JEAN-BAPTISTE LAGRANGE

COMPLEX CALCULATORS IN THE CLASSROOM: THEORETICAL

AND PRACTICAL REFLECTIONS ON TEACHING PRE-CALCULUS

ABSTRACT. University and older school students following scientific courses now use
complex calculators with graphical, numerical and symbolic capabilities. In this context,
the design of lessons for 11th grade pre-calculus students was a stimulating challenge.
In the design of lessons, emphasising the role of mediation of calculators and the devel-
opment of schemes of use in an ‘instrumental genesis’ was productive. Techniques, often
discarded in teaching with technology, were viewed as a means to connect task to theories.
Teaching techniques of use of a complex calculator in relation with ‘traditional’ techniques
was considered to help students to develop instrumental and paper/pencil schemes, rich in
mathematical meanings and to give sense to symbolic calculations as well as graphical and
numerical approaches.
The paper looks at tasks and techniques to help students to develop an appropriate
instrumental genesis for algebra and functions, and to prepare for calculus. It then focuses
on the potential of the calculator for connecting enactive representations and theoret-
ical calculus. Finally, it looks at strategies to help students to experiment with symbolic
concepts in calculus.

KEY WORDS: computer algebra, graphic and symbolic calculators, instrumental genesis,
pre-calculus, student behaviour

INTRODUCTION

Traditionally, computers and calculators are distinct technological tools in

the teaching and learning of mathematics. Early computer use in mathe-
matics teaching was through programming, but more recent use tends to
favour use of generic packages including software dedicated to algebra
or geometry. In the teaching and learning of algebra and calculus in
the last 10 years there have been many experiments using Computer
algebra systems, like MAPLE and DERIVE, see (Mayes, 1997). Over
these years, the use of increasingly sophisticated hand held calculators
has impinged on everyday life as well as on classroom activities. When
sophisticated numerical and graphical capabilities were added, it became
clear to students and sometime to teachers that calculators could play a role
in solving problems involving functions (see Tall, 1996, Trouche and Guin,
1996).

International Journal of Computers for Mathematical Learning 4: 51–81, 1999.

© 1999 Kluwer Academic Publishers. Printed in the Netherlands.
52 JEAN-BAPTISTE LAGRANGE

New hand held calculators offer, to some extent, a synthesis of

computer software and calculators.1 Like computers they have powerful
applications: computer algebra systems, geometric software and spread-
sheet. From calculators they inherit ergonomic characteristics (small,
disposable) and numerical and graphical utilities important to the study
of functions.
This paper presents an analysis of an attempt to integrate these powerful
calculators in the teaching of pre-calculus in France. This integration has
been carried out in four classes of the ordinary French scientific upper
secondary level (11th grade).2 In this paper, I do not seek to prove that
teaching and learning with calculators is definitively better than with
traditional paper and pencil. I merely assume that these calculators are
legitimate means of doing mathematics.3 From this assumption, this paper
provides reflection, based on theory and practice, on the changes that
these calculators may bring to the teaching and learning of mathematics,
and a search for efficient means to use them in order that students learn
meaningful mathematics.
This TI-92 experiment is a continuation of an earlier French experi-
ence looking at the integration of DERIVE into the study of algebra and
calculus. Working in close co-operation with a group of teachers supported
by the National Ministry of Education (DISTEN group, see Hirliman,
1996) to study the effects of this integration, we carried out a number of
classroom observations from grade 9 to grade 12 (Artigue, 1995, 1997).
We also questioned twenty five teachers and nearly five hundred of their
pupils.
From this research we compiled a number of interesting insights on
how technology may support the learning of mathematics, which will be
referred to later in this paper. However, an important limitation of the
DERIVE study was that students generally lacked the familiarity with
this technology necessary to really use it to support their mathematical
activities and learning. On many occasions, we saw students using their
own numerical calculator to try to solve a problem numerically, when
we expected them to solve it symbolically with the help of the computer
algebra system.
So, when ‘computer-like’ calculators became available we saw the
potential for easier student access to computer algebra technology which
might affect their everyday mathematical practices, and that we would
be able to observe more substantial changes. Therefore, an offer by the
National Ministry of Education to support a teaching experiment for
pre-calculus 11th grade classes where every student had a TI-92, was
stimulating and welcome. However, from the DERIVE experiment, we
 COMPLEX CALCULATORS IN THE CLASSROOM 53

