. Therefore, the learner may learn an inappropriate turning behavior
and may have trouble with the slotted screws because the blade pops out as the handle is rotated.

A similar prediction of distortion results if easy-to-hard examples are used for the spatial orientation of
screws. The easiest examples are those in which the screw head is up. Gravity tends to hold the screwdriver in
place. If the screw head is horizontally oriented, gravity tends to force the slotted screwdriver to slip out. If the
screw head is down, the learner must apply pressure to keep the screwdriver in place. By working on the easy
examples of screwing until the learner is proficient, we not only teach a slight motor-response distortion
(which results because of the gravity effect); we also set the stage for possible reversals. The learner may have
trouble when confronted with upside-down screws and may turn the handle in the wrong direction.

Illustration: SCUBA-diving. A program for using SCUBA equipment may be introduced in a way that
leads to possible problems if the sequence proceeds progressively from easy to hard examples. Underwater
breathing with SCUBA gear is achieved by inhaling through a tube held in the mouth. Breathing can be
accomplished when wearing a mask that covers the nose, or without a mask. If the instruction arranges
examples in the order of easy-to-hard, the first examples the learner practices involve wearing the mask. Like
other easy examples, this one permits the desired outcome with responses that are common to the entire range
of examples or with responses that work only for the easy example. In other words, the easy examples may
induce distorted responses. The possible distortion has to do with the involvement of the nose in breathing.
While the mask is worn, the learner has the option of inhaling so that only the mouth is involved in inhaling, or

 
breathing so that the nose is also involved. So long as the mask is fitted properly, the involvement of the nose
makes no real difference in the outcome. The underwater breathing is successful. The slight pull on the mask
caused by the involvement of the nose does not hinder the breathing. In fact, the learner may be quite unaware
of using the nose.

Breathing underwater while using only the air tube—no mask—is more difficult because any involvement
of the nose will lead to a nose full of water. Note that this example is more difficult because the outcome may
be achieved only if the learner performs within a narrow range of behavioral limits. Underwater breathing
while wearing a mask is easier because of the behavioral options it provides. However, it is possible to use
exactly the same set of behaviors to breathe in both situations. The objective of instruction would be to induce
this sameness in behavior as quickly and efficiently as possible. Teaching the behavior early is important
because unintended examples may come into the sequence. Let us say the learner begins with “easy” examples
that involve breathing with a mask and air tube. Suddenly, the learner’s mask fills with water. In effect, the
learner is presented with a difficult example—one that had not been scheduled in the sequence. Unless the
learner knows how to breathe in this situation, the learner may inhale water through the nose, panic, and have
an unpleasant experience.

The problem can be avoided if the sequence of examples teaches the behavior that is common to all
underwater breathing situations. The first examples in this program involve no mask. The learner sits in
shallow water with head under water, breathing with only an air tube. After demonstrating proficiency with
this example, the learner performs the same operation in deep water, while swimming and carrying out a
variety of maneuvers. Next, the learner is required to go underwater without a mask on, but with an air tube.
The learner is not allowed to clear the mask. (The mask remains full of water.) Finally, the learner puts the
mask on underwater and clears it (achieved by leaning back and exhaling through the nose). The skill of
breathing without a mask is reviewed by a water entry routine that the learner uses for some time. The learner
enters the water without a mask, goes underwater, puts on the mask, and clears it.

With this training the learner is not in jeopardy if the mask fills with water. The learner does not become
excited, because the type of breathing that the program teaches is the same for all underwater situations.

When we violate the easy-to-hard order of introducing examples, we do not overlook the fact that some
examples are harder than others. We do not place the learner in a situation that induces failure. Breathing
without a mask may be more difficult than breathing with a mask; however, there are gradations of task
difficulty for the no-mask situation. We could initially place the learner in deep water without a mask.
Obviously, this situation would be dangerous. We therefore try to make it easy by manipulating the setup.
When the learner sits in shallow water, the example has the properties of a good example. We can work on a
breath at a time. The quick examples of inhaling and exhaling can be juxtaposed. There is very little
interference that hinders this juxtaposition.

 
 We made the harder examples in the screwdriver program easier by changing the criterion of performance.
We made the harder example of the underwater breathing situation easier by changing the setup so that the
only behaviors required of the learner are those that are essential to breathing underwater. Swimming and other
behaviors that are produced simultaneously in some examples of underwater breathing were removed, leaving
only the behavior of breathing.

Both techniques are important for making harder examples easier. We follow this general guide:

• If possible, we eliminate the non-essential details of the harder example so that only the essential
details remain.

• If it is not possible to design such examples, we change the criterion for the harder examples.

Rehearsal and Transfer

Teaching some expanded chains involves establishing the chain in an appropriate instructional setting,
then transferring the behavior to a variety of other situations. The problem of transfer is particularly acute for
these chains because we must often construct practice situations for initial teaching that are not highly similar
to the transfer situations. This condition occurs when:

1. The transfer situation is not available for practicing the initial teaching.

2. The situation in which the learner is to respond occurs only infrequently.

3. The situation in which the learner is to respond does not permit juxtapositions that are appropriate
for initial teaching.

An example of (1) above would be behavior that we expect the learner to perform in the doctor’s office.
The office is not available for rehearsing the behavior; therefore, we teach the behavior in some situation in a
way that will prompt transfer to the doctor’s office.

An example of (2) would be the sheet-and-clothing-tearing behavior of a learner that occurs only at night
and only for a short period during the night. Rather than waiting for these infrequent situations, we must
construct practice or rehearsal situations that permit a massing of trials on some behavioral chain that should
transfer to the low-frequency situation. The rehearsal would involve following instructions to tear and not tear
things on command. The “no-tearing” command would then be transferred to the bedtime situation.

An example of (3) would involve teaching a blind person to respond to directions that tell how far to go
and in which direction. Because it is nearly impossible to carry out applications of this routine in 10 seconds or
less, the pattern of juxtaposition is poorly designed for initial teaching. Therefore, we must create a “model” or
rehearsal situation that permits manageable juxtapositions of examples. We expect the learner to transfer from

 
this situation to the “real life” applications.

Some teaching assignments may involve all three problems-unavailability of transfer situation, low-
frequency behavior, and poor juxtapositions.

Inducing Transfer

When we design instruction that will lead to transfer of an expanded chain to a situation different from the
practice situation, we must create sameness. The learner must be shown that some feature in the transfer
situation is also in the rehearsal situation and that this sameness triggers the same behavior in both situations.
The simplest technique is to design the practice so that the first steps in both the practice situation and the
transfer situation are identical. If the learner carries out the first steps, the learner will be strongly prompted to
carry out the remaining steps in the chain. In designing the first steps so they serve as a prompt for transfer, we
find things that are the same in the practice situation and in the application setting. We arrange the chain so
that it starts with these shared features.

We can illustrate this procedure with rehearsal for a visit to the doctor’s office. The purpose of the visit
may be that the learner is getting a shot. The purpose of the rehearsal situation is to give the learner a number
of things to do so that the shot will not be frightening. We can establish the chain in a way that will displace
the learner’s attention from the fear and provide the learner with behaviors that will lead to reinforcement.

For the chain to be efficient, it should begin with a detail that is common to both the doctor’s office and
the practice situation, possibly with the door to the doctor’s office.

“We’re pretending now. There’s the door to the doctor’s office. You open the door for me. You let me go
in first and then you go in . . .” This part of the chain is practiced and the directions are replaced with
questions. “What’s on the other side of the door? . . . Who’s going to open it? . . . Okay, let’s go.”

The beginning part of the chain is then linked with the other parts and practiced until the learner is
proficient. The learner is reinforced during the practice sessions for carrying out the various behaviors. To
practice the “shot,” the teacher pinches the learner on the arm.

The chances are good that the learner will perform appropriately in the transfer situation. By beginning the
chain the same way in the application setting, we imply that the behaviors practiced earlier apply to this
setting. “Okay, there’s the door to the doctor’s office. What’s on the other side of it? . . . Who’s going to open
it? . . . Let’s go . . .” By constructing the two situations so they are the same with respect to trivial details, we
imply that they must be the same with respect to the relevant details. If a reasonable parallel exists between the
situations the learner will probably perform appropriately in the application setting. (The learner will not cry
when getting the shot.)

 
 The chain with common first steps is very powerful for prompting the behavior because it establishes
compliance on details that are not critical to what happens in the application setting; however, since these first
steps are chained to subsequent steps, compliance on the first steps implies compliance on the later steps.

If the learner needs a prompt to begin the behaviors called for by the chain, this prompt is provided in
connection with details that are not critical to the operation. If the chain had been designed without a beginning
that involves details not critical to the central operation, the prompt would have to be given during the critical
part of the application, which is not an ideal time to try to establish compliance.

The following is a summary of the procedures for establishing the practice situations that promote transfer
to application situations.

1. Begin the chain with behaviors that are common to both situations but that are not critical to the
application situation (like opening the door).

2. Follow the first part with requirements for frequent responses. (Intermittent behaviors reduce the
probability of continued compliance.)

3. Practice the entire chain in the practice situation until the learner can perform with a minimum of
prompting.

4. Prompt the first part of the routine in the application setting (pointing out the details that trigger
the chain of behavior) and reinforce the learner for performing appropriately.

Creating Rehearsals for Low-Frequency Behaviors

To deal with inappropriate behaviors that occur infrequently, we create a situation in which we can work
on the behavior in a way that places it under instructional control (which means that the learner does it on
command) and also work on the opposite behavior, which is also placed under instructional control. The logic
of the program is that we can “tempt” the learner in this rehearsal situation. If the learner overcomes the
tendency to be tempted in the controlled situation, and if the learner consistently responds to the instructions
the teacher provides in this situation, the chances of the learner performing in the problem situation are greatly
increased.

We can illustrate the procedure with the tearing behavior of an autistic child that occurs only at night. We
design a situation in which we have many objects that can be torn or broken. (This training would occur in a
classroom or a room of the house other than the bedroom.) We present objects to the learner and give the
command, “Tear it,” or “Break it.” We insist that the learner must wait until we give the instruction to tear or
break. We provide no reinforcement for tearing or breaking. Also, we issue the command, “No tearing” or “No
breaking” when we hand the learner some objects. If the learner fails to comply (by tearing or breaking), we

 
present the learner with a long series of commands, “Stand up . . . sit down . . . stand up . . .sit down . . .” After
we present possibly 40 such commands (and the learner complies with at least the last 5), we return to the task
the learner had non-complied with. We again present the same object and say, “No tearing.”

Once the learner is firm on massed trials of “Tear it” and “No tearing,” we introduce “temptation tasks.”
We present the learner with the kind of objects that he apparently delights in tearing and we issue new
commands with these: “Hold it . . . no tearing,” or “Take it to Ms. Brown. No tearing.” We increase the degree
of temptation by making the tasks longer and increasing the time interval between tasks (following the same
progression that is used for shaping the context of a task). Any non-compliance is followed by a long series of
stand-up, sit-down tasks and repetitions of the task that had been non-complied with. Note that during the
“temptation” training the child is given no commands to tear, only to do different things with objects that are
tempting candidates for tearing.

When the learner is firm on “temptation” tasks, we present a series of similar tasks in the bedroom setting
before the child goes to bed at night. The child would receive a series of commands associated with the objects
that he typically tears. “Touch your pajamas. No tearing . . . Good. Hand me your pillow . . . Good. Touch your
blanket. Remember, no tearing . . . Good,” etc.

In this final stage, we are treating the bedroom setting in the same way that we had treated the original
training setting. The learner should therefore transfer to the bedroom situation. Of course, this transfer assumes
that we had brought the learner to a hard criterion of performance in the classroom and had presented tasks that
allowed the learner to be in the presence of tempting objects for long periods of time during which there was
no direct supervision from the teacher.

If tearing occurs during the night, the learner is put through a series of “stand-ups” and is then rehearsed in
the bedroom on “no tearing.”

Creating Improved Pattern of Juxtaposition

We have already illustrated the problem of poor juxtapositions. The illustration that we will present next
differs from those that we have dealt with because it involves creating a “model” for practicing a skill, but
doing so in a way that transfer to the application settings is strongly implied.

To teach a blind learner to respond to instructions for going places in the school, we would design a chain
that permits massed trials in the practice situation. We begin by presenting the learner with paper on which
there are tactually perceptible marks. The paper is oriented so the direction the learner is facing is at the top of
the paper. A dot in the middle of the paper marks the learner’s position. The bottom of the paper represents the
space behind the learner.

For the simplest exercises, we would inscribe lines from the center dot to different parts of the sheet. The

 
learner would first feel the line, then point in the direction shown by the map. (The pointing does not involve
the paper.)

The learner is next presented with applications that involve both distance and direction information. For
instance, an example such as the one illustrated above is presented. The learner is told the number of steps
(five) and is then presented with these tasks: “Show me the direction you’re going to move . . . Good. And how
many steps are you going to take in that direction? . . . Yes, five.”

Note that the learner does not move, but merely describes what will happen. Note also that the wording of
the tasks clearly implies that application situations will require the execution of these behaviors.

For the final set of rehearsal examples, we use a game format. “I’m going to put the ball some place in this
room. Then I’m going to use the map to tell you where it is. See if you can find it . . .” The objects are in place
before the game begins. We present the learner with various map examples and tell the distance for each.
“Show me the direction you’re going to go . . . Good. And how many steps are you going to take in that
direction? . . . Yes, seven. Do it . . .”

The practice situations present the same format that would be used in the transfer situations. The only
major difference is that the examples in the transfer situation cannot by presented rapidly. “Here’s a map that
shows where Mr. James’ room is. Feel it and then point in the direction you’ll go first . . . Point in that
direction. You’ll go about 50 steps. Then you’ll come to a door. You’ll turn at the door. Show me which way
you’ll go after you turn . . . You’ll go about 20 steps. Then you’ll come to another door on your right . . .”
Although the example is more complicated than those used in the rehearsal situation, it is composed of simple
components. (We assume that the learner has already been taught left and right.) If the learner performs in the
transfer situation, the learner should have no trouble dealing with the transfer applications.

Rehearsal for Field Trips

Often the procedures used for field trips are backward. The teacher expects the learner to recognize the
significance of things observed on a field trip and to retain this information. Actually, a more reasonable
procedure would be to rehearse what will occur on the field trip, point out the significance of things that will
be observed, and then use the field trip as both a transfer situation and a reinforcing experience. For instance, if

 
the learner is to go to a museum, the teacher describes things the learner will see and do. These occur in a
relatively fixed order. As part of the introduction, the teacher may use pictures.

“The first thing you’re going to do is get your ticket stamped at the ticket window. Then you go straight
inside and wait. Right in front of you, you’ll see two dinosaurs. Here are their pictures . . . What’s the name of
this dinosaur? . . . What’s the name of the other dinosaur? . . . And to your left, you’ll see . . .”

The verbal routine is repeated until the learner can answer a series of questions about the topic. “What’s
the first thing you’re going to do? . . . Where do you get it stamped? . . . Then where do you go? . . . There will
be something right in front of you. What’s that? . . . What’s the name of the biggest dinosaur? . . . What’s the
name of the other dinosaur? . . . If you’re standing just inside, point to where the dinosaurs would be . . . Now
point to where the African elephant will be . . . Good.”

Since the series of verbal questions can be presented in both the rehearsal situation and the transfer setting,
the teacher can assure transfer by presenting the same questions in the transfer setting. “Okay, what’s the
building we’re standing in front of? . . . And what’s the first thing you’re going to do? . . . And then what are
you going to do? . . .”

If the learner is pretaught, the transfer situation is reinforcing. The learner has performed successfully on
the various questions in the rehearsal situation; the learner has received reinforcement from the chain of
questions. The same chain is presented in the transfer situation; therefore, the transfer situation is established
as a reinforcing situation. Note that if a great deal of content is involved in the event that is to be presented, the
teacher might use a visual-spatial display for communicating this information. The technique can be combined
with the technique of acting out the first part of the chain. This variation would be appropriate for young or
naive learners.

Chains with Divergent Responses

Let us say that we want children to play a dress-up game in which the teacher “auctions” different articles
of clothing to a group of children. Each child selects those articles that would be worn by a particular character
(a cowboy, a dancer, etc.). If the child chooses to dress up as a cowboy, the child selects cowboy clothes.

Initially, an aide plays auctioneer while the teacher sits with the children and directs the activity. The
teacher models the behavior that the children are expected to produce, asks questions, and prompts them to
respond.

First, the teacher models a role that she will play. “I’m going to be a cowboy. Boy, I like a cowboy. What
are you going to be?” (A mailman.) “You’re going to be a mailman? That’s great.” The teacher models great
excitement. (If the teacher effectively models excitement, the learners will produce excited responses.)

 
 The teacher continues to ask the various children what they are going to be and to brag about her role until
all children have responded at least two times by telling what they will be.

Next, the game begins. The auctioneer holds up an article of clothing that the teacher will wear. The
teacher says, “Oh boy, there’s a cowboy hat and I’m going to be a cowboy. So I say, ‘I want it! I want it . . .’”
The auctioneer quickly gives it to the teacher. The procedure is repeated for the next object, chaps.

The teacher struts and shows off her outfit, making the point that she operated according to the rules of
getting the cowboy clothes. “I’m a cowboy. I got the cowboy clothes . . . Wow, look at me.” Quickly, the
clothes are returned to the auctioneer and without hesitation the game starts over. This time, articles of clothing
for the different roles are held up in an unpredictable order. For each article that is held up, the teacher says:
“Who wears that?” Then she asks the child who selected that role, “So what do you say?” (I want it.) “You got
it . . .” If the child does not respond immediately to both questions, the object is returned to the auctioneer.
“You can do better than that,” the teacher observes. The child is told to remember that object. It is returned to
the auctioneer and is presented after one or two other objects are presented. (This pattern of juxtaposition is
like level 2 of the three-level strategy. The learner has successfully responded to examples that are not
interrupted by interference. Now another activity is interpolated between the presentations of the object.)

The procedure is repeated. If any children give spontaneous comments about their role (“I’ve got the
fireman’s hat and I’m going to be a fireman”), the teacher reinforces the response. “He sure does look like a
fireman already . . . That’s great.”

After the children have their outfits, the teacher reinforces them and permits them to show off for a few
minutes. “Wow, I see a cowboy and a pretty dancing girl, and who is that big fireman over there? . . .”

The game is repeated with the teacher providing fewer instructions. First, the teacher makes sure that the
children have selected roles and remember their roles. “Remember what you’re going to be. Who’s going to be
the farmer? . . . Right. Who’s going to be the fireman? . . . Who’s going to be the cowboy? . . . Who’s going to
be the queen? . . .” The teacher selects a role, sits with the children, and models the sort of responses the
children are to produce. She talks to her neighbors. “I’m going to be the cowboy. So I get all the cowboy
clothes.”

The teacher praises the children. “Boy, they really know which clothes they’re going to get. They are
smart . . .”

As the children become proficient, the teacher no longer sits with the children. Children continue to
receive praise for selecting new roles and for selecting the clothes for that role.

Variations of the procedures indicated for the dress-up game can be used to establish any chain that
involves divergent responses. Although the procedure may seem quite different from those we have worked

 
with earlier, the introduction of the modeling is the only new technique. To establish behavior on a divergent-
response chain, we follow four steps:

1. We design the steps of the chain so that we can:

a. model the type of behavior we expect the learner to produce;

b. present questions and tasks to overtize what the learner does;

c. provide reinforcement for responding to the tasks and for spontaneous responses that are
consistent with the role.

2. We introduce the steps in the easiest context and then we proceed to more difficult contexts. We
follow the three-level strategy for controlling the context—first presenting repeated presentations
of a step (level 1), followed by presentations of that step when it is chained to one or more of the
other steps (levels 2 and 3).

3. When the learner can perform on the entire chain without requiring prompts or corrections, we
modify the chain by removing the tasks and questions. We continue to model (play the role of a
learner) and reinforce, but not to ask the structured questions.

4. After successful performance on this chain, we modify it again by removing the model. (The
teacher no longer plays the role of a learner.) We maintain reinforcement. The chain is now
operating as an independent activity or as one that does not require teacher involvement.

The most important ingredient in the procedures for establishing divergent responses is the modeling. The
reason is that there is something that is the same about all examples. For the dress-up game, the sameness is
that you are going to behave like a particular person. You are going to do what that person does. “I’m going to
be a ____. So I’m going to pick the things a ____ wears.” If the teacher makes this equation very strong for
one role, each of the other roles is a kind of transformation of the role the teacher models. Many behaviors are
the same for all roles-the excitement, the naming of what you are going to be, the selecting of things that the
selected character would wear. The only difference is the particular role that is selected and consequently the
specific articles that are selected.

Summary

Expanded chains consist of various discriminations and motor responses that are built around a central
physical operation. The physical operation is the “reason” for the chain.

The basic strategy that we follow is to teach parts of the chain and then join the parts together, forming an
increasingly longer chain. The two strategies for integrating the parts of the chain are the unit integration

 
(which establishes smaller chains that become juxtaposed units in larger chains) and the trunk integration
(which establishes a trunk that grows by the addition of single parts that are first taught in isolation and are
then added to the chain).

The trunk integration is usually the most manageable if we follow basic scheduling rules:

1. Do not introduce skills in the order in which they appear in the chain.

2. Begin instruction on the central operation as early as possible.

3. Teach other new responses as early as possible.

4. Try to make the program simple.

For chains that involve the introduction of many new skills, a given lesson would include juxtaposed
practice blocks for the various skills that have been taught. Skills being introduced are practiced in isolation.
Skills that have already been integrated into a chain are presented in variations of the chain (some variations
involving only some of the parts that have been taught and other variations involving all the parts that have
been taught).

The most serious problems associated with designing expanded chains are: inappropriate juxtapositions for
the initial teaching of a skill; a teaching emphasis that implies a false operation; and the ordering of examples
from easy to hard.

If the examples that are introduced for initial teaching require more than ten seconds (or possibly more
than six seconds for most skills), the juxtaposition of examples is not well designed for initial teaching. A
smaller part of the chain should be identified or a new setup should be designed so that the examples may be
presented more rapidly.

A false operation is implied if the skill being taught is a motor response and the introduction emphasizes
discriminations that are associated with the response. Also, a false operation is implied if the new skill is a
discrimination that is paired with a related motor response, suggesting that motor response is relevant to the
discrimination. The remedy for the false operation is to separate the discriminations and related motor
responses during initial teaching.

A progressive, easy-to-hard order of introduction for the applications of a physical operation creates both
distortion and stipulation. The remedy is to introduce harder examples early and to juxtapose them with easier
examples of the same operation. This juxtaposition conveys the sameness in the applications and therefore
reduces the stipulation and distortion.

Many expanded chains are practiced in one situation with the expectation that they will transfer to other
situations. The transfer may not be automatic because the rehearsal situation is different from the transfer

 
situation. The solution is to add sameness to both situations. The chain that is rehearsed should begin with
behaviors that are not critical to the central operation. The chain should permit a relatively high rate of learner
response. When the chain is applied to the transfer situation, the teacher prompts the first part of the chain.
Since this part is the same as the first part in the transfer situation, and since the rest of the chain always
follows the first part in the practice setting, the entire chain is implied for the transfer setting.

To design a chain for divergent responses, we should model a particular example and present the kind of
questions that permit the learner to produce a variety of responses.

Expanded chains are complicated. They present very little that is new, however, if we remember that the
order of events that occurs in a chain does not imply the order for teaching the skills or necessarily the form for
their initial teaching. We can usually identify sensible teaching for the skills. By following the basic rules for
demonstrating sameness, we can engineer transfer to specified situations.
 

 
 Expanded Programs for Cognitive Skills
Expanded programs deal with broad subjects or topics that imply more than a single discrimination or
cognitive routine. The subject may be expressive writing, solving elementary physics problems, basic fraction
operations, geology, etc. Expanded programs are the cognitive counterpart of expanded chains. An expanded
chain centers around a physical operation of some sort. (The central operation is never fully covertized.) The
central operation in an expanded program is a cognitive skill—a discrimination or cognitive operation. Like
the expanded chain, the expanded program consists of more than a single operation. New responses (motor
skill teaching) may be involved. An elaborate chain may be created to support the central operation; however,
in the end, the operation is a cognitive one and the chain of events is therefore an invention, created to make
the discriminations overt and obvious. The procedure may be complicated by the requirements to teach
particular vocabulary, facts, and behaviors. These requirements are subsumed by the central thrust of the
cognitive program, which is to teach an idea. Because of this somewhat strange structure, we have a strange
interplay of strategies. In one respect, we have great latitude about how we will teach the idea. We can adjust
the examples, group or create the subtypes, use prompts, and possibly even create unique wording and other
response conventions. In other respects, we are constrained by convention. In the end, the learner will be
expected to perform in the framework already established for the idea that we teach. We must therefore temper
our efforts to teach effectively with the facts about the conventions the learner must be taught.

Expanded programs involve all the formats that we have examined. Basic discriminations, joining forms,
subprograms for teaching different skills, new motor responses, cognitive routines, visual-spatial displays,
prompting-fading—all become tools in the expanded program. Although the total expanded program is largely
a coordination of smaller, simple programs, the total program cannot be reduced to the various components.
The expanded program deals with a unique organization that provides impetus for the components to come
together.

The focus of this chapter is on the strategies for organizing the various component programs (or tracks)
into an efficient whole. Although the rules for specifying what should be done are fairly simple, the execution
of the program is not. Sometimes years of testing the strategy takes place before the simple solution inevitably
emerges. The efficient solution is always a simple one. If the program is characterized by ambiguities,
vagueness, and “profundity,” the program is the product of poor analysis. There is order in what we wish to
teach, just as there is order in the pattern observed in clouds, sea shells, or traffic moving down a freeway. Our
task is to discover it and to communicate this order. If we do it properly, the development of the skills will
seem so easy that it might strike the naive observer as “cheating.” The program will not be difficult, will
involve a minimum of brow-furrowing discriminations, and will not present severe road blocks that preempt

 
all from learning but the intuitively gifted and the lucky.

Once we understand what the program ideally should be, we can judge whether the one that we have
created is a close approximation or not. Sometimes we may recognize that our emerging program is not what it
should be; however, we may not know how to change it. In these situations, we must judge whether it is better
than those already available, and if it is, use it with the full understanding that it is not an excellent program,
but a useful one. In time, and with observations of the problems that teachers and students have with the
program, we may be able to reshape it into a strong, simple pattern. (Techniques for such reshaping are
outlined in Chapter 28.)

Some aspects of an expanded program are similar to those of expanded chains; therefore, some basic facts
and strategies apply to both expanded chains and expanded programs.

1. Examples range from easy to hard, and the problems of sequencing examples exist for the
expanded program and the expanded chain. We create stipulation if we adhere to a rigid, easy-to-
hard sequence.

2. The strategies for integrating the newly-taught skills with the skills that have been taught derive
from the problem of integrating the skills of an expanded chain. The problem of creating
juxtapositions of appropriate examples for the newly-taught skill suggests that the skill should
first be taught in isolation and then integrated with the skills that have already been taught.

3. The development of the various cognitive skills runs parallel to the skill development for the new-
response programs. We can teach skills as removed components. We can present a non-essential-
feature program or we can present an essential-features program. Often, we use removed
components because we follow the procedure of teaching one thing at a time; however, we temper
teaching one thing at a time with the goal of achieving the program objective with the least
amount of teaching.

4. We design the activities to account for transfer, following the same basic rules that we use in
designing expanded chains. We create sameness in responses to show how applications are the
same. We modify the situations to reduce irrelevant details so that the apparent sameness is
increased. We create rehearsal situations so that we can control the juxtapositions of the events.
We then introduce the application or transfer situations in which the skill is to be used.

The Basic Strategy

The strategy that we will follow for designing expanded programs for cognitive skills is not complicated in
its major steps. Only three steps are involved:

 
 1. We first organize the content according to response dimensions, and we identify an operation for
communicating the idea that we wish to teach.

2. We next identify the range of examples for which the operation applies.

3. Finally, we identify the related operations for sequencing subtypes of examples and for processing
exceptions that are not handled by the “main” operation.

This three-step procedure becomes complicated because each step requires a very careful treatment,
analytically. First, we will outline the general procedure; then, we will apply it to different content problems.

Organizing Content by Response Dimensions

For initial teaching of discriminations and motor responses, we use an approach that juxtaposes different
examples involving the same response dimension. If we extend this approach to broad content areas, we
radically reorganize the traditional content because the traditional content areas are organized around topics—
not common operations. For instance, a topic in physics is optics. There are many facts about optics, each
implying different arithmetic operations. Traditionally, the learner studies optics and learns a variety of facts
and operations that are called for by this topic. Later, the learner may discover that some operations used in
optics are also used in other topics.

The response-dimension approach is different. Instead of trying to exhaust a topic, it deals with individual
operations, and it presents each operation so that it cuts across a variety of topics. Following the same basic
procedure used with simpler forms, juxtaposed examples of the operation are applied to a variety of contents to
show what is the same about all instances of the operation. In the end, topics emerge; however, the initial
teaching focuses on response-dimensions, not on events.

An illustration of the response-centered orientation is the operation of “squaring.” As the distance of an

object from the light source increases, the amount of light striking the object decreases by the square of the
object’s distance from the light source. This relationship, however, is not confined to optics. Springs work in
the same way. The amount of energy generated by a spring increases by the square of the “distance” that the
spring is compressed. Also, centripetal force of an object increases by the square of the increase in the object’s
velocity. There are many other applications of the squaring relationship. To maintain a response-dimension
orientation, we would introduce variations of a basic squaring routine and then apply it to problems from a
variety of topics.

The rationale for organizing content around a response-dimension is the same as that for the design of
simpler communications. By grouping examples of the same type, we can more effectively communicate
samenesses and differences. Instead of encountering one example of the squaring operation now and another
example perhaps weeks later, the learner encounters many examples immediately. The juxtaposition of the

 
examples prompts the operation and therefore makes it initially easier for the learner to perform. We face the
same responsibility for removing the prompts that we face with simpler communication forms. However, when
the unprompted practice is introduced, we know that the learner has the capabilities of performing in prompted
situations. The procedure for integrating the new skill is basically the same as that used for simpler forms. We
place the operation in increasingly difficult contexts.

Finding the Range of Examples

To approach content as a response-dimension endeavor, we must group operations. To group operations,

however, we must extend our analysis of examples.

The major difference between examples of simple discriminations and examples of operations (such as the
operation of long division or the operation of writing a good first sentence for a paragraph) is that the features
of operational examples are facts about the ways in which we can operate on the example.

To determine the “features” of an operation, we make up facts about examples of the operation. The
procedure is the same as that used in the subtype analysis. The difference is that the analysis is now applied to
examples of an operation. If we make up six facts for a given example of an operation, we are provided with an
operational definition for identifying other examples that are in the same class as the original operation. If the
six facts apply to another example of the operation, that operation has the same features as the original
operation and is therefore in the same class. If only five of the facts hold, we are dealing with two subtypes of
examples.

To identify examples that are in the same class:

1. We begin with an example that is a “mainline type,” one that is unambiguously an example of the
type we are concerned with.

2. We make up a set of facts about the operation.

3. Then we check the facts by seeing if they hold for other mainline examples.

4. If the facts do not hold, we adjust the set until we develop a set that holds for the mainline
examples.

5. Now we test the range of examples by using the set of facts we have developed as a criterion for
judging whether non-mainline examples are actually members of the set to which the operation
applies.

This procedure involves two phases. During the first phase (steps 1 through 4), we develop the criterion
for classifying examples. Step 5 is an application of the criterion to other examples. The assumption is that if
another example possesses all the features that are expressed by the criterion, the example is the same as the

 
mainline examples with respect to the operational details.

The procedure not only permits us to identify the range of variation of examples for a particular operation.
It also: (1) helps us clarify the operation by forcing us to refer to specific features of the operation; and (2)
provides a means by which we can discover unsuspected examples that are in the same class as mainline
examples. With the information about the range of examples, we are in a much better position to offer the
introduction of examples and types so that stipulation is reduced, relatively easy examples are presented early
in the program, and examples of a given type are juxtaposed in a manner that will clearly imply the operational
sameness that obtains across a range of example variation.

Identifying Subtypes

When we organize examples according to a common response dimension, we will identify some examples
that are relatively easy and some that are analytically more complicated. The more difficult examples will
involve more mechanical details and may require more knowledge.

In addition to the range of example variation, we will discover that some important types of examples do
not lend themselves to the operation that we have designed. Since these example types should be taught, a
variation of the operation must be designed to process them.

Both the range of examples to be processed by a single routine and the relationship between the main
routine and related routines to process specific subtypes of examples may be expressed as transformations. The
more difficult examples processed by the main routine are transformations of the simpler examples. The
modification of the routine that is needed to accommodate examples that are not processed through the main
routine involves a transformation. By viewing the problem of dealing with the full range of examples and the
full range of routines as problems in designing transformations, we are provided with specific guidelines about
how to achieve efficiency in design.

Illustration: Product Conservation

We can illustrate the procedure by referring to the physics concept of conservation. There is conservation
of energy, conservation of mass, and other conservations. These conservations are either the same from an
operational sense, or they are not.

To discover if an important sameness exists, we follow the procedures outlined above—starting with a
mainline example (an example of conservation of momentum, for instance) and making up non-trivial facts
about that example. A non-trivial fact is one that would not be true of everything or of a range of things far
beyond the example under investigation.

Here are some non-trivial facts about conservation of momentum:

 
 • Conservation is achieved through the product of values. (Simple momentum equals the rate times
the mass of the object.)

• If the product remains the same and if one value is made larger, the other value is made smaller.
(If the momentum of an object remains the same after the mass is increased, the object must have
slowed down.)

• The product changes only if something is added or taken away from one of the values. (If the
momentum is increased, the mass has increased or the rate has increased.)

We can use the set of facts that we have constructed as a criterion for identifying other examples. Each
statement applies to conservation of energy, so this type of example falls into the same set as conservation of
momentum.

By applying the set of common facts to a range of mainline examples, we can modify the set of facts—add
facts, change some, delete some—until we get what appears to be a good set for the mainline members of the
set—those examples that obviously seem to be the same in their operational structure.

We may also develop different ways of expressing the facts. For instance, we may express the product
conservation (one dimension increases as the other decreases) by this type of equation:

5
8
8 =
3 5
8
5
The idea is to change the pair of values on the left side of the equation into the corresponding values on the
right. The first number on the left changes into the first number on the right. The second number on the left
changes into the second number on the right. To change 8 into 5, we multiply 8 by 5/8. If we multiply the first
number (8) by 5/8, we have to multiply the second number (3) by 8/5 (not by 5/8) to find the missing value,
which is 24/5. We are multiplying the left side by 5/8 x 8/5, which equals one. Therefore, we are not changing the
initial amount on this side. That the answer is correct can be seen by cancelling the 5’s on the right side of the
equation (5 x 24/5), leaving 24, which is the value on the left side of the equation.

Although this equation holds for all simple product conservations, it is not the only way that we can
express the facts about conservation. It is perhaps the most operationally salient, however, because it permits
us to test the various examples with a precise procedure. Each example either shares the feature expressed by
this operational “fact” or it does not. If it does, it is an example of product conservation.

When we apply the set of criteria for conservation to other examples, we find that a rectangular display of
squares has all the basic conservation properties of the other examples. The rectangular display is the product

 
of two values (the length times the width). If the product remains the same and one value is made larger, the
other value is made smaller. For instance, if the total number of squares remains at 24, but the width is
increased from 2 to 3, the length must change from 12 to 8.

3
2
24
12 2= 3 3
2
3

The rectangular display does not possess irrelevant features of the other examples, only the operational
features. It is therefore an “easy” example of product conservation.

Ordering Example Types

With the knowledge of the range of examples, we can now sequence the introduction of example types.
We introduce each example in the same way (using the same routine) to signal the sameness in the examples.
We begin the program with relatively easy examples that are manageable. The rectangular display meets these
criteria. It also permits verification. The outcome or answer to each problem can be verified by arranging the
component squares according to the pattern suggested in the problem (12 x 2, for instance). Juxtaposed
examples involving this rectangular setup can be presented quite quickly. Therefore, the display could serve as
a primary type of example for the initial teaching of the concept.

Once the initial teaching is achieved, we require the learner to transfer the skill to various other examples.
Transfer is achieved if there are observable details of new problems that are the same as those of familiar
problems. To make the sameness as obvious as possible, we design the routine with the initial example in the
same way we design chains that are used in a rehearsal situation. We include unique details in the first part of
the chain. If the learner is prompted to treat the first part of an unfamiliar example in the same way that the
first part of a familiar example is treated, the learner will most probably transfer the operation to the new
example. The transfer will occur for the same reason the transfer occurs when we deal with well-designed
rehearsal chains. The operation is a chain of behaviors. The first part of the chain is prompted for both
examples. Since the first part holds for the new example, transfer for the remainder of the chain is implied once
the first part has been performed.

The remainder of the strategy parallels that used for expanded chains.

• We design a chain or a routine capable of processing all examples.

• We teach the preskills called for by that chain, scheduling the teaching of the motor skills early
and designing the teaching so that we can introduce applications of the entire operation as quickly

 
 as possible.

• We provide applications of the chain that permit juxtapositions appropriate for initial teaching,
examples that permit the learner to do the same thing with juxtaposed examples.

• We next shape the context in which the operation occurs by systematically introducing
interruptions between presentations of the operation.

• We try to include early examples of all subtypes that present no great mechanical problems.

• We delay the presentation of highly irregular examples until the learner is firm on the other
subtypes. Then, we introduce the irregulars either through a double-transformation program or
through some other procedure that permits the massed practice with the new type and integration
with the other types that have been taught.

Except for some details, the strategy is the same as that for expanded chains. The difference is that the
determination of range of examples for the expanded chains is much easier than for expanded programs. For
example, analyzing examples of using a screwdriver is relatively easy because:

1. We deal with only mainline examples as examples of using a screwdriver.

2. The variations in the operation that are possible can be discovered through observations of
physical manipulations.

When we deal with cognitive skills, we must focus attention on non-physical details of the examples and
functional samenesses. Once we have discovered a sameness (the operational features that are shared by
various examples) we can arrange the examples in much the same way we would do with examples of physical
operations.

Illustration: Averages

The operation of figuring out where a balance beam will balance illustrates how unsuspected examples of
an operation may be discovered. Let’s say that in our search for examples of the basic conservation
relationship, we experiment with the formula for levers: E x D = R x D1 (effort times its distance from the
fulcrum equals the resistance times its distance). We discover that this relation- ship is an instance of product
conservation. Consider this problem. If a force of 6 pounds moves 5 feet, 10 pounds of resistance would move
how far?

 
 E D=R D1
10
6

6 5 = 10 D1
6
10

D1= 30
10 or 3

The problem is solved using the same compensation operation used for rectangular displays.

When we move to another type of balance problem, we see that the same conservation relationship holds.
For these examples, the board does not move to create leverage. It simply balances. We put weights on one
side of the balance. We then figure out how much weight would have to be put on the other side of the balance
to effect a balance.

For instance, we start with this problem:

X
3 2 1 1 2 3

Three equal weights are placed 2 units from the fulcrum on the left. The X above the 3 indicates where we are
to place weights on the right to achieve a balance. The question is, how much weight goes in this position? The
same equation used for the preceding problem works for this problem: E x D = R x D1.

These are the values: 3 2= 3

2
3
The solution: 6
3 2= 3 3
3
2

The product of 6 is conserved on both sides of the balance; therefore, 2 units of weight must be placed at 3 on
the right side of the balance.

We discover that a different operation is needed when we try to find the balance for a beam that has
counters on it. For instance, where is the specific balancing point for this beam (given that the weights are of
the same value)?

 
 1 2 3 4 5 6 7 8

There are ten counters on the beam. Counting ten times at the unknown balance point is the same as counting
one time at 1, three times at 3, one time at 5, two times at 6, two times at 7, and one time at 8. In other words:

10 BP = (1 1) + (3 3) + (1 5) + (2 6) + (2 7) + (1 8)
10BP = 1 + 9 + 5 + 12 + 14 + 8
10BP = 49
BP = 4910 = 4.9

For problems of this sort, we add the various products. The number of counters tells the number of times
we must count at the unknown balancing point, BP, which is 4.9.

When we search for other problems that have the same operational features as this problem, we find other
types. Here is a problem with the same numbers as the problem above.

• Jane drank water on 10 occasions. On 1 occasion, she drank 1 glass. On 3 occasions, she drank 3
glasses. Once, she drank 5 glasses. On 2 occasions, she drank 6 glasses. On two occasions, she
drank 7 glasses. And once she drank 8 glasses. What is the average number of glasses that she
drank?

The solution:

10N = (1 1) + (3 3) + (1 5) + (2 6) + (2 7) + (1 8)
10N = 49
N = 4.9

Average problems and basic balance problems are the same because they have the same operational features.
Instead of being taught as logically separate types, therefore, they should by juxtaposed, included in the same
set of examples, and taught through the same operational steps (through a routine that prompts the sameness in
the operation and that stresses the conservation of the products). Unless such instruction occurs, stipulation
will take place and the learner probably will not have an intuitive understanding that balance-beam examples
and averages are the same.

Here is a routine that could be used to prompt the operational sameness. The problem involves ten
counters.

 
 Teacher Learner
We have to find the middle
of the distribution.
So what’s the first
thing we do? Count the counters.
Do it. (Counts.)
How many counters? Ten.
So what do you write first? Ten N equals.
Do it. (Writes 10N = )
Now what do you do? Add the products.
Do it. (Adds 1+9+5+12+14+8)
What’s the sum of the products? 49.
How do you find the middle Divide by the number
of the distribution? of counters.
Do it. (Divides 10 )49)
Where is the middle of
the distribution? 4.9

The wording, “We have to find the middle of the distribution,” serves as a prompt for all problems of this
type—including simple statistics problems. It permits the same operational steps to be applied to diverse
problems. The prompting, as noted above, is the same type used for expanded chains. Through the double-
transformation strategy, we could later introduce conventional wording. For example, the familiar set of
problems might refer to the middle of the distribution, while the new set refers to the “average” or the “mean.”
Once we have introduced an operation across the range of examples we have performed the most important
step, which is to establish the generalization.

Summary of Procedures

The conservation problems illustrate the basic approach to organizing content. First an operational
sameness is discovered. Examples that share this sameness are members of a set. Those that don’t share it are
members of another operational set. When we test examples for product-compensation features, we discover
that some unsuspected examples are in the set. Lever problems and some balance problems have the product
compensation features, expressed as:

 
 1
A C

A 1
B = C (A B C)

A C
1

Some problems that do not have this feature belong to a set of find-the-balance problems. The simplest
way to assure that the learner is shown the operational samenesses for each of the two operations we have
identified is to use a response-dimension approach and juxtapose various examples of a particular type, using a
unique routine to prompt the operational samenesses across the range of variation in examples. We would first
teach the basic product-conservation operation, then the discrimination for using the find-the-balance
operation.

Illustration: Expressive Writing

Expressive writing is one of the most difficult skills to teach and certainly one of the most thoroughly
mistaught. The problem seems to be that teachers expect the learner to simultaneously produce writing that
exhibits sentence variety, uses stylistically and grammatically appropriate expressions, follows spelling and
punctuation conventions, and adequately develops ideas. Obviously, any new learning situation that presented
all these criteria (or even half of them) would overwhelm most learners and produce the type of writing
behavior observed in many adults—a laborious attempt to try to create, compress, edit, and polish at the same
time.

To organize expressive skills, we must recognize that we will not achieve the final development in writing
skills at the beginning of the program. Just as the learner does not begin reading instruction by reading
something like Plato’s Republic, the learner should not start writing with applications that are too difficult.

We should not begin expressive writing by having the learner write about make-believe or imagined
events because:

1. We have no way of knowing whether learners are precisely reporting about the things they
imagine.

2. Therefore, we would not be in an ideal position to work on the better use of language.

If we begin with objective referents, we solve the problems associated with fantasy writing. We can
observe what the learner is referring to; therefore, we can determine whether the learner’s use of language is
precise, and we can help the learner work on better wording. Another advantage of an objective referent is that
it permits us greater control over the juxtaposition of examples.

 
 We begin the expressive writing by using action pictures as the common referent. The simplest example of
writing would involve a sentence that tells about something depicted by the picture. The simplest sentence is a
subject-predicate one. This sentence has the following operational features:

1. It starts by naming or identifying something shown in the picture.

2. It then tells more about that entity.

The operational features imply how to make examples relatively easy or relatively difficult. Easy examples
for the first part of the sentence would involve illustrated articles for which many labeling options are available
or labeling which involves very familiar words. Easy examples for the second part involve pictures of entities
that show a variety of possible actions or actions that are very familiar. More difficult examples are those that
call for descriptive precision. We use minimum differences in the wording of sentences to create negative
examples of the sentence. For example, a task requiring the learner to make up sentences using the word only
could be very difficult if the learner had to describe only the big boys in the picture and the only big boy
wearing a hat. A minimum difference in the wording of either examples creates a different meaning.