knew that the integration of symbolic facilities into the work of the student
was not an easy project. For that reason we had to develop a theoretical
approach to this integration and design lessons/activities which could be
applied in a wide range of settings: to students at various levels of attain-
ment, with varied attitudes to calculators and mathematics; to teachers
with distinctive epistemological views, teaching strategies and attitudes to
calculator use.
It appeared that we had to reflect on two linked set of issues.
• First, how can we conceptualise changes in the mathematical activity
in a classroom when every student has a powerful ‘computer-like’
calculator? To what extent can computer approaches in the teaching
of mathematics be used? How does our experience of using computer
algebra help us? What aspects of the work will be affected by the
personal character of the calculator?
• Second, what conceptualisation of calculator use, with its many multi-
level capabilities, arises in the teaching of a specific subject? What
help do the numerical and graphical utilities bring? Regarding support
for algebraic calculations, do the calculators help to build symbolic
definitions of concepts? How can we think the introduction of the
symbolic capability related with a concept (the key pressed to get a
derivative or a limit . . . )? Does it help students to conceptualise, if so
in what way? Are these capabilities a danger? Do students need to be
at a certain skill or conceptual level before using tools like this?
These issues, concerning both technology and mathematics, are of general
interest to those involved in mathematical education. The goal of this
paper is to reflect on these issues and explore outcomes from real teaching
situations.

THE EVOLUTION OF APPROACHES TO THE USE OF

COMPUTER TECHNOLOGY IN THE LEARNING
OF MATHEMATICS

Constructivist Approaches
When computers became available, many hopes were placed on the
autonomous cognitive activity that a learner could develop when faced
with specific tasks (Artigue, 1996). The general frame was a Piagetian
approach: acting in adequately problematic settings, the learner meets
insufficiency or inconsistency of his/her knowledge. Introducing computer
environment could help to create settings of this kind. The emphasis was
54 JEAN-BAPTISTE LAGRANGE

put on the role of purposeful action in the conceptualisation of knowledge

in opposition with the passive reception of meaningless mathematical
contents. Computer tasks appeared well suited to these conceptions.4
Another conception of the construction of mathematical concepts was
easily adapted to computers: many concepts, especially in algebra and
calculus, appear with two linked aspects, as a procedure and as an object.
Gray and Tall (1993) introduced the name ‘procept’ to describe this duality
in many areas of mathematics including calculus concepts. Computer
activity, especially programming, can give a sense of this duality. A func-
tion, for instance, can be defined by means of a programmed procedure,
then it will be considered and manipulated through the name of the
procedure.
Repo (1994) reports on an example of this approach in the learning
of calculus with the use of DERIVE. She blames the “quite algorithm
oriented” learning of mathematics prevalent in Finnish schools, and offers
six “critical activities” to activate prior knowledge of students, to inter-
nalise the concept of derivative, co-ordinate the representations of this
concept, generalise it and understand its reversibility. I will briefly study
Repo’s research because it had a significant influence on the view of
Computer Algebra Systems as “cognitive and didactic tool to engage in
reflective abstraction” (Mayes, 1997, p. 185). As for me, I see limits in this
approach and considering these will help to adjust my reflections. Repo’s
research design is that of a comparative study: a control group received
“standard mathematics teaching”, and the experimental group 50 lessons in
computer room based on the above critical activities. In an immediate post-
test, the experimental group performed significantly better on conceptual
items, and in a delayed post-test it showed better retention of algorithmic
skills.
My first criticism is that no evidence is given of the influence of
the computer on these improved performances. The control group had a
mainly algorithmic introduction to calculus, and a common consequence
is that they had a low understanding of calculus and a poor long-term
retention of algorithms, so the better achievement in the experimental
group may refer to the poor performances of the control group, rather
than to the computer activities. Repo’s approach stresses an opposition
between conceptual understanding and algorithmic skills, and the activi-
ties focus on understanding. Therefore, the way students acquired these
long-lasting skills is unclear. In France, approaches based on this opposi-
tion and on strong assumptions on the role of DERIVE to enhance the
conceptual learning have been tried. I argue (Lagrange, 1996) that there
is a gap between these assumptions and what actually happens in the
 COMPLEX CALCULATORS IN THE CLASSROOM 55