To teach the precise use of language with relatively easy examples, we begin by requiring the learner to
write only parts of sentences, not entire sentences. For instance, we present a picture showing two entities (A
and B) engaged in the same action (running). We require the learner to complete the sentence: “____ was
running” by telling about entity A. The learner must describe entity A in a way that will not permit confusion
of A and B. This activity builds the skill of identifying the entity, using descriptive words, using appropriate
class names, and referring to the details that make A different from B.

We use a variation of the activity to work on the predicate details. The picture shows identical entities (A
and B) doing different things. The learner is required to complete the sentence: “The little girl _____,” by
describing A’s action (not B’s action). The description must be accurate enough to avoid possible confusion of
A with B; therefore, the learner is required to use appropriate verbs and descriptions of position and objects
that are receiving the action.

Part of the operation that the learner follows in writing simple sentences about the picture is to tell more
about it. When we test to discover other possible examples that have this feature, we find that a variation of
these features is shared by the regular-order paragraph, by outlines, and by a variety of sentence types.

The simple, regular-order paragraph begins with a sentence that first tells what somebody was doing and
then tells more. For example, a paragraph for the picture in Figure 25.1 could begin with this sentence: The
children ran from the monster. This sentence names a topic. To create a paragraph, we simply tell more about
that topic—more about how the children ran from the monster. For initial teaching, we could refer to this
activity as “reporting.” For reporting, the writer is not permitted to interpret, merely to deal with the objective
facts shown by the picture. The sentences could tell what they did, what their expressions were, what they did

 
while running, and what happened as they ran.

Figure 24.5

The sentence that we begin with (“The girl ran down the pier”) is a good sentence because it efficiently
summarizes the action with a few words. We must recognize that this criterion for a good sentence is not
necessarily consistent with the traditional criteria (many of which require the learner to embellish the sentence
until becoming lost in its infractions). The use of two types of telling-more sentences make it clear which
aspect of the original sentence is being amplified. If the sentence tells about how the girl ran down the pier, we
underline the part that starts with the verb ran and we begin the part that tells more with another verb, and that
amplifies running down the pier. If the sentence tells more about the last part of the sentence, we underline the
pier and we begin the part that tells more with another word that describes the pier.

Here are the examples for the initial-teaching set that involve telling more.

 
 Sentences Paragraphs
Simple: She ran down the Simple with the sentence:
pier. She ran down the pier.
More about how she ran: Tells more about how she ran:
She ran down the pier, She kicked high into the air
trying to move as fast as she moved forward. She
as she could. glanced back at the monster.

More about the object: Tells more about the pier:

She ran down the pier, The pier was wet and
a structure made of slippery. It wobbled
posts and planks. under her feet.

The telling-more story strategy is also used in making an outline. A major heading provides the summary.
Points that are included under the heading tell more.

With this range of examples, we are able to sequence relatively easy examples (sentences) and harder ones
(paragraphs). We mix the task of writing sentences and writing simple paragraphs to induce the notion that
there is a sameness in the skill of telling more.

An actual writing program would be concerned with much more information than the central thrust of
telling more. Four basic skill tracks would be coordinated: sentences, paragraphs, mechanical details, and
editing. All new skills would be integrated into the editing track and ultimately into the paragraph track. The
editing track serves as a rehearsal track for the paragraph track. We expect learners to check the passages they
write for punctuation, sentence types, tense, and possibly other details. By presenting passages that need
editing, we provide easier examples that require the same basic behavior called for in writing. The editing is
actually a removed-component of the total paragraph-writing task. When writing paragraphs, the learners must
perform many other operations in addition to editing; however, if the learners are well practiced in reading
over material and scanning for errors, the probability is greatly increased that learners will be able to edit their
own writing.

One test of efficiency is the extent to which we can teach other writing skills as simple transformations of
skills that we teach earlier. If a transformation is involved, the later skills are subtypes of the skills taught
earlier. The extent to which we must engage in either teaching that contradicts what we taught earlier or
teaching that is new is the extent to which the earlier teaching was not efficient.

When we apply the transformation test to the skill of telling-more-through-subject-predicate sentences, we

can generate a large number of sentence forms through simple transformations of the subject-predicate type.

 
 By moving parts of the regular-order sentence, we can create sentences that do not start out by naming the
actor. Instead of writing, “They sat on the porch after they ate,” the learner later writes, “After they ate, they
sat on the porch.” Instead of writing, “The monster came out of the water, grabbing a post,” the learner later
writes, “Grabbing a post, the monster came out of the water.” A parallel transformation occurs for the
paragraph writing. Regular-order paragraphs start with the “topic sentence” and then tell more. To create a
simple transformation the learner deletes the first sentence, creating a description of a topic without first
identifying it.

Also, groups of paragraphs are created by instructing the learner to frame the topic as a problem or a
solution rather than a report of what somebody did. For example, the learner reads “Jane and Bill went on a
vacation to a place that was far away from any town. They had planned to reach their campsite by three
o’clock in the afternoon. But at 3 o’clock in the afternoon, they were not at their campsite because of a
problem. The picture shows that problem.” The illustration shows the car with a flat tire. Jane and Bill have
emptied the trunk. Bill is holding a jack handle, but his expression suggests that there is no jack.

The learner follows the same procedure used with simple paragraphs, except that the first paragraph begins
with a good sentence that tells about the problem that Bill and Jane encountered. (Their car had a flat and they
did not have a jack.) The rest of the first paragraph tells more about the problem. The second paragraph tells
how they solved the problem. (The picture contains details that suggest a solution—such as several long, thick
branches next to the road and a stump next to the car.) Note that this context permits the learner to interpret
(telling why the people did what they did to solve the problem). However, the context is safe. Inferences are
based on specific details of the picture and the scenario that frames the picture.

A final transformation occurs with the instructions of fantasy-writing assignments. Once we have reason to
believe that the learner has the ability to describe things that we can observe, we encourage the learner to apply
this skill to describing other things that we cannot observe. Fantasy writing is an extension of interpreting
pictured events. The difference is that the “pictures” are no longer available for our inspection.

The most difficult part of the program is not to teach the various skills, but to organize them so that: (1)
the learner uses what has been taught; and (2) the subsequent skills clearly build on what is taught.

Main idea. In addition to teaching writing skills, the program we have been describing teaches critical
skills for outlining and main idea. The teaching of main idea develops very naturally in the writing context.
The learner makes up a simple sentence about the picture. If the sentence succinctly expresses the “main
action” of the picture, the sentence expresses the main idea. The learner next tells more. The sentences that
serve this function are supporting sentences for the main idea. It is possible to teach main idea in different
ways—as a choice-response discrimination rather than a production-response skill. For such teaching, we
present the learner with a passage and with choices of sentences that express the main idea. The presentation is

 
fraught with potential communication problems and misrules. The major communication problem is that we
have trouble juxtaposing examples appropriately. The reason is that the examples are long. The learner must
read a passage before making the choices. The major misrule problem is that if we presented repeated
paragraphs of the initial type developed in the writing program, the main idea would be expressed in the first
sentence of every passage. The naive learner, therefore, might develop the spurious strategy of assuming that
the first sentence of a paragraph is the main idea. Only later, and painfully, would the learner discover that the
matter is more complicated. Teaching main idea in the writing context does not imply the misrule because the
learner designs the first sentence so that it meets certain specifications.

The analysis of sameness of applications of main idea discloses the problem with the traditional system
for teaching main idea as a reading-comprehension skill. Virtually all real applications of main idea require
the learner to make up the main idea or the summary, not simply respond to choices. In other words, it is a
language skill or a writing skill, not a reading skill. The simplest way to avoid distortion and stipulation is to
teach main idea as a production skill.

Certainly we can devise programs that teach main idea as a reading-comprehension skill. A set of
passages would be presented. All passages would: (1) deal with the same topic; (2) have many of the same
sentences; and (3) begin with the same sentence. The learner could not learn the misrule that the first sentence
expresses the main idea, because the first sentence never expresses the main idea. Some parts of the passages
change, and the main idea changes. Therefore, the details that change must account for the change in the main
idea.

Once the learner has mastered various sets of examples, we introduce passages in which the first sentence
expresses the main idea. (These would be integrated with the type in which the main idea is not the first
sentence.) This approach has a potential of working far better than that of barraging the student with passages
and hoping that the learner develops the “concept” of main idea. Beyond raising performance on achievement
tests, however, the approach has little justification. Main idea teaching may be used in reading; however, it is
most naturally taught through writing. The use of main idea is not only functional in the writing context—it is
essential.

Illustration: Analogies

Teaching analogies is far less complicated than teaching writing; however, the skill is elusive because
analogical reasoning occurs in many forms, and even the direct expression of analogies shows great variation.

All analogies can be reduced to a form that involves sentences.

A dog runs
as a fish swims.

 
That analogy transforms into:

A dog is to running
as a fish is to swimming.

We can introduce variations that involve different types of things and different classifications. For instance, we
can create analogies that name the class that the things are in:

A dog is to mammals
as a parrot is to birds.

Other analogies deal with location:

A dog is to the ground

as a fish is to the water.

Other analogies deal with analogous or homologous parts:

A dog is to lungs
as a fish is to gills.

Still other analogies deal with function:

A dog is to guarding things

as a horse is to carrying things.

All these analogies follow the same form. The names that appear at the beginning of the analogy are coordinate
members of the same class. For example, dog and fish are coordinate members of the class of animals.

The relationship between the name that appears first in the analogy and that which follows are correlated
in the same way. How are dog and running correlated? Running is an action that dogs perform when moving
from place to place. How are fish and swimming correlated? Swimming is an action that fish perform when
moving from place to place.

If we apply these facts, we see that many other types of analogy forms are simply analogies with unstated
parts. For example, “His secretary was a little bit of sunshine on a cloudy day.” The fully-stated analogy could
be expressed as: “His secretary is to a dismal environment as a little bit of sunshine is to a cloudy day.” Both
parts of the analogy begin with pleasant things. The parallel correlation is that both pleasant things exist in a
gloomy setting.

Here is a possible routine for teaching the basic analogies.

Example: A fish swims as a dog does something.

 
 Teacher Learner
This analogy is about a fish and a
dog. What class are a fish and a
dog in? Animals.
This analogy tells how the animals
move. What does it tell? How the animals move.
Name the first animal. A fish.
Tell how it moves. Swims.
Say the statement about a fish. A fish swims.
Now name the other animal. A dog.
What is the analogy going to tell
about that animal? How it moves.
How does a dog move? Runs. (Walks.)
Once more: What does the analogy
tell about the animals? How they move.
Say the whole analogy. First tell
about a fish, then tell about a
dog. Get ready. A fish swims, a dog runs.

The routine reflects the sameness revealed by analysis of the examples. The learner first indicates a class
for the fish and dog. The learner then follows the rule of telling how the animals move. The same routine is
used for other analogies:

Teacher Learner
This analogy is about a fish and a
dog. What class are a fish and a
dog in? Animals.
This analogy tells what the animals What the animals breathe in.
breathe in. What does it tell?
Name the first animal. A fish.
Tell what it breathes in. Water.
Say the statement about a fish. A fish breathes in water.
Now name the other animal. A dog.
What is the analogy going to tell What it breathes in.
about that animal?
What does a dog breathe in? Air.
Say a statement about a dog. A dog breathes in air.
Once more: What does the analogy
tell about the animals? What they breathe in.
Say the whole analogy. First tell
about a fish. Then tell about a A fish breathes in water;
dog. Get ready. a dog breathes in air.

A variation of the analogy would require the learner to make up analogies that tell how a bird, a fish, a
kangaroo, and a lion move. The learner could make up analogies that tell the material used to make a coin, a

 
bat, a belt, and a window. Also, the learner identifies the rule for various analogies.

Teacher Learner
Listen to these analogies. They
are going to tell different
things about vehicles.
Listen. A boat is to a boat dock
as a train is to a train station.
What does that analogy tell
about the vehicles? Where you get on them.
Listen. A boat is to water as a
train is to tracks as a bus is
to a street. What does that
analogy tell about the
vehicles? What they move on.
Listen. A boat is to sails as a
train is to a steam engine as
a bus is to a diesel engine.
What does that analogy tell
about the vehicles? What makes them move.
Etc.

Note that these examples are designed according to the rules for a single-transformation sequence.
Minimally-different examples show the learner how the rule changes as a function of the changes in the
analogy. After basic routines have been introduced, we present variations in which more divergent responses
are possible.

Teacher Learner
Here’s a strange analogy.
“Moving fast is to
something as moving
slow is to something
else.”
Name something that
moves fast. (Bullet, deer, etc.)
Say the first part of the
analogy. Moving fast is to a bullet.
Name something that
moves slowly. (Snail, turtle.)
Say the next part of the
analogy. Moving slow is to a turtle.
Say the whole analogy. Moving fast is to a bullet
as moving slowly is to
a turtle.

 
 After introducing variations of this type, the learner operates on applications that require identification of
how the things in an analogy are the same.

Teacher Learner
Listen. The word like tells you
that things are the same. The
hero moved like a bullet.
How could the hero and
the bullet be the same? They both move fast.
Her eyes were like the sky. How
could her eyes and the sky
be the same. They’re both blue.
The air inside the jungle was like
a sponge. How could the air
and the sponge be the same? They’re both damp.
The salesman was like a shark
that smelled blood. How could
that salesman and a shark be
the same? They’re both mean.

Cursive Writing

The next expanded program that we will overview deals with the teaching of cursive writing. Obviously,
writing in cursive is a motor skill, not a cognitive one. Therefore, the effective teaching of the skill must
involve practice, shaping, and techniques appropriate for teaching motor skills. Far less obvious, perhaps, is
that cursive writing may be presented largely as a transformation of skills that are known to the learner. To
achieve efficient transfer, therefore, the program should prompt the transformation.

To discover that a transformation is involved, we analyze examples of cursive letters and their manuscript
counterparts. This analysis reveals the following: The middle part of many cursive letters is the same as the
middle part of corresponding manuscript letters. By adding a stroke from the baseline and adding a tail, we
change the manuscript letters (see Figure 25.2).

 
 Figure 25.2

a a
c c
e e
g g
j j
m m
n n
o o
p p
t t
x x
y y
Another transformation involves the slant. By starting with manuscript letters that are straight up and
down and rotating them in the clockwise direction, we create manuscript letters that slant the same way that
cursive letters slant.

Original Set Transformed Set

a a
j j
We can achieve this transformation in slant most easily not by rotating the letters, but by rotating the paper on
which the letters are to be written. If we rotate the paper slightly in the counterclockwise direction and require
the learner to write the same up-and-down letters that are in the familiar manuscript set, the learner will make
letters that slant in the appropriate way:

 
 a j
Note that this transformation occurs whether the learner is right-handed or left-handed. We make this point
because a number of traditional cursive-writing programs require right-handed students to rotate their paper in
a counterclockwise direction (the correct way) and left-handed students to rotate it in a clockwise direction.
This incredible requirement is based on a complete misanalysis of writing skills. Left-handed students who
rotate the paper in a clockwise orientation must learn to write sideways to achieve the appropriate slant.

aj
The analysis on which this convention is based apparently does not take into account the idea that cursive
letters are simple transformations of manuscript letters and that rotating the paper is supposed to facilitate the
transformation. Instead, the analysis seems to be based on the idea that left is the opposite of right; right-
handers rotate in a clockwise direction; therefore, lefties should rotate in the opposite direction.

Stroke descriptions. Many traditional approaches to cursive also make serious mistakes with respect to
stroke descriptions. The purpose of stroke descriptions is merely to establish a basis for communicating with
the learner, not to “teach” the stroke. When the learner makes a mistake, communication is made much easier
if the teacher is able to say something like, “You didn’t go all the way up to the top line.” The stroke
description should be simple, not elaborate. Figure 25.3 shows the stroke descriptions for the cursive letters in
the E-B Press Cursive Writing Program. The letters appear in their order of introduction.

 
 Figure 25.3
Letter Strokes Lesson Letter Strokes Lesson
Up to the half line. 1 Start like an l. 34
i (i form) Down to the baseline and tail. h Trace up with a hump form.

i Make an i form with a dot.

Make an i form.
1
1 g Start like a c.
Close and finish like a j.
37

u Join it with another i form.

m Start like an n. Trace up
with another hump form.
40
Make an i form that touches 2
t the top line. Cross it.
Bend over at the half line. 4 sb Start like an i form.
Close and tail.
42

c Make a printed c.

Make a u, tail up to 8
Make an l. Shelf. 48
Start like a v. 50
w the half line. Shelf.
y Finish like a j.

(Hump Bend over at the half line and 11 Start like a c. 53

v form) tail.
Make a hump form, tail up to 12
d Close with an i form that
touches the top line.

v the half line. Shelf.

Start like an h. Close like 56

e Start like an i form,

come back, and tail.
14 k a printed r and tail.

Start like an a. 59

n Start like a hump form and

stop at the baseline.
16 g Go down to the subline.
Finish like an f.
Trace up with a hump form.
Make an n that sinks below 62
o Close a c at the half line. Shelf. 18 z the baseline. Finish like a j.

Make a hump form. 65

Start like an i form. Go down to 20
j the subline. Come around like a
printed j, and tail.
x Cross it going down.

Start like a j. 68
Make an e that touches 22 p Close and tail.
l the top line.
Start like a c. 24
For letters that pose possible problems, such as the
a Close with an i form.
Start like an l. Go down 28
letter f, the stroke description makes reference to a
known behavior—making an o. That behavior assures
that the student will finish like an o, which means
f to the subline. Finish like an o
at the baseline.
curve upward in a counterclockwise direction, not in a
j
clockwise direction (like the letter ) as students often do
Start like an i form. 32 when trying to make . f
r Shelf and tail.

The basic cursive forms referred to in the descriptions above are the i-form, the e-form, the c-form, the
hump-form, and the j-form. The letters finish in either a “tail” or a “shelf.” By referring to forms and letters
that have been taught, stroke descriptions for later letters are uncomplicated.

Sequencing cursive skills. Some forms are relatively easy for the learner to make—particularly i and e.
Working on these gives the learner a sense of success; however, if we work too long on these, the presentation
may induce distorted responses. The learner will make good i’s and e’s; however, the learner will have
difficulty making forms that are based on c, the hump-form, or possibly j. The most serious problem will

 
probably occur with c because c requires a response more complicated than the others. To avoid serious
distortion, we try to introduce c early. We also try to introduce the hump-form fairly early. The strategy for
introducing letters and integrating them involves the following steps:

1. Introduce the i-form with variations. Cursive u shows the repeated-stroke nature of cursive.
Cursive t shows the height variation.

2. Introduce c-form early. Integrate this form with the i-form. Present exercises in which the learner

writes joined letters, such as:

3. Introduce the shelf variation early and integrate. Give practice with joins such as:

4. Introduce hump-form and integrate. When this form is combined with the shelf, cursive is
formed.

5. Introduce the two other forms (e and j) as quickly as possible and integrate.

6. Intersperse letters of each type so that the order of introduction presents juxtaposed letters that are
not highly similar.

Once the learner has been introduced to at least one variation of a particular type, the probability of
distortion has been reduced. When c and the i-form letters are juxtaposed in practice, distortion is counteracted
because the learner must develop those responses that are common to writing both c and the i-form letters.

Joined letters. The final economy in the program comes from the introduction of joined letters. By
practicing particular joined letters, the learner is actually practicing a simple transformation of another cursive

letter. For instance, if the learner practices writing the , learner is actually practicing a transformation of

another letter. By bringing the parts together and dropping the dot from the i, we make the letter . The
strategy is to work on the joined letters ci and cu before introducing the a. The work with the joined letters
teaches the combination of behaviors called for by the letter. The introduction of a should go quite smoothly if
the preteaching has been effective. Figure 25.4 shows other joins that set the stage for later letters.

 
 Figure 25.4
Joins Set the Stage For

ci a
ii u
ct d
cj g
lr h
vi r
rj y
nj z
In summary, the teaching of cursive involves four major transformations.

1. The form transformation from manuscript letters to cursive letters.

2. The slant transformation from manuscript to cursive.

3. Transformations achieved by introducing early forms that can be modified to form more
complicated letters.

4. The joined letter to single-letter transformations.

The program is complicated only because it must proceed in two directions at the same time. If we initially
introduced all members of a given type (such as all i-form letters), we could make the practice relatively easy;
however, we would probably introduce serious distortion that results from protracted practice with a small
range of example variation. Therefore, we must trade-off the easy examples (the i-forms) with the harder
examples, particularly those that present generically new writing behaviors (the c-forms). By introducing one

 
c-form early, we buttress against distortion. At the same time, we continue to introduce the relatively easy
examples. The strategy is similar to that of the A-Z integration approach.

The introduction of examples is modeled after that of the screwdriver program or the bike-riding program.
Different practice blocks work on different skills, so that during a lesson the learner works on letters in
isolation, joined combinations, sentences, reading cursive sentences and possibly some tracing or activities that
require answering questions in cursive. Each activity is continuously integrated with the newly-taught skills.
The program will not eliminate the need for practice, or even reduce practice by a half, over a program not
ordered as carefully. However, the program will result in savings for both teacher and student.

Problem Solving

The theory of instruction that we have developed frames the psychology of learning within the domain of a
logical analysis. We have applied the theory to a wide range of skills, from the teaching of simple
discriminations to elaborate programs. The final illustration is the induction or “problem-solving” skills.

According to stimulus-locus principles, effective problem-solving programs derive from an analysis of the
problems to be solved, not from an analysis of the learner. A particular problem-solving strategy will hold only
for problems of a particular type. The problems that admit to the same strategy have a common quality, which
is their problem-solving features. By analyzing problems and grouping them according to common features,
we gain an understanding of these common features. This analysis is the same analysis of subtypes that we
have used for analyzing other cognitive skills.

Let us say that we want to teach the learner how to handle problems that involve identifying which of the
possible causes for failure in the system accounts for an observed failure. All problems that are in this group
have this common feature: outcomes could be the result of more than one possible cause. For instance, the sink
is plugged. The cause could be something in the drain, in the system of pipes that the drain feeds into, etc.
Another example is: Your car will not start. This outcome could result from failure of the distributor, the gas
line, or one of many other parts. Medical diagnoses thrive on examples of this problem-solving type. The
patient exhibits some symptoms. The observed symptoms could be caused by many possible malfunctions.

To apply the failure-diagnosis strategy to an automobile problem, the learner should understand that three
broad systems may account for the failure: The fuel system, the electrical system, and the mechanical system.
The learner should also understand the parts of the broad systems, such as the basic components of the
electrical system (battery, coil, distributor, wires, spark plugs, etc.). This information would be effectively
taught to the learner through visual-spatial displays that show the higher-order nature of the larger systems and
how the smaller systems make up the larger ones. (The electrical system would be a higher-order label, under
which would be the various components of the system.)

 
 To apply the problem-solving strategy to the diagnosis of diseases related to the circulatory system, the
learner should understand that there are three types of major sources of failure—blocks in the pipes, leaks in
the pipes, or malfunction of the pump (heart). The learner should also understand subtypes of each larger
class—for instance, the types of pump malfunction. As the language implies, the same diagnostic procedures
used for heart systems may be used for other hydraulic systems.

Understanding the major categories is important, because the strategy involves ruling out large categories
through binary tests or questions. One test or question (possibly two) should permit us to rule out whether the
problem with the car is an electrical failure. Ideally, one or two tests of the patient should rule out whether the
symptoms are caused by a failure of the pump (heart). With a very few tests, therefore, the learner would know
which major system was involved in either a heart problem or an automobile failure.

After determining the larger system that caused the failure, the learner rules out the various lower-order
possibilities until the lower-order system that caused the failure is identified. The same steps that were
conducted with the higher-order systems is now repeated with the lower-order systems. For example, if the
electrical system of a car is involved with the failure, the learner must now determine whether the problem was
caused by the battery, the coil, the distributor, etc. Ideally, we should be able to conduct one or two tests that
rule out each possibility.

At each point, we present a test that is binary. If the system passes the test, we know that the problem is
not caused by the component tested. For instance, our test of the battery determines either that the battery is the
cause of the problems or the battery is not the cause of the problem.

With an understanding of this procedure, we can design a general cognitive routine. Here is the first part of
a possible routine.

 
 Teacher wording Student response
Your car does not start.
What are the main systems
that might make your car
not start? Electrical, fuel, mechanical.
Which would you test first? Fuel.
How? First check the gas gauge.
Then check the gas at the
carburetor. Then check it at
the cylinder by removing the
spark plug.

What if the fuel seems okay? Test the electrical.

How? Check the spark at the spark plug.
What if there is no spark? Check the coil.
Etc.

Note that the learner has been pretaught all system information needed for the diagnosis.

A variation of the same routine would be applied to different systems, such as the circulatory system. Here
is the first part of a possible application.

Teacher wording Student response

A person suddenly becomes Blocked vessels, leak in the
pale. Pulse goes up. Person system, heart malfunction.
is dizzy. What are the main
cardio-vascular problems
that could cause these
symptoms?
Which would you test first? Leak in the system.
How? Blood pressure test.
What if the blood pressure
was near normal? Test the heart.
Etc.

This routine is the same one used for the car diagnosis. The only difference is the particular system that is
dealt with. The range of applications, therefore, is limited only by the learner’s understanding of different
systems.

A variation of the routine may be used in a highly technical field. The variation is not generically different

 
from the one above, simply more complicated because of the interaction of “systems” that can cause failure. If
we are dealing with medical diagnosis, for example, we might consider each disease a system that causes
failure. The symptoms are the outward manifestations of the diseases; however, symptoms are not uniquely
affined to specific diseases. A dozen diseases share many of the same symptoms (high fever, fast heart rate,
flushed appearance, dizzy, etc.). To deal with this interaction of symptoms and diseases, we present the various
symptoms to the learner and begin with this question: “What are the main diseases that could cause these
problems?” The learner must list all the diseases that have the symptoms. The learner then specifies tests for
ruling out the different diseases. A good test may rule out many of the candidates. The learner continues until
all but one of the diseases have been ruled out.

• The learner is presented with symptoms: 1, 2, 3, 4, 5.

• The learner would name the diseases that have this set of symptoms: diseases A, B, C and D.

• “How would you rule out disease B?”

• The learner indicates the test.

• The learner then receives information about the outcome.

• “Which disease would you rule out next?”

• The learner selects a disease, indicates the test, and then receives the results of the test.

The major differences between this routine and that for the simpler examples is that: (1) the same set of
“causes” for failure are not involved in all examples; (2) more than one “symptom” is involved in the
diagnosis.

With the simpler diagnosis, the first step involves the identification of the same three causes of failure—
leak in pipes, block in pipes, pump. The more sophisticated applications do not always involve the same causes
of failures and therefore require more knowledge. Symptom 1 may suggest that the “cause” is either disease A
or C. Symptom 2 may suggest a completely different set of diseases—A or B, for instance. The learner who
performs the diagnosis must know which diseases are suggested by which individual symptom or set of
symptoms.

The routine used for sophisticated examples implies preteaching quite different from that which is
traditionally provided. Instead of teaching various diseases with a list of symptoms for each, we would teach
different symptoms and the diseases that each symptom occurs in. (The learner would not memorize the
symptoms of chicken pox. The learner would learn the various diseases that have a specific symptom.) The
reason for treating symptoms in this way is that the learner will never encounter the disease, merely the
symptom. The categorization that lists symptoms for each disease, however, implies that all symptoms of the

 
disease will be observed. Only some symptoms are typically observed. The learner who had been taught the
implied relationship: If symptoms 1, 2, 3, 4, and 5 are observed, the disease is X, may be misled. A more
reasonable approach is to identify the set of diseases associated with a particular symptom or set of symptoms.
These suggest possible causes. By testing them, we can identify the single cause.

The point of this illustration is that even operations such as diagnosis can be approached following the
general guides for organizing expanded programs for conveying cognitive skills. Mainline examples of
diagnosis have a common set of features. If we express these features, we establish a criterion for classifying
other possible examples. Once we have grouped the examples, we can identify the preskills that are needed for
the learner to engage in the operation. We can express the operation as a routine. We can order the examples or
applications so they induce samenesses or a generalized understanding of how the same problem-solving steps
apply to a wide range of examples. Finally, we introduce variations (such as the medical diagnosis that
involves symptoms of different possible diseases) as simple transformations of the basic type.

Subtypes and Transformations

The programs that we have outlined illustrate the relationship between the subtypes for a given skill and
the organization of the expanded program. If we begin with the analysis of sameness, we find the features that
are common to the mainline example. We next try to find the simplest examples that have the set of features.
These are the most ideal for initial instruction. We then view the other subtypes as simple transformations,
which means that they share many features with the original set of examples, but not all features. If we view
these subtypes as transformations, we are provided with guidelines about how to introduce them. We teach a
type of example. By changing the various members of the set in the same way, we can create a set of
transformed members. The double-transformation program provides a model of how to communicate this
transformation. Therefore, the introduction of the later subtypes may be modeled after the double-
transformation strategy.

Summary

Expanded programs are the cognitive counterpart of expanded chains. At the center of the expanded
program is a cognitive skill. This feature distinguishes the expanded program from the expanded chain.

Expanded cognitive programs are created to teach “ideas,” or subjects such as the main idea of a passage,
the idea of analogical reasoning, or the idea of diagnosing failures in a system by referring to symptoms. All
skills, vocabulary, and specific behaviors required to teach idea or subject are part of the expanded program.
The most difficult part of the program, however, is the means by which the central idea is developed. The idea
must be analyzed, refined, and adjusted. The set of examples associated with the idea must also be identified,
refined, and adjusted.

 
 The procedure that we outlined for developing ideas and examples is to follow these steps:

1. Organize the content according to response dimensions (not according to traditional

classifications of “topics” or content”).

2. Identify the range of examples to which the operation applies.

3. Identify the important subtypes of examples that are not processed by the central operation.

Illustrations of this three-step procedure showed how content is organized in a new way if the content is
ordered around a central operation. The central product-conservation problems included rectangular displays,
level problems, and some balance-beam problems. The central operation could not accommodate problems
involving averages; therefore, a related procedure was needed to solve these problems (and their balance-beam
counterparts).

Expressive writing illustrated that both sentences and paragraphs are generated from the same idea of first
naming something and then telling more. A number of related routines permitted the generation of related
writing operations. The illustration pointed out that the related writing skills could be generated as simple
transformations of earlier-taught skills. The expressive writing illustration also pointed out that teaching of
main idea and outlining skills is very naturally presented as a subtype of naming and then telling more.
Expressive writing, however, is not a graceful extension of reading (the “subject” in which main idea is
traditionally taught).

Basic analogical reasoning was presented as a series of related operations that require the learner to
express parallel relationships, identify the rule that indicates how the parallel parts of the analogy are the same,
then use the knowledge of sameness to create similes.

Cursive writing was presented primarily as transformations from manuscript writing. The transformations
included transformation by adding “tails,” transformation in slant, transformation from earlier-taught forms,
and transformations created by practicing joined-letter combinations.

Finally, a paradigm for both general and specific diagnostic skill showed that more complicated diagnostic
procedures are simple transformations of easier ones. The major difference between the procedures is that the
one uses more complicated situations and requires more elaborate steps to eliminate possibilities. The more
complicated procedure was illustrated with medical diagnosis; however, the same procedure could be applied
to any system in which given symptoms could be caused by different maladies.

Although the procedures for identifying operations and organizing examples of operations is complicated,
it guarantees that the initial teaching will be efficient and will present the central operation so that it is
applicable to a full range of examples. It also guarantees that related operations are clearly framed according to
sameness and differences in operational steps. This type of organization ensures a smooth development of

 
related skills.

 
 Section VIII

Diagnosis and Corrections

This section presents topics that are related to the implementation of programs. When the teacher
implements in the classroom, mistakes occur. The program that the teacher uses may have reasonable
instructional provisions for buttressing against mistakes, but still mistakes occur. How the teacher corrects
mistakes when the program provides for corrections is the subject of Chapter 27. Chapter 26 deals with the
situation that occurs when the program does not have adequate provisions for buttressing against mistakes
and for specifying corrections. In this case, mistakes must be classified and appropriate remedies must be
developed. A diagnostic-remedial strategy is used.

Chapter 28 presents a paradigm for using field-tryout information to shape the instructional program.
 

 
 Diagnosis and Remedies
When we deal with problems of learning, we use two different diagnostic strategies. The first is a
diagnosis of instruction the learner receives. The goal of this diagnosis is to predict future learning problems.
The second diagnosis is a diagnosis of the learner. This diagnosis identifies current problems the learner is
experiencing and corrects them.

Diagnosis of the Problem

This diagnosis assumes that if the instruction has flaws in it, some learners will respond to those flaws
and will mislearn. Conversely, if the instruction is corrected, the kind of mislearning that was possible with
the original instruction would no longer be possible. This type of diagnosis is used to predict future problems,
even when the learner is apparently performing well, which is often what happens when the learner performs
on the tasks presented in a poorly-designed sequence. In this case, the problem is not that the learner is
failing, but that the learner probably will fail when presented with tasks that are not yet in the program.
Program diagnosis is logical, based on the communications the learner receives. Predictions rest on the
assumption that any misrules or stipulations conveyed by the communication will be learned, and will affect
performance on subsequent activities. Predictions become increasingly valid when learners are more naive
and when the flaw is more obvious. Suppose that we observe the teacher presenting a series of “following
instructions” tasks to a language-delayed child. The child performs on all items; however, the items we
observe occur in exactly the same order each time the sequence of tasks is presented (“go to the table . . . pick
up the glass . . . take it to the counter . . .” etc.). The communication is flawed because the learner could
perform correctly either by attending to the features of the instruction (the appropriate processing response)
or by memorizing the sequence of events (an inappropriate strategy).

If we observe such a flaw, we can test it in one of two ways:

1. We can eliminate the relevant stimuli and show that the same response persists.
2. We can change the context or setting for the relevant stimuli and show that the response does not
occur.

For the first type of test, we could present the learner with a series of nonsense commands, presented in
the same context as the originals (“Go to the table…flam over the glosser . . . umble um orgona . . .”). The
prediction would be that the learner will perform with the same responses produced for the original chain of
instructions (responding to “flam over the glosser” with the response the learner produces for “pick up the
glass,” and so forth). For the second type of test, we could present parts of the chain in a different context.

 
 “Hold this dish . . . put it down here . . . good. Now go to the table . . . come on back . . . take this to the
counter . . .”

The prediction for the second test is that the learner would fail to perform appropriately. The predictions
are not based on knowledge of the learner; rather the diagnosis is based on the communication the learner has
received. Note that this type of diagnosis is extremely difficult to achieve from a strict behavioral model. The
learner is performing adequately on the series of tasks that is presented. From what the learner does, the
problem is not obvious. It becomes obvious only if we consider the logical flaws in the presentation. These
flaws suggest a way that the learner could perform as observed and yet not understand what the teacher
supposes the learner understands.

The test of whether the learner has learned the misrule involves introducing tasks that are not part of the
program. We can either change the relevant stimuli and predict the same response, or we can change the
irrelevant stimuli (context details) and predict no response. We can derive the same sort of program-related
information without introducing new tests. We simply note any problems the learner experiences later in the
program and then attempt to confirm that each is caused by the earlier communications provided by the
program. The disadvantage of this approach is that problems are not predicted; rather, they are discovered
after they have become problems. If the learner later fails to generalize, we assume that the program provided
inadequate instruction to assure the generalization. If the learner tends to behave as if a subset of examples
had been stipulated, we can assume that the program in fact caused the stipulation. After making these
assumptions, we check the program sequence to determine whether the program is actually responsible for
the observed mislearning.

In summary, the procedures for figuring out what is wrong with the program involves two approaches:

1. The program-flaw-prediction looks for flaws in the program and tests to see whether these are
transmitted to the learner.
2. The learner-performance approach looks for inadequate performance and then attempts to verify
the causes by referring to the program.

A model for applying the learner-performance approach appears in Chapter 28.

Diagnosis of the Learner

The primary goal of diagnosing the learner is to identify the learning problem and correct it. This
diagnosis should imply an instructional remedy. Therefore, it must be expressed in terms that translate into
instruction. Regardless of how strange the learner may seem, we cannot refer to history, culture, neurology,
chemistry, or other variables that are not directly associated with instruction. If the diagnosis is to imply a
remedy, the diagnosis must express the learner’s problems as a skill deficiency.

 
 1. It expresses exactly what it is the learner is supposed to do; and
2. It expresses exactly how the learner actually performs.

The difference between the two statements is the problem—a specification of what the learner must be
taught. This difference may imply an intervention that is modest or one that is relatively extensive. We can
teach any skill by following the appropriate procedures. Therefore, we can specify a remedy for any
difference that is identified.

Six Learner-Problem Types

When we examine the range of situations in which a learner diagnosis is called for, we identify six
learner problem types. For each problem there is a remedy. The remedies for three problems involve
application of the stimulus-locus analysis; remedies for two types derive from the response-locus analysis;
the final remedy is appropriate for situations in which there is a question about whether the learner is actually
trying to respond appropriately.

Nearly all the techniques used in these remedies are variations of techniques discussed earlier in different
sections. The present classification provides a new focus for these techniques because it groups them
according to the type of failure that the learner exhibits (the difference between what is expected of the
learner and what the learner actually does). The following are the six types of learner problems.

Stimulus-locus problems
1. The learner does not transfer from one situation to another.
2. The learner does not differentiate between two situations or applications.
3. The learner can perform from some instructions, but not from the instructions provided by the
task under consideration.
Response-locus problems
4. The learner can produce a particular response in some context, but cannot do it without prompting
when it is presented in the context of the activity under consideration.
5. The learner cannot produce the response that is called for.
Possible compliance problems
6. Possibly, the learner is purposely not complying with the task requirements.

The following is a summary of the remedy implied by each stimulus-locus problem and an illustration of
the development of the remedy.

Stimulus-Locus Problems

The learner does not transfer from one situation to another. This problem implies that the instruction has
not been effective in showing the learner how the two situations are the same. The remedy is to increase the

 
 apparent sameness possessed by the two situations. The technique for showing that events are the same is to
treat them in the same way.

Illustration. The learner has been taught spelling behavior, and the learner performs well in the spelling
lesson. When writing reports for social studies, however, the learner does not use the spelling skills. The
learner’s behavior implies that the learner has not been effectively shown that the two situations are the same
and that they call for the same behavior. To show how the situations are the same, the teacher must treat the
situations in the same way. By treating the social studies content in the same way that spelling content is
treated, the teacher shows that both call for the same behaviors.

For instance, as part of the spelling lesson, the teacher passes out copies of a social studies paper. Some
of the words studied in spelling are misspelled in the paper. “See if you can find each misspelled word, cross
it out, and write the correct word above the misspelled word.”

Just as the teacher brings social studies into the spelling lesson, the teacher could also require spelling
behavior in the social studies lesson. The teacher assigns a paper. “Use at least 10 words from your spelling
list when you write about Holland. Make sure that you spell all words from your list correctly.”

To prompt the spelling behavior further, the teacher could require the students to make a box at the top of
their assignment sheet and check the paper for spelling. After checking the paper, the students are to make a
check mark in the box. The addition of this prompt facilitates establishing the same behavior (checking
spelling) in different content areas. The content areas become the same with respect to spelling because the
learner is required to treat them the same way.

Perhaps more instructional confusion centers around transfer (or lack of it) than around any other topic. If
the learner does not transfer spelling, the incorrect conclusion is that the spelling program failed because it
didn’t provide for the transfer. Actually, if the spelling behavior has been established in one context, the
program has not failed to achieve its primary objective of communicating the skill. The failure is one of
expansion. The remedy is to show the range of application for the skill. If a failure occurs in the social studies
context, the failure is that the teacher has not required the same behaviors that are performed in spelling
periods. The learner is behaving in a perfectly reasonable and predictable manner. Two situations are
discriminably different, and the learner is responding to this difference. The failure is that of not showing that
the two situations are the same in an important way.

The learner does not differentiate between two situations or applications. When we identify the specific
situations that the learner “confuses,” we identify a discrimination the learner has not been taught. The
remedy is to teach the discrimination and then to give the learner further practice on this discrimination in the
context of the original situation.

Illustration. The learner does not correctly work all division problems. The only problems the learner

 
 works correctly are those that have one-digit answers. The learner does not complete problems with two-digit
answers. Instead, the learner writes the first digit in the answer, multiplies and subtracts, and then stops. Since
the learner does not respond differentially to the two different problem types, the learner demonstrates a lack
of knowledge about how these types are different. To teach difference, we juxtapose examples that are
similar and treat them differently. To teach the discrimination—“Is the problem completed?”—we use
examples where some problems are completed and some are not. A correlated-feature sequence could be used
to convey the “rule” for determining whether an example is completed. Here is the first part of a sequence:

Examples Wording
Listen. If a number is written over the last digit
in the problem, the problem is completed.

7 What is the last digit in this problem? Is there

37 2950 a number over it? My turn: Is this problem
259 completed? No. How do I know? Because there
36 is no number over the last digit in the problem.
Your turn. Is this problem completed? How do
you know?

79 My turn. Is this problem completed? Yes. How

37 2950 do I know? Because there is a number over
259 the last digit in the problem. Your turn. Is this
360 problem completed? How do you know?
333
27
5 2950 Your turn. Is this problem completed?
25 How do you know?
45
45
0 Etc.

Note that the sequence is a correlated-feature sequence with some additional questions to assure that the
learner is responding to the appropriate numerals. The complete remedy must go beyond teaching the learner
to identify whether problems are done. It must teach the steps in working problems, which implies a cognitive
routine. The discrimination is contingently linked to a cognitive routine. If the judgment is that the problem is
not completed, the learner takes the conditional steps and completes the problem. The learner does not take
the steps if the problem is completed.

When the routine is taught and covertized, we may discover that the learner still does not perform. The
problem would not be a simple discrimination problem at this point, but one that implies a response-locus
remedy. When we deal with the original instruction problem, however, we respond first to the most obvious
problem, which is that the learner does not apparently know how to distinguish between problems that are

 
 complicated and those that are not. The immediate remedy is to teach this discrimination, then to attach a
conditional routine to it. We first try to make the communication faultless before we deal with possible
response-locus problems because we cannot be certain about classifying the problem as a response-locus
problem until we have ruled out the possibility that the communication is faulty.

All differentiation problems take the same form and imply the same remedial strategy. We identify a
detail that the learner is supposed to respond to. Teaching the discrimination involves applying one of the
familiar forms. The choice of sequences depends on the nature of the discrimination.

The learner can perform from some instructions, but not from those provided by the task under
consideration. The statement of the problem indicates that there is a single difference between the task that
the learner can perform and the task that is required. We can, in other words, transform the task the learner
can perform into the other task by changing only the instructions. Therefore, the remedy must show the
learner the relationship between the familiar and the new instructions.

The statement of the problem suggests three possible forms that we have studied:

1. The equivalent-pairing covertization technique.

2. The correlated-feature sequence.

3. The double-transformation program.

If we are dealing with a single item or if that item is embedded in a chain, the equivalent-pairing
technique is appropriate. The familiar and new instructions are paired first in the A-B order, then in the B-A
order. Finally, the familiar instructions (A) are dropped.

If we are dealing with generalized applications, the correlated-features sequence or the double-
transformation program may be adapted to the specific problem. Both provide means of showing the learner
the “rule” for using the new instructions.

If the learner does not understand the instructions, “Express the numeral as an expanded notation,” and
similar references to expanded notation, but is reasonably proficient at addition, we could present the
following correlated-features relationship:

My turn. What’s the expanded notation for 35? 30 plus 5. How do I know? Because 30 plus 5 equals
35.

My turn again. What’s the expanded notation for 71? 70 plus 1. How do I know? Because 70 plus 1
equals 71.

Your turn. What’s the expanded notation for 72? How do you know?

 
 Etc.

Various other correlated-features wording could be devised. Also, a short variation of the double
transformation program could be introduced. We would alter the program so that the familiar set was quite
small. Also, we would not integrate the sets. We would simply present a small familiar set, minimum
difference to the new set, and possibly generalization examples for the new set.

Below is a possible sequence:

Example Wording

35 Say the addition fact for this number.

(30 plus 5)
182 Say the addition fact for this number.
56 Say the addition fact for this number.
56 My turn to say the expanded notation for
this number: 50 plus 6. Your turn. Say the
expanded notation

182 Say the expanded notation.

35 Say the expanded notation.
279 Say the expanded notation.

In summary, all problems that require the teaching of a new “label” or signal for a familiar discrimination
are remedied by showing the relationship between the familiar and the new. Because the learner is not
required to learn the discrimination from scratch—but merely to learn the new label—the appropriate
techniques are those that establish the precise relationship between the familiar and the new.

Response-Locus Problems

All response-locus problems assume that there is a persistence in the learner’s behavior despite attempts
to change it. These problems may be linked with communication problems. For instance, the communication
may have seriously stipulated a behavior and the learner may have practiced this behavior until it became
habitual (highly resistant to extinction). The fact that a poor communication caused the problem is irrelevant.
We must remedy the problem. Merely showing the learner what to do does not work. The learner repeatedly
lapses into a behavior that had been stipulated, implying a response-locus remedy. Below is a summary of the
two response-locus problems and illustrations of the remedies implied by each.

The learner can produce a particular response in some context, but cannot do it without prompting when
it is presented in the context of the activity under consideration. This problem comes about in two different
ways. In some cases, the instruction the learner had received stipulated a particular behavior for particular

 
 situations. When the learner is required to produce a new response in a situation that had been stipulated, the
stipulated response persists. For example, if the learner speaks Spanish in particular situations, replacing the
speech with English involves replacing highly-stipulated behaviors.

For the other type of context problem, the instruction the learner had received is not responsible for the
problem. The learner simply cannot perform the response in the targeted context. The learner may “forget”
the response that is called for or may produce the wrong responses, even though the learner can produce the
response in other contexts.

The two types of context problems imply different responses to the program used to teach the learner. If
we identify that the program induces serious stipulation, we would try to change the program for future
students. For the type of context problem that is apparently a learner inability, no adjustment of the program
for future students is implied. Regardless of the program the learner received, the learner’s behavior shows
whether serious stipulation has occurred and whether the learner has been reinforced for producing the
stipulated responses. Merely showing the learner how to produce the new response in the different situations
does not change the learner’s behavior. The response is persistent. The instructional problem is one of
replacing an established response pattern with one that is incompatible. The introduction must provide not
merely for the establishment of the new response, but also for the extinction of the habitual response.

Note, however, the immediate remedy is the same for both types of context problems. The learner must
be taught to perform within the context of the targeted activity. The paradigm that is followed is that of
teaching the learner when to perform (by using the 3-level series.) Additional prompts may be useful for
pointing out to the learner that the behavior called for is the same as that used in other situations.

Illustrations. The first illustration deals with the transfer problem that results when established behaviors
must be replaced. The learner has been taught to speak English in school. Despite suggestions and
demonstrations, the learner continues to speak Spanish on the playground and in the corridors and lunchroom.
The teaching would be adequate to induce generalization if no behavior had already been established in these
situations. The teaching problem, however, involves replacing a set of behaviors with an incompatible set. As
soon as the learner becomes involved in an activity, the habitual behavior emerges.

To remedy this problem, we structure some playground activities in a way that permits us to present trials
rapidly and to monitor the learner. Perhaps we make up a list of the various things that we want the learner to
say. “Throw me the ball . . . Cover second base . . . Watch out . . . He’s going to pop up . . .” and so forth.
These are first practiced as symbolic exercises. The teacher describes the situations for which the various
utterances are appropriate: “What are you going to say when you want the ball? . . . What are you going to
say to make the batter miss the ball?” This context is something like a level 3 of the 3-level strategy. The
learner produces various utterances. Each occurs intermittently, separated by other utterances. The context,

 
 however, is quite a bit simpler than the one required on the playground.

Immediately before the learner goes onto the playground, the teacher reviews the various utterances and
reminds the learner to use these expressions during the game. The teacher then monitors the playground
situation and surreptitiously reinforces the learner for using the appropriate expressions. The teacher
continues to monitor the playground situation and to expand the range of expressions required on the
playground. Before a particular expression is used on the playground, however, it is first rehearsed in the
rehearsal context.

Note that the remedy involves increasing the number of ways that the playground situation and the
classroom situation are the same. The statements that are used in each are the same. A variation of the events
that evoke the responses are presented in both situations. The teacher describes those events during the
rehearsal sessions (“What are you going to say when . . . ?”). The same events will occur in the playground;
the teacher is in both situations.

In addition to the sameness, the instruction attempts to make the responses as fluent and easy as possible
by rehearsing them in a 3-level strategy until they are quite firm. (If they are weak, characterized by great
effort and latency, they will be very difficult for the learner to produce appropriately on the playground.)

In summary, the strategy involves four steps:

1. First firm the learner on the various responses in the 3-level strategy.

2. Frame the practice as rehearsal for the events that will happen in the application situation.

3. Monitor the learner in the application situation and provide reinforcement for using the targeted
behaviors.

4. Expand the range of application and reduce the degree of monitoring as the learner becomes
increasingly proficient.

This formula would apply to any situation in which the learner is to replace stipulated behaviors with
new ones. For instance, the corrective reader would be “rehearsed” in a practice situation that permits
controlled juxtaposition of words. This context is the list of unrelated words that does not prompt the learner
to guess on the basis of the syntax or story context in which the word occurs. It functions like level 3 of the 3-
level strategy. Following successful performance on this level, the learner engages in the reading of
connected sentences. This situation is analogous to a game. The learner will probably revert to guessing as
soon as the context becomes interesting, or as soon as the pattern of the sentences suggest what the next
words will be. The learner’s performance in this situation must be monitored carefully. The learner must be
reinforced for accurate reading, but must not be expected to change immediately. Change will occur as a slow
replacement of stipulated guessing behaviors with the new decoding behaviors.

 
 The second type of situation in which the learner cannot perform in a particular context does not involve
replacing stipulated behaviors with new behaviors. It involves making the learner facile in responses that are
difficult for the learner. The learner can produce the response in a simplified context, but not within the
context of the new activity.

For example, the learner gets on the bus, looks around, and apparently does not remember to put the coin
in the coin box. The response can easily be prompted by showing the learner how to respond; however, the
learner’s behavior does not change after repeated trials. The learner apparently forgets what comes next in the
expected chain of responses.

We cannot present repeated trials of this chain while the learner is on the bus. Also, the chain is too long.
If we repeated the entire chain, the behavior of putting coins in the box could occur once in about 15 seconds.
This interval is too long.

Since the behavior of putting the coin in the box always occurs within the context of a chain, we will use
the technique described in Chapter 24 for promoting transfer with expanded chains. We will begin with a
small chain and then we will lengthen the chain. The final chain that we design will start with a behavior that
can be produced in the practice situation or in the transfer situation. This behavior serves as a device to
demonstrate that the situations are the same.

For practice, we may construct a “mock bus,” which consists of a platform, steps leading to the platform,
something that resembles the coin box, and a person who is positioned next to the coin box and who plays the
role of bus driver.

The initial chain requires the learner to climb the stairs with coin in hand and place the coin in the
receptacle. Before each response, the teacher provides a verbal signal. This part of the teaching is very
important because it prompts the sameness of the practice situation and the real-life situation.

The basic signal could be, “Climb the stairs of the bus and put your money in the coin box.” The learner
does not have to understand precisely what these words mean because the practice will assure that the learner
discriminates this instruction from others. The instructions simply work as a unique signal.

The teacher’s assistant sits in the “driver’s” seat, wearing a driver’s cap and sitting behind a mock
steering wheel. Note that every attempt should be made to provide the practice situation with parts or features
that are shared with the real-life situation. The teacher directs the child. “Here’s a bus. There’s the driver.
Look—next to the driver there is a coin box.” The teacher then models the behavior. “My turn to climb up the
stairs of the bus and put money in the coin box.”

Immediately following the model, the child practices. Each trial is preceded by the instructions, “Climb
up the stairs of the bus and put your money in the coin box.” The money is placed in the child’s hand. After a

 
 successful performance, the teacher and “driver” praise the child. The child is returned to the “starting
position” and the next trial is presented. “Climb up the stairs of the bus and put your money in the coin box.”

After three or four successful trials, an intervening activity that requires about as much time as the trial is
introduced as a level-2 task (A > B > A). For instance, “That was really good. Let’s hear you count to 10. Go
. . . good job. Now let’s see you climb up the stairs of the bus and put your money in the coin box.”

After three or four level-2 tasks, the teacher can introduce a wider variety of intervening activities
(creating a pattern of A > B > C > D > A). “Let’s see if you remember how to do some of the things we’ve
worked on. Touch your head . . . good. Touch your foot . . . good. What are you touching? Say the whole
thing about what you are touching . . . Do you remember how old you are? . . . Good remembering. Now let’s
see you climb up the stairs of the bus and put your money in the coin box . . . Good remembering.”

After the learner has performed successfully on three or four trials of level-3 difficulty, additional parts
can be added to the chain. Each new part should be signaled by a new command. “I’m going to climb up the
stairs of the bus and put money in the container. Then I’m going to say hello to the driver . . .” (The activity
of saying hello can be practiced in isolation and then integrated with the other parts of the chain if the learner
makes a mistake.) The teacher gives a command for the new behavior, creating an interaction pattern T1-L1,
T2-L2. Later the interaction pattern becomes T1-T2, L1-L2. For example, at first the instructions are:
“Climb the stairs of the bus and put your money in the coin box.” After the learner drops the coin, “Good,
now say hello to the driver.” Later, the instructions are “Climb the stairs of the bus and put your money in the
container. Then say hello to the driver . . . Good job.”

This pattern is repeated with additional commands until the learner can perform an entire chain of
behaviors from a single chain of instruction. “First take the money from your pocket and hold it in your hand.
Then climb the stairs of the bus and put your money in the container. Then say hello to the driver and take a
seat near the front of the bus.”

After the learner is able to perform on level-3 context-difficulty for this activity when it is presented in
the mock bus situation, the learner is ready to perform in a real-life situation.

Transfer will probably occur to the real-life situation because:

1. A form of some things that are named in the instructions appear in both situations—coin, pocket,
hand, stairs, coin box, seat.

2. The behaviors are taught as a chain, which means that if the chain is started, it will tend to be
continued.

3. The first behavior that occurs in the chain is exactly the same in both situations—taking the
money from the pocket and holding it in the hand.

 
 4. The presence of the teacher and of the instructions in both situations further prompts the
sameness—particularly since these instructions are presented only in connection with a unique
chain of behaviors.

All that remains is for the teacher to prompt the learner that the situations are the same. “Look, here
comes the bus. You’re going to take your money from your pocket and hold it in your hand. Then you’re
going to climb the stairs of the bus and put your money in the container. Remember to say “hello” to the bus
driver. Then you’re going to take a seat near the front of the bus. Do it. Take your money from your pocket . .
. good. Now climb the stairs . . .”

The learner cannot produce the response that is called for. If the learner gives evidence of not being able
to produce the response that is called for, we must shape the response, starting with the behavior the learner
exhibits and moving toward the goal behavior. Before we begin the response-induction process, however, we
should be certain that the learner’s problem is one of not being able to produce the response. Specifically, we
must rule out the possibilities that the learner does not understand the instructions and that the learner cannot
produce the response in some other context.

Illustration. The learner is required to copy the word cat and the learner fails. To make sure that the
learner understands what the task requires, we could ask questions or model the behavior. Also, we would try
to reinforce the learner for trying, in an attempt to get a sample of the learner’s best effort. If the learner
apparently tries and apparently understands the instructions but cannot produce an acceptable variation of cat,
the problem is one of inducing a new motor response, a behavior that the learner has never produced before.

The specific program that we institute will depend on the level of response sophistication the learner
demonstrates. The program for the learner who has trouble holding a pencil would be quite different from
that for the learner who makes backward letters. Regardless of the program’s starting point, it would be
characterized by:

1. The use of shaping techniques—shifting the criterion for reinforcing the learners so that response
requirements change as the task remains basically the same.

2. Provisions for a great deal of supervised practice.

3. Possibly the identification of behavioral components or parts of the copying tasks.

4. Possibly an essential-response-features program that presents a variation of copying that requires

less behavior. (Perhaps the learner copies by using an index finger rather than a pencil.)

Everything the learner cannot do when trying to perform on the targeted task describes a specific skill
that we can teach. Therefore, if we carefully observe what the learner cannot do when trying something like
copying a word, we will be provided with the specific details for our program. For instance, the learner may

 
 have a habit of repeatedly looking at the model word and what is being written. This behavior suggests that
the learner does not remember what is to be copied. The deficiency implies a program that shapes the
memory. The teacher may make letters in the air. Immediately following the tracing of each letter, the learner
traces it in the air. When the learner becomes proficient at this task, the teacher requires the learner to wait a
few seconds before forming each letter. “Watch this . . . remember how that letter looked . . . your turn . . .”

As the learner becomes proficient at forming single letters that are delayed, the teacher introduces pairs
of letters, the teacher first forming oi in the air, then telling the learner to make both letters. The program
would continue until the learner was able to reliably form two letters following the teacher’s model.

At the same time that the air-tracing tasks are presented, the learner would continue to practice tracing
with a pencil. At first the learner would copy single letters. The teacher would permit the learner to look at
each; then the teacher would cover the letter and the learner would copy it below. This procedure would
assure that the learner does not continue to look back and forth between the model and what the learner is
trying to write. As an additional prompt, the learner could be directed to trace each letter before the teacher
covers it. Later the learner would be shown a series of letters, such as cat. The learner would be instructed to
look at each letter the teacher points to and to say that letter. The teacher would then cover all the letters. The
learner would be required to say the letters that are to be written, in order. When the learner is firm on saying
the series, the learner would write the series of letters. (This technique combines the procedure used for visual
spatial displays with the procedures for combining instructions T1, T2-L1, L2.) The general rule is that
everything the learner cannot do implies a specific intervention.

Possible Compliance Problems

The learner may not be trying. A diagnosis of the learner’s problem may be inaccurate if the learner is
not actually trying. There are many techniques for getting the learner to try. When working with low
performers or learners who exhibit a variety of inappropriate behavior, the best technique for assuring that the
learner is trying is the task of stand-up. The activity is based on juxtaposition prompting. We present a series
of stand-up and sit-down tasks to the learner in rapid succession. We continue until the learner performs
without hesitation on four consecutive trials. Immediately following the last trial, we present the task about
which we had questions. The learner’s performance on this task following the stand-up sequence may be
regarded as a sample of what the learner does when trying to perform. The rationale is as follows.

The series of tasks, “stand-up” and “sit-down” are presented in the same setting by the same person. All
are examples of the same instruction-following behavior (the teacher tells you what to do and you do it).
Presenting the tasks in rapid succession further assures the juxtaposition prompting of how the examples are
the same. By the time the learner has produced four behaviors in a row, the learner has demonstrated
compliance—willingness to perform on examples of the tasks in which the teacher says it and the learner

 
 does it. If another task immediately follows the fourth consecutive successful stand-up task, the probability is
overwhelming that the learner will treat it as if it is the same as the other tasks—another example of
complying with what the teacher says. The assurance of learner compliance may be further increased by
providing the learner with reinforcement when complying on stand-up tasks.

For instance: “Stand up . . . that’s good. Sit down . . . three more. Stand up . . . good. Sit down. Now
you’re doing it . . . Stand up . . . Sit down . . . Good job. Only one more. Stand up . . . Sit down. Good job.
Read this sentence . . .” Any mistake the learner makes at this point is not the result of indifference. The
learner is trying.

Diagnosis of a Complex Activity

The diagnosis of the learner may be difficult if the learner is engaged in a complex activity, particularly if
the teacher is “mother henning” the learner through the various parts of the activity. The teacher may be
prompting parts of the operation, helping the learner produce different responses, and generally obscuring
what the learner does not know.

For example, the learner may be performing a task such as putting on a coat. The teacher first directs the
learner to get his coat. The learner stares blankly. The teacher points to the coat rack and goes through some
wriggling motions as if she is putting on the coat. “Put on your coat. Your coat,” she says. The learner turns
around and looks toward the coat rack where other children are busily dressing. The learner runs over and
grabs the wrong coat. The teacher intercepts the learner and directs him to his coat. The learner then grabs the
coat and proceeds to put his left arm through the wrong armhole. As he stands there, somewhat puzzled about
why the coat is tending to go on backward, the teacher prompts some of the behavior—first orienting the
coat, then pushing the second arm through the sleeve, then operating the zipper.

Just as we teach only one thing at a time, we should test one thing at a time to discover what the learner
cannot do.

First test skills that are essential to the operation. Can the learner put on the coat unassisted if we start
the first arm through the appropriate armhole? This activity virtually removes all discriminations from the
task. What remains is part of the putting-on operation. If the learner cannot perform on this part of the
operation when it is not confounded with the discrimination of orienting the coat, we have identified the
central starting point.

To find out where the program should start, we may begin with an “easy” example—an over-sized coat
with one sleeve marked. The coat is on a hanger. The learner wears a wristband that is marked with the same
mark that appears on the sleeve. Without removing the coat from the hanger, the learner places the marked
sleeve into the marked armhole. Because the coat is over-sized, the learner is relieved of some problems

 
 associated with putting on a coat. The learner simply rotates and puts his second arm through the other
armhole.

It is important to test the central physical operation apart from the chain of events because (as we noted
earlier) this operation will probably require quite a few trials to teach if the learner has not mastered it. As
soon as the learner masters the central operation, independent (or semi-independent) practice can be started.

Each of the other discriminations should be tested separately, starting with additional motor operations
(in this case, zippering). Can the learner perform an easy example of zippering (a large zipper that has already
been started)? The discriminations should be tested in their simplest form.

If we show the learner his coat and then pause a few seconds, can he point to his coat? If the learner
performs on this task (a level-1 task on the 3-level series), we can test on levels 2 and 3 to determine the
learner’s ability to recall which coat is his.

In summary, to diagnose the learner’s performance on a complex activity that is responded to poorly we
follow these procedures:

1. If we are in doubt about the learner’s compliance, establish compliant behavior; then immediately
follow successful trials with the test.

2. If the learner fails this test, we test on the component discriminations in the chain.

a. Start with a simple variation of the central operation, stripped of discriminations (using
models or very simple signals to indicate the operation that is to be performed).

b. Test other motor responses in the chain and discriminations.

3. We design a program that starts with any central motor behaviors not mastered by the learner,
teaches the various discriminations, and integrates components and parts into a chain.

Perspective

Both this chapter and the one that follows, “Mechanics of Correcting Mistakes,” deal with identifying
specific learning problems and solving them. The difference is in the latitude of the remedy. The product of
the diagnosis developed in this chapter is a remedial program, something entirely new.

The perspective for mechanics of correcting mistakes assumes that a program already exists and that the
learner is going through that program. Because neither the program nor the learner are perfect, mistakes will
occur. How are they corrected in a way that assures that the learner will be able to take the next steps
specified in the program? The idea is not to make up a program, but to intervene in a manner that is quick and
effective and that permits the learner to continue in the program. Think of corrections as responses to

 
 unanticipated mistakes that the learner makes when being taught a program. The program is given and our job
is to help the learner through it using corrections. The procedure or diagnosing and remedying learning
problems does not assume that a program is given, and therefore creates new programs.

Summary

Diagnostic and remedial procedures may be directed at the instructional program. Diagnosis of
instruction identifies problems with the programs. The remedy that follows eliminates the problem. The two
approaches to diagnosis of the instruction are:

1. The program-flaw-prediction approach, which identifies flaws in the program and tests to
document whether these flaws have been transmitted to the learner.

2. The learner-performance approach, which documents specific learner problems and then
determines whether the instructional program was capable of creating the problem.

The program-flaw-prediction approach requires a test that is not part of the program. This test involves
either changes in the relevant stimuli (with the prediction that the learner will respond in the same manner) or
changes in the irrelevant stimuli (with the prediction that the learner will respond differently).

The learner-performance approach does not require the construction of additional tests; however, this
approach identifies problems after the fact. The approach, therefore, is well-designed to shape an instructional
program. (See Chapter 28.)

We may diagnose the learner as well as instruction. Diagnosis of the learner leads to an instructional
program.

The program derives from a description that states what the learner is supposed to do and what the
learner does.

The difference between the statements is the objective of the remedy.

The basic learner problems fall into six types, each implying a remedy.

1. The learner does not transfer from one situation to another. Remedy: Show how the situations are
the same.

2. The learner does not differentiate between two situations. Remedy: Show how the situations are
different.

3. The learner can perform from some instructions, but not from those provided by the task under
consideration. Remedy: Teach the transformation.

 
 4. The learner can produce a particular response in some context, but not without prompting in the
task under consideration. Remedy: Use the 3-level strategy to shape the context of the task.

5. The learner cannot produce the response that is called for. Remedy: Use shaping procedures to
shape the response.

6. Possibly the learner is not complying. Remedy: Use juxtaposition prompting with “stand-up”
tasks.

The first three problems suggest stimulus-locus remedies—discriminations that are to be taught.
Problems 4 and 5 are response-locus problems. Problem 6 is one of possible non-compliance. The form of
each remedy is quite simple. If the learner is not performing because of an inability to produce the response,
we use those procedures that will induce the response. If the learner’s problem is one of performing the
operation within a particular context, we shape the context starting with one that is relatively easy. If the
learner is not complying, we use prompting techniques that induce compliance.

When the learner makes a variety of mistakes on a complex operation, an involved program may result.
When attempting to formulate such programs, we test one thing at a time, starting with simple examples that
involve the central operation. We test the learner on other discriminations. We construct a program that
introduces the central motor response early.

The procedures for diagnosing the learner suggest that if the learner has a learning problem, the problem
can be described. If it is described, the remedy is implied. (The problem will fall into one of the categories,
and the direction of the remedy follows from the nature of the problem.)
 

 
 Mechanics of Correcting Mistakes
The learner may make a discrimination mistake, a response mistake, or a combination mistake (involving
both a mistaken discrimination and a faulty response). A discrimination mistake occurs when the learner is
capable of producing the response called for in the task, but produces another response. There is no question
about the learner’s ability to discriminate. A response mistake occurs when the learner cannot produce the
response. If you were told to do a backward somersault in the air and land on your feet, you would probably
have a perfect idea of what you are supposed to do; however, you would probably fail the task because you
are unable to produce the response that is called for. You have never produced the response and you are not
capable of producing it under any condition. To correct response mistakes, we shape. To correct
discrimination mistakes, we give information by showing the learner what controls the response.

Despite the differences in correction procedures, the analysis of communications suggests common
features of all logically designed corrections. These features are:

1. The correction should be designed to remedy one primary problem.

2. The correction should create juxtapositions that permit rapid repetition of the skill of
discrimination that is to be corrected.

3. The correction should provide for adequate practice.

4. The confirmation of the correction should follow the correction and should demonstrate that the
learner performs on the task in the context in which it had been originally presented.

The following is a brief rationale for the four common features of various corrections.

1. If the correction attempts to correct more than one thing, it may become so complicated that it
never ends. Suppose the teacher tries to correct several behaviors in a chain of responses; and the
learner then begins to make new mistakes, prompting the teacher to correct these. This elaboration
may prompt new mistakes.

2. The creation of adequate juxtapositions reduces the memory requirements for the task because the
learner does not have to attend to as much detail. The juxtaposition of examples that involve the
same response dimension prompt the sameness in examples. In terms of clock time (not trials), the
faster the examples can be presented, the faster the learner will produce the number of correct
responses required to learn the skill.

3. If the correction does not provide a sufficient number of trials, it may not provide enough training

 
 for the learner or enough information for the teacher. The latter point is particularly important.
The teacher who views an inadequate sample of the learner’s performance may conclude that the
learner is “firm” on the skill when the learner is not.

4. Finally, the justification for returning the learner to the precise context in which the mistake
occurred is that this context may present unobserved problems. If we correct the learner in a
simpler context without confirming that the performance transfers to the original context, we may
have presented only a partial correction. The learner’s failure on the original context and
successful performance on a simpler context (during correction) suggests that the final objective
of the correction must be to shape the context for the skill, changing it in the direction of the more
difficult original context. By ending the correction with the original context, we convey a very
important point to the learner: The correction prepares the learner for the situation in which the
learner had failed earlier. If we end the correction in a context other than that of the original
activity, the learner receives no compelling demonstration that the correction has such a purpose.

Correcting Discrimination Mistakes

A discrimination mistake occurs either within a sequence or within some other activity. The mistake is
either chronic or not chronic. The context of the mistake and the relative frequency of the mistake determine
the details of the specific correction procedures. We will illustrate both simple firming procedures and more
elaborate remedies.

Non-Chronic Mistakes Within Context

The first step in correcting a mistake is to firm it within the context in which it occurs. The procedure
involves these steps:

1. Model the correct answer.

2. Test on the missed item.

3. Back up several tasks or items and test the tasks in the order in which they originally occurred.

Illustration. The learner misses the first test example in a sequence for getting hotter (example 6). The
learner indicated that “it got hotter” when it did not. Here is the correction:

 
 Example Teacher Wording

a. 110° It didn’t get hotter.

}
b. 112° Let’s do it again
Repeat of example 6 Feel it now.
110° Did it get hotter?
…Good.

}
c. 55° Feel it now.
Repeat of example 4 Did it get hotter?
105°

112°
}Repeat of example 5 Did it get hotter?

110° }Repeat of example 6 Did it get hotter?

The teacher first modeled the answer to example 6 (step a). The teacher then repeated example 6 (step b).
To present this example, the teacher had to present the temperature of 112° (example 5). Finally, the teacher
presented example 6 within the context of examples 4, 5, and 6 (step c). Note that 4 and 5 were not tested
when the sequence was originally presented. These were modeled examples. They became test examples as
part of the correction.

If the learner performs adequately on example 6 when it is presented within the context of the other
examples, the teacher continues with the remaining examples in the sequence. If the learner again makes a
mistake, the teacher follows the same procedure—modeling the correct answer, testing on that item, then
backing up several examples and testing on the examples in the context in which they were originally
presented.

This firming procedure is used when the learner makes a mistake in an activity other than an initial-
teaching sequence.

Let’s say that the learner makes the mistake specified below:

 
 Teacher Wording Response
Which of these objects is taller? The glass.
What do you think is in the cup? Cocoa.
Which is hotter, the cup or the glass? The glass.

Here’s the correction:

1. The cup is hotter. The glass has ice in it, so it’s not hot
at all.
2. Which is hotter, the cup or the glass? (“The cup.”)
3. Good. Which of these objects is taller?...
What do you think is in the cup?
Which is hotter, the cup or the glass?
What is in the glass?
Are ice cubes hot? Etc.

The firming procedure is the same as that used for a mistake in the sequence. First, the correct answer is
modeled. Then, the learner is tested on the example missed. Then, the teacher backs up two or three tasks and
tests on the tasks in the order in which they originally appeared.

Rationale for basic firming procedure. As noted above, the purpose of a correction is to assure that the
learner can perform on the task when it appears within the context of a particular activity. The mistake
occurred within a particular juxtaposition of events. Therefore, it is important to back up and present the
missed item within the pattern of juxtaposition in which it originally occurred.

If the sequences are adequately designed, this correction procedure usually deals effectively with the
specific problems the learner experiences when responding to the sequence. The sequences are designed so
that the pattern of juxtapositions is carefully controlled. The learner is shown samenesses and differences.
Then, the learner is tested on mixed examples. If the learner makes a mistake within the context of items that
show samenesses, the learner is not attending to the basic sameness features. The remedy is to show how the
items are the same. This is precisely what happens if the basic firming procedure is followed. By backing up
several tasks and presenting the items in their original order, the missed item is now presented within a
pattern of juxtaposition that shows sameness. Similarly, if the learner makes a mistake on a minimum-
difference item, the most logical remedy is to show the difference, which is what the firming procedure will
do by repeating the missed item within a context that shows a single difference between the missed item and

 
 the item that immediately precedes it. Finally, if the learner misses items in the test segment, the response
implies that the learner is not using the information about the discrimination that had been demonstrated
earlier in the sequence. The most logical remedy is to require the learner to apply the discrimination to items
presented in the test context. (By returning to an easier pattern of juxtaposition, we are not providing the
learner with the needed information. The learner has already shown the ability to perform on items in the
easier context. What the learner cannot do is perform within the test segment, which is the context that should
be firmed.)

Remedies for Chronic Errors

The procedures above apply to simple firming. Any learner can be expected to make mistakes from time
to time, and these mistakes do not necessarily imply serious problems. Signs of serious problems are:

1. A high percentage of mistakes on a particular sequence or activity.

2. A high percentage of mistakes on a particular subtype or item within a sequence activity.

Corrections for chronic mistakes are more complicated than those for non-chronic mistakes. The Figure
27.1 shows the three basic procedures for chronic discrimination errors.

Figure 27.1
Remedies for Chronic Discrimination Mistakes
START

Does
a chronic Correction I
YES Are missed items NO
mistake occur within an
of the same subtype? a. Firm on original sequence.
initial-teaching
b. Firm on parallel sequences if necessary.
sequence?

YES
NO

Correction III Correction II

a. Firm on original task. a. Firm on original sequence.
b. Use the task in which the mistake occurred as the basis for design- b. Construct a sequence containing examples of the subtype of items
ing a sequence. missed.
c. Firm the learner on this sequence (and on parallel sequences if c. Firm the learner on this sequence and parallel sequences if
necessary). necessary.
d. Firm the learner on the activity in which the mistake originally d. Firm the learner on the sequence in which the mistake originally
occurred. occurred.

As the figure indicates, each remedy may call for the construction of additional sequences. If the mistake
is a general confusion (Correction I), the remedy is to construct sequences that are parallel to the original. If
the mistake is a subtype mistake (Correction II), a sequence containing only examples of the missed type is
presented. If the mistake does not occur within a sequence, but within the context of another activity

 
 (Correction III), the missed item is treated as a task that is used as a basis for designing an initial-teaching
sequence.

Remedies for Chronic Mistakes that Occur Within a Sequence

Corrections I and II present the remedial procedures for mistakes that occur within a sequence.

Correction I. Consider the percentage of errors on a sequence unacceptably high (suggesting “general
confusion”) if the learner initially misses 25 percent or more of the test items.

1. Firm on original sequence. Each time a mistake occurs within the sequence, follow the basic
firming procedures of modeling the correct response, testing on the missed item, and backing up
several tasks and repeating the tasks in order. Although the learner has been “firmed” on the
sequence at this time, we would still suspect the learner to make mistakes on new items if they
were presented.

2. Firm the learner on a parallel sequence. A parallel sequence is one that presents a different set of
examples and different specific juxtapositions, but that presents the same discrimination and the
same wording for test items. The parallel sequence is capable of providing us with information
about how firm the learner is and of providing the learner with the additional practice that will
firm the response. When we present this sequence, we model fewer examples than the original
and present fewer test items.

If the learner performs acceptably on the parallel sequence (making errors on less than 15 percent of the
test items), consider the learner firm on the discriminations. If the learner makes mistakes on more than 15
percent of the items, present another parallel sequence. Repeat the procedure until the learner meets criterion
of at least 85 percent correct on the test items.

Correction II. The correction for subtype errors involves four steps:

1. Firm on original sequence.

2. Construct a sequence containing examples of the subtype of items missed.

3. Firm the learner on this sequence and parallel sequences if necessary.

4. Firm the learner on the sequence in which the mistake originally occurred.

If the learner makes no more than 25 percent errors on the test items, and if the mistakes are of the same
subtype, use the subtype correction. If the errors exceed 25 percent, provide Correction I, even if the mistakes
are of the same subtype.

Like the remedy for a high percentage of mistakes, the subtype remedy begins by firming all mistakes in

 
 the original sequence. (We model the answer to each missed item, then test that item, then repeat the part of
the sequence in which the mistake occurred.)

The subtype correction requires the introduction of a new sequence (step b), not a parallel sequence. The
new sequence contains only examples of the missed subtypes (or a heavy predominance of these examples).

If the learner misses all no-change negatives in a sequence, the subtype sequence would consist
predominantly of no-change negatives. Figure 27.2 shows a possible sequence.

Figure 27.2
Example Teacher Wording

1 Watch the space. I’ll tell you if it gets wider.

2 Did it get wider? Yes.

3 Did it get wider? Yes.

4 Did it get wider? No.

5 Your turn: Did it get wider?

6 Did it get wider?

7 Did it get wider?

8 Did it get wider?

9 Did it get wider?

10 Did it get wider?

11 Did it get wider?

12 Did it get wider?

13 Did it get wider?

If the learner consistently fails to identify one of the members presented in a noun sequence (truck, for
instance), the subtype sequence would be a sequence in which that particular type predominated.

 
 Example Teacher Wording
red truck My turn. What kind of vehicle? Truck.
black truck Your turn. What kind of vehicle?
yellow truck What kind of vehicle?
car What kind of vehicle?
red truck What kind of vehicle?
blue truck What kind of vehicle?
car What kind of vehicle?
black truck What kind of vehicle?
car What kind of vehicle?
red truck What kind of vehicle?
yellow truck What kind of vehicle?

If the subtype occurs in a single-transformation sequence, the subtype would be presented in a single-
transformation sequence. It would be the only type in the sequence.

The following is a single-transformation sequence for teaching “numerical expansion.” Mistakes are
indicated with an asterisk.

Teacher Wording Response

My turn. What does fifty-
seven equal? Fifty plus seven.
My turn again. What does
seventy-one equal? Seventy
plus one.
Your turn. What does
seventy-two equal? Seventy plus two.
What does sixty-two equal? Sixty plus two.
What does twenty-six equal? Twenty plus six.
What does twenty equal? Twenty plus twenty.*
What does ninety-four equal? Ninety plus four.
What does thirty-three equal? Thirty plus three.
What does eighty equal? Eighty plus one.*
What does forty-five equal? Forty plus five.

The subtype is two-digit numerals that end in zero. Unlike the others in the sequence, names of these
numerals do not provide instructions for specifying the second digit. (The teacher does not say, “What does
fifty-zero equal?”)

The subtype sequence would consist totally of items of this type:

 
 Teacher Wording Response
My turn. What does thirty
equal? Thirty plus zero.
Your turn. What does twenty
equal? Twenty plus zero.
What does thirty equal? Thirty plus zero.
What does ninety equal? Ninety plus zero.
What does fifty equal? Fifty plus zero.
What does seventy equal? Seventy plus zero.

The following shows a different type of error pattern on the original sequence.

Teacher Wording Response

My turn. What does twenty-
six equal? Twenty plus six.
My turn. What does twenty-
seven equal? Twenty plus
seven.
Your turn. What does
twenty-eight equal? Twenty plus eight.
What does thirty-eight equal? Thirty plus eight.
What does eighty-three equal? Eighty plus three.
What does eighteen equal? Eighty plus . . *
What does forty-five equal? Forty plus five.
What does fifteen equal? Fifty plus . . *

The subtype is the teen numerals. This type is potentially difficult because the convention for naming
these numbers is the opposite of that for other two-digit numerals. Instead of naming the teen number first
(teen-eight), the convention calls for saying, “eighteen.” This sequence presents the teen subtype:

Teacher Wording
My turn. What does eighteen equal? Ten plus eight.
Your turn. What does nineteen equal?
What does fourteen equal?
What does twelve equal?
What does thirteen equal?
What does fifteeen equal?
What does eighteen equal?

Note. For single-transformation or double-transformation concepts, the subset sequence contains only the

 
 subtype that is missed. All other subtype sequences contain a predominance of the subtype item.

Step c in the subtype remedy calls for firming the learner on the sequence and on parallel sequences. We
use the same criterion of performance specified for Correction I. We consider the learner firm only when the
learner makes mistakes on no more than 15 percent of the test examples.

After the learner has performed adequately on the subtype sequence, we firm the learner on the sequence
in which the mistake originally occurred (step d). Note that this step has no parallel in the remedy for a high
percentage of errors. The reason is that we must return the learner to the context in which the original mistake
occurred. This step is taken naturally for the Correction I mistakes because the parallel sequences that we
construct present the same context as the original sequence. The situation is different with the subtype
sequence, because this sequence is not like the original.

Optionally, a more involved process of integrating a subtype with the familiar type may be designed.
This procedure is the integration procedure specified for integrating frequently missed worksheet items with
low-mistake items (Chapter 16). The integration is modeled after the double-transformation sequence. First,
the new set (subtype) is presented. Then examples of a familiar set are presented. Finally, the sets are mixed.
The value of this procedure is that it permits the learner to deal with the members of the new set first in a
relatively simple pattern of juxtapositions, then in a more difficult pattern. The procedure would be effective
for subtypes that are fairly large or fairly difficult.

Chronic Errors Outside of a Sequence

Correction III. A chronic error may occur outside of a sequence. To provide a remedy, the teacher first
firms on the missed task by modeling the correct answer, testing on the task, backing up several tasks, and
repeating the tasks in the same order they had been presented initially. The teacher then presents a sequence
designed from the task the learner missed.

To design a sequence from a task, we use the wording of the original task. We classify the task as either a
choice-response task or a production-response task. We classify the task as one dealing with a non-
comparative, a comparative, a noun, a transformation, or a correlated-feature relationship.

The cognitive routine below illustrates the procedure. Each step may be the cause of a chronic mistake.
We can classify each step as a task that clearly implies a sequence.

Teacher writes on the board: 9 - 14 =

1. “Read this problem.”

2. “How many are you starting out with?”

3. “Will the answer be positive or negative?”

 
 4. “How do you know?”

5. “Fourteen is five bigger than the number you’re starting out with. So what is the number in the
answer?”

6. “Read the problem and the answer.”

Step 1 requires the learner to “Read this problem.” The task is a production-response, single-
transformation task. (If the same task were applied to different problems, the response would be different.
Also, there are no negative examples for “Read this problem.”)

If the learner is chronically weak on this step, we would firm the learner through a single-transformation
sequence with the instructions, “Read this problem.”

Here is the first part of a possible test sequence:

Example Teacher Wording

9 - 9 = Read this problem.
9 - 10 = Read this problem.
10 - 9 = Read this problem.
10 - 4 = Read this problem.
10 - 14 = Read this problem.
9 - 21 = Read this problem.

Note that this sequence contains no models. The particular symbols that appear in the problems would
depend on the learner’s performance. If the learner had trouble with 9 and 10, the sequence above would be
appropriate; however, it would not serve the learner who confused 4 and 5.

Step 2 in the problem presents the task, “how many are you starting out with?” This is a single-
transformation task. If the learner identified the other numeral in the problem as “How many you start out
with” (saying, “fourteen” instead of “nine”), we could use this sequence.

 
 Example Teacher Wording
9 - 7 This problem starts out with nine. How many
does this problem start out with?
8 - 7 This problem starts out with eight. How many
does this problem start out with?
8 - 6 How many does this problem start out with?

6 - 8 How many does this problem start out with?

12 - 5 How many does this problem start out with?

1 - 3 How many does this problem start out with?

5 - 4 How many does this problem start out with?

9 - 4 How many does this problem start out with?

14 - 1 How many does this problem start out with?

24 - 3 How many does this problem start out with?

Step 3 (“Will the answer be positive or negative?”) is followed by step 4 (“How do you know?”). The
pair of items presents a correlated-feature relationship (“When the number you minus is greater than the
number you start out with, the answer will be negative.”).

Here is a sequence:

 
 Example Teacher Wording
9 - 8 My turn. Will the answer be positive
or negative? Positive. How do I know?
Because we’re minusing less than nine.
9 - 10 My turn. Will the answer be positive or
negative? Negative. How do I know?
Because we’re minusing more than nine.
9 - 11 Your turn. Will the answer be positive or
negative? How do you know?
12 - 11 Will the answer be positive or negative?
How do you know?
11 - 12 Will the answer be positive or negative?
How do you know?
5 - 12 Will the answer be positive or negative?
How do you know?
14 - 12 Will the answer be positive or negative?
How do you know?
15 - 3 Will the answer be positive or negative?
How do you know?
6 - 4 Will the answer be positive or negative?
How do you know?
4 - 6 Will the answer be positive or negative?
How do you know?

Step 5 is a variation of a correlated-feature relationship with part of the relationship not stated. (“Fourteen
is five bigger than the number you’re starting out with. So what is the number in the answer?”)

 
 Example Teacher Wording
10 - 14 My turn. Fourteen is four bigger, so what’s
the number in the answer? Negative four.
13 - 14 Fourteen is one bigger. So what’s the
number in the answer?
14 - 14 Fourteen is zero bigger. So what’s the
number in the answer?
13 - 14 Fourteen is one bigger. So what’s the
number in the answer?
0 - 14 Fourteen is fourteen bigger. So what’s the
number in the answer?
8 - 14 Fourteen is six bigger. So what’s the
number in the answer?
11 - 14 Fourteen is three bigger. So what’s the
number in the answer?
-1 - 14 Fourteen is fifteen bigger. So what’s the
number in the answer?
-4 - 14 Fourteen is eighteen bigger. So what’s the
number in the answer?
7 - 14 Fourteen is seven bigger. So what’s the
number in the answer?

The same task analysis procedure is applied to the illustration in Figure 27.3. However, for this
illustration the firming sequences do not derive as directly from the routine in which the chronic mistake
occurred.

Figure 27.3
1. Look at this clock. Start at the top and say twelve.
Count and write the number in the first circle and in
12
the next circle. 11 1
2. Touch the little hand. Remember, if the little hand is
between numbers, we write the first number. Is the 10 2
little hand between numbers?
3. What number do we write for the little hand? 9 3
4. Write that number on the line.
8
7
6

There are two problems with the routine:

1. The tasks involve writing, which means that a sequence based on them would be slow and
possibly would be failed if the learner is a poor writer.

 
 2. A deduction is involved in steps 2 and 3; however, it is not set up in the standard correlated-
feature form.

The solution to these problems involves modifying the tasks before creating the sequences. The final
step, however, would be to return the learner to the wording in the routine.

Step 1. A possible juxtaposition problem occurs if the learner has trouble writing numerals. Since the
writing is central to the operation, we would first test the learner on this skill as a component removed from
the context of the routine. “Write the numeral 5 . . . write the numeral 6 . . .” The procedure of removing the
discrimination and testing the motor response in a direct manner is basic to diagnosing the learner. (See
Chapter 26.)

To test the learner’s understanding of which numeral to write, we would change the sequence to permit
quick juxtaposition of examples. Figure 27.4 shows a possible sequence.

 
 Figure 27.4

Example Teacher Wording

You’re going to start at 12. Move to
the circled numbers. Say the numbers
that go in the first circle and the next
circle.

11 12 1
10 2
Say the numbers that go in the first circle
9 3
and the next circle.
8 4
7 6 5

11 12 1
10 2
Say the numbers that go in the first circle
9 3
and the next circle.
8 4
7 6 5

11 12 1
10 2
Say the numbers that go in the first circle
9 3
and the next circle.
8 4
7 6 5

11 12 1
10 2
Say the numbers that go in the first circle
9 3
and the next circle.
8 4
7 6 5

11 12 1
10 2
Say the numbers that go in the first circle
9 3
and the next circle.
8 4
7 6 5

11 12 1
10 2
Say the numbers that go in the first circle
9 3
and the next circle.
8 4
7 6 5

11 12 1
10 2
Say the numbers that go in the first circle
9 3
and the next circle.
8 4
7 6 5

The wording for the original task is modified. The word say has to be substituted for write. The sequence
is a single-transformation.

 
 The sequence above could be treated as the familiar set for a double-transformation sequence. The
transformed set would consist of the same examples, but different instructions: “Count and write the number
in the first circle and in the second circle.” Upon completion of the transformed set, the learner would have
performed on the instructions that appear in the original task and would have done so with a sufficiently large
sample of examples to imply adequate understanding of the concept.

Step 2. “Is the little hand between numbers?” The task is a choice-response task, and it suggests a non-
comparative sequence that presents the discrimination between numbers and not-between numbers. Figure
27.5 shows a possible sequence.

 
 Figure 27.5

Example Teacher Wording

11 12 1
10 2
9 3 My turn. Is the hand between numbers? No.
8 4
7 6 5

11 12 1
10 2
9 3 My turn. Is the hand between numbers? Yes.
8 4
7 6 5

11 12 1
10 2
9 3 Your turn. Is the hand between numbers?
8 4
7 6 5

11 12 1
10 2
9 3 Is the hand between numbers?
8 4
7 6 5

11 12 1
10 2
9 3 Is the hand between numbers?
8 4
7 6 5

11 12 1
10 2
9 3 Is the hand between numbers?
8 4
7 6 5

11 12 1
10 2
9 3 Is the hand between numbers?
8 4
7 6 5

11 12 1
10 2
9 3 Is the hand between numbers?
8 4
7 6 5

11 12 1
10 2
9 3 Is the hand between numbers?
8 4
7 6 5

To make the response stronger, we might use two questions with examples that show the hands between.

 
 “Is the hand between numbers? . . . Which numbers?”

Steps 2 and 3 are linked through a rule: “Remember, if the little hand is between numbers, we write the
first number . . . Is the little hand between numbers? . . . What number do we write for the little hand?”

The rule creates a correlated-feature link between the little hand being between numbers and writing the
first number. Although the questions are not presented in a standard form, we can use the wording of the
question that occurs in step 3 (“What number do we write for the little hand?”) and add the question, “How
do you know?” (See Figure 27.6.)

 
 Figure 27.6

Example Teacher Wording

11 12 1
10 2 My turn. What do I write for the little hand?
9 3 Four. How do I know? It’s the first number.
8 4
7 6 5

11 12 1
10 2 Your turn. What number do we write for the
9 3 little hand? How do you know?
8 4
7 6 5

11 12 1
10 2
What number do we write for the little hand?
9 3
How do you know?
8 4
7 6 5

11 12 1
10 2
What number do we write for the little hand?
9 3
How do you know?
8 4
7 6 5

11 12 1
10 2
What number do we write for the little hand?
9 3
How do you know?
8 4
7 6 5

11 12 1
10 2
What number do we write for the little hand?
9 3
How do you know?
8 4
7 6 5

11 12 1
10 2
What number do we write for the little hand?
9 3
How do you know?
8 4
7 6 5

11 12 1
10 2
What number do we write for the little hand?
9 3
How do you know?
8 4
7 6 5

The sequence contains only examples in which the hand is between numbers. A variation could be

 
 presented in which some hands were not between numbers. The second question could be dropped for these
examples.