classroom. Using symbolic computation in the teaching of mathematics

requires teachers and researchers to think in depth about the relationship
between the conceptual and the technical part of the mathematical activity
rather than opposing them.
On a wider reflection Noss and Hoyles (1996, p. 21) stress the potential
productivity of the constructivist approaches, but also their limits. First,
when knowledge is built trough actions in a given computer context, pupils
are able to produce powerful reflections on objects in this context to solve
difficult problems, but it is not clear that this knowledge helps with tasks
outside the computer context. It appears, therefore, very contextualised,
and the decontextualisation is a problem. Second, the ‘procedure-object’
approach is sometimes a too rigid way for building concepts. There
is no permanent necessity to consider first an ‘operative’ (Sfard, 1991)
approach of concepts. In contrast, computers now offer a range of views
(or windows) on a concept wider than just the procedure-object duality. For
instance, the graphical utility is one between many views of the concept
of function in a computer environment, and the resulting plot can be
considered as a procedure (tracing the plot) and as an object (the global
properties of the plot).

The Computer’s Role in the Mediation of Students’ Activity

Noss and Hoyles (ibid., p. 54) point out the dialectic between human
culture and technology. A ‘cognitive’ tool is made from human cogni-
tion and it has an effect on the cognitive functioning of a person who
uses it. In this way, Noss and Hoyles stress, a computer application may
operate as a linguistic tool, and they emphasise programming as a tool for
expressing and articulating ideas. In their approach to the teaching of a
topic like proportionality (p. 75), they combine paper and pencil problem
solving, a computerised ‘target game’ and work on Logo procedures for
drawing objects in proportion. The power of expression of the computer
helps to broaden students’ conceptions of multiplication and, working on
Logo procedures, students act on the relation of proportionality and on a
formalisation of this relation. In off-computer activity students are able to
refer to the Logo formalism for explanation and evaluation.
So, in Noss and Hoyles’ view, the computer environment is not only
a field for students’ purposeful actions. The computer offers special
means for interacting with objects. Using the means, students enlarge their
conceptions of the objects, especially towards generalisation and formal-
isation. Therefore, Noss and Hoyles introduce this mediation as a major
role for the computer in the student’s process of abstraction. This idea
of mediation of instruments in the mental sphere of human activity was
56 JEAN-BAPTISTE LAGRANGE

initiated by Vygotskii. Primarily, the mediation is the use of properties of a

given object to act on another for a given task.5 The point is that mediation
changes the nature of the action of human over objects. In the psycho-
logical sphere, Vygotskii’s assumption is that “language, (. . . ) algebraic
symbols, (. . . ) and all possible signs and symbols” are instruments which
change the mental activity.6
This idea of mediation is useful in our project because a purely
constructivist view of the use of computers is insufficient to analyse
the interaction between the user, his/her instrument and the objects in
the settings. A constructivist view assumes that the computer settings
will provide the means for a predictable and meaningful interaction.
What actually happened when we observed the use of DERIVE was
different: interaction situations of the students and DERIVE were often
less productive than teachers’ expectation. Teachers generally expected
that students would build mathematical meaning from DERIVE’s feed-
back. Students’ reactions and reflections did not have this meaning because
their perception of the feedback was influenced by the operation of the
software (Lagrange, 1996). For instance 9th grade students with little
familiarity with DERIVE, were asked to observe the result of the Expand
command on the square of algebraic sums. The teacher expected that
the students would concentrate on regularities in this expansion like, for
instance, the relation of the number of terms in the sum and in the expan-
sion. In contrast students reflected deeply on the order of the terms in the
expansion, which is a regularity only linked to the software. Mediation
accounts for this phenomenon because students perceived the mathe-
matical settings through DERIVE, and being unaware of the properties
of this instrument, they could not understand that the regularities that
they found had no mathematical significance. In contrast, the teacher was
an expert both in mathematics and in DERIVE, and did not mind this
regularity.7
How do contemporary instruments like computers and calculators fit
with a theory of mediation? A computer, as considered by Noss and
Hoyles, is an instrument in two dimensions: a physical object with a
keyboard, a screen and so on, and an abstract operative language. Noss and
Hoyles focus on the abstract dimension of the Logo language, and there-
fore meet Vygotskii’s view of mental instruments. In the use of complex
calculators that I intend to analyse, this view seems less effective, particu-
larly in the phase where the user is learning new capabilities. In this phase,
a user sees the internal capabilities through the features of the interface
(for instance, with a TI-92,8 the different capabilities for solving are seen
through various entries of the algebra menus). This perception of the calcu-
 COMPLEX CALCULATORS IN THE CLASSROOM 57