Note that the sequence contains a fair sample of examples involving numbers 7 through 12, usually the
most troublesome. When the learner deals with numbers 1 through 6, the first number is the one on top.
When dealing with 7 through 12, the first number is on the bottom.

For the last step in the routine, the learner writes the number for the little hand on a line. A variation of a
single transformation sequence firms this skill.

Example Wording
50 Write that number on line a.
0-five Write that number on line b.
0-one Write that number on line c.
10 Write that number on line d.
25 Write that number on line e.
50 Write that number on line f.
15 Write that number on line g.
35 Write that number on line h.

We would adjust the sequence to the particular writing problems the learner was experiencing. (In the
sequence above, we stress two of the common confusions—0-one versus 10, and confusions on 50, 15, and 0-
five.)

The final illustration of creating corrections sequences for chronic mistakes that occur outside a sequence
comes from an intermediate math series.

 
 a. Write: 2 5 =
7
LET’S CHANGE THIS MIXED NUMBER INTO A FRACTION. HOW MANY
PARTS WILL BE IN EACH WHOLE? “Seven.”
b. Write: 2 5 =
7 7
HOW MANY WHOLES DO WE HAVE? “Two.”
c. A FRACTION EQUALS TWO WHOLES WHEN THE TOP IS HOW MANY
TIMES BIGGER THAN THE BOTTOM? “Two.”
d. WHAT IS THE BOTTOM OF THE FRACTION WE’RE WRITING? “Seven.”
e. SO THE TOP MUST BE TWO TIMES BIGGER THAN SEVEN. TELL ME
THAT NUMBER FOR THE TOP.
f. Write: 2 5 = 14
7 7
WE USED THE TWO WHOLES. HOW MANY PARTS ARE LEFT? “Five.”
g. Write: 2 5 = 14 + 5 =
7 7
AND HOW MANY PARTS DO WE HAVE ALTOGETHER? “Nineteen.”
h. Write: 2 5 = 14 + 5 = 19
7 7
AND HOW MANY PARTS ARE IN EACH WHOLE? “Seven.”
i. Write: 2 5 = 14 + 5 = 19
7 7 7

We could use a variation of the same sequence for firming each step—the single-transformation
sequence. However, it is also possible to combine the steps that involve deductions and present them as
correlated-feature sequences.

Below are the first parts of the various single-transformation sequences. In step a, the teacher asks: “How
many parts will be in each whole?” Here are the first few examples in a possible sequence.

Example Teacher Wording

2 5 How many parts in each whole?

7

2 4 How many parts in each whole?

7

2
7 How many parts in each whole?
4

2
7 How many parts in each whole?
5

In step b, the teacher asks, “How many wholes do we have?” Here are the first few examples.

 
 Example Teacher Wording

2 5 How many wholes do we have?

7

2
7 How many wholes do we have?
5

1
7 How many wholes do we have?
5

21
7 How many wholes do we have?
5

In step c, the teacher asks, “A fraction equals two wholes when the top is how many times bigger than the
bottom?”

Here are the first examples in a possible firming sequence.

Teacher Wording
A fraction equals two wholes when the top is how many
times bigger than the bottom?
A fraction equals three wholes when the top is how many
times bigger than the bottom?
A fraction equals thirteen wholes when the top is how
many times bigger than the bottom?
A fraction equals nine-R when the top is how many times
bigger than the bottom?

Step d asks, “What is the bottom of the fraction we’re writing?” Here is the first part of a possible
sequence.

Example Teacher Wording

5
What’s the bottom number of the fraction
2 7 we’re writing?

7 What’s the bottom number of the fraction

2 5 we’re writing?

In step e, the teacher says: “So the top must be two times bigger than seven (the bottom number). Tell me
that number for the top.” To simplify this task, we could change the wording slightly.

Teacher Wording
If the top is two times bigger than seven, what’s the top number?
If the top is two times bigger than six, what’s the top number?
If the top is three times bigger than six, what’s the top number?
If the top is five times bigger than four, what’s the top number?

 
 In step f, the teacher says: “We used the two wholes. How many parts are left?”

Example Teacher Wording

2 5 If we use the wholes, how many parts are left?
7

2 4 If we use the wholes, how many parts are left?

7

7
2 4
If we use the wholes, how many parts are left?

7 If we use the wholes, how many parts are left?

5 14

3
8 9
If we use the wholes, how many parts are left?

Etc.

In step g, the teacher asks: “And how many parts do we have altogether?” If the learner has been taught
an operation for adding, this step could be firmed through a sequence like the one below.

Example Teacher Wording

14 + 5
7
How many parts do we have altogether?

14 - 5
7
How many parts do we have altogether?

14 - 5
1
How many parts do we have altogether?

14 + 5
1+2
How many parts do we have altogether?

7
J+3
How many parts do we have altogether?

The learner’s problem may not be with the individual steps, but with the relationships between the steps.
For most of these relationships, it would be possible to create correlated-feature sequences; however, each
would do some violence to the wording or the order of the steps in the routine. A far more manageable
alternative is to identify clusters of steps and schedule them according to the procedures specified in Chapter
19, Scheduling Routines and Their Examples. For instance, if the learner tended to make mistakes on both
steps b and c, we would use the wording of steps b and c to create a cluster.

 
 Example Teacher Wording
5
2 7
=
7
How many wholes do we have?

A fraction equals two wholes when the top

is how many times bigger than the bottom?
5
3 7
=
7 How many wholes do we have?

A fraction equals three wholes when the top

is how many times bigger than the bottom?

It is possible to design other clusters and to incorporate the two steps above into a longer cluster. Also,
the same step may be involved in more than one cluster. For instance, we might design a cluster that involves
steps c, d, and e in addition to the one that involves b and c. The step c is common to both clusters. Here’s
the c-d-e cluster:

Example Teacher Wording

2 =
7
A fraction equals two wholes when the top is
how many times bigger than the bottom?
What is the bottom number of the fraction
we’re writing?
So the top number must be two times bigger
than seven. Tell me that number.

2 =
8 A fraction equals two wholes when the top is
how many times bigger than the bottom?
What’s the bottom number of the fraction
we’re writing?
So the top number must be two times bigger
than eight. Tell me that number.

Following the practice with the clusters, the entire routine would be reintroduced. Note that we would
follow the cluster remedy whether or not the program had provided for the teaching of clusters before
introducing the routine. However, if the program had provided for such teaching, our correction of the
chronic mistake would simply involve returning to the part of the program in which the clusters had been
taught and firming the learner on them.

Summary for firming chronic errors that occur outside a sequence. We have presented a large number of
examples of correction problems that occur outside an initial teaching sequence to demonstrate that the
procedures for analyzing tasks and classifying discriminations will permit a fairly rigorous treatment of any
discrimination mistake. The mistake occurs in response to a specific task or series of tasks. The constraints

 
 are the wording of the task, form of the example, and the “meaning” of the particular discrimination that is
intended.

Creating the sequence involves classifying the type of discrimination called for by the task and creating
an initial-teaching sequence for that type. In some cases, we alter the wording somewhat or change the nature
of the examples. However, we return to the original wording after the learner is firmed on the variant task
form. (We can use a variation of the double-transformation program for achieving this transition.) The final
step of the firming is to return the learner to the activity in which the chronic mistake originally occurred. The
proof of the firming is that the learner can perform within the juxtaposition pattern of tasks presented by the
original activity.

Correcting Chronic Response Problems

The situations in which response firming is needed parallel those in which discrimination firming is
implied. The chronic mistake may occur within a sequence or activity that is designed for initial teaching, or
in some activity that is not designed for initial teaching. When a chronic problem occurs in an initial-teaching
sequence: (1) the learner is not performing acceptably on at least 70 percent of the trials; and (2) the poor
performance is persistent. When the response problem occurs outside the initial-teaching situation, the learner
does not perform an activity acceptably because a particular response is: (1) omitted from the activity or (2)
produced unacceptably. The problem is persistent.

The remedies indicated for the various cells in Figure 27.7 are not new. They have been outlined in
earlier chapters. Their specific application to correction situations, however, is new. The particular remedy
that is provided must be designed not only to correct the problem, but also to return the learner to the activity
in which the problem occurred.

 
 Figure 27.7
Procedures for Chronic Response Problems
START

C A B
Classify problem as
to when to produce
response or how to
produce response. Does the mistake
Create juxtapositions NO occur within a YES Can the learner
appropriate for initial response-teaching produce the response?
teaching. Provide program?
Correction I (for
how) or Correction II
(for when). Then
return to original
activity. YES
NO

Correction I Correction II

Change criterion or Shape the context.

change the response. Increase sameness in
the context.

Remedies for Errors that Occur Within Response-Teaching Programs

As Correction I indicates, the problem may be that the learner cannot produce a response. There are two
possible avenues for remedying this problem. We may change the response by simplifying it, or we may
change the criterion for reinforcing the response. We should look first at the possibility of changing the
response. We do this by:

1. Eliminating any complicated or unnecessary discriminations from the response chain.

2. Shortening the chain to the part that immediately precedes the part on which the error occurs.

3. Substituting the examples presented with those that are easier (in the sense that they can be
processed through a wider variety of response options).

The elimination of the unnecessary discriminations improves the juxtaposition of practice examples and
focuses the activity so the learner works with only that part that presents a problem. Shortening the chain
improves the juxtaposition of practice trials. The use of easier examples assures that the learner will perform
the operation more quickly on some examples.

The final step in the remedy for motor response problems is to shape the response from the simplified
version described above to that which occurs in the response-teaching sequence.

The two avenues indicated by Correction I (change the criterion or change the response) are not

 
 exclusive remedies. We should consider the possibility of doing both. By changing the response, we redesign
the activity so that it creates a greater possibility of success. In addition, however, we should adjust the
criterion of performance so the learner succeeds (receives reinforcement) on at least 70 percent of the trials.
The procedures for establishing such criteria are described in Chapter 22.

Correction II indicates the remedial steps implied if the learner can produce the response that is called
for, but the response fails to occur in the program that is designed to teach the learner when to respond. The
remedy involves two steps: shaping the context and increasing the sameness elements in the context. We use
a variation of the 3-level strategy to shape the context. To increase the sameness across applications, we add
unique behaviors to all examples. These behaviors may be perfectly irrelevant to any task. “Let’s say that I
walk up to you and say, ‘Hi, what’s your name?’ . . . The introduction of this sameness (“Let’s say that I walk
up to you and . . .”) has two functions. The first is to prompt sameness. (A unique and obvious behavior is
attached to each presentation of the task; therefore, each presentation task must be the same and must call for
the same type of response.) The second function of the added behavior is to create a non-failure set for the
activity. Remember, the learner had failed this task repeatedly. The task is likely to be punishing. If our
remedy is to be effective, we must create a reinforcing set. By using the added-behavior prompt with
examples that are easy, we associate the prompt with reinforcement and success. If we present the same
prompt with more difficult examples (those increasingly similar to the original activity with respect to context
difficulty), we imply that these tasks are also relatively reinforcing. They have the same feature that predicts
reinforcement with simpler tasks. After the learner has performed successfully with the sameness prompt, the
prompt is dropped. Reinforcement continues and a positive set is created for the original activity.

Remedies For Problems That Do Not Occur Within Response-Teaching Programs

The problem that occurs outside the response-teaching program is either a problem of when to produce
the response or how to produce the response. Therefore, the first step in effecting a remedy is to classify the
problem. We then use the basic procedures specified for either Correction I (for how-to-respond problems) or
Correction II (for when-to-respond problems). Finally, we return the learner to the original activity (which
may happen automatically as part of the procedure for Correction I or Correction II).

Implying false operations. Perhaps the biggest potential problem that occurs if we do not create
appropriate juxtapositions for firming responses is that we may imply false operations. False operations are
implied if the juxtaposition of events suggests we are dealing with a motor response rather than a
discrimination.

Consider this situation. The teacher points to an object and says, “What is this?”

The learner replies, “Du u car.”

 
 The teacher then works on the response. “Listen. This is a car. Say it with me . . . again . . . again . . . All
by yourself, what is this? . . . Yes, what is this? . . . Yes, what is this . . .”

The correction implies a false operation. The question “What is this?” is repeatedly presented in a context
of saying, not of discriminating between car and not-car. Therefore, the question “What is this?” implies
saying words—not saying words that are based on the feature of the examples. The naive learner can
therefore be expected to learn serious misrules about the nature of the response.

By creating appropriate juxtapositions, we solve the problem. We separate the discrimination from the
elaborate response:

Teacher: What is this?

Learner: Car.

Teacher: Good. Say the whole thing.

Learner: Du u car.

Teacher: Not bad. Listen to me: This is a car. Listen again: This is a car. Say it with me. Get ready . . .
This is a car. Again with me: This is a car . . . Pretty good. All by yourself. Say the whole thing . . .” etc.

The signal for the discrimination is “What is this?”

The signal for the motor response is “Say the whole thing.”

The repeated trials of “Say the whole thing” can be presented without implying a false operation. (If the
learner becomes confused, a series of “What is this?” tasks can be presented to demonstrate that the response
for these is a single word or phrase.)

The inappropriate juxtapositions of responses and discriminations characterize many teaching activities.
The learner may be reinforced for sitting properly, looking at the teacher, and other “proper” behaviors.
When these responses are worked on repeatedly, a false operation is implied. The implied goal of instruction
appears to be to sit with both feet on the floor, to look at the teacher, and to engage in other irrelevant
activities. If the response of looking at the teacher is to be shaped in a way that does not imply a false
operation, it should be framed in a way that requires the learner to look at the teacher. The formula is fairly
simple:

1. Place a highly reinforcing contingency on the discrimination or skill that is to be taught.

2. Design the task so that the learner can perform only if the learner observes what the teacher does.

For instance, the learner may be inattentive while the teacher is pointing to different objects and asking

 
 “What is this?” Here is a possible solution:

“I’m going to go very fast. I’m going to point to different things and ask the question. If you get them all
right, you get 10 minutes of extra recess. Here I go . . .” Now the learner has some functional reason for
watching the teacher. Watching becomes a necessary step in getting extra recess. Given that the learner wants
extra recess, the learner will watch. Of course, the teacher should adjust the task so that if the learner does
watch and respond, the learner will succeed.

In summary, the problem of implying false operations is largely solved if we create juxtapositions that
are appropriate. If the learner has problems in producing the response, work on the response production
through tasks that do not imply discriminations (as in “Say the whole thing”). If the problem is one in which
the learner can produce the response, but is not producing it, introduce a context that provides the learner
with the appropriate reason for producing the response (as in the case of watching the teacher).

Creating a success set. Chronic response failures that occur in activities other than initial-teaching
situations tend to make these activities punishing. For example, if the learner has had problems saying a
series of addition statements such as, “Five plus one equals six; five plus two equals seven; five plus three
equals eight,” the learner may develop symptoms that are associated with chronic failure. These include: long
response latency; frequent checks of the teacher’s expression for confirmation of the response’s correctness;
and various superstitious or nervous behaviors.

Part of the correction should be to establish a positive set for the statement-saying. A solution for this
aspect of the correction is an add-on. In addition to simplifying the operation and creating adequate
juxtapositions, we create a new basis for treating all examples in a positive manner. For instance, we may
press the learner’s palm with one finger as a signal to say the first statement in the sequence; two fingers for
the second statement; and three for the third. Or we may introduce some other behavior that is perfectly
extraneous to the task, but that can be produced with all examples of saying a series of addition facts. For
instance, “Hold your hands way over your head and get ready to say some statements.” Or, “Close your eyes
and get a picture of the statements you say.”

The general rule is that if the easier examples and harder examples are the same in some observable
ways, we can create add-on behaviors to imply that they are the same in other ways that are not observed.
The diagram below shows how the implication is created:

Easy Examples Hard Examples

Involve new behavior A. Involve new behavior A.
Lead to reinforcement. Therefore: they must lead to reinforcement.

Some traditional remedial procedures that involve using a new modality or combination of modalities

 
 tend to work, but not because of any direct influence of the new modality. When we introduce a new
modality, we are actually requiring the learner to perform a new behavior (new behavior A). If the new
behavior happens to be associated with success (which is what happens when the learner is taken back in the
arithmetic program and is taught the new facts through multiple-modality inputs), the basic requirements for
easy examples has been established. When more difficult examples are presented, particularly those that were
formerly associated with chronic failure, the probability is increased that the learner will respond to the new
prompt and approach these problems in a way that does in fact lead to success. The cause of the success is
then falsely attributed to the introduction of new modalities, when the program simply introduced a new
sameness that permitted the classification of all applications as “reinforcing applications.”

Summary

The correction of a mistake occurs as an immediate response to the learner’s inappropriate response to a
task. The correction is designed to assure that the learner will perform correctly in subsequent presentations
of the activity in which the mistake occurred. The correction may involve non-chronic or chronic mistakes.
Different procedures are used for each type of mistake.

For correcting non-chronic discrimination mistakes, the teacher:

1. Models the correct answer.

2. Tests on the missed item.

3. Backs up several tasks or items and tests on the tasks in the order in which they were originally
presented.

Step 1 provides the learner with information about the correct response; step 2 provides the most direct
tests on whether the information had been effectively communicated to the learner; step 3 tests whether the
learner is able to perform on the item when it is again placed in the context in which it originally occurred.

The 3-step firming procedure is used when a mistake occurs in any activity—an initial teaching sequence
or some other activity.

For discrimination mistakes that are chronic, a more elaborate procedure is called for (because the simple
firming procedure has apparently failed).

For correcting mistakes that occur within a sequence, we first determine whether the mistakes are
predominantly of a particular subtype. If they are, we first firm on the original sequence, firm on one or more
sequences composed entirely of the subtype (or predominantly of the subtype) and finally return to the
original sequence.

 
 If the mistakes are not of a particular subtype, the learner tends to miss items of different subtypes, which
means that the learner has a general lack of understanding about the discrimination. To correct chronic
mistakes of this type (high percentage of errors that do not involve the same subtype), we firm on the original
sequence and firm on parallel sequences if necessary. Parallel sequences are those that require the same
discrimination, and use the same wording, but present a different juxtaposition of items.

If the mistake occurs outside the sequence, we firm on the task in which the mistake occurred (using the
3-step firming procedure). We then construct a sequence that uses the task as the basis for the teacher
wording and that presents the same meaning as the task. (If the task is a choice-response, the sequence is a
choice-response. If the task is a production-response, the sequence presents production-response tasks.) The
learner is firmed on the sequence and parallel sequences, after which the learner is returned to the original
activity in which the mistake occurred.

If the mistake occurs within a cognitive routine, the firming may become elaborate and may resemble the
preskill teaching that should precede the introduction of a cognitive routine. Juxtaposed steps of the routine
may be presented as a cluster. More than one cluster may be designed. When the learner is firm on the
clusters, the entire routine is presented and firmed.

In addition to discrimination mistakes, the learner may make response mistakes. The two basic types of
response problems are those of not being able to produce the response acceptably, and those of not producing
the response when it is called for. We have examined both types in earlier chapters. The new element that is
introduced when we deal with corrections for these responses is that of potential punishment. If the learner is
consistently failing at a high rate, the activity is likely punishing and the learner has probably developed
avoidance behaviors and superstitious behaviors for dealing with the activity.

If the mistake occurs within a response-teaching activity, we should follow the usual procedure of
shaping the response or the context. We should also try to change the activity so that it is reinforcing. The
basic procedure for doing this is to add some unique behavior to all examples. If we start with easy examples,
the unique behavior serves as a signal for a reinforcing activity. The implication is that when the behavior
occurs with other activities, these activities are also reinforcing.

The same strategies apply to situations that are not designed to teach responses. We can create a success
set by adding unique behavior to the tasks that involve the response and by assuring that the learner achieves
a fairly high percentage of successful trials with these novel tasks.

The problem of appropriate juxtapositions is also important for response-teaching. The reason is that the
context of a complex activity may imply a false operation. The repeated work on the response may convey
the idea that the production of the response is the operation called for by a discrimination task. The remedy
for this problem is to make sure that the part of the operation that involves a response is removed from the

 
 other parts and that there is a discriminable instruction for working on the response and a different instruction
for “figuring out the answer.”
 

 
 Program Revision and Implementation
Implementation involves a number of issues: (1) the field testing of material; (2) the training of teachers
and aides; and (3) the use of data to monitor teacher and student performance. A full discussion of these
issues is beyond the scope of this work. However, program implementation is relevant to the full
development of programs. This chapter, therefore, describes a system of logically-based principles for
achieving consistent implementation.

The Logical Ingredients

For implementation to occur, three ingredients are essential: (1) a standard; (2) teacher behavior that can
be compared to the standard; and (3) expected student outcomes.

The standard is a very precise statement about what the teacher is supposed to do (according to the
program) and the anticipated student outcomes. Note that the standard is not necessarily correct. The
specified teacher behaviors may be inane and the predictions or assumptions about student outcomes
preposterous. The standard serves as a standard only in the sense that it provides a basis for comparing the
actual teacher behavior with that suggested by the standard and for comparing the actual student outcomes
with those implied by the standard. The student outcomes may not be stated by the standard; however, they
are implied by the activities the students are expected to do. If students work independently on a particular
skill, the teaching provided for that skill by the program is assumed to be adequate.

Teacher behavior is the actual behavior we observe in the classroom. If the program is fully
implemented, the assumption is that we would see teaching that corresponds precisely to the standard.

Student outcomes are the actual changes in student behavior. If the standard is accurate, the student
outcome should be consistent with the predictions of the standard.

A model of an ideal implementation would look like this:

Standard
No discrepancy
Teacher behavior
No discrepancy
Student outcome

This ideal shows no discrepancy between the standard and the teacher’s behavior and no discrepancy
between the teacher behavior and the student outcome. In other words, the teacher behavior that we observe

 
 is perfectly consistent with the standard (predicted in every detail by the standard) and the student outcomes
are those that are predicted. If this implementation were observed, the standard would be perfect in every
respect and so would the teacher’s performance.

If there are discrepancies between the standard, the teacher behavior, and the student outcomes, we can
draw inferences about the nature of the problem. The following diagram shows a situation in which there is
no discrepancy between the standard and the teacher behavior, but a discrepancy exists between the teacher
behavior and the student outcomes:

Standard
No discrepancy
Teacher behavior
Discrepancy
Student outcome

This situation shows that the standard does not work. The teachers are performing in accordance with the
standard; however, the predicted student outcomes do not occur.

Both diagrams above are greatly simplified, because they do not deal with the fact that both teachers and
students exhibit a range of variation in performance. The following diagram shows a possible range of
variation in teaching behavior.

Group A Teachers Group B Teachers

Standard
Discrepancy No discrepancy
Teacher behavior
Discrepancy No discrepancy
Student outcome

This model shows that the teachers who perform according to the standard (Group B) achieve the desired
student outcomes; those teachers who do not perform according to the standard do not achieve the desired
student outcome. The standard is valid. Those teachers who are not performing according to the standard
need training.

The following pattern of teacher behavior leads to the conclusion that the standard is poor:

Group A Group B
Standard
Discrepancy No discrepancy
Teacher behavior
Discrepancy Discrepancy
Student outcome

 
 Although there is a range of teacher behavior, the performance of students did not change in the
predicted manner for teachers who follow the standard (Group B); therefore, the standard is poor.

Another model shows variability in student outcomes within teachers who implement:

Standard
No discrepancy
Teacher behavior
Discrepancies with
some students
Student outcomes

When teachers perform according to the standard, they achieve the desired student outcomes with some
students, but not with others. The implication is that the standard is not fully valid. It must be modified if it is
to work with all students. Note that the modification of the standard does not necessarily involve a
prescription for all students. It may be that the revised standard specifies procedures for students who have
trouble with particular tasks presented in the unrevised standard. The standard then specifies some new
procedures for those students (specific “firming techniques,” different forms of practice, etc.).

Another variation in the model shows that some activities within the program are invalid:

Standard
No discrepancy
Teacher behavior
Discrepancy on
some activities
Student outcome

All students tend to succeed on some tasks and fail on others. The implication is that specific parts of the
standard are faulty. These parts do not achieve the outcomes predicted from them, even though the teacher
behavior is in strict accordance with the standard.

From all the situations we have described, we can draw specific implications about the validity of the
standard. The reason is that in all variations, the teacher behavior corresponds closely to the standard. If the
teacher behavior deviates from the standard, we are left with ambiguous information about the validity of the
standard. Consider this situation:

Standard
Discrepancy
Teacher behavior
No discrepancy
Student outcome

 
 Exactly what implication does the student outcome hold for the standard? The outcome does not show
that the standard is invalid, merely that it is possible to achieve the desired student outcomes through a
different standard (or the variety of standards that describe the observed teacher behavior). We do not know
how effective the stated standard is because the standard is not tested in this implementation. The outcome,
therefore, is not of great value for learning about the standard or about procedures for training teachers to
achieve the specified behavior for that standard.

In the following model the teacher behavior does not correspond to the standard:

Standard
Discrepancy
Teacher behavior
Discrepancy
Student outcome

The fact that the students do not achieve according to the standard provides us with no information about
the potential of the standard to effect the desired student outcomes. The standard was not tested. The teachers
used some other standard and the student outcomes show that the standard they used was ineffective in
achieving the student outcomes predicted by the original standard. It would be appropriate, however, to judge
the observed standard(s) (based on teacher behavior) on the basis of the observed student outcomes (just as it
would be appropriate to judge the observed standards for the preceding example as being successful).

Is the Standard Manageable and Trainable?

Let us assume that we begin with the situation depicted in the last diagram. To test the standard in
question, we train the teachers and provide whatever kind of monitoring of teachers we judge necessary to
maintain their behavior in the classroom. We must use our best judgment about how to train. At this point, we
are not immediately concerned with the student outcomes, merely with establishing teacher behavior that is
specified by the standard.

By observing the variability in teacher behavior after training, we can draw conclusions about the
effectiveness of our training remedy. Let us say that we exhaust different training and monitoring possibilities
and none has a substantial effect on the teacher behavior. The teachers uniformly do not perform according to
the standard.

Standard
Discrepancy
Teacher behavior

Although a possibility exists that we are very poor at training, a more compelling conclusion is that our
standard, regardless of its potential for teaching the student, is unmanageable and must be discarded or

 
 revised. There is little point in retaining a standard that we cannot reach through training.

After training on the standard, however, a different result may occur. Some teachers perform according
to the standard and others do not.

Group A Group B
Standard
Discrepancy No discrepancy
Teacher behavior

This situation raises questions about our standard and our teaching procedures. The fact that the training
we used worked with some teachers permits us to take the next step of looking at student outcomes for the
no-discrepancy teachers. If these students are not performing according to predictions, we would not retain
the standard in its present form. If the teachers who teach according to the standard achieve predicted student
outcomes, however, the implication is that the standard is valid, but that the training procedures do not work
for all teachers. We would try to find better training procedures for the teachers who did not respond to the
previous efforts.

A situation similar to the variability in teachers occurs if, after training, teachers generally perform on
some activities or parts of the standard, but not on others.

Part A Part B
Standard
Discrepancy No discrepancy
Teacher behavior

This outcome suggests that the teacher training for some components is weak. Possibly the weakness
results from poor training techniques, possibly from an unmanageable or unreasonable standard. Before
refining the standard or the teacher-training procedures, however, we should observe the student outcomes. If
they are the outcomes predicted by the standard, refinement of weak parts of the standard or of teacher-
training behavior for these parts is implied. However, if student outcomes for the non-discrepant students are
not as predicted, the standard should be discarded.

The basic implementation strategy followed for all situations is to create a situation in which there is no
discrepancy between the standard and the teacher behavior with at least some teachers or at least some tasks.
Then we:

1. Compare the student performance on the no-discrepancy components with the predicted
performance.

2. Respond to the information provided by these observations (producing teacher-training

refinements, refinements in the standard, or both).

 
 Without detailed knowledge of both the teacher performance and student outcomes, we cannot
intelligently deal with specific possible weaknesses in either the training procedure or the standard.

Implementation Designed to “Shape” the Standard

Some implementations involve programs and training procedures that are not fully developed. These are
field tryouts of programs. Simple procedures based on the discrepancy model can be used to “shape the
standard” until it is workable. We save a great deal of time, however, if we begin with two assumptions about
the standard. These assumptions are expressed as two constraints.

Constraint 1: The standard must specify or clearly imply teacher and student behaviors. It should specify
the exact tasks the teacher is to present or the exact procedures that are to be executed. Unless the details of
the standard are explicated in a manner that is apparent to a trainer and teacher, the standard will need further
“shaping.” If the standard merely indicates that the teacher will “carry on a discussion about a particular
topic,” there will be an unacceptable range of variation among teachers. The reason is logical. It is possible to
“discuss” a topic in a way that will not induce the student outcomes called for by the activity. Therefore, the
specification that the teacher should discuss is not sufficient.

Constraint 2: The standard must analytically account for the induction of the various student behaviors.
In addition to being specific, the standard must be non-magical. We should be able to inspect it and determine
that there are provisions for achieving the student outcomes. The program should provide teaching
demonstrations, clear communications, tasks that deal with the topic, and provisions for practice. We should
be able to see that preskills of complex tasks are identified and pretaught, and that the teaching at any given
time does not seem burdened with new skills.

If the initial development of programs provides the degree of standardization and specificity suggested
by the two constraints, many problems of implementation or field tryout are obviated, and the program will
be close enough to being adequate to benefit from the field tryout.

Designing Field Tryouts

Field tryouts of instructional programs can be used to shape details of the program further and to provide
information about what is needed for adequate teacher-training. To be successful, however, the tryout must
not follow traditional procedures which use standardized tests, large samples, and modify the program
according to learning tendencies of students. The traditional approach is inefficient and usually incapable of
providing the designer with the type of information needed to modify the program intelligently.

Logical strategies for gathering tryout information are based on two assumptions about the nature of
implementation:

 
 1. The primary focus of all information gathering must be on failures, not on successes, and the
information must permit us to investigate the cause of the failure. We must know the qualitative errors
students and teachers make, because the only changes we make will be qualitative. All changes we will make
will be designed to improve the program. We therefore need information about where the program fails. We
will manipulate parts of the program that are associated with observed problems. We cannot manipulate the
program intelligently if we know only that students tend to achieve a particular score on an achievement test,
or even that they tend to miss a particular item. This information is merely a signal for us to secure more
precise information about the mistake. Once we know the precise mistake in the context of the instructional
situation, we can identify the causes of the mistake.

2. Our second assumption is that relatively lower performers make all the mistakes that higher
performers make and additional mistakes that higher performers tend not to make. It follows that if the
mistakes made by the lower-performing groups tend to include all mistakes made by higher performers, the
lower groups provide the best information about program problems. Therefore, tryouts should concentrate on
these groups. Low performing groups make the program problems and teacher-performance problems far
more obvious than higher groups do. Higher groups may yield the error information at such a low rate that we
cannot easily determine the pattern.

The two assumptions about implementation—the focus on faults and the idea that lower performers
provide the best information about program weakness—determine the general orientation of the tryout. The
design, perforce, will be different from that suggested by classical statistical evaluations. We are interested in
qualitative problems. Statistics direct us to quantitative problems. We are concerned with the context in
which a particular mistake occurs, because our assumption is that the events correlated with the mistake are
possible causes of the mistake (or provide information about the causes). Statistical procedures (as they are
traditionally used in tryouts) direct us away from such information. The manipulations that we will institute
to correct mistakes have to do with specific wording changes and specific changes in responses, but
aggregated data give us little information that helps us identify the specific types of changes that we should
make.

The Too-Difficult Program

We can design a field tryout so that a relatively few teachers yield a great deal of information about the
program. This goal is achieved if the program provides a high information yield and if the tryout employs
efficient data-collection techniques.

To achieve a high information yield, we violate our best guess about what would be perfectly appropriate
for the learner. We try to design the program so it errs in the direction of providing too little repetition and
structure and of moving too fast for the “average” student who will use the program.

 
 This strategy is important because of the paradox of the perfect program. If we design a perfect program
and the field tryout information suggests that no students have trouble with the program, we do not really
receive information that the program is perfect. We know only that it is easy for the students who use it. But
why? Is the program far too easy for them? Would they have been able to progress at twice the rate? We have
no basis for answering these questions. If we design the field-test program so that it does move too fast for
some students, we have a basis for identifying the optimum rate of presentation for the final program.

By following a similar strategy, we can determine the extent to which preservice or inservice training is
necessary for adequate teacher performance. Instead of providing all the information and training that we
think should be necessary to perform in the program, we initially provide teachers with less information, so
that we again err in the direction of providing what we feel is too little information. If all teachers perform
perfectly following our training, we will not know whether our training was far too laborious or whether it
was appropriate. If the training leaves some teachers in trouble, we receive precise information about areas in
which training is needed.

Analyzing Failures

If teachers have trouble teaching a particular type of exercise, there are two solutions: (1) change the
activity, or (2) provide training that permits the teachers to perform better.

A clue about whether training is needed is provided by teachers who follow the program closely. If these
teachers succeed, the standard (program) is apparently manageable. Therefore, the problem probably has to
do with training. If the only teachers who succeed deviate from the program, the standard is highly suspect.
(There is still a possibility that training would remedy the situation; however, the successful teachers provide
us with a solution that is probably more effective: change the standard.)

If most tasks in the program work as anticipated, teachers who succeed on some tasks and fail on others
provide very articulate feedback about which tasks or activities are not well designed. Their performance on
successful exercises in the program provides the teacher with baseline information about how well-designed
activities should work. The teacher presents the exercises; the students respond as specified. Most students
have no serious problems with the task, so no great amount of repetition and firming is needed. The poor task
stands in stark contrast. The teacher cannot maintain good pacing on the activity, either because the task
wording is awkward or because the students are not producing the desired responses. The task requires much
repetition and firming. Because these tasks stand in contrast with successful trials, the teachers are generally
quite accurate at identifying them as poor tasks.

The strategy of erring on the “too little” side does not mean that we leave the program or the teacher
training this way. We want to find the minimum set of training activities and program activities that serves
most teachers. We must revise as we learn about specific failures in the program.

 
 Designing Lesson Blocks for Tryout

An efficient way to respond to weaknesses in the program is to develop the program in small chunks,
perhaps 20 lessons at a time. If lessons are taught every day, 20 lessons should last teachers about one month.
If problems with a particular strand or track within the program become evident early in the 20-lesson block,
we can provide the teacher with replacement activities for those in the suspect tract. Providing substitute
lesson parts for perhaps 15 lessons is far easier than providing replacements for an entire school-year
program (140 to 160 lessons). If the problem emerges near the end of the 20-lesson block, we can design the
lessons in the next block so they compensate for the problem. (Often, the type of compensation that we
provide at this time is stop-gap. Our primary concern is that the teachers have lessons for every day they need
them. Later, when we revise the program, we can integrate revised parts of lessons into a streamlined
sequence.)

Some activities or procedures specified by the tryout program may have to be discarded or changed, even
though the teachers who follow the specifications for these activities succeed. The program interacts with the
training. We may discover that the amount of training needed to remedy the problem is impractical. Our
choice is to modify the program or accept the fact that most teachers will not perform according to the
standard. The most reasonable solution is to modify the program, because the activities are of no value if they
are not communicated to the student.

Tryout Data

The goal of data in a tryout is to help identify flaws in a product, not to discover a population tendency.
To identify flaws we use the type of tools that are implied by quality control. We inspect instruction through
direct observations. We also augment this effort with data-collection tools. We respond to observed faults in
instruction by providing quick responses, either in the form of replacements for parts of the program, or
training.

Observations. During field tryouts, we observe the teacher working with groups of students. The
observer should note:

1. Whether the teacher is following the specified procedures, and if not, what types of deviations
occur.

2. Which activities are apparently unsuccessful because of task wordiness or some other problem
with the standard.

3. Which activities are apparently unsuccessful because of poor student responses and other teacher-
delivery problems.

 
 4. Which activities are apparently unsuccessful because of sequencing or programming errors.

Note that errors in the program are inevitable. The program may schedule an introductory activity for too
many days. The examples presented for some activity may not precisely fit the wording of the task.
Necessary teacher directions may not have been included. Careful observations of a few teachers should
disclose nearly all these errors. The simplest procedure is for the observer to sit with a copy of the program
and follow along as the teacher presents. The observer marks each place at which the program is “lumpy.”
The observer does not try to solve the problems, but merely to report them. Observations of the problems
predictably shape programs in the following ways:

1. We modify procedures that require the teacher to present a series of examples using some material
not provided by the program (presenting a series of examples using a pencil to show slanted, not-
slanted, for example). Although teachers can be trained to present these sequences, the training is
difficult because tasks introduce variables not controlled by printed-page examples. Replacing the
examples with a simplified presentation that appears on the printed page is a much more
manageable alternative (even though the sequence may not be as precise for communicating with
the students).

2. We reduce the number of variations in wording for a series of fades or covertizations of a

particular activity. It is generally better to have fewer covertizations and to have each run longer
than it is to have more frequent covertizations. From the students’ standpoint, a greater number of
covertizations is desirable; however, teachers have mechanical problems learning the subtle
variations for activities that are highly similar.

3. We present a given procedure longer than is needed to teach the students. Two problems exist: (a)
students are absent, and (b) teachers should be reinforced for following the program. If the
program activities change too frequently lesson to lesson, teachers tend to be punished and often
stop following the program precisely. If the program is as tightly sequenced as it should be for
optimum instruction, students who are absent for a day or two tend to fall behind. In a traditional
setting, absence does not present a problem because the program does not permit teaching to
mastery. A tight program sequence, however, makes the returning student’s deficiencies apparent
and frustrating. The situation can be appeased somewhat if there is unnecessary repetition in the
program.

Error data. The basic assumption about diagnosing errors is this: If the teacher follows the program and
if the students have trouble with specific items, the program is faulty and must be changed. For error data to
be of value, therefore, we must know that the teacher follows the specifications of the program. The teacher
must not tell students answers when they are working independently. If the teacher helps students, their

 
 worksheet performance does not reveal performance problems; furthermore, it does not imply what students
know about attending to the teacher presentation, retaining the information, and applying it to independent
work.

The analysis of errors on the worksheet reveals items that tend to be missed more frequently than other
items. The more frequently missed item may be poorly worded or may present “unusual” examples. The
students may have answered correctly, but the scoring key may not consider their answers correct. This type
of problem is common in field-test programs. For example, a story for a field-test teaching program told
about Joe Williams, who is a red, number 4, wide, felt-tipped pen. The item: “What was Joe Williams?” The
item is poor because the scoring options are enormous. To solve the problem, a series of items replaced the
original: “What color was Joe Williams? What type of tip did he have? . . .” A range of acceptable responses
still exists for the second question; however, the range is substantially reduced.

The best instrument for recording error data on worksheet exercises is a copy of the student worksheet.
The use of a coding procedure helps the recorder to classify the different types of errors. Separate tallies
should be kept for NR (no response) and for each substantive type of error. Those errors that cannot be
classified as a type should be recorded. The worksheet page in Figure 28.1 shows error summary for ten
students on a worksheet. All items that are missed by more than 20 percent of the students receive scrutiny
(item 9).

 
 Figure 28.1

From Reading Mastery, Workbook IIIB, by Siegfried Engelmann and Susan Hanner. Copyright ©SRA/McGrawHill 1982. Reprinted by permission of the publisher.

 
 A second check on item problems comes from the analysis of the high-performing students. The
probability is high that any items they miss are weak items. If we use six high performers as our target group,
we simply note any items that are missed by two or more students in this group. Usually these will be items
missed by many other students. Responses of higher performers often give dues about the cause of the
problem.

The percentage of all students missing an item provides a general warning signal that an item may be
poor. However, the fact that a high percentage of students misses a particular item does not imply that the
item is poor or that it should be changed. A 25 percent error-rate may not represent a serious problem,
particularly if it occurs on the first day that a new skill has been transitioned to independent work. Often,
students’ error-rate on similar items drops to 10 percent on the following day. The 25 percent item might
warrant no more than bonus points for students who make no mistakes on the newly-transitioned items
(which are marked with stars on the worksheet).

Teachers’ verbal reports about program problems. In addition to direct observations of teaching and
error data, teacher verbal reports provide information about program problems. An interesting phenomenon
associated with programs that are well-designed is that teachers who have never taught effectively before
being involved in the tryout can, with a minimum of training, tell about tasks or activities that are relatively
poor. They may be wrong about the cause of the problem; however, their identification of weak tasks is
surprisingly accurate and reliable.

The reason is that the teacher is provided with a baseline against which to measure the effectiveness of
various tasks. Given that most of the tasks in the program are reasonably well-designed and that the teacher
follows them fairly closely, the teacher learns very quickly about how students respond to these tasks. For the
poor tasks, the pacing is wrong, the students do not respond in the specified manner, and corrections are
awkward. The teacher does not receive good information from traditional programs because these programs
do not provide well-designed tasks that serve as a baseline for judging poor tasks.

Summary: Designing Field Tryouts

If we start with the assumption that the teachers and students will provide us with information about the
weaknesses in our program, we can use field-tryout information to shape the program. We must design our
information-gathering techniques so that they are sensitive to problems. The techniques must then identify
them quickly and accurately. We must then respond by remedying the program.

Weaknesses in the standard or program specifications are indicated by unpredicted student outcomes. If
the students do not learn the intended discriminations or skills, the program does not work. For this judgment
to be valid, however, we must know that the teachers are following the program specification, or trying to. If
they are trying but are unable, we must consider redesigning the standard so that it becomes more

 
 manageable. By designing the tryout so that both teachers and students will tend to make more errors
(because the program moves too fast or provides insufficient information), we will discover precisely where
additions are needed and what kind are needed. By analyzing error data, we identify weak parts of the
program. We revise specific tasks and items. Through careful observations of teachers, we can shape the
training for specific skills and skill clusters. If we continue to pursue the fairly simple procedure of attending
to program failures, assuming that these failures are caused by inadequacies in the program, and changing the
program, we will “shape” the details of the program until the program approximates a faultless one. Of
course, it will never actually become faultless; however, the various details will have been improved and the
program will certainly perform far better than the original.

The Role of Trainers

Training teachers is a microcosm of implementation. Like the larger implementation, training involves a
standard, behaviors, and outcomes. The difference is that the standard is the specifications for preparing the
teacher to teach; the behaviors are the behaviors of the trainer who works well with the teacher; and the
outcomes are the changes in teacher behavior caused by training.

The same problems that characterize the larger implementation characterize the training program, and the
same sorts of information shape the training program. Also, the same constraints are introduced to make the
instructional program more manageable and to facilitate the training.

The training program is primarily concerned with inducing the desired presentation behavior in the
teacher. The standard should provide clear specifications about how the teacher is to perform after training.
The training should pass the test of logical inspection with respect to provisions for inducing the behaviors
that are required. The training should be standardized with provisions for working with different types of
teachers.

The most sobering constraint for training is time. Typically, a trainer must work with as many as 20 to 30
teachers. This situation is parallel to one in which one teacher must work with 20 to 30 students. The
available time must be used efficiently if substantial teaching is to occur.

Effective training is possible only if the instructional programs used by the teachers are standardized
across all teachers. We will arrive at this conclusion if we seriously pursue training in a situation that permits
each teacher to use a different program to achieve instructional objectives. Here are facts about this situation:

1. If the tasks designed by the teacher are poor, they must be changed before the trainer works on
specific presentation behaviors. We cannot work on presentation behaviors if the best possible
presentation will lead to student failure. Working with poor tasks will provide the teacher with
very poor information about presentation variables and about what the students are capable of

 
 learning. The tasks must have the potential to communicate unambiguously.

2. The probability is great that the trainer would have to repeat the same basic program remedy with
different teachers. If one teacher has problems designing a program to teach a specific skill, other
teachers will experience the same problem. The mistake patterns for teachers parallels that for
students. If one tends to make a particular mistake, others will make the same mistake.

3. When different program sequences are used in different classrooms, the trainer is provided with
no warning signals about which teachers are falling seriously behind in the rate at which they are
teaching skills.

4. The possibility of the trainer responding to problems quickly is further reduced because of the
time associated with remedying the teaching problems. First the trainer must specify program
modifications; then train the teachers in the execution of appropriate procedures; then monitor the
tasks. If this process is to work with teachers who use various programs, extensive monitoring is
required in each classroom to determine precisely what each teacher is teaching and the effective
rate of skill teaching.

5. The work that the trainer does on program modification does not relieve the trainer of working on
specific presentation behaviors.

In this situation, the trainer will fail. The trainer will not receive adequate information about the nature of
problems, will not be able to work on common training problems, and will certainly not be able to fix up the
sequences the various teachers design.

If the program is standardized, the situation becomes far more efficient. Ideally, the program is divided
into lessons, each of which can be handled by the average teacher in a specified period of time. Ideally, all
tasks in the program are manageable, which means that the teacher should be able to achieve the responses
that are specified without having to repeat them an inordinately great number of times. Ideally, all responses
and teacher behaviors are specified in a way that presents no ambiguity.

The advantages of a standardized program are:

1. The trainer has pre-knowledge of tasks in the program that are typically failed by teachers. The
trainer knows that teachers need training in the specific behaviors called for by these tasks. The
trainer also knows that the teacher’s presentation of potentially troublesome tasks should be
monitored in the classroom.

2. Efficient classroom monitoring is possible because the trainer does not have to look at entire
lessons, merely parts that provide the most information about the teacher’s problems. The trainer
can use the lessons as a guide for scheduling classroom observations.

 
 3. With the removal of program-development responsibilities, the trainer’s basic job is now reduced
to working on presentation behaviors; a job that is made even easier if the program is well
designed and does not have a great many rough spots.

4. The content of in-service training can be responsive to the more common problems the teachers
are experiencing. If teachers tend to have trouble with specific procedures or tasks, the trainer can
work on these relatively effectively in a large group, rather than individually. The group work
represents a great time savings.

5. The fact that the rate of teaching can be reliably expressed in terms of lessons taught implies that
projections can be made on the basis of lessons. By comparing the number of lessons taught by a
teacher with the number of available school days, the trainer can quickly determine whether
teachers are progressing at the projected rate.

6. Assessing the progress of teachers (and students) is easier because lessons serve as mastery tests.
If we wanted to know, for example, whether a student has skills that were taught by lesson 60 in a
program, we would simply sample skills that appear in a lesson (tasks similar to those presented
on lesson 60). Formal mastery tests can be designed so they indicate the lesson range in which a
student would be appropriately placed; however, a quick observation of the teacher reveals the
mastery level of students.

Training is necessary for good implementation. If the district or school does not provide for relevant
training, no vehicle exists to assure that teachers are taught to the standard of the program, or that they are
required to meet criterion of student performance.

Implementations of “Final” Programs

The purpose of the field tryout is to shape the standard (the program). After the program has been revised
and the training procedures have been refined, the program is implemented on a broad scale. This
implementation is similar to the field tryout in some ways. The focus remains on failure and on efficient
techniques for identifying failure. Usually, the remedies provided for failure will not involve great changes in
the standard, but will be teacher-training remedies. For some students, the program may move too slowly,
implying acceleration of part or all of the activities. Some groups may need firming or repetition in addition
to that provided by the program. These and similar problems constitute the main focus of the implementation;
however, modification of program sequences is also undertaken. (The program may have mistakes that
somehow slipped through the field tryout—a situation that is more common than program designers like to
see.)

The major difference between the typical on-site implementation and the tryout of an instructional

 
 sequence is the availability of knowledgeable trainers. If we are to use titles as a guide, we would conclude
that the school districts have people who can serve as trainers (curriculum specialists, etc.). However, these
people typically have no training expertise and no precise knowledge of program development. If training
and monitoring are to succeed in typical school settings, most of the monitoring and training functions must
be performed by personnel who have limited training skills—building supervisors, school psychologists, and
people in similar roles. Here are general guides for effectively using available personnel and outside trainers.

1. If possible, use experienced trainers who have detailed understanding of teaching the program for
formal training sessions (preservice training). Preservice should give the teachers practice in the
actual behaviors or procedures that are called for by the first part of the instructional program they
are to use. (Note that there is no pressing need to train teachers in skills or procedures that occur
late in the instructional program. The teachers will probably forget the procedures before using
them. Over practicing the early program procedures is more efficient. If these procedures are
nearly automatic—at least in mock-presentation situations—the teachers will be in a far better
position to present in a classroom situation.) Preservice training should not be handled by people
who are unfamiliar with the program. Inexperienced people cannot identify potential problems of
teaching pacing, corrections, praising, and various mechanical details.

2. Train on-site personnel for performing classroom monitoring and specifying routine remedies.
Principals and others not familiar with the program can provide a very useful program-
implementation function by monitoring classrooms and observing what the teachers do. The facts
below suggest how an inexperienced person can serve a useful program-implementation role.

a. The skills that are observed tend to be those maintained by the teachers. If the pacing of
teachers is observed on a regular basis, teachers will tend to maintain better pacing. If the
teachers’ scheduling of time is observed, teachers will tend to maintain more exact time
schedules.

b. Observations do not always support teachers’ verbal reports. Some teachers report that
things are going quite well; however, observation discloses serious presentation problems. If
we assume that things are going well in a particular classroom and therefore do not monitor it,
we will make serious mistakes.

c. Many performance problems result from violations of very basic procedures. Perhaps the
most typical problem causing poor student performance is an inadequate daily schedule. The
teacher is trying to teach the program in unreasonably short or infrequent periods. Often, the
remedy is appallingly simple and involves starting earlier, specifying large-group instruction,
or opening activities, etc.

 
 As the facts suggest, people who are not familiar with details of teacher-training or program design can
be effective in three broad areas:

1. They can demonstrate there is real interest in specific areas of teachers’ performance. They can
reinforce teachers who attend to these areas.

2. They can deal with simple problems—scheduling, basic classroom management procedures, and
mechanical problems.

3. They can identify problems that are serious (without solving them). The monitor can report these
problems to a trainer.

Unless the monitor observes frequently, the monitoring role is not very effective. When behavior
problems have escalated or students have been mistaught a difficult skill, remedies are awkward. To assure
that the monitoring goes quickly, each classroom should be set up so that basic program information is
immediately accessible to the monitor.

Key information should be posted in the classroom. The posted information includes the daily schedule
and error data of students (or points earned) for each major subject. This information tells the observer:

• What the teacher is supposed to be doing at the time of the observation (according to the
schedule).

• Whether the teacher has been teaching a “lesson” each school day (inferred from the number of
lessons on which there is error data for the students).

• Which individual students or which instructional groups tend to have trouble (or are progressing
relatively slowly).

• Which activities tend to be missed by a relatively high percentage of students.

To augment the information that is provided by the posted data, the classroom monitor should identify
possible serious flaws in the teacher’s presentation. By watching the teacher teach, the observer can identify
these types of gross problems:

• The teacher is not doing the activities called for by the schedule.

• The teacher does not follow lesson script.

• The teacher’s pacing is poor.

• The teacher ignores student errors (instead of correcting immediately).

• The teacher does not reinforce students.

 
 • The teacher has serious management problems.

Some problems call for a trainer. Others, however, can be solved by another teacher. If the monitor
knows that teacher B works with the same sorts of children as teacher A, and that teacher B does not
experience some of the problems that teacher A has, the monitor may be able to arrange for teacher B to take
over A’s group. After B has taught them (and A has observed the teaching), A receives assignments about
holding the students at the level of performance established by B.

If the problems observed are serious and cannot be handled by on-site teachers, outside trainers should be
brought in as quickly as possible. (Sometimes they can specify remedies on the phone; however, they cannot
usually provide a perfectly adequate remedy unless they observe the problem.)

Performance Testing

Traditional evaluation with standardized achievement tests has only an oblique place in systematic
instruction. The tests are quite poor because they are designed from an aptitude model, which means that the
items on a particular test bear no necessary relationship to what is taught in the classroom. Items are included
on the test if they spread the distribution and exaggerate the individual differences (creating a normal
distribution curve). This type of test may play a role in categorical funding (determining on a “norm” basis
whether a particular student qualifies for a program for mentally retarded). The standardized nature of the
tests may also serve to provide a district with some universal yardstick for comparing the performance of
students in the district with others. (We cannot readily compare the students in different districts if we use
different tests or if we used tests that used with one group do not correlate highly with the tests used with the
comparison group.)

Norm-referenced evaluation (the type provided by the standardized tests) is not appropriate for designing
quality-control measures. Primary measures used to evaluate instruction must identify failures that may not
be obvious to the monitors. If a monitor makes efficient use of time, the monitor will certainly miss some
problems, because the monitor will not observe all teaching activities. The teacher may be “helping” the
students on their independent work. The error data are low, which suggests the students are learning well, and
observations in the classroom may not disclose the problem.

To identify problems that may be missed by monitors, we use criterion-referenced measures. These
measures should be designed so they test what is taught and should be correlated with the program the
teachers use. For instance, if a skill is introduced on lesson 50, it would be possible to test the skill before the
students reach lesson 60. The test items should be the same type of examples provided by the program. This
type of test is a continuous progress test. It is the most useful test for program implementation because it
identifies specific problems and clearly implies what must be done to remedy these problems.

 
 1. It suggests whether the teacher is inadequately firming students. If the posted lesson in the
classroom is always ahead of the performance on the continuous test, the teacher is not adequately
firming students. For example, the posted information shows that the teacher has taught lesson 60.
The test of student performance, however, reveals that the students generally cannot perform on
skills introduced after lesson 40 in the program. The teacher is teaching 20 lessons beyond the
performance level of the students. The implied remedy is for the teacher to return to the lesson at
which the students perform and to teach to a higher criterion of performance.

2. Test results also imply whether individuals within a group are being firmed. If one or two children
in a group consistently do not score at the lesson range the teacher is presenting, these children are
not being taught in the group. The remedy is to provide additional teaching for these children—
perhaps transferring them to a different group.

3. Also, the tests can be used to draw conclusions about specific tracks or groups of related exercises
in the program. If students score many lessons behind the lesson being taught in one track, the
teacher must be provided with techniques that permit more consistent firming within this track.

4. A parallel set of inferences derive for skipping groups of students ahead in the program, skipping
individual students, and for skipping specific tasks in the program. If tests show that the group is
ahead of the lesson on which the teacher is teaching, the group may be skipped. (Note that this
remedy does not always follow from the fact that the group performs on a higher lesson than the
teacher is presenting. Some skills, such as reading, are not merely “teaching” new information,
but are providing practice. If the practice is to facilitate reading fluency, the material that is read
should be relatively easy, which means that the students would be able to perform on a higher
lesson than the teacher is presenting. Typically, students should perform 20-40 lessons beyond the
one that is being taught. Moving students to the point at which the learner can just manage the
material is not a good strategy.)

5. Trends across different teachers show teacher problems and difficult tasks. If teachers tend to
have problems with a particular task, a group remedy is implied. Teachers should be trained to
teach the track. If only a small percentage of teachers have problems with a particular track, an
individual remedy is implied. Some one-on-one training is needed and perhaps it can be provided
by another teacher.

In summary, continuous tests do not provide norms, but document the extent to which the program
delivers the skills it is supposed to deliver. The tests provide trainers and monitors with back-up information
about problems or failures. If used properly, continuous tests can provide timely information about teachers
who need additional training, groups that are weak, individual children who are not properly placed in the

 
 program, and parts of the program for which additional training is needed. They can also be used to provide
long-range projections of student performance. By extrapolating on the basis of the teacher’s effective rate of
teaching (lessons mastered during so much time), we can determine how far the teacher will get by the end of
the year. (If a group has mastered 60 lessons in 80 school days, we would project the group to master 130
lessons by 160 days, the end of the year.)

Other tools help the implementation and monitoring processes. Monitors should keep records of when
they see teachers and what sorts of comments or assignments they provide the teacher. The monitor’s
summary of teacher observations should be kept in a log, so that the trainer or monitor can follow up on a
regular basis and can provide those teachers who need more help with more frequent visits.

Copies of assignments or comments should be left with teachers, perhaps on a type of teacher-
performance summary form. Figure 28.2 shows a possible form. Note that it has places for assignments and
for checking the various behaviors that are being monitored.

Figure 28.2
TECHNICAL ASSISTANCE FORM

Observation of: Date: Subject: Arithmetic Language Reading

Observer: Lesson: Level: I II III

IMPLEMENTED Formats Reinforcement Individual turns

WELL! Signals Correct non-responding End-of-lesson
Pacing Correcting errors Individual test
Firm-up
Will check back (date)
Assignment:

Checked back on:

Additional tools and more sophisticated ones may be used in implementation; however, their purpose
must not be subverted. Continuous test information is valuable if it is received currently. Remedies based on
problems that have existed for months are not intelligent remedies. Teacher performance forms and
summaries of monitor’s visits are useful only if they are used to identify problems and to solve them. All
tools and procedures must reflect the basic theme of implementation, which is to identify failures and
problems—and to respond to them quickly.

The Corrective Reading Program—A Case History

 
 The development of SRA’s Corrective Reading Program illustrates the use of tryout strategies to refine
the program and to identify teacher-training needs.

Phase 1 of the program’s development involved four groups of students, most of whom were in junior
high schools. Four co-authors of the program taught groups of students and trained and monitored several
other teachers.

The initial version of the program was designed to provide information about how much the student
behavior would change if students received directed practice in reading stories. This version of the program
contained only the simplest instructions about management, and no teacher-directed word-attack exercises
(work with words not in the context of the stories). Stories were developed according to the error analysis of
students. The early stories focused on regular words (those that have clear sound-symbol relationships for all
letters). Later, stories attempted to provide practice on specific words the students confused, such as big and
dig. (Several stories about “The Bug That Dug” focused specifically on b and d.) A series of stories
involving “Chee” presented fairly heavy practice on those words and strategies that were most difficult for
students. Chee was a dog who spoke in gibberish when she became flustered, uttering such things as, “Oh, to
do so if go what then there who where.” The words that appear in these sentences are those the students
tended to miss the most.

The stories failed to change some aspects of the students’ behavior. Additional stories were interpolated
into the original sequence of stories and a second wave of students began the program. When the error
patterns of the students showed which behaviors were still changing at an inadequate rate, word attack
exercises were introduced. These exercises focused on the most frequently missed words that the students
read in the context of the stories. The addition of the word attack exercises represented an educated guess
about what would be needed to change some of the behaviors that persisted.

The next tryout wave focused on complete training for teachers—a management system with points,
complete word-analysis exercises, and training presentations on management.

Later research by Linda Meyer (1979) suggests that some of the correction procedures specified in the
edition of the Corrective Reading Program (the fourth major revision of the program) are not entirely
satisfactory from the standpoint of teacher training or of student performance. Other improvements in the
sequences, the stories, and the word-attack exercises have been suggested by specific program problems. In
all, however, the strategies used to develop the program were quite efficient. The initial tryout disclosed the
number of errors students made on words like a and the, what and that. It showed that, typically, more than
a thousand practice trials were needed before the students performed at a highly accurate level when reading
these words in sentences. The tryout showed the consistency of progress across students of different ages
(showing that the older students require more practice to learn how to decode accurately). The tryout

 
 suggested words and word-families that should be practiced. It demonstrated the typical problems that the
teachers had in presenting exercises to students, and it implied the kind of training or orientation that teachers
needed to present the program. The field tryout, in other words, shaped the specific content presented to the
students, shaped the kind of tasks the teacher was to present, and shaped the communication needed to train
or orient a teacher.

The tryout for the Corrective Reading Program relied heavily on empirical data because there is no
prima-facie or logical way to determine the difficulty students will encounter when trying to relearn accurate
decoding. The initial development of a new-teaching program relies far less on empirical information and
more on the analysis of the skills; however, a new-teaching program should go through the same sort of
shaping received by the Corrective Reading Program. It should be put through the type of test that shows
which parts of it work, which do not, and why some parts do not. The tools needed to document the
performance of the teachers and students are those that show specific errors and when they are made. Tools
needed for the trainer are those that permit the fastest, most effective observations of teachers.

Microcomputers

If a program is standardized, we can identify problems with the program (inadequacies of the standard)
and deficiencies of the teacher behavior. Without standardization, we lose information. With the loss of
information goes a loss of ability to respond intelligently to problems. When the program is standardized we
know where it is weak, which parts present problems, and where both teacher and student failure will most
likely occur. Standardization begets simplified procedures for teaching and training.

Microcomputer technology is capable of achieving greater standardization, of eliminating many teacher-

presentation problems, and of providing a better medium than the printed page. The computer terminal can
display continuous changes, pace presentations very precisely, and respond to a variety of contingencies that
are not easily handled through the medium of the printed page.

The model below shows how the computer simplifies implementation:

Standard-teaching

Student outcomes

No discrepancies can exist between the standard and the teaching because the standard and the teaching
are inseparable. A variable has been eliminated from the implementation paradigm. The microcomputer has
incredibly great potential for achieving standardized excellence, if the technology of instructional
communication is intelligently combined with the computer technology.

At the present, computers are used primarily for drill and practice. This application may be reasonable,

 
 but certainly does not exploit the potential that computers have for eliminating teacher-presentation variables
(including training variables) for exercises that are difficult to present. Higher math, physics, sophisticated
comprehension skills, and similar activities are typically very difficult for teachers to present. However, a
very realizable possibility is a computer presentation that could convey these difficult communications in a
nearly faultless manner.

Summary

For field tryouts to be successful, they must identify problems and they must do so efficiently. By
identifying the standard, the teacher behavior, and the student outcomes, we become aware of the logical
problems associated with drawing conclusions about the program.

• If teachers do not perform according to the standard, the outcomes cannot tell us about the
program that is being implemented.

• If extensive training does not bring the teachers’ behavior into compliance with the standard, we
can assume that the standard is unreasonable and should be changed.

• If teachers who perform according to the standard achieve the predicted student outcomes, the
standard is reasonable.

• If parts of the standard are consistently failed by teachers who perform according to the standard
on other parts, the standard must be revised or training must be provided for the weak parts.

By taking into account the pattern of teacher behavior and the pattern of outcomes, we can draw
inferences that shape the standard and shape the training program. The field tryout will tell us which parts of
the standard are weak, and it will suggest specifically how those parts are weak. Similarly, the tryout will
imply what kind of training is needed.

Training procedures are shaped in the same manner that the program is shaped, because the
implementation of training parallels that for student instruction. Like the instructional program, the training
procedures become standardized as we receive information about the types of training problems that exist and
the range of individual variation among the teachers to be trained. The outcomes suggest the parts of our
standard that are weak and the parts of the trainers’ behaviors that are deficient.

The data collection tools used in a sensible tryout are those that provide the most detailed information
about what is going on. If we observe problems, we assume that they are the product of our standard or of our
teacher training. Without observing the events that are correlated with specific problems, however, we will
not have the type of information needed to draw qualitative conclusions about problems.

If we know only that the students missed an item, we may assume that the program caused the problem,

 
 but we do not know the cause because we do not know the specific mistake.

If we know the type of error the students made, however, we have much more specific information about
how the program or training failed.

To secure the maximum amount of information, we try to work with teachers who will disclose problems
at a reasonably high rate. These are relatively unsophisticated teachers who work with lower-than-average
students.

We also design the program so that it is slightly less structured and faster-moving than we think it should
be. To establish a reasonable rate for the revised program, we must have this information about the program
limits.

For implementation of a program that has already been shaped through field tryouts, training and
monitoring are needed. Both must be designed from the standpoint of quality control. They must focus on
failure. They must be designed to identify problems quickly and in a way that implies a remedy.

A great deal of monitoring can be handled by a staff that does not have extensive teaching or training
skills. Their classroom observations of a lesson disclose whether:

1. The teacher has been teaching at the prescribed rate (based on the number of the lesson being
taught).

2. The teaching follows the standard (based on the teacher performance).

3. The students are responding as the program predicts.

The tools that are needed for such monitoring are fairly simple. They include performance forms that are
given to the teachers and information that is posted in the classroom—the schedule and the error data for
different subjects being taught.

To identify problems that an observer misses, criterion-referenced tests are needed. The tests should be
administered frequently. Their results provide information about trends and problems. When used with
classroom-observation data, they permit projections of how far students will progress during the school year,
which groups of children need special work, and which teachers need additional training.

When effective quality control measures are used, a school district has a basis for serving teachers and
students. The formula works, however, only if all quality-control ingredients are present. The instructional
programs must be standardized and must work well. Training must be provided for the teachers. Some form
of classroom monitoring and continuous-progress testing must be used to provide current and accurate
information about specific problems. Finally, there must be immediate responses to the observed problems.
As ingredients are removed from this formula, the probability of a quality implementation decreases, very

 
 quickly reaching an unacceptably low level.
 

 
 Section IX

Research and Philosophical Issues

This section answers two important questions that have not been dealt within preceding sections:

What database is there to suggest that the principles and procedures derived from the stimulus-locus
analysis are valid?

To what extent are the philosophical underpinnings of the analysis consistent with the assumptions of
science (inductive reasoning) and logical analysis?

Chapters 29 and 30 present summaries of studies that are based on the major details of the stimulus-locus
analysis. Note that these chapters provide only brief summaries of the studies and do not include all the
available studies.

Chapter 29 deals with the details of communications (juxtapositions, range of example variation,
overtness of steps for complex operation, and related variables). Chapter 30 presents:

• Studies that diagnose instruction and identify learner problems that are implied.

• Studies that achieve outcomes that would not be predicted from normative data of subjects.

• Studies concerned with the teaching of generically new skills and that therefore involve
application of both principles for inducing new responses and stimulus-locus principles.

Chapter 31 discusses key theoretical issues that influence details of the stimulus-locus analysis—the
departures from a strict behavioral analysis, methods of analyzing sets of stimuli, the relationship of the
analysis to historically prominent theory, and the theory’s relationship to current theories of cognition and
learning.
 

 
 Instructional Research: Communication Variables

Basic-Form Communications

The studies in the first section of this chapter clarify the role of specific communication variables on basic
learning outcomes. The studies are based on the logical assertions of the stimulus-locus analysis. The analysis
provides assertions about the role of positive examples, the role of negative examples, the nature of
differences between positives and negatives, the prediction of generalizations on the basis of samenesses, the
effects of stipulated sets of examples, and the importance of consistent teacher wording.

Only Positive Examples

A stimulus-locus assertion is that it is impossible to convey a given sensory discrimination through the
presentation of only one concrete example. A related fact is that it is very difficult to convey only one
interpretation if only positive examples are used. To test this assumption, Williams and Carnine (1981)
compared a positive-only treatment (consisting of 12 positive examples and no negatives) with a positive-
negative treatment that involves the same number of examples (8 positives and 4 negatives). Twenty-eight
subjects received training on the discrimination “Gerbie,” positive examples of which are angles between 0
and 110 degrees. The transfer test included both positive and negative examples, none of which had been
presented to either group during the training. The positive-only group scored 50 percent correct on transfer
items; the positive-and-negative group scored 88 percent correct. The difference is statistically significant.

Sameness

According to the principle for showing sameness, if juxtaposed examples differ greatly and are treated in
the same way, interpolation is implied. A study conducted by Carnine (1980a) tested this principle by varying
the range of positive examples presented to different groups. The stimulus-locus prediction is that the group
receiving the widest variation would perform best on a test involving new (generalization) examples.

Forty-seven students in grades 2-5 who had some knowledge of fractions, but not of decimals, were
randomly assigned to two groups. Both groups received the same number of demonstrations about how to
convert fractions with denominators of 100 into their decimal equivalents. The Full Range group received
demonstrations with three types of fraction-to-decimal conversions of this form:

! !! !!!
to .OX to .XX to X.XX
!"" !"" !""

 
 XX
The Restricted Range group received demonstrations of only this type of conversion to .XX
100

The transfer items were fractions not presented during training with a denominator of 100 and 1, 2, and 3
digit numerators.

The Full Range group successfully performed on 80 percent of the transfer items, whereas, the Restricted
Range group performed correctly on only 36 percent of the transfer items (a highly significant difference).

Correlation Between Same Behavior and Same Features

According to the stimulus-locus analysis, sameness is shown by treating examples that are the same in
the same way. The same treatment prompts the learner to discover how the examples are the same. If this
tenet of the analysis is correct, communications that are not precise in treating examples of the same concept
in the same way will not communicate as articulately as presentations that clearly treat the examples in the
same way.

Williams and Carnine (1981) investigated the hypothesis with 22 preschool students divided into two
groups. All received the same positive and negative examples of diagonal (called “Blurp”). Twenty pairs of
examples were presented, each on a separate page. For one treatment, students were presented the same
wording for all test examples: “Is there a blurp on this page?”

The other treatment presented different questions with different examples of blurp. “Is there a blurp
here?” or, “Do you see a blurp?” or “Are there any blurps on this page?” or “Can you draw a circle around
the thing that is not a blurp?” The prediction was that this treatment would tend to obscure the basic
sameness.

Students receiving the treatment in which the same response was produced for all examples reached
criterion (ten consecutive correct responses) in half as many trials as students who received different
instructions—a highly significant outcome.

Continuous Conversion

According to the analysis of communicating with the learner, continuous conversions increase the
obviousness of the difference between positive and negative examples. Gersten, White, Falco, and Carnine
(1982) compared continuous and non-continuous conversion sequences in four different studies. In the first
study, 40 preschoolers were randomly assigned to a continuous or non-continuous presentation of the concept
diagonal. In the second study, 40 different preschoolers were randomly assigned to continuous or non-
continuous presentations of convex.

The third and fourth studies were replications of the first except the students were handicapped. The

 
 examples in all four studies were the same for both the continuous and non-continuous groups. In the
continuous-conversion treatment for diagonal, a line segment was rotated to generate positive and negative
examples. In the continuous-conversion treatment of convex, a wire was moved to create positive and
negative examples of convex and non-convex.

The continuous conversion groups met the criteria of performance in significantly fewer trials. For
example, the continuous-conversion non-handicapped group in the diagonal study achieved criterion
performance in 20.6 mean trials, whereas the non-continuous conversion group required 56.4 mean trials. The
continuous-conversion non-handicapped group for convex required 10.5 mean trials, whereas the non-
continuous conversion group required 15.8 mean trials. Again, all differences were statistically significant.

Effect of an “Irrelevant” Difference

The stimulus-locus analysis asserts that if only one difference occurs between positives and negatives,
that difference must account for the labeling of the examples as positives or negatives. Stated differently, the
presence of the same features in both positives and negatives logically rules out possible interpretations.
Carnine (1976a) investigated the extent to which college students respond to these logical implications.
Thirty-eight students were assigned to three treatments. The same artificial concept was presented in all three
treatments. Geometric figures with one, two, or three points were treated as positive examples; and examples
with four or five points were treated as negative examples. In one treatment, a dot pattern appeared in both
positive and negative examples. Since this feature appeared in both positive and negative examples, the dot
pattern could not logically serve as a basis for determining whether an example was positive or negative. In a
second treatment, the dots appeared only in positive examples (figures with one, two, or three points). In the
third treatment, the dots appeared only in negative examples (figures with four or five points). Transfer test
performance indicated that subjects learned the generalizations consistent with the different presentations.
When dots appeared in both positive and negative examples, dots were ruled out as a basis for the concept on
almost 100 percent of the trials. Subjects responded only to the number of points. When dots appeared only in
positive examples, almost 100 percent of the subjects responded to the presence of dots and responded at a
chance level to the presence of points. Similarly, almost all subjects in the third treatment group (dots present
only in negative examples) responded to the absence of dots on a transfer test (treating items as negatives
only if they had dots). The findings indicate that subjects responded to the logical possibilities. Related
studies have been conducted by Gibson and Robinson, 1935; Harris, 1973; Overing and Travers, 1967;
Rosenthal and Zimmerman, 1973; Samuels, 1973; Williams, 1969.

A Test of the Minimum-Difference Principle

A negative example logically rules out the maximum number of interpretations when the negative
example is least different from some positive examples. To test the extent to which subjects responded to this

 
 logic, Carnine (1980a) developed five sets of examples for the concept on. (In the study a nonsense label was
used for the concept.) All had the same positives, but different negatives (see Figure 29.1). Set A showed the
minimum difference between positive and negative, while Set E contained no negative and therefore
generated the largest number of possible interpretations.

Table 29.1

Five groups of 13 preschoolers (65 children aged 4 to 6 years) received training, each group working on
one of the sets of examples. After receiving a fixed number of demonstration examples, all children received
a transfer test. Carnine found a significant linear trend between the similarity of positive and negative
examples and correct response on a transfer set. Children presented with Set A examples responded correctly
to 10.2 transfer items, while children presented with Set E responded correctly to only 5.0 transfer items. The
trend clearly suggests that the greater the number of possible interpretations consistent with the set of training
examples, the greater the probability that some students will learn an interpretation that leads to an
“inappropriate” generalization (from the teacher’s point of view).

The Setup Principle

Continuous conversion is logically superior to non-continuous conversion because if one example is

converted into the next, only some details of an example are changed to create the next example. A number of
features remain unchanged from example to example. If a change in example leads to a change in label (from
negative to positive), whatever details remain the same are irrelevant to the change in label. If a change in
example does not result in a change in label (the example staying either positive or negative), whatever
details change are irrelevant to the label.

 
 The setup principle is derived from this logic. This principle implies that a communication will be most
precise if the maximum practical number of details remain the same in all positive and all negative examples
within the initial communication.

Tennyson, Woolley, and Merrill (1972) and Tennyson (1973) have suggested a different procedure, one
that changes the setup features after every second example (two examples with the same setup features are
presented, followed by two examples with the next setup features). To determine the relative effectiveness of
different setups, Carnine (1980b) taught the discrimination of 90° or more to 30 preschool children using
three different variations in the setup features. For the continuous conversion group, all examples were
created through continuous conversion to assure that a maximum number of details remained the same. For
the non-continuous conversion group, the same set of examples and the same order of examples were used;
however, the examples were shown on cards and were not created through continuous conversion, thus, the
sameness from example to example would be less obvious. For the paired-setup group, as suggested by
Tennyson, et al, the same setup features appeared in two juxtaposed examples and were replaced by different
setup features in the next pair of examples.

Following initial training, all students were trained to a criterion on the same set of transfer examples.
These examples were non-continuous and contained setup features that were found in none of the earlier
training examples.

The most efficient procedure proved to be the continuous conversion presentation. Mean trials to
criterion for students in this group was 10.6. The non-continuous procedure was the next-most efficient (with
mean trials to criterion of 15.8.) The least efficient procedure was the paired-setup presentation. Students
required an average of 26.0 trials to meet the specified criterion of performance. The linear trend was
statistically significant.

Juxtapositions for Showing Differences

The difference principle asserts that when minimum-difference examples are juxtaposed and treated
differently, the intended interpretation is made most obvious. Granzin and Carnine (1977) conducted two
studies that compared the effects of minimum-difference juxtapositions of positive and negative examples. In
the first study, 44 first graders were randomly assigned to two treatment groups. Both groups received the
same set of positive and negative training examples. The only difference was the juxtaposition of the
examples. Minimally different positive and negative examples were juxtaposed for one treatment; maximally
different examples were juxtaposed for the other treatment. A conjunctive concept (positive examples
requiring the presence of more than one feature) was taught to all students.

The number of training trials needed for students to reach a specified criterion of performance was
significantly lower for the minimum-difference juxtaposition group. This group required 17.4 mean trials to

 
 meet criterion. The students in the maximally different juxtaposition group required 29.9 mean trials to meet
criterion.

In a second study conducted by Granzin and Carnine (1977), a disjunctive concept (the presence of at
least one of a possible set of features) and a conjunctive concept were taught to second graders. The pattern
of significant differences was the same as that for the first-grade study. Children who received the minimum-
difference-juxtaposition treatment reached the training criterion for the conjunctive concept in significantly
fewer trials than the children in maximum-difference juxtaposition treatment (means of 8.1 versus 19.1).
Differences were also significant for the disjunctive concept. The minimum-difference children required 21.1
mean trials while the maximum-difference children required 37.1 mean trials.

Stipulation

Sprague and Homer (1981) investigated stipulation with six severely handicapped subjects of high school
age. Subjects were first taught through procedures that stipulate behaviors that are relevant to operating a
vending machine. Following stipulated practice, subjects worked on a second set of examples sequenced to
demonstrate sameness across all vending machines. Three additional vending machines were practiced in this
set. The machines differed greatly from each other with respect to those features that are relevant to operating
the machine (the location of the coin slot, selection buttons, etc.). Throughout the training, subjects were
tested on ten “probe” machines that were judged to sample a very wide range of variation in vending
machines.

The results are consistent with the stimulus-locus prediction. No subject successfully operated more than
two of the ten probe machines following their practice with the first machine or practice with machines
highly similar to the first one. Furthermore, the specific errors subjects made in trying to operate the ten
probe machines were fully accounted for by the feature differences between the probe machines and the
machine(s) that had been practiced. On the final probe following the second set of examples (designed to
show sameness across different machines), no subject successfully operated less than eight of the ten probe
examples.

Homer and McDonald (1981) used a similar multiple-baseline design to test the effects of stipulation and
sameness training for the skill of crimping biaxle capacitors. Subjects were first trained on a set of capacitors
that were highly similar in size, shape, lead, color, etc. Following work on this stipulated set, subjects worked
on a sameness set—a small number of capacitors that differed greatly from each other in size, shape, lead,
color, etc.

The generalization of performance was tested through probes. Each probe presented 20 capacitors that
had not been presented during training. During the training on the stipulated set, none of the four subjects
correctly responded to more than five of the generalization items. During the sameness training, no subject

 
 responded correctly to fewer than 15 generalization items on any probe. The length of time that subjects
worked on the stipulated set had no apparent effect on the number of generalization items responded to
correctly during this phase of the training.

A specific prediction based on stipulation is that if all examples of decoding of connected sentences are
immediately followed by comprehension activities, the association of comprehension with the decoding of
connected sentences is strongly stipulated. The procedure, in other words, will stipulate that connected
sentences are to be read for meaning. The most direct way of assessing such stipulation would be to ask the
student questions and see if the decoding has resulted in “understanding.” An indirect way is to analyze the
type of errors the students make and determine the tendency to “self-correct” on the basis of syntax. Errors
students make should tend to “make sense” if students read for meaning. The tendency should be quite
independent of whether the children learn to read through “sounding out” or through “look-say.”

To test this assumption, Linda Carnine (1980) applied Goodman’s (1965) miscue analysis to 55 children
in grades K through 3, who were taught DISTAR® reading, which teaches sounding-out and which stipulates
strongly that sentence reading is reading for understanding. Goodman’s analysis shows the extent to which
the errors make sense grammatically and contextually. The percentage of errors that made sense
grammatically and contextually for the kindergarten children was 21; in first grade it was 24; and in third
grade it was 84. This trend suggests that although the children continue to work on the analysis of words in
isolation, the stipulation provided by sentence reading leads to acquisition of a “set” for anticipating meaning,
self-correcting words that apparently make no sense, and making the type of mistakes that are probable only
if the meaning context is understood.

Joining-Form Communications

A common prediction for communications is that if the presentation is relatively faultless, it will lead to
superior learning. The following studies deal with this prediction as it relates to the teaching of correlated-
features and transformations.

Correlated Features

The suggested tasks presented with each example of a correlated features task ask first about the
correlated feature and then ask, “How do you know?” The answer to this second question relates the
correlated features to the features of the example. The answer shows which aspects of the example predict
this correlated outcome.

It is possible to present the same set of examples as basic discrimination tasks (without making the
correlation overt) by asking only the first question. The stimulus-locus analysis predicts that this procedure
would probably be ambiguous, because it would not show which features of the examples predict the

 
 correlation. The more overt procedure would show the correlation unambiguously.

To study the effect of overtly expressing correlations, Ross and Carnine (1982) taught the binary duality
relationship to children in grades 2 and 3 and in grade 5. The same set of examples was presented to all
children. The only difference was the overt treatment of the correlation.

The 44 children were divided into three groups. For the Discovery Group, the teacher presented each
demonstration example this way: “Five times six equals thirty. Is thirty a binary duality? No.” No reason was
given to suggest why not. The Rule-Plus-Discovery group received the same presentation of examples;
however, before receiving this communication, the group said the rule: “A binary duality is the answer you
get when you multiply two numbers. One of the numbers you multiply must be exactly two more than the
other number.”

The Rule-Plus-Overt Steps Group received this communication: “Six times five is thirty. Is thirty a
binary duality? No. How do you know? Because six is not exactly two more than five.”

The fifth-grade comparison showed that the Discovery Group was significantly poorer than either the
Rule-Plus-Discovery Group or the Rule-Plus-Overt-Steps Group. Only 26.6 percent of the Discovery Group
children met the training criterion; 92.8 percent of the Rule-Plus-Discovery Group children met the criterion;
100 percent of the Rule-Plus-Overt-Steps Group met the criterion. There was no significant difference
between the Rule-Plus-Discovery and the Rule-Plus-Overt-Steps groups.

With children in grade 2, however, the difference between the Rule-Plus-Discovery and Rule-Plus-Overt-
Steps groups was significant. Only 20 percent of the children in the Rule-Plus-Discovery group achieved the
training criterion, whereas 83 percent of the Rule-Plus-Overt-Steps children achieved criterion. Neither the
fifth-grade children nor the younger children tended to discover the relationship, even though: (1) the range
of examples logically needed to induce the generalization was provided; and (2) the examples were massed,
which should make discovery of the rule easier.

Rules and Applications

The stimulus-locus approach to the teaching of complex tasks is to teach the preskills that could affect
learner performance and to apply the rule to a range of examples so that sameness is demonstrated. Two
studies investigated the extent to which this approach improves learner performance.

In the first study, Kameenui, Carnine, and Maggs (1979) presented the following rule to first and second
graders: “The lower you eat on the food ladder, the more protein you get.” The students were divided into
three groups, two containing 12 students each, and one containing 24 (12 average and 12 above-average
students). The Rule-Only Group (12 students) received practice in saying the rule. The Rule-Plus-Concept
Group (24 students) received practice in saying the rule and exercises designed to teach the component

 
 concepts expressed by the rule (lower on the food ladder, protein, etc.) The Rule-Plus-Concept-Plus-
Application Group (12 students) received the training presented to the Rule-Plus-Concept Group and also
received exercises in applying the rule to different factual situations.

The test following instruction consisted of ten application items, such as “Which has more protein, a
worm or a fish?” No test item had been presented to any group during training.

The results show very little difference between the two groups that did not practice application (4.67 and
5.13 mean items correct). The group that practiced applications of the rule achieved 8.75 mean items correct.
There was very little difference between the average and above-average students (as defined by test
performance) on the transfer test. The means were: 5.08 for the average students and 5.17 for the above-
average students. This lack of difference suggests that for these children, knowledge of the component
concepts and the ability to state the rule were insufficient to induce understanding of how the rule applies to
examples. A corollary is that although the presentation of applications items may require slightly more time,
the items are more likely to assure that the communication will be effective.

In the second study, Ross & Carnine (1982) investigated the relationship between the clarity of a rule and
learning. Stimulus-locus analysis predicts that a more clearly-stated rule would lead to better performance.
Ross & Carnine presented two rules to different groups of children. One rule made the application steps fairly
explicit. “A binary duality is the answer you get when you multiply two numbers. One of the numbers you
multiply must be exactly two more than the other number.” The other rule did not specify the steps as clearly:
“A binary duality is the answer you get when you multiply one number that is exactly two more than the
other.” The prediction would be that the children would tend to learn better from the rule that makes the steps
more obvious.

The study was conducted with 26 fifth-graders. Ninety percent of the children who received the more
explicit rule met the training criterion; fifty percent of the children who received the less explicit rule met the
criterion. This highly significant difference suggests the importance of a clear statement of steps to the
learner. Related research has been conducted by Anderson and Kulhavy, 1973; Bourne, 1979; Bourne and
O’Banion, 1971; Crist and Petrone, 1977; Feldman, 1972; Johnson and Stratton, 1966; Klausmeier and
Feldman, 1975; Klausmeier and Voerwerk, 1975; Markle and Tiemann, 1974; Merrill and Tennyson, 1971;
Swanson, 1972; and Voerwerk, 1979.

Transformations

Transformations are important to the credibility of the stimulus-locus analysis because they are among
the more difficult “concepts” to describe verbally, which leads to the intuitively compelling conclusion that
they are unteachable or difficult to teach. The stimulus-locus analysis contends, however, that if the
presentation of examples show the basis for the transformation, the desired generalization will occur.

 
 To test the importance of juxtapositions in inducing a single-transformation relationship, Carnine and
Stein (1981) presented juxtapositions of arithmetic facts to two groups of six preschoolers (four to six-year-
old children). The groups received instruction on eight sets of related facts. Each set contained three
consecutive problems: 6 + 1, 6 + 2, 6 + 3, 7 + 7, 7 + 8, 7 + 9, etc. One set was introduced at a time, followed
by a review of all facts that had been taught. The children were tested on the facts within a set by presenting
the facts in random order. Criterion of performance for mastering a set was ten consecutive correct responses.

For the Transformation Group, the facts were presented in order. The rationale is that the systematic
changes in the presentation show the learner more about: (1) what is the same about the various facts; and (2)
the parallel between response changes and changes in the problems. For the Non-Transformation Group, the
facts within each set were presented in random order.

Although the Non-Transformation Group completed the study in less total time (50 minutes compared to
74 minutes), the Transformation Group performed significantly better on a delayed test of the facts. The
mean number of correct responses for the non-strategy group was 6.0 correct responses, compared to 13.8
correct responses for the strategy group. The ratio of number correct over time of instruction is one-and-a-
half times greater for the Transformation Group. This ratio indicates the number of correct delayed-test items
that are accounted for by each minute of instruction. The efficiency of the transformation instruction is
therefore considerably greater than that of the non-transformation instruction. This outcome suggests that
once learned, the transformation permits the learners to place individual facts in a framework rather than
having to remember all details of all facts.

The procedure for communicating a double-transformation relationship is to first present a series of

examples that show within-set sameness and then present examples that show across-set differences. An
alternative approach would be to present examples in pairs. The learner responds first to an example from one
set and then responds to a corresponding example from the transformed set. This approach was rejected
because it does not juxtapose examples in a way that shows how examples within a set are the same. It also
prompts a possible “elimination strategy.”

To test the importance of juxtaposing examples, Carnine (1978) taught the transformation “Make a
statement” and “Make a command” to preschool children. For each example, the child was presented with a
picture and was told to “Make a statement” or “Make a command.”

Three groups received the same amount of instruction. For the Across-Set Difference Group, the child
was always required to perform on two tasks with each picture “Make a statement . . . Make a command . . .”
The order of the tasks was randomized.

The juxtaposition of examples for the Within-Set-Sameness Group was designed to show sameness about
examples within each set. Children first worked on a group of different examples in which they made a

 
 command, and then on a group of examples in which they made a statement.

The Transformation Group received the sequence that shows first within-set sameness and then across-
set differences and then a mixture of examples from both sets.

The Transformation Group and the Within-Set Sameness Group performed equally well on a transfer test
to new examples (60 percent correct for the Transformation Group and 61 percent correct for the Within-Set
Sameness Group). The Across-Set Difference Group performed significantly more poorly (41 percent correct
on transfer items).

The treatment received by the within-set sameness group is the same as the first part of the double-
transformation sequence. The results suggest that when within-set sameness information is clearly provided,
the transformation is mastered and generalizations occur. If this information is not presented clearly, the
desired generalization tends not to take place.

Complex Form Communication

Visual-Spatial Displays of Fact Systems

According to the present analysis, the communication of facts that are related to a topic are most
efficiently communicated through a visual-spatial display that shows the coordinate relationships of entries,
higher-order relationships, and important “details.”

Engelmann and Niebaurer (personal communication) compared two methods for transmitting study-skill
information to students in grades 4 through 6. One method involved a program that presented various routines
for showing students where to write their name on the paper, where to write the title, how to indent, etc. For
each skill, a rule and applications were presented. The other method presented the same information, but
consisted of a single visual-spatial display that showed the position of the name, date, title, procedure for
indenting paragraphs, lines skipped, etc.

A test of application followed responses to questions about the display. The stimulus-locus analysis
prediction is that if the students understood the information presented through the visual-spatial display, the
display would be a more effective teaching procedure. The use of the verbal rules and operational steps
provided by the routine are unnecessary. The results confirmed the prediction. Over 90 percent of the
students in both groups reached criterion (appropriate application). The Visual-Spatial Group reached
criterion in an average of 20 minutes. The Overt-Routine Group required an average of more than 180
minutes.

To investigate the relative effectiveness of the visual-spatial presentation over a presentation that was
visual but not spatial, Sprick (1979) conducted an experiment with two groups of children, each composed of

 
 24 children in grades 4, 5, and 6. Both groups received the same presentation of information and tasks. The
only difference was the nature of the displays. For the Visual-Spatial Group, visual-spatial displays were
presented. For the Visual-Only Group, each cell was presented in isolation, so that subjects did not receive
information about the spatial relationships of one cell and other cells. To provide subjects in this group with
information that paralleled spatial information, each cell was keyed with a symbol. If three cells were
coordinate, each would have a coordinate symbol (possibly an orange strip in the comer). The symbols also
suggested differences between cells. (No two coordinate cells were marked with the same shade of orange.
Cells that were not coordinate were marked with another color or another symbol.)

Although the information provided by the two treatments was logically equivalent, the Visual Spatial
Group showed a significant learning advantage on a test of information that immediately followed each
presentation, but not on a retention test one week later.

Another study of visual displays was conducted by Darsh & Carnine (1981). The purpose of the study
was to investigate the importance of three instructional variables when combined into various instructional
packages designed to teach factual information in five content areas (i.e., weather, geographical information,
etc.). The subjects were 86 sixth graders. Subjects were randomly assigned to one of four instructional groups
by the school principal. The instructional package for Group 1 contained: (1) a teacher script (the teacher
carefully followed the script; (2) a visual display (graphic display which presented the material in a logical
form); and (3) an instructional game which required the students to separate into groups and review
information in a highly motivating context. Group 2 was taught exactly the same as Group 1 except that the
instructional game was replaced with self-study. Students were required to practice on their own, using their
visual displays. Students taught in Group 3 read four written texts adapted from the scripts that were used
with Groups 1 and 2. Included in the instructional presentation was the graphic display information. Finally,
students were asked to self-study the visual display. The students taught in Group 4 were taught with the
same text as presented to Group 3 and these students were required to self-study from the text material rather
than the visual display. (Group 4 modeled traditional instruction. Several teachers evaluated these
instructional units and considered them to be very powerful for teaching the material.) Instruction lasted 25
minutes a day for 15 days for each of the treatment groups.