lator does not distinguish between the interface and the internal logic.
This phase of learning is what I want to analyse because, in this phase,
cognitive processes are likely to appear, involving both the calculator and
mathematics.
For this reason, I prefer to consider a calculator as a complex instru-
ment like those existing in the area of professional working (for instance a
computerised system to pilot a process) rather than to reduce it to an addi-
tion of a neutral interface and an internal algebraic language. An advantage
of this approach is that it is easier to think about the changing relation of
the user and his/her calculator: in this relation, the user discovers together
the characteristics of his/her calculator together with the mathematical
underlying features.

The Role of the Instrumental Schemes

The process of development of new uses of an instrument, and the asso-
ciated cognitive changes, have been analysed by psychologists in terms
of conceptualisation. In a ‘study of thought in relation to instrumented
activity’, Verillon and Rabardel (1995)9 stress that a human creation, an
‘artefact’, is not immediately an instrument. A human being who wants to
use an artefact builds up his/her relation with the artefact in two direc-
tions: externally s/he develops uses of the artefact and internally, s/he
builds cognitive structures to control these uses. After Piaget, Verillon and
Rabardel describe these structures in terms of schemes, which are mental
means that a person creates to assimilate a situation. When a person acts on
settings trough an instrument his/her behaviour has a specific organisation.
For that reason, the authors10 introduces the notion of ‘instrument utilisa-
tion schemes’. These utilisation schemes have the properties of adaptation
and assimilation of the schemes and direct the uses of the instrument by
the person. Being mental structures of a person, utilisation schemes are
not given with the artefact. They are built in an ‘instrumental genesis’
which combines the development of uses and the adaptation of schemes:
when developing the first uses, a person pilots the artefact through existing
schemes, then this primitive experience is the occasion of an adaptation of
the schemes, and the better adapted schemes are a basis for developing new
uses, and so on. This genesis is both individual and social: a person builds
his/her own mental structures, but, generally, an instrument is not used by
only one person and therefore the process of adaptation takes place in a
social context.
58 JEAN-BAPTISTE LAGRANGE

Schemes in Calculus Using a Complex Calculator

Verillon and Rabardel’s cognitive approach to instruments shares many
aspects of Hoyles and Noss’ view of the computer in mathematical activity:
the instrument is not something neutral, it has an effect on the cognitive
functioning of a person who uses it. The cognitive approach describes this
effect as the development of specific schemes, and organises this devel-
opment in a genesis. This approach was stimulating for our project of
integrating ‘computer-like’ calculators because they are complex devices
with a lot of capabilities, each of them implying many specific schemes
that the user has to co-ordinate to achieve a given task. The idea of genesis
is useful because our project took place over a year and we had to think
through the development of the uses and the schemes, together with the
progression of the mathematical topics.
As an example, Figure 1 displays various schemes, calculator oriented
or not, algebraic, graphic or symbolic that a user of a TI-92 can use to
search for the variations of a function like x +x+0.01
2
x
. The schemes have
several dimension of functionality: decisional, they organise and control
the action; pragmatic, they act on the settings; interpretative, they help to
understand the settings.
Being adaptive mental constructs, schemes cannot be entirely described
in a rational form. In Figure 1, some of them are approached by their nature
and by features of the above three dimensions. Many other schemes exist
and are more difficult to describe. For instance, in the Graph window,
a student often develop exploratory zooming based on his (her) private
knowledge and previous experience. Moreover, schemes in Figure 1 are
made of a number of ‘sub-schemes’ more difficult to explicit.1 Never-
theless, the brief description of schemes in Figure 1 accounts for the
complexity of an action with this complex instrument, and from this
description, I will show what relation students may have with these
schemes.
The first scheme (graphing in the standard window) is prevalent among
most students. In the initial stages of learning calculus very few students
are able to produce critical interpretations such as those in the second
scheme, even when they have the algebraic knowledge to do so. The more
able students develop schemes where graphical action is linked with alge-
braic and analytic interpretation: they see, in the graph, properties that they
anticipate from an algebraic analysis of the function. This co-operation of
schemes of different nature gives them a new efficiency.
Transforming the expression of the function like in the third scheme is
not a spontaneous action. Most students initially choose the transformation
randomly among the TI-92 capabilities rather than from rational reflection.
 COMPLEX CALCULATORS IN THE CLASSROOM 59

Figure 1. Schemes in a search for the variations of a function.