The dependent variables for this study were: (1) a 15-item posttest which included a sampling of the
factual material covered during instruction, and (2) a 4-item generalization test developed from content taken
from a social studies text. This test was used to evaluate the students’ ability to take textual information from
traditional texts that included a visual display, and answer comprehension questions after study time. Planned
comparisons were conducted with both tests on Group 1 versus each of the other three groups. For the
posttest, all three comparisons were significant (percent correct for the groups were: 1 - 85%; 2 - 67%; 3 - 62
%; 4 - 70%). For the generalization test, only the comparison between Group 1 and Group 4 was significant

 
 (67% versus 48% correct).

These outcomes suggest that fact systems may be more readily conveyed if they are presented as displays
that show the relationships in a fact system. However, more work is needed to carefully unravel the variables
affecting teaching outcomes using visual displays.

Cognitive Routines

The stimulus-locus analysis of cognitive operations implies that communications are best if: (1) each step
is made overt to provide evidence that the learner is attending to relevant dimensions of the problem; (2)
feedback is provided; (3) the same operational steps are applied to a range of examples; (4) examples are
presented in a manner that implies generalization; (5) the preskills involved in the operation are taught; and
(6) the routine is systematically “covertized” (modified so that covert steps replace the overt ones).

Feedback for cognitive operations. Stimulus-locus analysis suggests that cognitive skills are generically
different from physical operations with respect to the feedback provided by the environment. While the
environment is capable of correcting or providing feedback with physical operations, no such feedback can
logically occur with the naive learner who is trying to learn a cognitive skill. Therefore, independent practice
with a cognitive skill may actually produce decrements in performance (because the learner may practice
doing the wrong things and never receive feedback from the environment).

To test this position, Williams & Carnine (1981) presented a passage to two groups of 13 beginning
readers. The children in both groups read the passage orally a total of three times. On the second and third
readings, the Feedback Group received corrections as they read. The other group read with no feedback on
these readings.

Errors on familiar words for the Feedback Group dropped from 4 percent to .7 percent between the first
and third reading. Errors for the No-Feedback Group increased from 3 percent to 5.4 percent from the first to
third readings. The differences are highly significant and tend to confirm analytical assumptions that
independent practice on newly-learned cognitive skills seems to be ill-advised (as stimulus-locus analysis
suggests).

In another study conducted by Carnine (1976b), a reversal design was employed during the teaching of
arithmetic facts to 4 preschoolers. The treatments alternated from the Feedback Condition (during which
mistakes were corrected) and No-Feedback Condition (during which no feedback was provided).
Performance of all children varied predictably as a function of the condition used. During the Feedback
Conditions, correct responses (both during the lesson and on posttest) averaged above 70 percent correct,
while during the No-Feedback Conditions, the averages were below 20 percent correct. This difference is
highly significant. As stimulus-locus analysis suggests, cognitive operations (in contrast to physical

 
 operations) are not learned through practice alone.

Research on the role of overt responding and feedback, not necessarily restricted to cognitive routines,
has been conducted by Abramson and Kagan, 1975; Bourne, Ekstrand, and Dominowski, 1971; Durling and
Schick, 1976; Frase and Schwartz, 1975; Tobias and Ingber, 1976.

Overtness of steps. The stimulus-locus analysis suggest that if the example has many features, it is
possible for the unguided learner to perform correctly on a particular example by attending to inappropriate
features of the example. If the learner is required to respond overtly to tasks that demand attention to relevant
details, it is less possible for the learner to learn a “misrule.” The study by Ross & Carnine (1982) discussed
earlier suggested that the overt procedure tended to work best in teaching correlated features of examples
(although the procedure was not significantly better than use of a rule followed by examples with fairly
sophisticated learners. Does the same relationship maintain for cognitive operations that are more involved
than correlated features? Studies were undertaken to answer the question: two studies involving arithmetic
operations, one involving decoding, and several involving complex comprehension skills.

While these studies deal with instruction on overt cognitive routines, other studies have gathered
descriptive data on the effects of different task variables, like vocabulary (Kameenui, Carnine, & Freschi,
1982). Data on these variables served as a basis for designing instructional interventions that are consistent
with learner responses to the specific variables.

For each arithmetic study, two groups were identified. The group sizes were 12 to 13. One was taught a
highly overtized operation; the other was taught the same skills and same set of examples through an
“intuitive” procedure specified in an adopted mathematics textbook. Single-digit subtraction was taught to
first-graders (Stein & Carnine, 1980). Long division was taught to fourth-graders (Kameenui & Carnine,
1980). The outcomes of both experiments were in accord with the stimulus-locus predictions.

The covertized subtraction group achieved a significantly higher percentage of correct answers on the
training problems, transfer problems, and delayed test problems. The overtized long-division group achieved
a significantly higher percentage of correct answers on the training problems, but not on posttest problems.
(All students performed reasonably well on the posttest after several days of training. However, the overtized
division group learned the skill more quickly, as indicated by the scores in daily tests.)

In another study comparing overt steps with covert procedures, Carnine (1977) taught two groups of
preschoolers (15 in each group) to read a set of regularly-spelled words. They were later tested on
generalization to both regularly-spelled and irregularly-spelled words (none of which appeared in training).

One group received instruction on component skills and instruction on performing a “sounding-out”
operation for identifying words. The other group received only “look-say” practice on the words in the
training set. The Sounding-Out Group reached the training criterion significantly faster (116.5 average

 
 minutes compared to 132.4 average minutes). On the transfer test, the Sounding-Out Group performed
significantly better than the Look-Say Group—on both regular words and irregulars. The sounding-out
subjects averaged over three times as many correct words as the look-say subjects.

The study clearly implies that if overt responses are made for the various elements that make one word
different from another (the individual letters), the communication will more precisely communicate
generalizable attack skills.

In a fourth experiment on the overtness of steps, Patching, Kameenui, Colvin and Carnine (1979) taught
intermediate-grade students to identify three types of faulty arguments-faulty generalization (“Just because
you know about a part doesn’t mean that you know about the whole thing”); faulty causality (“Just because
two things happen together doesn’t mean that one thing causes the other”); and invalid testimony (“Just
because a person is an expert in one field doesn’t mean that the per- son’s testimony is valid in another
field”). Classifying and dealing with examples of these arguments assumes a number of steps; therefore, the
prediction is that the highly-overtized routine would probably provide the best communication.

In the Overtized-Routine Group (consisting of three subjects), students went through a series of steps to
identify the conclusion, the evidence, and to state any faults. This group performed significantly better than
either the Feedback Group or the Comparison Group (26.8 mean correct for the Overtized-Routine Group
compared to 17.5 for Feedback and 17.4 for Comparison). Note that the feedback and comparison groups
perform almost identically, suggesting that the examples are consistent with many possible interpretations
and therefore feedback was not sufficient to suggest the specific steps the learner would have to take to deal
with the examples.

In the fifth experiment, conducted by Woolfson, Kameenui and Carnine (1979), 36 fifth-graders were
presented with passages that were judged complex on the basis of variables that had been demonstrated to
increase the “difficulty” of the passages. Information presented in these passages was separated so that the
learner could not refer to a single place in the passage to answer particular questions; the material did not
directly answer some questions, but provided only the evidence that could be used to draw the appropriate
conclusions; the material also contained distractors, information that was not relevant to the questions asked,
but appeared to be relevant; and the material did not begin with a statement about the point or the passage of
the question that should be answered.

The Overt-Routine Group was taught a procedure for:

1. Finding out what the problem is.

2. Identifying the information or evidence that is relevant.

3. Transforming this information through deductions.

 
 4. Answering the question (solving the problem).

The Feedback Group received feedback on the same set of passages presented to the Overt-Routine
Group.

The Overt-Routine Group performed significantly better than the other groups (mean = 1.9 out of 3
correct compared to .7 for the Feedback Group, and .6 for the Comparison Group). It seems that the feedback
condition did not work well because the examples were too complicated with too many possible features to
suggest an appropriate strategy.

A study by Adams, Carnine and Gersten (1982) taught study-skill strategies to fifth-graders who had
adequate reading (decoding) skills, but deficiencies in study skills. Forty-five (45) fifth-graders with reading
scores on a standardized achievement test of less than one year below grade level, and scores of less than 50
percent on two individual tests of study skills, were randomly assigned to one of three samples. One group
was taught to extract information using a cumulative-summary procedure. The performance of this group was
compared to that of two comparison conditions. The Independent-Seatwork Group was presented with the
same material and had the same opportunity time as the Cumulative-Summary Group to study the material.
The Independent-Seatwork Group read the work passages independently, did worksheet items, and received
feedback. The third group in the study received no instruction.

After four days of training, students were given a passage to study from a fifth-grade social studies text.
They were then asked to retell important elements of the passage, and received a short-answer test on
important facts in the passage. Two weeks later, the testing process was repeated. Results indicated students
receiving systematic instruction in study skills performed significantly higher on a factual short-answer test
on both occasions. No significant differences were found on the retell measures.

Dommes, Gersten and Carnine (1982) taught comprehension of pronoun-referent structures (identifying
pronoun antecedents and answering factual questions based on this knowledge) to 45 fourth-grade students
(identified as skill-deficient by a screening test) who were randomly assigned to one of the three treatment
groups: Pronoun Specific Group, Retell Group, and a No-Intervention Group. Students in both the Pronoun
Specific and Retell Groups received 20 minutes of individual instruction per day for three consecutive days.
On the day following treatment, all three groups were given two tests, one assessing the ability to identify the
word to which a targeted pronoun referred, and the second requiring the child to answer factual questions
based on the comprehension of pronoun referent structures. Maintenance tests were given two weeks later.
The Pronoun Specific Group performed better than the other groups on the Pronoun-Antecedent
Identification Test.

In a final study on the overtness and organization of steps for complex operations (Fielding, 1980), a total
of 42 high school students, 33 juniors and 9 seniors, were taught a two-week unit on the constitutional rights

 
 of youth. To assure equivalence of content across groups, all students received the same reading materials
(summaries of legal cases) and viewed the same three filmstrips. Students were divided into the Direct
Instruction Group and the Inquiry Treatments Group. The training for the two groups differed in teacher
questions and review.

After six hours of instruction, all students completed two multiple-choice tests, an essay test of legal
problem-solving, an “opinion” test, and an attitude questionnaire. The multiple-choice and essay tests were
re-administered two weeks after the initial posttests as retention measures. All tests were reviewed by two
third-year law students to establish content validity. These law students also rated the quality of subjects’
essays.

Direct Instruction students significantly outperformed Inquiry subjects on the knowledge test and on the
composite multiple-choice test. Direct Instruction students did significantly better on the initial essay, but not
on the retention essay. No statistically significant differences were found between treatments on the opinion
test, in which students wrote their views on legal policy issues relevant to the cases studied.

Related research on presentations for teaching complex operations has been conducted by Anastasiow,
Sibley, Leonhardt and Borick, 1971; Baker and Brown, 1981; Brown, Campione, and Day, 1981; Egan and
Greeno, 1973; Francis, 1975; Fredrick and Klausmeier, 1968; Gagne and Brown, 1961; Gordon, 1980;
Hansen, 1981; Hansen and Pearson, 1982; Klausmeier and Meinke, 1968; Raphael, 1980; Rosenthal and
Carroll, 1972; Rosenthal and Zimmerman, 1972; Tagatzh, Walsh, and Layman, 1969; Tennyson, Steve, and
Boutwell, 1975; Tennyson and Tennyson, 1975; Wittrock, 1963.

Non-functional routines. Several experiments discussed have shown that if the learner is sophisticated, a
presentation of the examples and a rule tend to work nearly as well as an overt strategy. For the less
sophisticated learner, the overt steps make a significant difference in performance.

To investigate possible non-functional routines with reading comprehension skills, Coyle, Kameenui, and
Carnine (1979) used three conditions to teach 36 intermediate-grade students a strategy for figuring out the
meaning of words presented in the context of passages. The word to be analyzed was always underlined in
the passage. One group received an explicit routine for dealing with this word. Students were told to look for
other words in the sentence that could tell about the hard word. Students were also provided with a scanning
strategy and with practice. A second group received the practice and feedback, but with no overt series of
steps. A third group received no training. The analysis of the task suggests that:

1. The underlined word served as a stable signal for the learner to do something.

2. The other words in the sentence served as a high-probability source of information about the
meaning of the underlined word.

 
 3. The feedback on errors provided information about different strategies.

4. Therefore, the overt routine was probably not necessary.

The Overt Routine Group performed slightly better than the Feedback Group (6.25 to 5.33, a non-
significant difference). Both groups were significantly better than the No Training Group, which scored 3.0.
While the overt strategy had some effect, the information provided by the examples apparently served as
primary influence in shaping the discriminations.

The three conditions used in the study above were also used to test the effectiveness of strategies for
determining a character’s motives (Clements, Stevens, Kameenui, & Carnine, 1979). The results were
comparable, with the Feedback-Only Group performing slightly (but not significantly) better than the Overt-
Routine Group. Both were significantly better than the No Training Group.

Preskills for clear communication. The stimulus-locus analysis implies that a communication is precise
only if the learner “understands” the information that is being conveyed. The learner should therefore be
pretaught those components of complex cognitive routines that possibly affect the communication. Other
research related to preskills has been conducted by Gagne, Mayor, Carstens, and Paradise, 1962; Gollin,
Moody, and Schadler, 1974; Jeffery and Samuels, 1967; Mayer, Stiehl, and Greeno, 1975; Royer and Cable,
1975.

To test this assumption, Carnine (1980d) taught 15 first-graders (divided into two groups) a complex
cognitive routine for solving simple multiplication problems. The routine calls for the learner to “count by”
the first number in the problem the number of times specified by the second number in the problem. To make
the operation overt, the learner is taught to hold up the number of fingers indicated by the second number and
then count by the first number. The answer is the number named as the last finger is counted.

An experienced Direct Instruction teacher taught both groups. The No-Preskills Group was taught
everything as it came up in the operation. Each mistake the children made was corrected and firmed. The
teacher modeled what the children were to do, led the children through the steps of the routine for each
problem, and repeated the problem until the children performed adequately on it. The Preskills Group was
taught in stages, with one new skill introduced at a time. The children were first taught to count by different
numbers. Later, to count so many times (holding up the appropriate number of fingers), and taught how to
translate written problems (such as 2 x 4) into the counting operation (“Count by 2, 4 times”).

Both total teaching time to a specific training criterion and transfer tests were used to measure the
relative effectiveness of the two approaches. The Preskills Group required significantly less total teaching
time 105.5 mean minutes compared to 137.6 mean minutes for the No-Preskills Group). The difference on the
transfer test was 6.5 items correct for the Preskill Group and 4.9 for the No-Preskills children. Similar results
were obtained with older children who were taught a borrowing operation (Kameenui, Carnine & Chadwick,

 
 1980).

Precise corrections. An assumption of the stimulus-locus analysis is that the more precise the feedback,
the less ambiguous the correction and therefore the better the communication. Carnine (1980c) studied this
assumption with three groups, each containing three preschool children who had learned to read by sounding-
out regularly-spelled words and then identifying them. Before the experiment, all children had been taught to
sound-out and identify five words. New words composed of sounds (letters) familiar to the children were
introduced each day. The initial correction used for all groups was for the teacher to say the correct word and
require the children to repeat it. A multiple baseline, single-subject design was used so that at different times,
the groups were introduced to a new correction procedure. The second correction required the students to
sound out the missed word, then identify it. The sounding-out correction is judged by the stimulus-locus
analysis to provide a more precise communication because it shows which details of the word relate to the
pronunciation of the word.

The results of the experiment corresponded closely to the analytical predictions. The performance of the
groups showed great improvement only when the sounding-out correction was implemented. The longer the
period of practice before the sounding-out correction was introduced, the more gradual the improvement. The
group that remained in the whole-word correction the shortest period of time achieved 95 percent correct by
the end of the experiment and the group retained in the whole-word correction for the longest period achieved
only about 50 percent correct by the end of the experiment.

Covertization. An assumption of the stimulus-locus analysis is that the teaching of a highly overtized
operation does not imply that the learner will perform problems of the operation covertly unless the learner
receives the sort of communication that shows how a variation of the overtized operation can be performed
covertly.

To investigate the necessity of specific covertization teaching, Paine, Carnine, White, and Walters (1982)
used a multiple-baseline design in which three third-graders were given an opportunity to generalize an
operation to a situation that was different from that in which the operation was taught. The children were
taught a highly overtized procedure for solving column multiplication problems. During part of the lesson,
the teacher directed the children in the problem-solving steps of various problems. During another part, the
children worked worksheet items independently. This latter condition provided them with the opportunity to
generalize the teacher-directed procedure even though this procedure was inadequate to imply such
generalization.

Following the Opportunity Condition, children were directly taught how to handle the problems covertly.
The worksheet performance of the students served as a dependent variable for the two phases of the
experiment.

 
 During the Opportunity Condition, the worksheet performance averaged 21 percent correct. By the third
day of the covertization training for the group, worksheet accuracy improved to 76 percent correct. Following
the formal covertization training, the performance went to 90 percent correct, suggesting that the children had
generalized the overtized operation to independent applications.

Other Important Communication Variables

This book focuses on those instructional-design principles that relate to the content of communications,
not to the behavioral details of communicating with a learner. Although we do not deal with specific teacher
behaviors that affect the clarity of the communication, these variables may be considered as possible pre-
empting variables. If we assume that the communication is faultless and has the potential to induce the
desired generalizations, some things may happen during the transmission of information to the learner that
effectively reduce the potential of the communication. Some of these variables are based on analytical
consideration; some are based on empirically derived principles about the learner.

If the teacher does not juxtapose the examples in the manner called for by the communication, at least
part of the communication’s potential is pre-empted. If the teacher does not secure the learner’s attention, the
message may be further attenuated. If the teacher uses inappropriate wording, corrections, or reinforcement,
further attenuation is likely for some learners. Even details about use of group turns and individual turns and
the seating of the learners have the potential of overriding the message.

Becker, Engelmann, and Thomas (1975) have summarized experiments that show the relationship
between the use of reinforcement, praise, behavioral rules, and other techniques and appropriate classroom
behavior. Becker also observed (Madsen, Becker & Thomas, 1972) that the use of appropriate behavioral
techniques is not enough to induce cognitive skills and that good management practices do not totally
compensate for inadequate instructional sequences.

Teacher attention. One experiment demonstrated that the teacher can (through critical attention) actually
increase the behavior he is trying to eliminate. This experiment involved the controlled use of disapproval
statements in a reversal design (Thomas, Becker & Armstrong, 1968). When the rate of the teacher’s
disapproval statements to off-task behavior was high, the rate of on-task behavior was reduced to 65 percent.
When the teacher reduced the rate of disapproval and gave approval to on-task behavior, relevant behavior
went up to 85 percent. The disapproval statements apparently were reinforcing off-task behavior.

Teacher pacing. Carnine (1976b) showed that when teachers presented tasks at a higher rate to low first-
graders, both on-task behavior and correct responses increased. When the teacher asked approximately 12
questions per minute in the fast-rate condition, the students answered correctly about 80 percent of the time
and were off-task only about 10 percent of the time. When the teacher asked about 5 questions per minute
(the slow-rate condition), the students answered correctly about 30 percent of the time and were off-task

 
 about 70 percent of the time.

Good-bad teaching behaviors. Carnine (1981) conducted a study using a reversal design in which the
alternating conditions presented generically different teacher behaviors. One condition was judged
analytically good because it controlled for pacing, signals, praise, and corrections. The other presentation was
judged analytically “bad” because it presented contraindicated pacing, signals, praise, and corrections.
Correct responding was 95.5 percent for the good condition versus 80.6 percent for the “bad.” On-task
behavior was 85.1 percent for the good condition and 50.6 percent for the “bad.”

These and similar studies tend to confirm the general maxim that if a detail can be analytically
demonstrated to affect the communication, that detail will be empirically demonstrated to be relevant to the
effectiveness of the communication.

Summary

The stimulus-locus analysis is based on the idea that all details that are analytically capable of affecting
the transmission of a communication will have an effect on learning rate, generalization of information, or
retention. This chapter summarized studies that have investigated the effects of these details.

Studies on basic communication variables confirmed the analytically derived procedures for showing
sameness and difference. They also demonstrated the effects of stipulated example sets across a wide range of
applications.

Studies dealing with joining-form concepts (transformations and correlated-feature relationships)

confirmed that the sequence of examples and the specific procedures for directing responses to the examples
led to significant differences in learner performance.

The studies on cognitive routines demonstrated that practice without feedback does not improve
performance, that the overtness of the steps greatly affects performance, and that preteaching critical
information and systematically covertizing the operational steps improve learner performance.

As a group, the studies tend to show that a great deal of information about how learners will perform on
an instructional sequence may be predicted before the fact by reference to the details of the presentation and
by using as guidelines the principles of juxtaposition and the facts about the nature of cognitive learning.
 

 
 Instructional Research: Programs
Chapter 29 presented studies that suggest an empirical basis for the communications that derive from the
stimulus-locus analysis. Those studies focused on specific communication issues. The studies that are
presented in Chapter 30 deal with issues that are more complex. These studies are divided into three groups:
those dealing with diagnosis of complex communications, those concerned with unpredicted normative
outcomes, and those investigating the teaching of new skills and behaviors.

The diagnostic studies are based on the idea that it is possible to predict specific learning problems the
student will encounter through an analysis of the communications the learner receives. (If the stimulus-locus
analysis is useful, it should permit us to make such predictions.)

The studies dealing with normatively unpredicted outcomes suggest that some causes of student failure
may not reside with students, but with the instruction they receive. If students receive instruction capable of
teaching specific skills, students who would normatively not be expected to learn specific concepts should
learn them.

The final group of studies, those involving new skills and behaviors, clarify the interactions between
behavioral principles (based on the characteristics of the learner) and stimulus-locus principles (based on
analysis of communications and skills). According to the stimulus-locus analysis, it should be possible to
identify the influence of communication variables even if the new learning involves responses that are
generically new to the learner and that require a great deal of shaping.

Diagnosis of Complex Communications

The stimulus-locus diagnostic assumptions are that:

1. Flaws in teaching communications can be identified through analysis of the communications.

2. Any flaws that are identified will be learned by some learners.

Studies have been conducted in which inappropriate learning is induced through presentations that have
obvious logical flaws. In one study, Kyzanowski and Carnine (1980) designed a flawed drill sequence. The
drill involved letters, which were identified by sounds. Two groups of preschoolers were identified. Each
child was presented with 60 items (letters). Two target sounds were in the set—the letters i and e (responded
to by the short letter sound). Each group received a different sequence of items. Both sequences presented e
15 times and i 15 times. However, the distribution of the letters differed for the two groups. In the e-
Discriminated Group, the letter e was distributed (never followed by another letter e) and the letter i was

 
 always blocked (a group of i’s occurring as juxtaposed examples). The set of letters presented to the i-
Discriminated Group reversed the roles of e and i. The i’s were never juxtaposed to other i’s, while the e’s
were always juxtaposed.

A diagnosis of this communication for possible misrules suggests that the communication for the non-
distributed letter in each block should not be learned as well as the distributed letter within each block. The
reason is that the type of response called for within the non-distributed letter is that of repeating the letter
sound again and again. Stated differently, the only letter that must be discriminated when the letters appear in
blocks is the first letter in each block. Since the non-distributed letters were presented in only four blocks, the
learner receives 15 discriminated trials for one letter and possibly only four effective trials for the other.

The results conform to the prediction. Although the distributed items were missed more frequently during
training (56 percent correct versus 73 percent correct for the blocked items), they were identified correctly
more frequently on a posttest (72 percent versus 30 percent). This pattern was consistent for both letters.
Saying things over and over in a non-discriminated context is not an efficient procedure for teaching
discriminations.

Time-on-Task and Program Variables

Rosenshine and Berliner (1978) and others have suggested that time-on-task (engaged time in teaching)
is the primary variable that accounts for differences in effectiveness of teaching sequence. The stimulus-locus
analysis predicts that time-on-task becomes a highly relevant variable only if the programmed sequences are
designed so they effectively communicate the skills being taught. A related prediction is that bringing the
learner to a high level of performance on training examples will not induce desired generalizations if the
communication is flawed. The additional practice may simply result in stipulation.

To test the relationship of engaged-teaching time and program variables, Darch, Carnine, and Gersten
(1981) taught four groups of low-performing fourth graders (N = 74) procedures for solving multiplication
and division story problems. Two program sequences were used. The one adopted from Distar and
Corrective Mathematics was judged to be capable of communicating the skills needed to solve the problems.
The other program sequence was adapted from commercial programs listed on the Oregon State-adopted list
of approved programs. This sequence was judged to be ineffective. Two treatments were provided for each
sequence. One received a fixed number of practice problems (60). For the other, additional practice was
provided to all students who did not achieve 90 percent accuracy on a 10-item test following the program
sequence.

Highly significant differences were observed on the 24-item posttest that followed instruction. Both the
fixed-trials and mastery groups in the Direct Instruction sequence scored over 90 percent correct on the
posttest (93.1 percent correct for fixed trials and 94.2 percent correct for mastery). The students receiving the

 
 traditional program achieved 69.2 percent correct for the fixed trials and 69.2 percent for the mastery. Note
that the additional work with the mastery students (up to three extra lessons) did not produce improvement in
the group’s performance. Clearly, engaged time was not the primary variable associated with mastery in this
experiment.

Another diagnosis study investigated the fact that the experimental approach used by Guess (1969) for
teaching plurals to low-performers was consistent with possible misgeneralizations. The learners were always
presented with two groups of objects. One group always contained only a single object, while the other group
contained more than one object. All objects in the pair of groups had the same label and were from the same
class (one dog and three dogs, for instance, never one dog and two cats and a dog). The implied
misgeneralizations:

1. The repeated presentation of examples stipulates that the smaller group is singular.

2. The comparison always involves the same label for the pair of groups.

To study which of these misgeneralizations are learned, Flanders (1978) replicated the procedure used by
Guess to teach a language-delayed child. Following the instruction, the child was presented with transfer
tests.

Flanders found that when the child was presented with two groups of like objects, both of which were
plural:

1 2
dog dog
dog dog
dog

the child tended to respond to the smaller group as a singular. This outcome is consistent with the stipulation
that the smaller group is always singular.

When tested with “spurious” plural groups:

1 2
dog dog
cat

the child tended to identify the second group as a plural (calling it either “dogs,” or “cats”).

Flanders corrected the presentation misrules by modifying the examples and overtizing the procedure for
naming the individual objects before identifying the labels for the groups. If a traditional diagnostic
procedure had been used rather than an analysis of the communication, the learner would have been labeled

 
 as one who fails to generalize, when in fact the presentation was guilty of teaching the inappropriate
generalizations.

An elaborate diagnosis of instruction was conducted by Steely and Engelmann (1979). The analysis dealt
with the specified instruction for teaching reading comprehension skills in grades 4 to 6 provided by the four
widely-used basal reading programs—Holt, Scotts-Foresman, Ginn, and Houghton Mifflin. The diagnostic
strategy used in this study involved:

1. Analyzing the material presented to teach different skills and determining the extent to which the
material had flaws.

2. Analyzing the actual teaching provided by the teacher and diagnosing it for possible flaws.

3. Comparing the program specifications, the actual teaching, and the performance of the students
who received the communication.

The analysis of the instructional programs and of the teaching focused on features that are logically
important and that have been shown through empirical studies to affect the clarity of the communication. For
instance, consistency in wording is logically relevant to the communication and has been empirically
demonstrated to be a factor affecting the communication. Other variables relevant to the clarity of the
communication were analyzed, including: the set and range of examples presented, the number and types of
examples, the possible misinterpretations implied by a set of examples, possible spurious prompts, the
covertizations procedures that relate the structured presentations to the independent work.

Table 30.1 summarizes the analysis of the programs for teaching main idea. The summary is based on
the analysis of all the exercises involving main idea that were presented over three years (grades 4 through
6).

 
 Table 30.1
Program Analysis Results Across Programs
Means Ideal
Across
Program

Percentage of questions misleading or

specified answers wrong 12 0
Percentage of questions relevant to
concept, teacher presentation 62 100
Percentage of questions relevant to
concept, student workbook 75 100
Number of possible interpretations
based on examples presented for
each skill 4 1
Percentage of visual distraction, student
workbook 25 0
Percentage of academic distraction,
teacher presentation, student
workbook 31 0
Percentage of responses spuriously
prompted, teacher presentation 24 0
Percentage of responses spuriously
prompted, student workbook 49 0
Days since two examples of a topic
were presented 62 ?
Total number of examples over 3-year
period 66 ?
Total number of student examples
presented on same day as teacher-
presentation examples 9 ?
Percent of student examples on same
day as teacher-presentation example 14 ?
Total number of lessons 22 50-80
Percentage of examples for which
correction is specified 0 20-40

The analysis discloses an almost total lack of concern with the logic of communications. The set of
examples presented to the students for any given lesson supported four possible generalizations. (For

 
 instance, all the two-paragraph, main-idea examples presented in a reading series might have the “topic
sentence” at the end. The examples are consistent with the generalization that if two paragraphs are involved
in the example, the topic sentence appears as the last sentence.)

Spurious prompts were lavishly provided, particularly in the student material (which means that the
students could answer items correctly by attending to the prompts and not to relevant features of the
examples). Over three years, a total of only nine examples of main idea appeared in student materials on the
same day that the teacher taught about main idea. The assumption seems to be that if the students are told
once, they should understand. Perhaps the most telling feature of the programs is the extent to which
corrections are specified. No program provided specific corrections.

The analysis of the programs leads to the conclusion that if the teacher follows them, they will not teach,
which means that a large percentage of the students receiving the communication will not learn (although
there is nothing to prevent any given student from “figuring out” the intended discriminations related to main
idea).

To determine how the teachers taught, 17 teachers who had used one of the programs at least one year
were videotaped for two half-hour periods. Different topics were taped to determine the consistency of
teacher behaviors across topics. A comparison of what the teachers actually presented and what the program
specified for a given lesson disclosed that the teachers tended to follow the programs very faithfully. Every
teacher presented every example the program specified and presented every question and discussion topic.
Their greatest deviation from the program was in the type and number of questions they asked. They asked
151 percent more questions than the programs specified; however, nearly all of the questions they added were
judged irrelevant to the topic. Therefore, the teachers consistently taught slightly worse than the instructional
programs specified. The instructional program, in other words, served as a model or limiter. The fact that
teachers followed the programs much more closely than they reported following it strongly suggests that they
did not know how to teach the various skills and relied heavily on the material.

Table 30.2 gives a summary of the teachers’ behavior:

 
 Table 30.2
Teacher Behavior Data Across Programs

Grand Ideal
Mean

Total percentage of questions for which

errors were made 27 0-12
Total percentage of questions that
were group tests 16 25-60
Total percentage of questions that
were individual 84 40-75
Total percentage of errors that were
corrected 37 100
Total percentage of errors that were
corrected and retested 10 100
Total percentage of tasks that were
models or leads 34 0-25
Total percentage of responses that
were given general praise 44 0-10
Total percentage of responses that
were given specific praise 2 15-30
Total percentage of responses that were
given negative feedback 1 0-2
Rate of tasks per minute 4.4 8-15

The analysis of the teacher material and the analysis of the communication provided by the teachers lead
to the same diagnosis, which is that they would not teach the skills they purport to teach. They have far too
little regard for the examples, the sequence, the tasks, and the manner in which the information is conveyed to
the students.

To determine the extent to which the communications did actually teach, the students (middle-class
children) were tested on the videotaped lessons immediately following the teaching. Eight topics were tested.
The tests did not introduce anything new. The wording was as simple as possible; the generalizations were
limited to those explicitly taught or assumed by the presentation; and the responses required by the students
were straightforward. The results of the tests are summarized in Table 30.3.

 
 Table 30.3

Student Performance
No. of
Presen- Mean Percentage of Students
Topic tations At or Above Criterion Of:
90% correct 75% correct 50% correct
Main Idea 8 10% 33% 58%
Key Words 3 8% 32% 65%
Map Skills 3 30% 33% 56%
Inferences 2 15% 30% 62%
Context
6 0% 0% 15%
Clues
Relevant
2 24% 82% 99%
Details
Cause
1 10% 30% 60%
Effect
Fact/
1 0% 25% 70%
Opinion

Means Across
All Topics 12% 30% 55%

Although there is great variation in the success of the various communications, the pattern of
performance suggests that the teaching is well-designed to maximize individual differences. About half the
children tended to fail half the items. An analysis of the mistakes that the students made revealed that the
mistakes were largely consistent with the communication the students received.

Normatively Unpredicted Outcomes

These studies are based on the idea that if the communication is controlled to teach skills, it should be
possible to teach skills to children who would not be expected to learn them, as indicated by normative
trends. Inferences that derive from these studies are particularly strong because the predicted outcomes are
highly unlikely. For instance, it is highly unlikely that preschoolers can be taught formal operations or
arithmetic concepts that are typically not learned before the third grade. Because the outcome is highly
unlikely, the probability is great that the intervention correlated with these outcomes was the cause and the
only cause of the outcome.

 
 Studies based on this rationale have demonstrated that disadvantaged preschoolers and middle-class
preschoolers were able to learn concepts that are normatively unpredicted, that disadvantaged school-age
children learn skills at a rate that is not normatively typical of these children, and that low-performing (low
IQ) children learn skills at an unpredictably high rate.

Teaching Formal Operations to Preschoolers

According to Piaget’s developmental hierarchy (1952), formal operations (the most sophisticated mental
structures that develop) typically occur during puberty, although Piaget does not assert that everybody passes
through this stage. According to Piaget, formal operations require the learner to make up propositions about
propositions.

The stimulus-locus interpretation is that although the same skills do not normatively appear until a
certain age, they could be induced at a much younger age if the communication with younger children is
made precise. If the stimulus-locus proposition is accurate, it should be possible to induce formal operational
“thought” or reasoning in children much younger than teenagers—possibly in preschoolers.

Through such an experiment, Engelmann (1967b) taught two groups of preschool children (a middle-
class group and a disadvantaged group) a program designed to induce formal-operational generalizations.
Daily instruction presented different examples of problems that required the same chain of reasoning. The
test involved solving a new problem that had obvious differences from any problem practiced during training.
The prediction was that the children would treat this problem in the same way they treated the training
problems and successfully solve it. The training involved the following activities:

1a. You start with two lines that are the same size but when you come back to look at them, they look
like this:

What are the two things the guy who changed them could have done? (Made the top line shorter
or the bottom line longer.)

1b. If the guy did not touch the top line, what did the guy do? (Made the bottom line longer.)

2a. You start out with the bar sticking out just as far on either side of the wall. When you come back,
the bar looks like this:

 
 Wall Bar

What are the two things the guy who moved the bar could have done? (Pushed in on the left or
pulled out on the right.)

2b. Half the bar is painted (the darker part). If the guy who moved the bar did not get paint on his
hands, what did the guy do? (Pushed in on the left side.)

The children worked a variety of these problems, following the same two steps with each problem. The
test problem was also the same in some feature. The outcome that was shown could be achieved in two
different ways:

You start out with the teeter-totter level. When you come back, the teeter-totter looks like this:

What could the guy have done? (Pushed up on the left or pushed down on the right.)

The top of the teeter-totter is painted. If the guy who moved the teeter-totter got paint on his
hands, how did he move the teeter-totter? (Pushed down on the right side.)

Note that the children could not merely memorize a rote response pattern because the problems differed
with respect to how the display had been changed (which side was down) and with respect to the information
about what the person who changed the display had done (operated on either the top or the bottom).

Four of the five middle-class children passed the test of generalization; two of the five disadvantaged
children clearly passed it; one possibly passed it; and two failed it. The middle-class child who did not pass
the problem was three years old. The children were tested later on conservation of substance. Only one child
passed.

A few weeks of daily instruction had induced a non-trivial generalization in children who were, by
“developmental standards,” preoperational. Their performance is accounted for only by the communication
they received.

In addition to learning this generalization, the members of the middle-class group also learned to solve

 
 problems of relative direction and of reflection. The training on the reflection problems provided a good
indication of how inductive reasoning is possible, if information about sameness is provided. The investigator
first taught the children to predict the “path” of a ball that rolls against a wall. The investigator used this type
of diagram:

After the children had become proficient at predicting the angle of the ball’s reflection, the investigator
used a similar diagram for a new problem.

The investigator . . . drew a map of the room and indicated a mirror against the wall. He asked the
children if they could predict the path of their vision if they stood off to one side and looked into
the mirror. He indicated the path of vision from the diagrammed child to the mirror. They were
asked to indicate the reflected path. They could not. The investigator indicated the reflected path.
He did not tie the problem in with the familiar rules about rolling balls. Instead, he wanted to see
how long it would take for the members to see the analogy. He presented another problem (with
the diagrammed person standing in a different position) and asked those who thought they knew
the answer to raise their hand. All members raised their hand. The one called on to indicate the
reflected path indicated it correctly. The investigator asked the other members to indicate where
they would have to stand in relation to different mirrors if they wanted to see themselves. All but
one answered correctly . . . (Engelmann, 1964, p.43).

In another experiment, Engelmann (1971) set out to teach a group of six-year-olds the skills needed to
pass the Piagetian tasks of conservation of substance, speed, volume, weight, and the test of specific gravity.
The test of specific gravity (an indicator of formal operational thought) was modified by the addition of a
series of questions about mercury. After the children had worked with the two steel balls in water (predicting
whether each ball would float or sink), the children were asked the same set of questions about whether the
balls would float in mercury. (They first observed the smaller ball float and were asked what the larger ball
would do. Next, they were asked to figure what was heavier, the mercury or the water. They were not
permitted to touch either container.)

Engelmann attempted to teach the skills in a way that violated all those principles Piaget suggested were
necessary for the induction of cognitive structures.

 
 1. No real-life objects were presented because Piaget assumed manipulation of real-life objects to be
necessary for cognitive growth.

2. No manipulation was presented during instruction.

3. No process of change was shown (either real or diagrammed) because Piaget suggests that
knowledge of the process is necessary.

4. No long-time period for assimilation and accommodation was permitted (with the entire training
involving less than five hours).

The children were taught logical rules about the relationship of things. The rules were rote. Their
application was not, however, because the test required the children to apply these rules to situations quite
different from any encountered during training.

The prediction that the children would handle the mercury problem as readily as the water problem was
based on the fact that the instruction provided the rule for handling objects in any medium—water, air,
gasoline, or whatever. The basic rule was: “It will sink if it is heavier than a piece of medium the same size. It
will float if it is lighter than a piece of medium the same size.” This rule implies applications such as, “A pin
floats in air. So what do you know about the pin? . . . It’s lighter than a piece of air the same size.”
Conversely, “A balloon is lighter than a piece of water the same size. So what’s the balloon going to do in
water? . . . That balloon is heavier than a piece of air the same size. So what’s the balloon going to do in air? .
. .”

A variation of the rule about sinking permits conclusions about the weight of the medium. “If the cork
moves up in gasoline, which is heavier, the cork or a piece of gasoline the same size? So what do you know
about the gasoline? . . . It’s heavier than a piece of cork the same size.”

Since the rule is framed to show what all examples of floating have in common, the prediction is that the
children would be able to solve the problems correctly if they correctly perceived that the demonstration was
providing information about the object and the medium.

Three of the five children in the experiment answered all primary questions about specific gravity
correctly. They did not know, of course, that the small steel ball would float in mercury. Once they saw it
float, however, they predicted that it would float again, and that the large ball would float. They also
concluded that the mercury is heavier than the water.

One girl was absent on the day that the rules for compensatory change were introduced. These rules were
designed to help children pass the test of conservation. The girl failed the test; however, during the Piagetian
test of specific gravity, she showed that she was quite capable of applying what she had been taught in a
perfectly new and creative manner. The experimenter asked whether a large candle would float or sink in

 
 water. The girl indicated that it would sink. The experimenter proceeded to saw it into two pieces of unequal
length. The experimenter stopped sawing and asked the child whether the longer piece would float or sink.
“Float,” the child indicated. The experimenter asked about the shorter piece and received the same response.
The examiner then asked about the entire candle. The child indicated that the whole candle would float.
When asked why she had changed her mind, the girl explained that while the examiner was sawing the
candle, a piece of the candle had landed in the pan of water. It floated, which means that the whole candle
would float and that any part of it would float.

Another interesting relationship between the communication and the performance of the children was
demonstrated on the test of the conservation of speed. It was impossible to teach anything about speed
without showing movement. Therefore, the children were not taught about speed. They uniformly failed the
test of conservation of speed. These aberrations in performance are explicable if we view learning as a
reasonable response to communications. From a developmental standpoint, however, the children’s
performance was enigmatic.

In another experiment, Engelmann (1967a) demonstrated that the skill of conservation could be induced
in non-conserving children by teaching them the “compensation argument” or the logic of how a change in
one dimension is accompanied by a compensating change in another direction. After 54 minutes of
instruction that did not involve real life objects, 10 of the 15 experimental subjects successfully passed the
generalization test for conservation of substance.

Teaching Academic Skills to Preschoolers

Therese and Siegfried Engelmann (1966) taught preschool children academic skills, including reading
and arithmetic skills. Four-year olds were taught to solve simple algebra addition problems (4 + □ = 7),
multiplication, subtraction, fractions, and applications of multiplication, such as solving the area-of-rectangle
problems. Although these children had not been taught addition or subtraction facts, they could solve
problems by decoding each problem as a set of instructions for carrying out a counting operation and then
carrying out the operation.

Bereiter, Washington, Engelmann, and Osborn (1969) presented four-year-old and five-year-old
disadvantaged children with an intensive half-day academic curriculum which taught language skills,
arithmetic, reading, and singing. The demonstration performed with the first group (entering as four-year olds
in 1964) was that:

1. They could learn to read through the application of logical rules.

2. They could learn basic arithmetic operations that are typically not mastered by children in the
third grade.

 
 The second group of children (entering in 1965) received a more intensive language program, and the
emphasis was switched from showing the extent to which they could learn sophisticated operations to
implementing curricula that were effective in teaching the foundation skills in arithmetic and reading.

The results with the first group showed that two of the three instructional groups could learn the same
kinds of arithmetic operations that had been taught to middle-class preschoolers.

The results with the second group (1965 to 1967) showed large IQ gains on the Stanford-Binet achieved
during the children’s first year (15 points), and additional gains of 9 points during the second year. The
children’s achievement test performance in arithmetic was grade level 2.5, and their reading grade level was
2.6. Upon graduating from the preschool, the children had an average IQ of 121. Middle-class children taught
in the preschool achieved higher IQ scores and larger academic gains.

Stein (personal communication) taught 10 four- and five-year-old preschoolers in a half-day program.
These children ranged from “behavior problems” to very gifted children. Their performance at the end of one
year was grade level 4.3 in reading and 3.0 in arithmetic on the Wide Range Achievement Test.

Teaching Academic Skills to School-Age Children

In 1978, Engelmann taught a class of 4th, 5th, and 6th-grade average children for 16 sessions (12 hours
total teaching time). At the end of these sessions the students averaged over 90 percent correct on
independent problems of these types:

! !
• a=
! !

! !
• a=
!

• If two tanks fill in 1.45 hours, how long will it take to fill 3 1 2 tanks that are the same size?

• The elm tree is 15 meters tall and its shadow is 11 meters long. How long is the shadow of an oak
tree if the oak tree is 6 meters tall?

• If it takes 4 men 3 days to build 5 houses, how long will it take 7 men to build 6 houses?

The results of the USOE Follow Through study provide crude measures of how the Direct Instruction
approach (used by Engelmann, Becker, and Carnine) compares with other approaches to teaching
disadvantaged children. Different sponsors worked with participating school districts, with a given district
implementing a particular sponsor’s program under the auspices of USOE Project Follow Through. The
Direct Instruction Program was implemented in a variety of settings—large city (New York, Washington,
D.C.), rural (Williamsburg County, S.C., Smithville, Tenn.), Indian (Cherokee, N.C., Todd County, S.D.),

 
 Spanish (Las Vegas, N.M.), Mexican (Uvalde, Tex.). The implementation of the program in most sites was
far below the standards that could have been achieved if school districts were oriented toward performance
and were responsive to classroom problems. This situation, however, was not one unique to the Direct
Instruction Program. Sponsors of programs far different from Direct Instruction reported similar frustrations
with attempts to implement (Nero, 1975).

Despite the problems of implementation, the USOE comparison of the Follow Through sponsors showed
that the third-grade Direct Instruction students scored higher than all other approaches in reading, arithmetic,
spelling, and language. The performance of D.I. students on measures of self-image suggested stronger self-
images. They excelled both in academic skills designated as “basic” and those designated as “cognitive
conceptual.”

Figure 30.1 summarizes the third-grade performance of kindergarten-starting disadvantaged children for
the major sponsors.

 
 Figure 30.1
Comparison of Third-Grade
Follow Through Children on the
Metropolitan Achievement Test

1 2 3 4 5 6 7 8 9
60

50 LS
M S

40 R

R
Percentile

S S
31
R
M S,R
S
R

R
23 L
S L
L R L
L
S M L M S,R
L M
16
R,M
M M

L
11 M

Open Education
Lab (SEDL)

Curriculum
Bank Street
Instruction

Southwest

Education

Cognitive
Responsive

(Arizona)
Education
Behavior
Analysis

TEEM
Parent
Direct

(EDC)

Total Reading 41 15 24 34 30 28 21 26 18

Total Math 48 15 14 28 19 17 11 18 14

Spelling 51 19 32 49 32 28 22 27 18

Language 50 18 20 22 23 18 12 22 19

Data summarized from Stebbins St. Pierre, Proper, Anderson, and Cerva (1977). Median standard scores by
site were analyzed and the average converted to a percentile equivalent.

The line at the 20th percentile indicates the expected performance for disadvantaged children (and

 
 corresponds closely to the pretest performance level of the Direct Instruction students). The figure is divided
in quarter-standard-deviation units. The difference between the 31 percentile and the 40 percentile, for
instance, is one-fourth standard deviation. The University of Kansas Behavior Analysis is clearly the closest
competitor. However, the Direct Instruction students are nearly one-fourth standard deviation above the
Behavior Analysis Model in reading, one-half standard deviation in math, and three-fourths standard
deviation in language. The other models, which tend to be based on more the current idiom of instruction,
reach the 32nd percentile in only one skill—the rote skill of spelling.

Another analysis of the Follow Through comparison shows how the various sponsors performed on basic
skills (those subtests identified by Abt Associates as being relatively heavy in their rote-learning component)
and on cognitive-conceptual skills (those that involve drawing inferences and using chains of reasoning that
involve more than one step). The results are summarized in Figure 30.2 (after Stebbins, et. al., 1977).

 
 Figure 30.2
ISOs for Basic Skills (B),
Cognitive Skills (C), and
Affective Measures (A)
Index of Significant Outcomes

-600 -500 -400 -300 -200 -100 0 +100 +200 +300 +400 +500 +600

Direct
Instruction

Parent
Education

Behavior
Analysis

Southwest
Lab (SEDL)

Bank Street

Responsive
Education Basics
Cognitive
TEEM Affective
(Arizona)

Cognitive
Curriculum

Open
Education
(EDC)
-600 -500 -400 -300 -200 -100 0 +100 +200 +300 +400 +500 +600

The centerline of the figure is the null outcome, e.g., no difference between a Follow Through model and
the comparison subjects. The further right the bar goes from the centerline, the higher the proportion of sites
in which a model showed both educationally and statistically significant effects. Some models had positive
impacts (bars on the right side of the baseline). Many had negative results. The Direct Instruction approach

 
 created the highest degree of positive impact in both cognitive-conceptual skills and in basic skills. Note that
the values for the cognitive-conceptual skills are greater than those for basic skills. These findings strongly
militate against the interpretation that Direct Instruction is “rote learning.”

Significant gains in IQ are also found for DI students, and are largely maintained through third grade.
Students entering the program with IQ’s over 111 do not lose during the Follow Through years, although one
might expect some regression phenomena. The low IQ children, on the other hand, display appreciable gains,
even after the entry IQ has been “corrected”—students with IQ’s below 71 gain 17 points in the entering-
kindergarten sample and 9.4 points in the entering-first sample; gains for the children with entering IQ’s in
the 71-90 range are 15.6 and 9.2, respectively (Gersten, Becker, Heiry & White, 1981).

Studies of how low IQ students (under 80) perform under Direct Instruction show the program is clearly
effective with students who have a higher probability of failure. As indicated in Figures 30.3, , and 30.5,
these students gain nearly as much each year in reading (decoding) and math, as students with higher IQ’s—a
gain of more than a year per year on the WRAT, and a year for a year on MAT Total Math.

Figure 30.3

WRAT Reading
Longitudinal Progress by IQ Block for Children in Entering-Kindergarten (EK) Sites (N=692)

140

130

120
Mean Standard Score

110

100 50th Percentile

90

80

70

N=8 N=108 N=194 N=214 N=156 N=18

a
Grade: EK K 1 2 3 EK K 1 2 3 EK K 1 2 3 EK K 1 2 3 EK K 1 2 3 EK K 1 2 3
Std.Score: 75 99 104 104 108 78 101 110 106 106 82 104 112 108 108 88 112 119 115 116 93 120 124 119 121 105 130 139 135 139
S.D.: 15.2 16.3 13.8 13.4 23.0 14.2 15.4 15.9 16.8 18.6 13.3 13.6 15.8 16.8 18.6 14.2 16.6 16.8 18.4 18.9 14.5 17.1 17 17.8 17.9 10.5 13.9 11.1 15.2 15.1
%ile: 5TH 47TH 61ST 61ST 70TH 7TH 50TH 75TH 66TH 66TH 12TH 63RD 79TH 70TH 70TH 21ST 79TH 90TH 84TH 86TH 32ND 90TH 95TH 90TH 92ND 63RD 97TH 99TH 99TH 99TH
G.E.: - K8 1.9 3.1 4.3 - K9 2.2 3.2 4.1 K0 1.0 2.3 3.4 4.3 K2 1.3 2.7 3.9 5.0 K4 1.6 2.9 4.2 5.4 K8 2.0 3.8 5.3 7.1
IQ: Under 71 71-90 91-100 101-110 111-130 Above 131

a
All testing performed at end of academic year, except for EK

 
 Figure 30.4
MAT Total Reading
b
Longitudinal Progress by IQ Block during 2nd and 3rd Grade for Children in Entering-Kindergarten (EK) Sites (N=1,082)

80

70
Mean Standard Score

60

50

40

30

N=19 N=181 N=271 N=310 N=265 N=36

a
Grade: 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
Std.Score: 33.9 46.6 47.4 40.1 50.6 54.3 43.1 52.7 56.0 46.0 54.5 59.6 49.0 57.1 63.2 55.4 65.1 72.6
S.D.: 7.7 7.0 13.1 8.2 6.5 8.4 8.6 7.5 8.6 8.4 8.1 8.5 9.3 8.7 9.6 8.1 9.2 7.7
%ile: 30TH 22ND 11TH 50TH 41ST 29TH 66TH 51ST 34TH 78TH 59TH 44TH 88TH 70TH 58TH 94TH 88TH 81ST
G.E.: 1.6 2.2 2.3 1.9 2.5 2.8 2.0 2.7 3.0 2.2 2.9 3.4 2.4 3.1 3.7 2.9 3.9 4.9
IQ: Under 71 71-90 91-100 101-110 111-130 Above 131

a
All testing performed at end of academic year
b
During Grade 3, the highest IQ block gained significantly more than the lowest block (which reflects
the verbal content of the SIT and the Total Reading subtest of the Elementary Level MAT)
Grade 1 National Median
Grade 2 National Median
Grade 3 National Median

 
 Figure 30.5
MAT Total Mathematics
Longitudinal Progress by IQ Block during 2nd and 3rd Grade for Children in Entering-Kindergarten (EK) Sites (N=1,056)

80

70
Mean Standard Score

60

50

40

30

N=18 N=176 N=265 N=301 N=262 N=34

a
Grade: 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
Std.Score: 29.2 46.1 62.6 35.8 51.4 67.7 40.0 55.2 69.7 42.6 56.9 73.5 45.6 60.2 76.4 54.4 69.2 85.1
S.D.: 7.3 10.1 9.5 8.0 8.5 10.6 9.0 8.2 9.7 9.2 8.8 10.7 10.8 10.0 11.6 10.6 8.5 8.1
%ile: 10TH 10TH 24TH 30TH 22ND 39TH 46TH 38TH 47TH 51ST 50TH 61ST 57TH 63RD 69TH 83RD 84TH 88TH
G.E.: 1.2 2.0 3.2 1.5 2.3 3.5 1.7 2.5 3.7 1.8 2.7 4.0 2.0 3.0 4.3 2.5 3.6 5.3
IQ: Under 71 71-90 91-100 101-110 111-130 Above 131

a
All testing performed at end of academic year
Grade 1 National Median
Grade 2 National Median
Grade 3 National Median

On reading comprehension (MAT Total Reading), students with IQ’s of 80 or less show as much gain
from the end of first grade to the end of second grade as other IQ groups, but not as much from the end of
second grade to the end of third grade. This latter effect may reflect the change in the Elementary Level MAT
reading test to an uncontrolled (adult level) vocabulary (while the earlier tests used a controlled vocabulary).
IQ is strongly influenced by one’s vocabulary, which relates closely to parents’ education.

Probably the most remarkable feature of Figures 30.3, 30.4, and 30.5 is that they show nearly parallel
gain functions for the various IQ groups, with the one exception noted above. These parallel functions imply
that how much a child can learn under Direct Instruction is largely unrelated to IQ (in the ranges shown).
Children with lower IQ’s start at lower levels and end at lower levels, but gain as much as the others in nearly
every case. It should be noted that data from another 750 entering-first grade students (with more than 50 in
the lowest IQ grouping) show remarkably similar results.

These results serve as a low-probability demonstration, particularly when they are compared with other

 
 large-scale attempts to implement effective programs for disadvantaged children. McLaughlin’s (1975) book
Evaluation and Reform summarizes the abortive results achieved by Title I implementations. Virtually all
Title I programs that were evaluated showed no positive results. This theme recurs consistently in the
evaluation of bilingual programs and other special programs for the disadvantaged. Against this background
of failure, the Follow Through evaluation stands as perhaps the only one that suggests what might be possible
with a complete implementation of resources. The results are low-probability in these respects:

1. The total time available would not predict superiority in every subject unless there was some
generically different approach used to teach the various skills.

2. The relative uniformity of the outcomes across different school settings and administration types
suggests that the approach is capable of making a range of teachers and administrators more
effective in delivering instruction to children.

3. The relative uniformity across children suggests that the program is effective with lower
performers as well as middle-class children.

The summaries above are based on poor children only. Within each Follow Through site is a percentage
of middle-class children. These typically enter somewhat higher and progress at a somewhat faster rate. Table
30.4 gives a summary of the test percentile means for non-poor kindergarten-starting children in the Direct
Instruction model at the end of the third grade. (Data from Becker & Engelmann, 1978.)

Table 30.4
Mean Standard Scores Converted to Percentiles
on the Metropolitan Achievement Test
Total Total
N Reading Math Language Spelling
Poor Children 1800 40 52 50 47
Middle-Class
Children 250 62 69 75 60

To test the applicability of Direct Instruction to other hard-to-teach populations, Alex Maggs and his
associates conducted a series of over 30 studies in metropolitan and rural areas of Australia, using Direct
Instruction programs and teaching methods (Distar, Corrective Reading, Morphographic Spelling). Subjects
included retarded, migrant, aboriginal, learning-disabled, and economically disadvantaged.

Alex Maggs and Robyn Maggs (1979) conclude their review of this work:

There is no other major output of acceptable educational research in Australia that has shown

 
 the results obtained by this body of Direct Instruction research. There is concern being expressed at
all levels of the Australian community that the schools are not making children sufficiently
competent in their critical basic skills. The dramatic results of the Direct Instruction findings raise
important questions of effective teaching and educational accountability. There is now a body of
empirical data upon which decisions can be based. This evidence provides administrators,
principals, and teachers with a basis for a choice of teaching strategies that have proven to be
effective (p. 32).

New Skills and Behaviors

The stimulus-locus analysis assumes that the learner is a perfect receiver of information. In situations that
require the learning of generically new skills, including new motor responses, however, the learner is not a
perfect receiver. Primary questions associated with generically new learning have to do with the effectiveness
of specific techniques and with the influence of the program on the learner’s performance.

Tactual Hearing Experiments

The tactual vocoder is a device that transforms sound into patterns of tactual vibration, in much the same
way sound is transformed when it travels on a long-distance phone connection. The tactual vocoder creates an
analog of sound through frequency-to-locus patterns. The display used in experiments conducted by
Engelmann and his colleagues consists of 24 vibrators arranged in order from high to low frequency. When a
word is said verbally into a microphone, the device produces a series of vibratory patterns that theoretically
should be a tactual counterpart of the word as it is experienced auditorially. Increases in loudness are
expressed as more vigorous vibration; changes in pitch are expressed by a change in position; changes in
phonemes result in changes of the vibratory pattern; changes in rate are reflected in rate of change of
vibratory patterns.

In the first experiment four adult hearing subjects were taught a 60-word vocabulary (Engelmann &
Rosov, 1975). Subjects also worked on identifying words that rhyme with known words, matching pitch and
inflection of verbal patterns, identifying individual “sounds,” and identifying novel sentences composed of
words from their vocabulary. Three subjects had banks of vibrators placed on their forearms; one had the
vibrators on her fingers.

The format for instruction was for the instructor to stand behind the subject and say words. The subject,
who wore headphones that transmitted a high level of white noise, responded by trying to repeat what the
instructor had said.

Subjects who practiced on the device for at least 20 hours learned to identify words, individual sounds,
and sentences. They also learned to match pitch and inflection. Although the vocabulary presented to the

 
 subjects contained a number of minimally different words, as well as those that are not highly similar, the
subjects generally achieved 90 percent accuracy on words when they were presented in a random order.

On about 40 percent of the trials that involved new sentences (sentences that had not been presented
before), the subjects identified all words correctly on the first repetition. These sentences were spoken at a
normal speaking rate. On about 40 percent of the trials, the subjects would request the sentence to be repeated
(or part of it) and would then identify it correctly. On about 20 percent of the trials, they would make a
mistake (usually on words at the beginning of the sentence).

The subjects could match unique musical inflections of vocabulary words, such as the name “Lori Skill-
man” presented as a novel melody. The subjects consistently matched such unique patterns without deviating
by more than a half-note on any part.

The learning of sounds, words, and sentences was not linear in difficulty. Words were not generally more
difficult than sounds; sentences were somewhat more difficult than words, but not much more difficult.

The nature of what had been learned was further verified by location transfer tests. Subjects were tested
on vocabulary words when the vibrators were placed on their legs, which placement had never occurred
during training. For the subject who had worked with the vibrators on her fingers, performance on the transfer
test was very poor (less than 20 percent accurate). The pattern of vibrators on her legs was not a perfect
analogue to the pattern on the fingers. For subjects who had performed with the vibrators on their forearms,
the transfer performance was about 85 percent of what their average performance had been with the vibrators
on the arms, suggesting that a pattern-identification skill had been learned that was easily transferred to the
legs.

A series of training studies was conducted with profoundly deaf subjects. These subjects predictably
learned more slowly than the hearing subjects, because a much greater amount of new learning was implied.
Instead of learning the transformation from a familiar spoken code, these subjects had to learn the spoken
code (how to say the words); learn how to classify similar sounding words; learn to discriminate between
them; and learn how to remember the association between a given tactual pattern and the spoken counterpart.

The data for one deaf subject appears in Figure 30.6.

 
 Figure 30.6
First Trial Word Identification
Performance for Deaf Subject #1.
Words Correct
Percent of
Week Number

Number of Words
15 30 45 60 75 90 105 120 135 150 165

100
50
50
67 Number of Words in
50
Subject’s Tactual
5

100
63 Vocabulary
86 Number of Words
80
Correct on First Trial
94
10

69
95
90
92
93
15

82
75
61
58
63
20

50
78
41
53
49
25

41
90
79
36
66
30

76
77
79
80
75
35

86
78
77
86
78
40

91
84
No Data
85
81
45

80
90
84

The fact that the learner learned faster as the set of words became larger implies that the learner learned
the system of sameness and differences that characterizes the material.

The plateau between 35 and 60 words (weeks 18 to 31) is typical of other deaf subjects. When the
vocabulary reaches a certain size, the learner must apparently work out new “rules” for identifying words.
Until this process is completed, learning is erratic. Once the consolidation has been achieved, however,
learning proceeds very rapidly.

The general learning trend shown in Figure 30.6 is consistently observed in new-learning situations.

 
 Even work with autistic children is characterized by a “multiplier effect” with later members of a given set
being mastered faster than earlier members of the set.

Sherman and Lorimer (personal communication) presented object-naming tasks to five- and six-year old
autistic children. The children were taught to identify different objects. A large number of trials (of ten over
3000) was required to achieve the discrimination of the first-taught objects; however, the number of
repetitions required to achieve discriminations dropped to an average of 60 after the sixth object had been
discriminated.

An interesting fact about the performance of these children is that their progress was quite irregular and
was characterized by large regressions (objects whose name learning required unusually high numbers of
repetitions). In every case, these “difficult” items were associated in a container relationship observed in
everyday situations. All regressions involved discriminations like a glove (which bears a container
relationship to hand), a cup (which bears a container relationship to juice or milk), a shoe (which is related to
foot), etc. This outcome may illustrate a basic information-processing problem confronting the autistic child.

To provide a more precise understanding of the interaction between learner variables and communication
variables in learning new skills, Williams, Engelmann, Becker, and Granzin (1979) designed a vocoder study
that compared “communication predictions” with “learner predictions.”

A set of 12 words was taught to four groups of hearing adults. The set contained minimally different
words as well as some that were more greatly different. The criterion of performance the subjects were
required to achieve and the sequence of the words were varied across the groups. The criterion was high or
low. A high criterion required the learner to perform correctly on a larger consecutive number of items before
a new word was added to the learner’s set. A low criterion of performance required fewer consecutive correct
responses. The sequence was designed either hard or easy. The hard sequence presented minimally different
words early in the sequence, while the easy sequence distributed the difficult words, introducing one member
of a minimally-different pair early and the other quite late.

Here are the words and the two sequences:

Hard sequence: eating, meeting, mile, smile, lag, log, toe, patrol, implement, geese, rung, up

Easy sequence: toe, eating, patrol, mile, implement, lag, geese, smile, rung, log, up, meeting

A pure stimulus-locus analysis would predict that the best sequence is the one that presents the high
criterion of performance and the hard sequence. The rationale is that the high criterion assures that the learner
is not responding to irrelevant features of the words. The hard sequence shows the learner quite early the
minimum ways the words differ from each other. Therefore, it should more readily induce learning in a way
that does not imply misrules for later words. (If the learner can discriminate between mile and smile, the

 
 learner will not confuse mile with patrol.) Note that the stimulus-locus analysis would be modified
somewhat if the fact is given that the learner will proceed very slowly through the sequence. This situation
will induce possible stipulation problems (working too long on the same small set of members).

A pure analysis of the learner leads to a different conclusion about which treatment should be best. This
analysis would hold that the criterion should be relatively low because the learner is apparently incapable of
attending to the degree demanded by a high criterion. For the hearing subjects in the initial vocoder study,
mastery was inversely related to the criterion, with the subject working on a 60 percent correct criterion
performing best; those on a 70 percent criterion performing next best; and the subject working on an 80
percent criterion performing most poorly. Not only do facts about the learner suggest that the low-criterion
condition should work best, but also that the easy sequence should work best.

The group receiving the low criterion and the hard sequence performed the best by a considerable
margin. Table 30.5 summarizes the means for the four groups. As the table shows, the sequence variables
interact with the criterion variables. The same sequence (hard) is the best sequence when the criterion is low
and the worst when the criterion is high. The treatments that involve the easy sequence are not so obviously
affected by the change in the criterion, which means that these sequences are relatively “safe,” if not capable
of producing spectacular results.

Table 30.5
Mean Total Trials to
Criterion by Condition
Criterion Sequence
Easy Hard
Low 1593 984
(N=4) (N=4)

High 2361 2602

(N=5) (N=5)

P Levels
Sequence (S) .49
Criterion (C) .001
Interaction (SXC) .05

All groups were given a transfer test in which the vibrators were placed on the subjects’ legs and a
sequence of 36 words was presented (12 words each three times in random order). The average performance
was 72 percent correct (which was 85 percent of the performance achieved when the vibrators were placed on

 
 their arms) with no significant differences among the four groups, suggesting that modalities are not as
important as the information. Transfer is predicted from the fact that the learners had generally mastered the
words and could therefore perform on a tactual presentation that was analogous to the original.

The study shows that learner variables interact with communication variables. Even in situations that
involve highly unfamiliar discriminations (those that are achieved only after an average of over 2,000 trials),
the logical aspects of the presentation have some predictive value.

Easy-to-Hard Sequences

According to the logical analysis of skills, the easy-to-hard sequence will lead to distorted responses and
stipulated responses. (The easy-word sequence in the vocoder study presented a sequence that proceeded
from easy discriminations to hard ones.) To investigate the effect of the easy-to-hard sequence on learning
new motor skills, Colvin (1981) predicted the generalizations that would occur if profoundly retarded
students went through a program that taught using screwdrivers. This program required students to meet
certain criteria on easy applications before moving to more difficult ones. Subjects worked first with socket
drivers (for square-headed bolts) and last with slotted screwdrivers. Also they worked first on screws that
were oriented vertically, with the head pointing up (0°), and they then worked on screws at different angles.

As Figure 30.7 shows, the work at 0° did not facilitate the work at 60° with the slotted screwdriver
(lesson 6). The variation in angle had not been implied by the earlier work. As a result, stipulation occurred
and the learners produced responses that did not readily transfer to screw driving on an angle.

 
 Figure 30.7
Mean Number of Trials to Criterion
for Tighten Components Across Lessons (N=3)

90

80

70

60
Number of Trials to Criterion

50

40

30

20

10

0
Lesson 3 4 5 6 7 8 9
Plane 0° 0° 0° 60° 90° 90° 120°
Screw
Type
Square Phillips Slot Slot Hex Slot Slot
Mean Number of Trials to Criterion for
Tighten Component across Lessons (N=3)

Figure 30.8 shows a different aspect of an easy-to-hard sequence. In lesson 1, the learners were taught to
hand-turn screws that were already set in a board. In lesson 2, learners were required to place screws without
turning them until they were hand-tight. Learners required many trials to meet criterion on placing in lesson 2
simply because they placed the screw and then hand-tightened it. If placing had been introduced in lesson 1,
followed by placing and turning in lesson 2, this problem would probably have been avoided. Although it

 
 may seem to be analytically easier to place without turning, this skill is actually harder for low performers if
they have already been taught to hand-turn the screws.

Figure 30.8
Mean Number of Trials to Criterion
on Basic Skills

Turn

Place

180

160

140

120

100
Mean Number of Trials to Criterion

80

60

40

20

0
1 2 3 4 5 6 7 8 9
Lesson
Mean Number of Trials to Criterion (N=3)
on Basic Skills across Lessons 1-9

Teaching Generalized Compliance

According to the stimulus-locus analysis, generalizations are induced by demonstrating sameness across

 
 a range of applications. Generalizations are predicted for any example that clearly falls within the range.
Engelmann and Colvin (1981) developed procedures based on this analysis for inducing compliance in highly
non-compliant subjects. The training was designed not to induce specific compliant behaviors, but to teach a
“compliance set”—a generalization about how a range of adult-learner situations are the same.

To induce the generalization, the program first established compliance to the commands “stand up” and
“sit down.” The purpose of this training was to demonstrate that compliance is required whether or not the
learner wishes to comply. Subjects received no reinforcement for following the commands and were
physically made to comply if they did not respond. Each presentation of this set of commands continued until
learners met a criterion of performance (such as following four consecutive commands without requiring
physical assistance or acting-out in any way).

In addition to this set of tasks, the program provided for a range of other tasks for which reinforcement
could be received (such as “come here,” “touch your nose,” “walk to the chalkboard,” and so forth).
Appropriate performance on these tasks was reinforced. Also included in this set were tasks that related to
behaviors expected of the learner in different situations (such as the lunch room). If the learner failed to
perform on any task presented in this reinforcement set, the learner was taken through a series of “stand-up”
tasks, followed by repetition of the task that had not been complied with.

Successful performance on the range of tasks for which reinforcement is provided implies that the learner
will comply to receive the reinforcement. Successful performance on the stand-up tasks implies that the
learner has learned that compliance is required. When the number of stand-up series diminishes to nearly zero
and the learner performs consistently on the reinforcement activities, the learner’s performance suggests that
the learner will generalize compliant behavior to any situation that would fall within the range of tasks
presented in training.

Figure 30.9 shows the summary of tearing and breaking behavior performed by an autistic child. At the
beginning of training, the child broke things at the rate of about three items a day at home and two items per
day at school. At night, he would tear up his pajamas, sheets, even the bed. The “stand-up” procedures were
modified for this subject so that breaking was performed on command. (For instance, the trainer would hand
the child a stick and say, “Break it,” or “Don’t break it.”) The transfer on “No breaking” was generalized
from the training situation to situations in which the learner was not supervised. As Figure 30.9 shows, the
program was first implemented in the home, resulting in an immediate change in the rate of breaking and
tearing. However, no transfer occurred to the school setting. (If anything, the rate of tearing in school
increased.) When the program was implemented in the school, the rate quickly dropped in school. On only
one day after the implementation of the compliance training was the rate at home or at school more than one
break per day.

 
 Figure 30.9
Reduction in Breaking/Tearing Behavior Through Compliance Training

10
9

8
broken/torn at home
Number of items

7
X=2 8 X=0 2
6

10

8
broken/torn at school/bus

7
Number of items

X=2 3 X=0 4
6

9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 1 2 3 4 5 6 7 8 9 20 21 22 23 24
SEPTEMBER OCTOBER
1980

Similar results have been consistently achieved with a variety of non-compliant children. The program
taught what was the same about compliance and showed that compliance was required for a range of
situations. Knowledge of compliance transferred to situations beyond the specific tasks involved in training.

Introducing Similar Members to a Set

The vocoder experiment reported earlier by Williams, et al. (1979) showed that both criterion of
performance and sequence of introduction are important. The experiment dealt with highly unfamiliar
discriminations (evidence of which is the high number of repetitions needed to achieve the learning). In
situations that do not involve highly unfamiliar discriminations, which are far more typical of the kind of
learning that occurs in academic situations, minimally different members are not juxtaposed in their order of
introduction. Rather, they are separated, so that the learner first receives practice on one name that has a
highly similar counterpart before the second name is introduced. For instance, in Distar Reading I
(Engelmann & Bruner, 1975), d is introduced early (on lesson 27) but b is not introduced until lesson 121.
The idea is that the relationship of b and d is made most obvious if d is quite familiar when b is introduced. If
the learner is not firm on d, the learner may become confused about which shape goes with which sound. The
similarity between the members may lead to chronic errors (reversals).

To investigate the importance of separating highly similar members in their order of introduction,

 
 Carnine performed two studies. In the first (Carnine, 1976c) similar members were introduced successively to
one group of first-graders and preschoolers and non-successively to another group (with the two similar
members separated by a series of non-similar members). The children were taught the short sounds for e and
i. When these two letters were separated, first graders made fewer errors during training (33 percent error rate
for the Separated Group versus 52 percent for the Successive Group). Preschoolers reached criterion in fewer
total trials when the letters were separated. (178 versus 293).

In the second study, the context was shaped (Carnine, 1980e). Children were required to match letters, all
of which were highly similar (p, q, b, d). The children matched the letter shown by the teacher with letters on
a “choice sheet.” For the separated-members treatment, the children first worked from choices that presented
only one highly similar member (one was correct; one was highly similar; two were not similar). Later the
number of highly similar choices increased until all four were highly similar to the one presented by the
teacher. The choices presented to the Successive Group were all highly similar. The Separated Group
required 31 trials to reach training criterion while the Successive Group required 69 trials.

This experiment suggests that if it is possible to “shape” the context in a way that creates no serious
misrules, the shaping may prove to be the most effective procedure. Note that even for the Successive Group,
the number of trials is relatively small compared to that required by highly unfamiliar sets of discriminations
(such as those in the tactual vocoder experiments). Related research has been conducted by Cheyne, 1966;
Gruenenfelder and Borkowski, 1975; Houser and Trublood, 1975; O’Malley, 1973.

Teaching the second member of highly similar pairs. When highly similar members are introduced to the
learner, the introduction of the second member is very important, even if the members are separated in time.
The introduction of the second similar member can either involve both the similar members (new and
familiar) or one of them (new). If the members are similar in features and similar in name, the problem of the
spurious transformation may occur. If the learner has mastered b and the letter d is introduced, the names are
similar and the shapes are similar.

To compare the effectiveness of the separated-set approach and the integrated-set approach with highly
similar members, Carnine (1981a) introduced concepts labeled bif and dif to four groups of preschoolers. Dif
referred to a left-leaning parallelogram, and bif referred to a right-leaning parallelogram. First bif was taught
to four groups of preschool children. All groups were then taught dif. One group was immediately required to
discriminate between examples of bif and dif. The second group was introduced to dif in the context of a
circle and triangle. This introduction did not initially require discrimination of bif and dif in the same
sequence.

For the third group, dif was not initially identified as dif, but as not-bif. Students were immediately
required to discriminate between bif and not-bif. Later, students were taught the other name for not-bif (dif).

 
 For the juxtaposition of the fourth group, bif and dif were included in the same sequence (following the
introduction of dif). However, triangle and circle also appeared in this sequence. Bif and dif were seldom
juxtaposed in this sequence.

The results showed that one treatment was significantly poorer than the others. That was the sequence
that presented only bif and dif. The children performed about the same on the others. This study shows that if
there is a strong basis for a spurious transformation, such as the one prompted by the relationship between the
parallelograms and the names bif and dif, the similar names must be separated.

Semi-specific prompts. Darwin and Baddeley (1974) and others have conducted experiments showing
that similarity in names causes the identification problem more than similarity in examples. The problem
seems to be that the learner is aware of specific features of the example, but is unable to “associate” those
features with the appropriate name. The appropriate name becomes “mixed-up” with the inappropriate one.

To investigate this phenomenon, Engelmann and Granzin (1980) conducted a reversal experiment with a
deaf girl who was learning speech through tactual-vocoder practice. The words taught to the girl were not
“highly” similar on an absolute scale; however, they were functionally highly similar, which means that the
girl had a great deal of trouble discriminating between them. On the words most recently taught to her, she
averaged 50 percent correct when she was required to “listen” to words (presented through tactual vibration)
and identify them. Her performance with words that had been introduced earlier (those words not in the most
recent 10) were identified correctly on over 80 percent of the items.

To determine the extent to which the learner’s problem was one of recognizing the words, but not being
able to produce the response that signals recognition, the investigators presented a semi-specific prompt. The
last 10 words introduced were written on the chalkboard. The learner (who could read) was permitted to
observe these words during the test. She was tested on all words that had been taught (as many as 85 words),
10 of which were the most recently introduced. The test words were presented in an unpredictable order. If
the learner recognized a particular word as one of the last 10, but did not know how to pronounce it, she
could look at the list, find the appropriate word, and use the spelling as a guide for pronouncing it.

During the semi-specific prompt condition, she performed at 65 percent correct on the most recently
introduced words. Removal of the prompt resulted in the accuracy of these words dropping to 20 percent. The
experiment showed that her problem was not one of recognizing the words presented tactually, but one of
associating a word with a specific response and producing that response. Since she was not practiced in either
phase of this operation, she tended to perform poorly on words that had not been practiced many, many times.

Related research on prompting has been conducted by Archer, 1962; Drotar, 1974; Imai and Garner,
1965; Lyczak, 1976; Restle, 1959; Silver and Rollins, 1973; Trabasso, 1963; Trabasso and Bower, 1968;
Warren, 1954.

 
 Summary

This chapter dealt with demonstrations of low-probability learning outcomes, studies that show the
correlation between the diagnosis of flaws in programs and learning outcomes, and studies concerned with
the interaction of stimulus-locus variables and learner variables.

The demonstrations of low-probability learning outcomes show that when communications are carefully
controlled, different types of learners are consistently capable of learning much more than they would under
“normal” circumstances. Demonstrations with preschoolers showed that they could learn both cognitive-
conceptual skills (such as understanding of specific gravity) that were not normally learned by young
children, and academic skills (reading, arithmetic, logic) far beyond normative expectations. Disadvantaged
preschoolers performed on the second-grade level in academic subjects while their peers performed below the
age norm for five-year olds. Low-probability demonstrations were also provided with school-age children.
The most extensive was the Direct Instruction Follow Through Model. Children in this model outperformed
all comparison children in all academic areas and in self-esteem measures. The children were taught from
kindergarten through third grade. Those taught by other methods performed around the 20th percentile (on
the average), while the Direct Instruction children performed near the 50th percentile. These outcomes were
achieved in different geographical areas with children from different types of backgrounds. Low-IQ students
(under 80) exhibited unpredictably large achievement gains, as did middle-class children.

Studies that deal with diagnosis confirm that when a teaching communication has analytically observable
flaws, some learners will learn inappropriately (responding to the flaws). If programs are analytically
incapable of providing a clear communication, they tend to be ineffective with learners. Furthermore, the type
of learner failure that is exhibited tends to be predictable from an analysis of the communication.

Studies that deal with learner variables confirm that learning is characteristically slow. Even within the
framework of a learning experience that requires hundreds of trials, however, the learner tends to be logical.
Stated differently, although the overriding necessary condition for generically new learning is a large number
of practice trials, different outcomes are predictable from the “logic” of the communication. The studies in
this group tended to show that by applying the basic analysis of flawless communications to the learning of
generically new skills, we can predict generalizations for diverse behaviors such as compliance and new
motor responses. We can identify where problems will occur in a program that proceeds rapidly from easy to
hard examples.

Singly, the studies cited in these research chapters are inconclusive. As a group, however, they lend
strong support to the basic stimulus-locus premise, which is that the learner responds to the information
conveyed by the communication. Therefore, it follows that unless the communication is controlled by making
it as nearly flawless as possible, we will seriously misdiagnose learning problems and may falsely conclude

 
 that the learner is responsible for failing when in fact the learner responded imperfectly reasonable ways to
the communication.
 

 
 Theoretical Issues
This chapter outlines the primary philosophical views that influenced the analytical system we have
developed. The chapter also addresses more specific theoretical issues—the relationship of our analysis to
traditional learning theories, the logical basis for some reinforcement and punishment phenomena, the logic
of learning to learn, the theoretical problems associated with strict behavioral objectives, and the role of
humanism in instruction.

Philosophical Underpinnings

Scientific theories rest on basic philosophical assumptions about knowledge, meaning, and rigor. The
present work is obviously influenced by philosophical orientations quite different from those assumed by
either strict behaviorism or cognitive developmental theories such as the one formulated by Piaget.

Behaviorism is perhaps best described as an extension of pragmatism. It seems to be heavily influenced

by the rule presented by “the father” of pragmatism, C.S. Peirce (1935). The criterion that Peirce used to
determine whether an idea was useful is: Does the difference make a difference? For example, two people
express different positions on a point and may argue. There is an apparent difference. But does this apparent
difference make a real difference in their lives or in some significant outcome? If not, Peirce would contend
that the difference is not really a difference, because it does not result in any difference that is substantive.

By applying this principle of “differences that make a difference” to learned behavior, we might come up
with a scheme quite similar to that developed by strict behaviorists. What are the important differences?
Behaviors. Does it matter what goes on inside the head to produce given behaviors? No. Therefore, these
internal differences do not make a significant difference, and we can deal with problems of human behavior
in a direct, uncomplicated way by considering only the relationship between those differences in stimuli that
cause differences in behavior. By following this line of reasoning, we are left with a system in which stimuli
are the agents that make a difference, and the differences that are made are the responses.

The position adopted in the present work is that there is nothing wrong with behaviorism as far as it goes.
It simply does not go far enough in permitting us to deal theoretically with stimuli. The guidelines that we use
for analyzing stimuli are logical. When we say that they are logical, we use the term in a very broad sense. If
something is not possible either through “induction” or “deduction,” we will consider it impossible and will
further assume that a learner could not learn it.

 
 Our first problem in dealing with stimuli involves sets of stimuli. We know that we can present different
shades of red to a naive learner and that the learner will appropriately classify new stimuli as red after
exposure to the different shades we had presented. Since the new stimuli are not identical to any stimulus
presented during the training, it is impossible for the learner to “memorize” a color. No such color was
presented during the training. If it is impossible for the learner to simply “associate” one particular stimulus
value with the word red, we must postulate a mechanism that makes the learner’s performance possible. The
simplest such mechanism is the capacity to learn what is the same about all the stimuli in the set of
demonstration examples that were identified as red. For the learner to perform this feat, the learner must
abstract this sameness from the various stimuli and must be able to use this abstracted sameness to identify
new examples of red.

Interestingly enough, this particular problem of color generalization played an important role in the
history of philosophy. David Hume (1975) had developed a highly mechanical system that explained how
thinking took place. He suggested that the mind simply recombined sensory impressions, enabling the thinker
to imagine such things as a golden tree, although the person had never experienced a golden tree. The person
simply “combined” the idea of gold (which was familiar) with the idea of tree (which was also familiar).

Hume did demur about the perfectly mechanical nature of ideas and impressions, however. He indicated
that he had little doubt that a person would be able to identify a new shade of blue, one that had never been
experienced before. Hume added that this example was singular and unimportant. However, it proved to be
important enough for Immanuel Kant (1899) to destroy Hume’s theory. Kant pointed out that ideas and
impressions are not totally determined by experience. Some preconditions must exist for the experience to be
possible. The question of ideas, therefore, had to be considered within the framework of these prerequisites.

Although the details of Kant’s system of prerequisites are not highly relevant to our discussion, Kant
clearly pointed out the fallacy of trying to explain “ideas” by referring only to experiences (simple responses
to stimuli). Some intermediary is necessary to explain how it would be possible for a person to identify
something that had never been experienced.

A very good analysis of stimuli was developed by an English philosopher who came after Kant. This
analysis dealt with sets of stimuli and provided specific guidelines for developing sets of examples so they
would rule out all but one possible interpretation. The philosopher was John Stuart Mill (1950). The analysis
that he developed was not designed as an educational theory. Instead, it addressed the problem of scientific
inquiry. Perhaps the most interesting aspect of Mill’s analysis is the failure of later investigators to recognize
that it indeed provided a basic structure for a theory of instruction. Although educational investigators seem
quick to adapt theories from various fields outside education (from information theories to developmental
theories), the single theory that carried the most significant implications for instruction has had virtually no
impact on instruction. Apparently, investigators generally failed to recognize the close correspondence

 
 between the instructional settings and one in which the investigator is trying to determine the precise cause of
various outcomes. In both situations, we must deal with concrete examples and draw inferences from these
“particulars.” In both cases, the analysis must therefore focus on the features of the examples. In both cases,
sets of examples are needed if we are to communicate the precise relationship between the features of the
examples and the “outcome” (which is the behavior that accompanies the examples in the instructional
setting). In both cases, the problem is one that involves inductive reasoning—proceeding from specific,
concrete instances to a “concept” that applies to instances not observed as part of the communication that
induces the concept.

Mill developed methods for showing sameness, for showing difference, for showing the relationship
between sameness and difference, for dealing with concepts that are related to familiar concepts, and for
presenting correlated-feature concepts. Below is a brief summary of these principles.

By substituting the word feature for circumstance, we can see the similarity between Mill’s principles
and the principles of juxtaposition that we postulated.

Mill’s first principle is the principle of agreement (or sameness).

If two or more instances of the phenomenon under investigation have only one circumstance in
common, the circumstances in which alone all the instances agree is the cause (or effect) of the given
phenomenon (Mill, 1950, p. 214).

Stated differently, if the examples are different except for a common feature and if the outcome is the
same for all instances, the only possible cause of the common outcome is the common feature. This principle
is logically irrefutable (if the cause or effect is associated with only a single feature). The fact that the various
examples share only this feature rules out the possibility that another feature is a cause (or effect).

The juxtaposition principle of sameness is based on the same logic. The cause in question is the basis for
labeling examples in the same way. If the examples vary greatly and have only one common feature, the set
of examples communicates unambiguously what this cause is. If the examples are highly similar, however,
the communication is ambiguous because it presents many possible causes.

Mill’s second principle is the principle of difference.

If an instance in which the phenomenon under investigation occurs and an instance in which it does
not occur have every circumstance in common save one, that one occurring only in the former, the
circumstances in which alone the two instances differ is the effect, or the cause, or an indispensable
part of the cause of the phenomenon (Mill, 1950, p.215).

Stated differently, if the positive and negative examples of a given outcome are the same in all features
but one, the single feature must be essential to the outcome. Again, this principle parallels the juxtaposition

 
 principle of difference. If there is only one difference between two examples and if the outcome is different in
any way, the difference in the examples must cause the difference in the outcome. (The cause here is the
objective basis for treating the two examples differently.) If we violate the difference principle, we create
communications that are guilty of implying false causes. If two examples are greatly different and are treated
differently, the different treatment may be attributed to any single difference in the examples or any
combination of differences. This principle, like the sameness principle, is logically unassailable (given a
single cause is associated with the observed differences).

In some experimental situations, the cause may be complex enough to warrant the use of both the
principle of agreement and the principle of difference. Mill refers to this principle as the joint method of
agreement and difference.

If two or more instances in which the phenomenon occurs have only one circumstance in common,
while two or more instances in which it does not occur have nothing in common save the absence of
that circumstance, the circumstances in which alone the two sets of instances differ is the effect or the
cause, or an indispensable part of the cause, of the phenomenon (Mill, 1950, p. 221).

This principle describes the logic followed by the stimulus-locus analysis for demonstrating concepts in
basic sequences. Minimum difference examples show the single difference between positive and negative
examples while juxtaposed positive examples that differ greatly show the basis of sameness or agreement of
the positive examples.

Mill’s principle that suggests transformations is called the method of residues:

Subduct from any phenomenon such part as is known by previous inductions to be the effect of certain
antecedents, and the residue of the phenomenon is the effect of the remaining antecedents (Mill, 1950,
p.223).

Although this principle does not contain the detail necessary to derive the procedures for communicating
either single-transformation or double-transformation concepts, the basic logic of demonstrating “residue” is
used by both these sequences. The progressive minimum differences at the beginning of the single
transformation sequence are designed so the part that is known from previous inductions is large, and the
residue (or difference from example to example) is small and specific. In this way, the nature of the “residue”
is made clear. Also, the double transformation is based on the idea that the transformation for one set is
known. The procedures for juxtaposing instances of this set with instances of the new set shows the nature of
the across-set transformation (or residue).

Mill refers to the principle for correlated features as the method of concomitant variations.

Whatever phenomenon varies in any manner whenever another phenomenon varies in some particular

 
 manner is either a cause or an effect of the phenomenon, or is connected with it through some fact of
causation (Mill, 1950, p. 227).

The goal is to show the correlation between two sets of events—one set of antecedent events and a set of
outcome events—by determining “whether we can produce the one set of variations by means of the other”
(Mill, 1950, p. 227). Note that this principle describes the correlated-feature logic. Two sets of events are
related by a factual or causal link. By presenting examples of one set and requiring the learner to apply the
causal link to each example in this set, the learner creates the instances of the correlated set.

When designing the principles of juxtaposition, we did not refer to Mill’s logical principles. In fact,
although we had studied Mill’s scientific method years before, we were not consciously aware of the parallel
until after the fact. The similarity of Mill’s principles and our principles of juxtaposition occurs because
instruction is inductive in nature. We therefore face the same induction problems that occur in the
experimental situation—the problems of showing the single basis of sameness across the positive examples,
showing the difference between positives and negatives, and providing special arrangements of examples for
complicated concepts. The principles of juxtaposition (or some approximation of them) logically follow once
we recognize that concepts are not the property of the mind, but are the inductions described by possible sets
of examples.

Mill’s analysis has apparently stood the test of time. Although the examples that Mill gave of correlated
events were weak and although the principle of residue could have been amplified to deal with sets of
examples, the system is basically sound, which implies that the principles of juxtaposition are logically
sound.

The method that we used for analyzing concepts for their sameness was suggested by a philosophical
school that developed a unique method for providing alternative explanations. The two most important
proponents of this school were G.E. Moore (1903) and A.C. Ewing (1947). These philosophers were not
interested in scientific investigation, but in the relationship between definitions or descriptions and reality.
They focused their analysis on ethics or “the definition of good.” Moore suggested that good was like the
property yellow. This inference is based on the fact that both good and yellow are the same in some
observable ways; therefore, a reasonable hypothesis is that they are the same in other ways. Specifically:

Yellow

1. Yellow is learned through experience.

2. We know that something is yellow through intuition.

3. Yellow is an irreducible property of events.

Good

 
 1. Good is learned through experience.

2. We know that something is good through intuition.

3. Good is an irreducible property of events.

If the first two sentences are accepted, the third becomes compelling. Note that the reference that “yellow
is known through intuition” is not a mystical statement. It means simply that there is no logical means by
which one can explain the person’s basis for identifying something as yellow. If a person says, “I know it’s
yellow because that’s what I learned,” the person is making a statement of history, not of one that explains
how the person knows that it is yellow. If the person says, “I know it’s yellow because it’s a particular wave-
length,” the person has simply sidestepped the question. How does the person know that the object is of a
certain wave-length if the person simply looks at the object? In the end, the person will say something to the
effect of, “I know it’s yellow because I know it’s yellow.”