Teaching can help to develop this reflection. Switching back to the graph
window, as in the fourth scheme, is quite natural. Some students anticipate
immediately the required zooming, while others take considerable time
over this decision. The latter may use trial and error processes, productive
for some but unproductive for others.
The calculus approach in the fifth scheme may derive from a teaching
method. I observed, however, that this scheme is activated only when
the function is similar to standard functions considered in the teaching.
When a student is perplexed, because of an unusual function, this scheme
is not likely to appear. It may not appear with the example of Figure 1,
because variations are not perceptible in a standard window. It certainly
does not appear when a student meets a new type of function, for
instance a trigonometric function when the student is used to rational
functions.
The sixth scheme is about limits. It illustrates how specific an instru-
mental scheme may be. In ordinary paper and pencil practice, the notion
60 JEAN-BAPTISTE LAGRANGE

of left and right hand limit is difficult because their computation implies a
reflection on the sign of sub-expression which is not familiar to students.
With the TI-92, the scheme described in Figure 1 works well on most
functions and contributes to give sense to this notion. However, this
sense is often partial, because most students have difficulties in inter-
preting the values of the limits in term of asymptotical behaviour of the
graph.
In this brief description of features of schemes appearing in a calculus
task, and their apprehension by students, the question of genesis appears
with some complexity. The development of utilisation schemes by students
appears to be linked to the development of their mathematical know-
ledge. But what is the nature of this link? Schemes appear to be more
or less influenced by teaching. But what is this influence, and how is
teaching to be oriented to help the development of suitable schemes,
their generalisation and their co-ordination? These questions call for
theoretical and practical reflection that I will undertake in the following
section.

AN APPROACH OF TEACHING WITH INSTRUMENTS

Schemes for Building Knowledge

Rabardel and Vérillon’s approach is an ergonomic one: finding a better
way of conceptualising human-instrument relations. Hoyles and Noss’
concern, as well as ours, is slightly different: to try to conceptualise how
the use of instruments intervenes in the learning of mathematical topics.
With respect to this aim we can go back to Vergnaud’s (1990) work on the
role of schemes in conceptualisation: schemes organise the behaviour of
a person in a class of problems and situations representative of a field of
concepts and are a basis for knowledge in this field. A given concept, from
this viewpoint, can be seen in relation to the set of problems to which it
provides a means of solution, and knowledge of this concept derives from
the schemes that a person builds to solve these problems.
When a person learns mathematics with an instrument, his (her)
schemes organise behaviours related to the use of the instrument as well as
more general conducts. Interpreting Noss and Hoyles’ study of a teaching
of proportionality, I can see that Logo programming is an instrumental
practice for manipulating a formalisation of proportionality in a problem
of expanding given patterns. Acting with this instrument, students develop
utilisation schemes, for instance rules of transformation of Logo expres-
sions to maintain the shape of a pattern. These schemes are specific and
 COMPLEX CALCULATORS IN THE CLASSROOM 61