Our purpose in presenting the argument about good is not to argue for it, but to point out the strength of
the analysis. Any two things, no matter how different, are the same in some ways. (If two things were
completely different in every possible dimension, they would both be in the class of things that were
completely different from at least one other thing. Therefore, they would be the same in one way.) By
constructing a set of statements that express the various samenesses, we are provided both with a profile of
how the things are the same, and with the possible basis for drawing inferences about how these things may
be the same in some unobserved ways. For example, if object A and object B are the same in eight ways and
if object A has a ninth feature that is unobserved, but not contradicted in B, there is a possibility that B may
also share this ninth feature. The only way to determine whether this inference is valid is to perform some
sort of test or investigation. The analysis, however, suggests the test and suggests both the kind of evidence
that would support the inference and the kind of evidence that would provide for possible falsification of the
inference. Because the method permits us to develop possible classification systems for events that are
superficially quite different, we used this analysis as the basis for formulating the classification of knowledge.
For example, the concept of book and the concept of moving faster are apparently quite different from each
other. Yet, by applying a variation of the sameness analysis suggested by Ewing and Moore, we can
determine the extent to which they are the same.

Books

1. There are concrete examples of books.

2. There is a label for books.

3. There are negative examples of books.

4. The naive learner would not know the basis for the label book unless we present concrete

 
 examples of book.

5. The naive learner would not know the basis for not-book unless we present examples of not-
book.

6. The naive learner would not know which examples were actually books and which were not-
books unless we treated the examples differently.

7. The learner would not know what is the same about various books and what is different about
books and not-books unless we design examples to show same- ness and difference.

Moving faster

1. There are concrete examples of moving faster.

2. There is a label for moving faster.

3. There are negative examples of moving faster.

4. The naive learner would not know the label for moving faster unless we present concrete
examples of moving faster.

5. The naive learner would not know the basis for not-moving faster unless we present examples of
not-moving faster.

6. The naive learner would not know which examples are moving faster and which are not-moving
faster unless we labeled the examples differentially.

7. The learner would not know what is the same about various instances of moving faster and what
is different about moving faster and not-moving faster unless we design the examples to show
sameness and difference.

We could certainly add to this list. In the end, however, we are provided with a set of common features
that suggest both a common classification (basic forms) and common procedures (the labeling conventions,
the setup convention, the principles for sameness and difference, the test of generalization).

By extending this analysis, we are provided with a precise basis for determining how the examples are
different. A statement that can be made about one example but not about the other describes a specific
difference. After determining a difference (such as the fact that examples of moving faster are not absolute),
we can use this feature as a basis for finding other examples that share the relative feature observed in
moving faster. With information about enough other examples, it is possible to determine a taxonomy that is
reasonably precise for accommodating all knowledge and skills.

 
 Both the analysis provided by Moore and Ewing, and that suggested by Mill, address the problem of
dealing with sets of examples. But later philosophers have tended to shy away from the analysis of concrete
examples in favor of an analysis of language.5 For example, Wittgenstein referred to the use of examples in
teaching. For Wittgenstein, the examples do not show the learner what is common about the examples, but
show the learner how to use the words. Pitkin explains Wittgenstein’s position about using examples:

The place where explanation fails and training is called for is where the pupil lacks knowledge of how
to use the word. And that kind of knowledge is completely contained in the examples . . . (Pitkin,
1972, p. 48).

But Wittgenstein seemed to miss the most important point about the examples, which is that we can show
the common feature of the examples that are treated in the same way. Wittgenstein says,

One gives examples and intends them to be taken in a particular way. I do not, however, mean by this
that the learner is supposed to see in those examples that common thing which I—for some reason—
was unable to express; but that he is now able to employ the examples in a particular way. Here giving
examples is not an indirect means of explaining—in default of a better. For any general definition can
be misunderstood too (1968, p. 208).

The final approach that influenced the stimulus-locus analysis was that of Sigmund Freud. Although our

                                                                                                                         
5
The movement from analysis of concrete examples is evident in the philosophy of science. Although the
assumption of concrete examples is implicit in any discussion of hypotheses and the procedures by which they are
corroborated or falsified, primary emphasis is on language. Popper (1959), for instance, argues that hypotheses must
be designed so that they provide for their falsification. The reason is basically logical. It is never rigorously possible
to show a cause or a concept by presenting only positive examples. On the other hand, it is possible to show with
great rigor that something does not cause a particular outcome. Therefore, by designing the hypothesis so that it
indicates the precise conditions (or set of conditions) that would lead to a negative outcome, the hypothesis provides
a compelling basis for ruling out different possible causes and for lending greater support to the conclusion that the
unique antecedents common to the examples in which the positive outcome occurred describe the sole cause of the
positive outcomes. Although this idea is quite valuable, it can be directly derived from Mill’s analysis. By using the
joint method of agreement and difference, the investigator receives information about both positives and negatives,
and the information should be quite precise about which antecedents were relevant to the different outcomes.
Mill’s joint method of agreement and difference therefore suggests an operational model for constructing
hypotheses that provide for possible falsification and that are very specific. If the investigator specified a possible
set of examples to be tested, indicated a range of possible outcomes, and showed how the various examples insured
that the specific cause would be evident regardless of the outcome, the “hypothesis” would be required to remain
closely associated with the concrete realities of the experimental situation. It would not become a language problem
that is at least partially divorced from the features of the experimental instances. This hypothesis would clearly
demonstrate each possible basis for falsifying the hypothesis.
Our purpose here is not to get deeper into the philosophy of science, merely to point out that this field is
preoccupied with the propositions or statements. Certainly, these are very important. However, the propositions are
important only because they relate to possible concrete situations that may be misunderstood or interpreted in a way
that will lead to possible problems (and paradoxes) unless particular language is used. So long as one understands
the concrete bases for the language conventions, the conventions present no problem; however, the emphasis of
modern theories of science tend to focus so strongly on the propositional level that the concrete examples are all but
lost. And these theories often fail to suggest that there is probably always more than one linguistic solution to a
specific concrete problem.

 
 analysis is quite different from Freud’s with respect to specific details, we used two aspects of Freud’s (1938)
approach to formulate theory:

1. Assuming an observation of an unexpected phenomenon is lawful.

2. Postulating an explanation for the phenomenon that is based on evidence, not on conscious
impressions or introspection.

Freud would typically observe one instance of a strange behavior. Instead of treating this instance as an
aberration, he would use evidence (in the form of the subject’s reactions to different topics) to formulate a
possible explanation. This explanation would typically run counter to the subject’s conscious impressions of
the situation.

Freud’s approach involved first attending to the qualitative details of the subject’s responses, because
these details indicated what the subject was “hiding.” The second step was to postulate a general mechanism
that would describe the phenomenon (such as displacement, sublimation, etc.). Freud’s tacit assumption was
that if one subject used this mechanism, the mechanism was probably a universal one. This reasoning
assumes that the subject’s behavior is lawful.

A problem parallel to that confronting Freud is encountered in instruction. A tacit assumption of many
investigators is that the specific nature of the mistakes a learner makes is relatively unimportant and that
conscious impressions of how one thinks leads to valid assumptions about learning or teaching. Our analysis
assumes that the mistakes the learner makes are perfectly lawful and are caused by the communication, not
by aberrations of the learner. We are therefore quite concerned with these qualitative details of instruction,
because these details are critical to understanding the learner’s problem and must account for the specific
error the learner made.

The stimulus-locus analysis further assumes that our conscious impressions play a very minor role in
analyzing learning. Evidence plays the major role. And this evidence often flies in the face of conscious
impressions. Stated differently, we cannot specify the basis that we use for identifying something as
“yellow.” We probably cannot verbally describe what we do when we walk. We are not clearly aware of the
standard that we use to indicate when something is slowing down.

A weakness in Freud’s theory was the suggestion that the unconscious was primarily associated with
emotional problems. He failed to recognize that most of what we do when we perform the simplest acts of
cognition are perfectly unconscious. Our impressions of what we learn are highly unreliable. Experiments
with blind people who could avoid bumping into objects disclosed that these people gave a variety of
explanations about the mechanism that they used to avoid these objects. The suggested mechanisms ranged
from the use of the ears, to fingers, to sensitive spots on the forehead. Despite the conscious impressions
provided by these subjects, an experiment revealed that all used only one mechanism—the ears. Just as their

 
 conscious impressions were basically unfounded in evidence, most of the other strong beliefs based on
conscious impressions of learning are unfounded. If we learned long division a particular way, we may have
the conscious impression that our way is the only natural way. However, we must recognize that any method
that had been used may have induced the same impression.

The reason that we have indicated the major philosophical underpinnings of the analysis we have
developed is that any theory rests on a foundation of philosophical assumptions, and the theory will be as
weak as its foundation. A theory of learning or instruction cannot glibly ignore the issue of inductive
reasoning, because, if the learner learns through examples, inductive reasoning is the only means by which
the learner can learn a generalization. The theory cannot ignore the issue of the learner’s capacity to organize
particulars into integrated wholes by trying to reduce the question of learning to one of pragmatism and
functional relationships. While the functional approach will permit some modest advancements in
knowledge, it will ultimately try to reduce questions of logic (the logic implied by the examples and the
communication the learner receives) to empirical facts about how learners respond and to apparent “laws”
and “sublaws” that purport to be behavioral, but that have nothing to do with behaviors. Unless the theory
provides an analysis that is philosophically sound, the theory will suffer in very predictable ways.

Relationship of the Stimulus-Locus Analysis to Theories of Learning

Traditionally, instructional theories have been step-children of psychological and medical theories.
Nearly every development in psychology, neurology, or medicine has found its way into education. Theories
of the brain have prompted instructional procedures. So have theories of intuition, theories of psycho-sexual
development (such as the one formulated by Freud), learning theories, and theories of personality. Actually,
however, an appropriate analysis of instruction is not subsumed by psychological theories of learning and
development. The analysis of instruction provides the specific rules for changing behavior in virtually any
situation—whether we are trying to induce new motor responses or provide some type of psychotherapy. In
psychotherapy, the learner will be presented with specific stimuli. The design of these stimuli is assumed to
exert a causal influence on the learner’s behavior. The primary theory for designing such influences must
provide guidelines about how to design communication capable of communicating the intended “concepts.”
The theory must provide a basis for providing adequate practice and describing an adequate test of
generalization. Although different specific procedures will be applied to deal with specific new learning
situations (such as learning a new motor response or learning new behaviors that relate to emotionally laden
situations), and although the use of reinforcement and punishment plays an important role in these situations,
any procedure will be relatively ineffective unless it is: (1) designed as a program that communicates what
the learner is expected to do, and (2) follows the rules for unambiguously conveying this information. The
theoretical foundation for this endeavor is not to be found through investigations of behavior, but through the
analysis of concepts and communications.

 
 Learning Theories

The stimulus-locus analysis is not continuous with the works of Skinner, Guthrie, Tolman, Hull, or other
learning theorists (Hilgard & Bower, 1975); however, parts of it are compatible with aspects of these theories
and with aspects of developmental explanations of learning. For example the response-locus analysis that we
propose is largely compatible with Skinner’s analysis of behavior (1953). Great departures from Skinner’s
orientation come about when we move to the sphere of discriminations or cognitive learning.

Of the prominent learning theories, Tolman’s (1967) expectancy theory is probably the closest to the
stimulus-locus analysis. Tolman suggested that when learning occurs in an operant learning situation, the
expectancy is conditioned, not specific behaviors. Tolman produced some experimental data to support this
notion. For instance, in a study conducted by Tinklepaugh (1928), a monkey who received a lettuce reward
when he had been accustomed to receiving banana rewards reacted with a tantrum, although lettuce served as
a reinforcer to the monkey in other settings. This outcome might suggest that indeed the monkey had an
expectation that was not satisfied; however, the problem of operationalizing the notion of expectancy is not
easily solved. If we try to express expectancy in terms of internal states, our definition relates to unobservable
features and is therefore mere speculation. Another possibility is to describe expectancy in terms of
observable signs. If we begin to interpret glances, expressions, or partial responses as expectations, we must
provide some convincing validation that these “signs” are not mere accidents, that they are predictive, and
that we can reliably identify them. In the end, we are required to develop a calculus of signs. This
understanding would run into serious problems if we try to make the notion of expectancy operational,
objective, and procedurally reliable.

If we explain expectancy in terms of units of behavior larger than signs, we merely play a word game in
which the word expectancy is used in place of the word behavior. For instance, if the learner responds more
rapidly, we say that the “expectancy” increased or strengthened. The notion of expectancy is now circular.
How do we know that the expectancy changed? Because the behavior changed. What made the behavior
change? An increase in the expectancy. Clearly, this sort of description is not rigorous.

To make the notion of expectancy rigorous, we must be able to describe it so that: (1) a particular
expectancy is unambiguously manifested in behaviors of the learner; and (2) the expectancy must be
described in non-behavioral operational terms. We must be able to predict the behavior that we will see by
following specific operational steps. The description of an expectancy, therefore, would take the form: If we
do operations A, B, and C, we will observe behaviors X, Y, and Z.

The stimulus-locus analysis provides the means for operationalizing expectancy. When this analysis is
used, the investigator is not required to “hunt for signs” that are interpreted as expectancy. Rather, a large
segment of behavior is predicted. This behavior is the generalization, or set of generalizations, implied by the

 
 sequence. The cause of the “expectancy” behavior (or generalization) is clearly specified and is described in
non-circular terms. The method of analyzing the sequence to determine the specific expectancy (or the set of
expectancies) that will be induced are reasonably reliable.

Furthermore, the probability is nearly zero that the learner could respond in the manner predicted by the
stimulus-locus analysis unless an “expectancy” rather than specific stimulus-response associations had been
conditioned by the presentation. Only an expectation or a concept could enable the learner to “anticipate”
what is critical about the positive examples and generalize to new examples that possess this critical feature.
For the series of generalization responses to occur by accident is hundreds of times less likely than the
probability of the learner producing a “sign” or partial responses that may be spuriously interpreted as an
expectancy. The predicted sample of generalization behavior is large; the causes of the generalization are
singular; and the tests for determining the type of expectancy that will occur are reasonably rigorous.
Therefore, the stimulus-locus approach tends to solve the most serious problems associated with Tolman’s
formulations.

Cognitive-Developmental Theories

Current emphases in cognitive theory that have developed from Piaget’s developmental theory have the
same sort of problems that Tolman’s theory has. The emphasis of these approaches has been to describe the
internal process by which the learner approaches specific cognitive problems. The explanations, in other
words, suggest the expectancies that the learner possesses. The problem with this orientation is very basic.
The internal scheme must be validated through observations of some types of behavior. The problem
becomes one of designing these behaviors so they are: (1) consistent with the predicted scheme, and (2)
consistent with only the scheme. The problem is basically the same one encountered with Tolman’s theory. A
monkey’s response may indeed be consistent with the notion that the monkey had a particular expectation;
however, merely being consistent with this possibility is not precise enough for scientific endeavors. The
probability must be near zero that the response did not occur for any other reason. The only design that
verifies expectation is one in which the behavior occurs if and only if a particular expectancy or schema or
mental organization is present. The Piagetian model and those modeled after it are not constructed from the if
and only if standpoint. Rather, only a plausible explanation is provided. If data that are consistent with the
explanation are provided, those data are interpreted as supporting the explanation, even though it is possible
to account for the observed outcomes with alternative explanations.

The stimulus-locus approach helps solve many problems associated with mapping the learner’s
knowledge, structures, and concepts by permitting us to induce these in a naive learner. If we deal with
concepts for which the learner’s only exposure is our experimental intervention, we can create a situation in
which the probability is very low that the observed pattern of generalization was caused by anything but our
intervention. The learner’s pattern of behavior may now be used as a fairly reliable indicator of a cognitive

 
 structure. By plotting the learner’s pattern of responses to a wide variety of tasks, we can create a fairly
extensive map of the learner’s cognition. By repeating this procedure with various learners, we can derive
extended laws of cognitive learning and mental structures. Unless we start with a rigorous basis for inducing
a given concept or expectation, however, our attempts to map knowledge will be characterized by false
conclusions and abortive remedies for cognitive deficiencies in a learner.

Reinforcement and Behavioral Theory

The attraction that current behavioral theory has for dealing with problems of changing the learner’s
behavior is that it is simple, operational, and fairly rigorous. It suffers from none of the problems that
characterize Tolman’s and Piaget’s explanations. The difficulty in determining what the learner is thinking is
not a problem for the behaviorist because thinking is not an observable behavior and therefore is beyond the
purview of behavioral theory. Skinner (1953) rejected definitions and substituted for them actual responses or
behavior. This bold treatment assured that the behaviorists dealt with observables and that they did not
interpret these speculatively. No interpretation was permitted beyond the observation of whether responses
occurred, the rate at which they occurred, the consequent and antecedent conditions correlated with their
occurrences, and possibly the changes in topography of the responses. This system has a wide range of
applications (socio-political behavior, instructional behavior, animal behavior, and therapy). It also reduces in
a straightforward way to the use of numbers—a clear indication that it is in the same league as other number-
oriented sciences. It permits us to express learning with numbers that tell us about rate, response latency,
intensity, proximity, and the other legacies of Newton and the English empirical philosophers.

Although behavioral theory has much to recommend it and is applied well to situations that involve new-
response learning, it is obviously not designed to deal with cognitive learning. The learner’s behavior
becomes the criterion that determines whether appropriate learning has occurred. Because the theory cannot
predict generalizations, however, it provides no basis for designing the teaching in a way that assures the
appropriate generalization. If we follow a strict behavioral orientation in teaching the concepts on and under,
we may discover that when we use a cup and a table for examples of on and use a pencil and a chair for all
examples of under, we can induce what we describe as appropriate behavior in fewer trials than we can
achieve if we use the cup and the table for both on and under. We are using a spurious prompt, but to
understand the problem, we must abandon a strict behavioral orientation and analyze the possible
generalizations or stipulations that are implied by the communication (the common sameness of the examples
that determines responses that have not yet occurred). Unless we employ this analysis of stimuli, it is not at
all obvious that the teaching of on and under with different objects is inappropriate. Granted, the investigator
may discover the problem by trying to apply on and under to new objects. Rules of “behavior” may result
from such studies; however, the problem is that they are not actually rules of behavior, but rules of
communication. By treating them as rules of behavior, we introduce circuity that both conceals their actual

 
 nature and displaces the problem from one of logic to one of brute empiricism.

If we further pursue the notion that the learner responds to the implied logic of the communication, we
discover alternative explanations for some learning phenomena that are treated as behavioral laws. Certainly
these phenomena are legitimately expressed as behavioral laws; however, they may be viewed more basically
as predictable outcomes implied by the communication. We will present two examples of such alternative
views—schedules of reinforcement, and punishment procedures.

Schedules of reinforcement. The communication for a fixed-time schedule of reinforcement shows the
learner that every so often reinforcement for producing a response occurs (every 30 seconds, for instance).
All examples of reinforcement have the same quality, which is that they come a half-minute after the
preceding reinforcement had been presented. The generalization implied by the communication is that only
these responses produced a half-minute after the preceding reinforcement will be reinforced. A law of
behavior is that when fixed-time schedules are presented, the learner tends to respond at a higher rate
immediately preceding the presentation of the next reinforcer. This behavior is predicted by the
communication, implying that the law is not actually a behavioral law. It is a stimulus-locus corollary.

The common feature of reinforcers that are presented on a variable schedule is that the presentation of
reinforcement is unpredictable with respect to the preceding reinforcer. The law of behavior is that this
schedule induces a high, constant rate of response and the responses resist extinction. (The learner continues
to produce the response after the reinforcers have been discontinued.) Both outcomes are perfectly
predictable on the basis of the common quality of the reinforcers presented through the communication. If the
communication demonstrates that the learner may produce as many as 20 responses (or some other “large”
number) before receiving a reinforcer, a series of more than 20 trials without reinforcement provides the
learner with one example that the schedule has changed and that the response is now on extinction. If the
learner had been on a schedule in which reinforcement was presented continuously, after each response, each
response that is produced during extinction has a quality of not leading to reinforcement. Therefore, the task
of learning about this sameness of extinction trials logically requires fewer responses (20 responses for each
variable-schedule extinction trial versus one response for each continuous-schedule extinction trial).
Therefore, extinction is predictably slower following variable schedules.

Punishment phenomena. Just as the phenomena associated with extinction and schedules of
reinforcement have a logical basis, punishment phenomena are grounded in the logic of the communication.
Punishment tends to change the naive learner’s behavior very quickly; however, it may inhibit the learner
from producing responses in various situations and may condition the learner to avoid entire situations in
which punishment occurs. By diagramming a situation in which either reinforcement or punishment occurs,
we see that there is a generic difference between punishment and reinforcement with respect to the
information that is conveyed to the learner.

 
 A

B
C

The letters refer to the behavioral context of a response. A is the most immediate behavioral context.
Context A consists of the behaviors the learner produced immediately before the reinforcing or punishing
consequence. Context A occurs within the behavioral context of B (which means that the learner must be
doing B before the learner performs A). B occurs within the broader behavioral context or situation C. (The
learner must engage in C, then B, before A is possible.)

When we reinforce A, our communication is always relatively precise in indicating the behavior that is
reinforced. The behavior is one that occurs in the behavioral context of B-C; however, the B-C context itself
is not sufficient to “cause” reinforcement. Some behavior within the B-C context must be produced.
Although the learner may not immediately learn that behavior A causes the reinforcer, the learner discovers
that the reinforceable response occurs within the B-C context and that some behavior within the context
causes the reinforcer. Given this information, the learner experiments until discovering that A or A in
combination with some other behaviors will lead to reinforcement. (The learning of A in connection with
some other behaviors is the learning of a superstitious behavior.)

When reinforcement is used, the learner must respond with A and only with A to receive reinforcement,
which means that the communication shows the precise cause of the reinforcement. However, when
punishment is involved, the communication is consistent with more than one possible interpretation.

Consider the same context of A-B-C, but assume that A is a behavioral context for punishment. When the
learner responds with A, punishment follows. The communication implies that A should be avoided. It also
implies that B should be avoided and that C should be avoided. Note that all possibilities will lead to the
successful avoidance of the punishing consequence. By avoiding context B, the learner successfully avoids
the punishing consequence because A is avoided when B is avoided. By avoiding context C, the learner also
avoids punishment.

According to an analysis of the communication, learning from punishment is logically easier than

 
 learning from reinforcement because the punishment communication permits the learner many possible
alternative concepts of strategies. So long as A is avoided, punishment is avoided. But the learner may learn
to avoid A by avoiding B or avoiding C. To learn that A is reinforcing, on the other hand, the learner must
learn to produce A and only A.

The punishment situation is like a very easy example in a new-response teaching program. The example
is relatively easy because it permits a variety of behaviors that lead to a successful outcome. Like other easy
examples, the punishment example should produce faster learning because it presents a much wider range of
behavioral options than the analogous reinforcement situation. The punishment situation, like other “easy”
examples, may teach a successful response to the situation that later proves to be distorted and requires
unlearning. Note that the punishment may lead to a non-distorted response just as an easy example of
buttoning may lead to a non-distorted response. However, because the learner is provided with behavioral
options, we may later encounter a problem if we want the learner to produce responses other than A in the
behavioral context of B-C. If the learner avoids B, the learner cannot discover that some behaviors in the
context B-C lead to reinforcement.

Learning to Learn

Harlow (1949), Kohler (1925), and others have demonstrated that the learner learns to learn. We have
discussed this issue earlier; however, it deserves a final mention because it is the quintessence of
generalization. As the learner learns different members of a set, the learning rate increases so that later
members are learned far faster than earlier ones. (Obviously, this trend does not continue indefinitely.) The
stimulus-locus explanation for the trend is: the savings in time for the learning of later members is possible
only because they are the same in some way as the earlier members. The quality that is the same does not
have to be relearned for the learning of each member; therefore, later members are learned in less time.

The proper analysis of the phenomenon focuses on the quality that is the same and on how to
communicate this quality so that learning proceeds smoothly. Let’s say that we wish to teach the learner to
“generalize” a puzzle-solving skill to various puzzles that require fitting pieces together in either a two-
dimensional or three-dimensional display. The faster program may be one that introduces greatly different
puzzles, such as a simple jigsaw and a simple, three-dimensional “barrel” puzzle. The sequence could then
possibly alternate between three- and two-dimensional displays, each track progressing slowly in difficulty
and providing plenty of review (the learner working variations of the same puzzles many times as new ones
are introduced).

The juxtaposition of greatly different examples would demonstrate the common quality shared by the
puzzles. Once this quality is learned, the learner would be able to solve various puzzles that share the quality.
The learner would not have to “relearn” the common quality of the various examples. This savings in the

 
 amount that must be relearned from puzzle to puzzle would be reflected in a decrease in the amount of time
needed to solve new puzzles, until ultimately the learner would develop a “generalized” puzzle-solving skill
and would be extremely proficient at solving any new puzzle that has the quality of the puzzles presented in
the set of training examples. This prediction is possible by reference to the set of examples and the features
common to all examples.

Behavioral Objectives

The stimulus-locus analysis implies that the use of “behavioral objectives” is appropriate only after a
series of steps have been taken by the instructional designer. A behavioral objective describes a task that is to
be taught. According to convention, the task must specify the behavior or response the learner is to produce.

In Chapter 12, we introduced procedures for analyzing tasks. We indicated the problems associated with
treating tasks as immediate criteria for devising instruction. We suggested creating transformed tasks that
deal with a broader range of application than a single task. The transformed task describes a variety of
specific, concrete tasks of the same form. The transformed task therefore avoids miscommunication problems
that may result if we simply accept the task as presented and teach it to the learner. In the chapters dealing
with cognitive routines, complex responses, and diagnosis, we have used the general transformed-task
strategy to create instruction that teaches generalized responses and that therefore avoids stipulation.

Our purpose here is to provide a more rigorous basis for demonstrating the problems associated with the
use of non-transformed tasks, particularly for cognitive skills. The demonstration shows that although it is
possible to use non-transformed tasks as a possible checklist or test of whether instruction is effective, they
should not function as directives for specific teaching.

The two problems that may occur if we simply teach a task without considering how that task interacts
with other tasks are:

1. We may stipulate that the response is limited to a particular context when in fact it should occur in
a variety of other contexts.

2. We may stipulate that a particular context calls for only a certain response when in fact the
context should support a variety of responses.

We can use diagrams to show the extent to which there are problems with the context and the response
specified (or implied) by a stated task or behavioral objective. Each diagram consists of two circles that
overlap—one circle representing the stated context and one for the stated response. The overlapping part of
each diagram is shaded. The larger that part is, the less there is a problem with the context or the response.
Conversely, the unshaded part shows the extent to which a problem occurs if the behavioral objective is
taught as stated (without transforming it).

 
 Figure 31.1 shows three diagrams. The context circle is largely unshaded in the first diagram, implying
that if we teach the objective as stated (without transforming it), we will have problems with the context that
the response occurs in. The second diagram shows the response circle largely unshaded, implying that if we
teach the objective as stated we will later encounter problems with the response. The third diagram show very
small, unshaded areas for the response and the context circles, implying that if we pursue the task as stated,
we will not create problems with the response or with the context.

Figure 31.1

Context Response

Context Response

Context Response

To determine problems with the context, we ask: Could the context that is specified in the behavioral
objective reasonably be expected to support responses other than those specified by the objective? If the
answer is “Yes,” part of the context circle is unshaded (indicating potential context problems). If the answer
is “No,” all (or nearly all) of the context circle is shaded.

To determine problems with the response that is stated, we ask: Would the response specified in the
behavioral objective reasonably be expected to occur in contexts other than the context indicated by the

 
 objective? If the answer is “Yes,” part of the response circle is unshaded (indicating a potential response
problem).

Illustration. Here is a behavioral objective: The learner will put her material away at 3:30.

The context is 3:30. The response is putting things away.

Context question: Could the same context that is specified in the behavioral objective reasonably be
expected to support responses other than those specified by the objective? Yes, in other situations the context
of 3:30 might function as a signal for the learner to catch a bus, to start dinner, to go to the movies, etc.
Therefore, the context circle has an unshaded area (indicating that a context problem would occur if we teach
only a single response in the context of 3:30).

Response question: Would the response specified in the behavioral objective reasonably be expected to
occur in contexts other than the context indicated by the objective? Yes, we would expect the learner to
understand how all situations of “At time X put your things away,” are the same. We would expect the
learner to produce the response of putting things away within the context of 9:00, 12:30, 4:10, etc. The
response circle of the diagram therefore would have an unshaded area.

Response:
Context
Putting
at 3:30
things away

The large amount of unshaded area shows that the objective does not take into account the fact that: (1)
the context should support other responses, and (2) the response would occur in other contexts. The
implication is that if we actually provide a communication that is capable of achieving the stated objective,
the communication may create serious stipulation problems with respect to both the context and the response.

To communicate the objective “At 3:30 the learner will put away her things,” without regard to other
objectives that involve the context or the response, we would condition the learner to respond at 3:30.
Possibly, strong adverse stimuli would be presented if the learner did not respond at 3:30 and a strong
reinforcing consequence would be provided if the learner responded within a particular time span. The time
span would be narrowed until the learner had been conditioned to produce the response of putting things
away across a wide variety of situations. When 3:30 occurred, the learner would put things away, regardless
of what the learner was doing at the time, what day of the week was involved, and where the learner was.

 
 Obviously, this sort of instruction is absurd. The learner should be taught the skill as a “cognitive” skill,
not through a mindless conditioning program. Note, however, that the conditioning program is implied if we
accept the behavioral objective and apply behavioral technology to achieving it. To correct the instruction,
we recognize that the learner should use the response in context other than 3:30 and should not be pre-empted
from producing responses other than putting things away in the context of 3:30.

Just as the diagram of the non-transformed objective depicts the nature of the problem, it suggests what
must be done to transfer the objective so that it is reasonable, which means adjusting it so there are no
unshaded areas in the diagram.

There is an unshaded area in the context circle of the diagram, indicating that the context should support
other responses. By listing other possible responses that we might expect to observe in the context of 3:30,
we are provided with a list of things that should occur within the context.

Similarly, by listing other contexts in which the response of putting things away might be expected to
occur, we fill in the unshaded area of the response circle. The various contexts that we list for the response
suggest that the learner will perform at some specified time (not merely at 3:30). The various responses that
are reasonably expected within the context of some specified time are best described as, “Some specified
behavior.” Here is the transformed objective:

At some specified time, the learner will perform some specified behavior. The statement of objective
generates a number of specific applications, such as:
At 9:00 the learner will sharpen her pencil.
At 9:00 the learner will turn on the TV.
At 1:00 the learner will turn on the stove.
At 3:30 the learner will call Mrs. Anderson.
At 3:30 the learner will turn off the stove.
Etc.

The new objective implies appropriate teaching for all examples. Furthermore, the program that is used
assures that the learner is taught the cognitive skills needed to perform the behavior. The learner is taught to
tell time, to estimate time, and to do the various behaviors that are called for (turning off the stove, calling
Mrs. Anderson, etc.).

The type of faulty diagram shown above (with two unshaded areas) results whenever a cognitive skill is
expressed as a specific behavior. If we teach the behavioral objective without transforming it, there will be
later problems with both the response and the context. Behavioral objectives, in other words, are perfectly
inappropriate as mandates for communicating cognitive skills. A cognitive skill is best described not as a
response, but as a pattern of potential responses that occurs regardless of the specific responses that are called

 
 for. When we view cognitive skills in this manner, we see that the basic description that must be used as an
objective for teaching is a general statement, such as the transformed objective of doing things at some
specified time. The general statement is an inclusive one that implies various concrete applications. It is,
therefore, consistent with the nature of the cognitive skill. It leads to the teaching of specific behaviors;
however, the type of teaching provided is consistent with the need to communicate the pattern or quality that
is common to all applications. In the case of doing some specified behavior at a specified time, juxtaposed
examples would be processed through a variation of the same cognitive routine to imply how the various
situations are the same and how the same strategy applies to all—estimating time, anticipating what must be
done, and carrying out the specified behaviors.

Analyses that are based merely on behavioral objectives may well achieve the desired objectives. The
cost, however, becomes evident when we attempt to teach objectives that involve the response or the context
of the objective presented earlier.

Humanism

Perhaps the greatest impediment to intelligent instruction comes from investigators who purport to be
humanistic. Socio-linguists (e.g., Labov, 1972), and those espousing a natural-learning or general-stimulation
approach to learning (e.g., Weikart, in Maccoby & Zellner, 1970), see the use of instructional technology as
contrary to humanistic beliefs. A long-standing trend that dominates the post-sputnik era tends to reduce the
problems of instruction to understanding the culture and language of the learner, to feelings of empathy, and
to telling the truth. If there is a guaranteed formula for failure, that is it. Furthermore, it is probably no
accident that this belief is prominent. It serves an inept educational establishment by reassuring teachers and
administrators that there is no need to deal with the concrete realities of the teaching failures that occur in
nearly every school. The humanistic position conveniently views these failures as non-failures. The
philosophy further assumes that those who deal with instruction on a technical level—not an amorphous
one—are ignorant of “theory,” unenlightened about the facts of humans, and reactionaries concerned with the
suppression and manipulation of children rather than with growth and creativity.

This attitude is not humanistic because it overlooks the basic fact that instruction is manipulation and
instruction occurs through communications. However, communications are presented and communications
teach whether they are designed by intent or whether they are the product of accident. The humanist in the
classroom does not have the luxury of “not teaching.” No matter what the teacher does, a model will be
presented; the behavior of the teacher will suggest rules about the relative importance of particular material.
The teacher is responsible for achieving student outcomes. If the teacher permits the children to progress “at
their own rate,” and “in their own style,” some children may demonstrate slow rates and poor styles. In that
end, a self-fulfilling prophecy is realized. Some children will indeed “prove” that they are slow, and the
teacher will believe—out of ignorance, not humanity—that these children would have been slow no matter

 
 what type of instruction had been provided. From our perspective, many classroom demonstrations provided
by people who express great concern for humanity are no more humane than the practice of using leeches to
bleed diseased patients.

If we are humanists, we begin with the obvious fact that the children we work with are perfectly capable
of learning anything that we have to teach. We further recognize that we should be able to engineer the
learning so that it is reinforcing—perhaps not “fun,” but challenging and engaging. We then proceed to do
it—not to continue talking about it. We try to control these variables that are potentially within our control so
that they facilitate learning. We train the teacher, design the program, work out a reasonable daily schedule,
and leave NOTHING TO CHANCE. We monitor and we respond quickly to problems. We respond quickly
and effectively because we consider the problems moral and we conceive of ourselves as providing a
uniquely important function—particularly for those children who would most certainly fail without our
concerted help. We function as advocates for the children, with the understanding that if we fail the children
will be seriously pre-empted from doing things with their lives, such as having important career options and
achieving some potential values for society. We should respond to inadequate teaching as we would to
problems of physical abuse. Just as our sense of humanity would not permit us to allow child-abuse in the
physical sense, we should not tolerate it in the cognitive setting. We should be intolerant because we know
what can be achieved if children are taught appropriately. We know that the intellectual crippling of children
is caused overwhelmingly by faulty instruction—not by faulty children.

Because of these convictions, we have little tolerance for traditional educational establishments. We feel
that they must be changed so they achieve the goals of actually helping all children.

This call for humanity can be expressed on two levels. On that of society: Let’s stop wasting incredible
human potential through unenlightened practices and theories.

On the level of children: Let’s recognize the incredible potential for being intelligent and creative
possessed by even the least impressive children, and with unyielding passion, let’s pursue the goal of assuring
that this potential becomes reality.
 

 
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 Subject Index
Academic studies, research on teaching – Ch. 30
Analysis of examples versus learner – Ch. 23
Attention to misbehavior, research on – Ch. 29
A-Z integration procedure in teaching related subtypes – Ch. 10
Basic forms of knowledge – Ch. 3, Sec. II
Behavior, analysis of – 1
Behavioral objectives – Ch. 31
Choice-of-example worksheet items – Ch. 16
Choice-of-label worksheet items – Ch. 16
Choice responses, in designing routines – Ch. 17
Cognitive-developmental theories – Ch. 31
Cognitive problem-solving routines chaining components of – Ch. 17
comparisons to physical operations – Sec. 6
components of – Ch. 17
conditional parts – Ch. 19
construction of – Ch. 17-19
descriptive rules – Ch. 17
illustrations of – Ch. 18
overviews – Sec. 6
range of examples – Ch. 17, Ch. 19
research on – Ch. 29
scheduling of – Ch. 19
Communicating through examples, rules about – Ch. 4
Communications
about events – Ch. 3
analysis of – Sec. I
design of – Ch. 1
faultless – Ch. 1, Ch. 2
Comparatives – Ch. 3, Sec. II, Ch. 7
Complete teaching, Overview – Sec. V
Compliance training – Ch. 26
research on – Ch. 30
Complex forms of knowledge – Ch. 3
research on – Ch. 29, Ch. 30
Complex responses (see Response teaching)
Components of responses – Ch. 23
Concepts, basic – Ch. 2
Context shaping – Ch. 3, Ch. 22
Continuous conversion – Ch. 2, Ch. 4, Ch. 6
research on – Ch. 29

 
Coordinate-class programs – Sec. IV, Ch. 10
Corrections, Ch. 27
discrimination errors
chronic – Ch. 27
non-chronic – Ch. 27
in cognitive routines – Ch. 21
in joining forms – Ch. 9
response errors – Ch. 27
Correlated-feature joining forms
(see also Fact statements)
and-or – Ch. 9 and J.S. Mill’s method of concomitant variation – Ch. 31
correcting mistakes – Ch. 9
non-comparative sequences – Ch. 9
overviews – Ch. 3, Sec. 3
programs for – Ch. 9, Sec. IV
representations – Ch. 9
research on – Ch. 29
sentences – Ch. 12
single-dimension sequences – Ch. 9
Covertizing cognitive routines
choice of method – Ch. 21
corrections – Ch. 21
dropping steps – Ch. 21
equivalent pairs of instructions – Ch. 21
inclusive instructions –, Ch. 21
overview – Ch. 3
programs for – Ch. 21
regrouping – Ch. 21
Cumulative review
in program design – Ch. 10
in hierarchical programs – Ch. 11
Diagnosis
of learning failures – Ch. 2
of instructional programs – Ch. 26
research on – Ch. 30
of the learner – Ch. 26
Difference principle – Ch. 4, Ch. 31
Differences, how to communicate – Ch. 2
Disadvantaged, school research – Ch. 30
Discriminating members of a set – Ch. 10
Distortion of responses – Ch. 3, Ch. 23
Divergent responses – Ch. 9

 
Double transformations
criteria for – Ch. 13
familiar set – Ch. 13
overview – Sec. IV
programs for – Ch. 13
structure of – Ch. 13
transformed set – Ch. 13
Easy-to-hard sequences – Ch. 24
research on – Ch. 30
Enabling responses – Ch. 3
Enabling-response-features program – Ch. 23
Equivalent meanings, how to teach – Ch. 9
Essential-response-features program – Ch. 23
Event-centered tasks – Ch. 15
Expansion activities – Ch. 10
in programs – Ch. 15
Expanded chains of behavior – Ch. 24
Expanded cognitive programs – Ch. 25
Expanded teaching – Sec. V, Ch. 15
Expectancy theory – Ch. 31
Extrapolation – Ch. 2, Ch. 14
Fact statements – Ch. 9
involving transformations – Ch. 9
Fact systems
designing displays – Ch. 14
designing scripts – Ch. 14
overviews – Ch. 3, Sec. IV
programs for – Ch. 14
research on – Ch. 29
types of – Ch. 14
Fading – Ch. 3
programs for – Sec. IV
of prompts in examples – Ch. 20
Follow Through research – Ch. 30
Fooler games – Ch. 15
Formal operations, teaching preschoolers – Ch. 30
Generalization – Ch. 1, Ch. 2, Sec. II
Hierarchical-class programs – Sec. IV, Ch. 11
coordinate classes – Ch. 11
lower-order sequences – Ch. 11
Higher-order noun sequences – Ch. 6
structure of – Ch. 11

 
Humanism – Ch. 31
Implied-conclusion tasks – Ch. 15
Indirect sequences, for comparatives – Ch. 7
Initial-teaching sequences – Sec. II, Ch. 5, Ch. 6, Ch. 10
Interpolation – Ch. 2, Ch. 14
Irregulars (see Subtypes)
Joining forms, overviews – Ch. 3, Sec III
Juxtaposition principles – Ch. 4
research on – Ch. 29
Juxtaposition prompting – Sec. V
Knowledge
analysis of – Sec. I
types of – Ch. 1, Ch. 3
Learning mechanism, assumed – Ch. 1
Learning sets – Ch. 2, Ch. 31
Manipulative tasks – Ch. 15
Match-of-label-and-example worksheet items – Ch. 16
Minimum differences – Ch. 5, Ch. 6
in examples and labels – Ch. 10
in program sequences – Ch. 10, Ch. 31
introduction of – Ch. 10
involving same responses – Ch. 8
misrules – Ch. 22
progressive minimum differences – Ch. 8
research on – Ch. 29, Ch. 30
within subtypes – Ch. 8
Narrow-range noun sequences – Ch. 6
Negative-first sequences – Ch. 5, Ch. 7
No-change negatives – Ch. 7
Non-comparatives – Ch. 3, Sec. II, Ch. 5
Non-continuous conversion sequences, Ch. 5
Noun concepts – Ch. 3, Sec. II, Ch. 6
Operations
cognitive, Ch. 3, Sec. VI
physical, Ch. 3, Sec. VI
Ordering rules for set members – Ch. 10
Pacing, research – Ch. 29
Positive-first sequences – Ch. 5
Preskills, contained in fact statements – Ch. 12
Problem solving – Ch. 25
(see also Cognitive problem solving)
Production-of-examples worksheet items – Ch. 16

 
Production-of-labels worksheet items – Ch. 16
Production responses in designing routines – Ch. 17
Program types
derived from tasks – Sec IV, Ch. 12
for coordinate members – Sec. IV, Ch. 10
for double transformations –, Sec. IV, Ch. 13
for hierarchical classes – Sec. IV, Ch. 11
for teaching fact systems – Sec. IV Ch. 14
overview – Sec. IV
to reduce prompting and stipulation – Sec. IV (see also Complete teaching)
Programming, choosing alternative strategies – Ch. 9
Programs
Implementation – Ch. 28
field tryouts – Ch. 28
research on – Ch. 30
Prompting
by grouping examples – Ch. 20
difference prompts – Ch. 20
of examples – Ch. 20
programs to fade – Ch. 9
sameness prompts – Ch. 20
semi-specific prompts – Ch. 20
Rehearsal – Ch. 24
Related concepts – Ch. 2
Removed-component-behavior program – Ch. 3, Ch. 23
Responses
complex – Ch. 3
simple – Ch. 3
Response classes, types of – Ch. 3
Response-locus analysis
overviews – Ch. 1, Sec. VII
traps – Sec. VII
Response teaching – Ch. 22
analysis of complex responses – Ch. 23
complex responses – Ch. 23
selecting one objective – Ch. 22
simplifying the form – Ch. 22
teaching how – Ch. 22
teaching when – Ch. 22
teaching when and how – Ch. 22
Sameness
how to communicate – Ch. 2

 
 research on – Ch. 29
Sameness principle – Ch. 4, Ch. 31
Setup principle – Ch. 4, Ch. 5
research on – Ch. 29
Shaping
context – Ch. 3, Ch. 22
form – Ch. 3, Ch. 22
Simultaneous components of responses – Ch. 23
Single-dimension concepts
comparatives – Ch. 7
non-comparatives – Ch. 5
Starting point, for comparative sequences – Ch. 7
Stimulus-locus analysis – Ch. 1, Sec. VII
philosophical, under pinnings of – Ch. 31
relations to learning theories – Ch. 31
Stimulus qualities – Ch. 1
Stimulus similarity – Ch. 7
Stipulation – Ch. 2, Ch. 5, Sec. V, Ch. 20, Ch. 23
counteracting it – Sec. IV, Ch. 10, Ch. 23
research on – Ch. 29
Subtypes
in expanded cognitive programs – Ch. 25
in transformation sets – Ch. 8
difference approach in teaching related subtypes – Ch. 10
Tactual vocoder, research – Ch. 30
Task analysis – Ch. 12
transformed tasks – Ch. 12
Task-based programs – Ch. 9, Ch. 12
Temporal parts, of complex responses – Ch. 23
Testing principle – Ch. 4
Transfer, how to induce – Ch. 24
Transformations – Ch. 3, Sec III
(see also double transformations) and J.S. Mill’s method of residuals – Ch. 31
divergent tasks for transformations – Ch. 15
double transformation sequences – Ch. 13
research on – Ch. 29
single transformation sequences – Ch. 8
Two-choice tasks – Ch. 5
Verifying conclusions, in teaching facts – Ch. 9
Wide-range noun sequences – Ch. 6
Wording principle – Ch. 4
Worksheet items – Ch. 15-16

 
 design of – Ch. 16
functions of,
expansion – Ch. 16
integration – Ch. 16
review – Ch. 16
types of – Ch. 16
use of in programs – Ch. 16

 
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