not directly transferred in a non-Logo context. But, together with other

schemes, they are a frame for students’ conceptual reflection, and they
make specific contributions to that reflection.
At this point, comparing earlier approaches where mathematical know-
ledge is thought to be built from situations involving personal interaction
with the computer, the potential contributions of computers and of calcu-
lators appears different: technology acts as a mediator for the action of
students. In this mediation technology is by no means neutral: students
have to elaborate utilisation schemes, a nontrivial task.
This approach is consistent with Hoyles and Noss’ view of the role of
technology in building mathematical meanings. In addition, I focus on the
development of uses and utilisation schemes because in our project the
students use a complex calculator over the course of a year as a everyday
support to their mathematical practices. Given this, adequate utilisation
schemes of ’hand held’ technology are a condition for this support. In
turn, the development of schemes (the instrumental genesis) is dependent
on students’ progressive understanding of the calculus. For this reason I
emphasise this genesis and its role in students’ learning.
The genesis is, however, problematic. Mathematical meaning and
knowledge grow with the multiple schemes that students develop when
doing tasks in a domain, but not all schemes are productive of adequate
knowledge in all situations. Consider, for instance, the limit of a rational
expression at a finite or infinite point. In an ordinary ‘non computer’
context, students may apply the following reasoning to, say, limx→∞ x1 :
‘one over a large number will be small . . . , therefore, the limit is
0’. When the expression is more complex they may transform it, e.g.
change limx→∞ (x − x 2 ) into limx→∞ (x(1 − x)). Numerical and graphical
approaches may contribute to students’ progressive understanding of this
task.
In contrast, with a calculator like the TI-92 or algebraic software like
DERIVE, students are able to associate the idea of limit with a single
scheme: pressing the ‘limit’ key of the calculator and reading the output
on the screen. This scheme is effective for the task but, as Monaghan et al.
(1994) observed, it may result in giving students a narrow understanding
of the notion of limit. Comparing students who made extensive use of
DERIVE with other students, they found that the latter had more varied
representations of limits including infinitesimal approaches, whereas the
DERIVE students focused solely on limits as objects. Viewing their
report from the perspective of my theoretical framework, I say that the
scheme associates too closely the idea of limit with the limit capa-
62 JEAN-BAPTISTE LAGRANGE

bilities of DERIVE and this scheme generates a restricted mathematical

meaning.
On the other hand, the scheme for right and left hand limits in the
example of Figure 1 is very close to the above ‘key-stroke limit scheme’.
I said above that it is productive when giving students a sense of the exist-
ence of the limits, otherwise hard to grasp, because of the difficulty of
calculation.
So, depending on their co-operation with other schemes or meanings,
schemes of use of the TI-92 or DERIVE are productive or not. Therefore,
for the support of the technology to be effective teachers must control
students’ development of utilisation schemes and their co-ordination with
the advancement of mathematical knowledge. However, there might be
a contradiction here, because schemes are mental structures built by
the student, rather than objects for the process of communication, like
teaching. I thus examine the role of teaching in the context of the use of
technology by students.

The Role of Tasks and Techniques

I look at the teaching of techniques and at the relationship of this with
the instrumental genesis, as this is a key point in the use of technology to
teach and learn calculus. In Repo’s (1994) research we saw above that
approaches of this use may pretend to favour students’ higher concep-
tual thinking, in opposition with the usual training to algorithms in the
paper/pencil context. More precisely, authors and teachers assume that the
symbolic capabilities in this technology are means to lessen the stress on
techniques which, they consider, restrain students’ reflection on concepts.
This view was clearly present in teachers’ expectations in the French
DERIVE experiment, and reflecting on this was useful in establishing the
limits of this excessively conceptual approach (Lagrange, 1996).
First, the technical work did not vanish when doing mathematics using
Computer Algebra. Not all students welcomed the relief from the usual
pen and paper skills: some of them considered these skills as important
for success in Mathematics. It also appeared that using Computer Algebra
itself required specific techniques. For instance, when a student obtains
an output using the system, this output is not always the usual expres-
sion generally accepted in the pen and paper context. In this situation few
students could transform the system’s output to obtain the usual expres-
sion. A consequence is that although most students thought of Computer
Algebra as a helpful tool for ‘double checking’, they generally lacked the
techniques to perform effectively this double check.
 COMPLEX CALCULATORS IN THE CLASSROOM 63

Understanding mathematics with the help of Computer Algebra was

not a view that students generally considered. Even when they enjoyed
the new classroom situations they experienced using Computer Algebra,
they generally did not recognise that these situations could bring a better
comprehension of mathematical content because the situations focused on
conceptual aspects of a subject, and not on the usual techniques associated
with this content.
This observation was a starting point for a reflection on the relation-
ship between the technical and conceptual part of mathematical activities.
Chevallard (1992, 1996) stresses the links between techniques and theory.
Every topic, mathematical or not, has a set of tasks and methods to perform
these tasks. Newcomers in the topic see the tasks as problems. Progres-
sively they acquire the means to achieve them and they become skilled.
That is how they acquire techniques in a topic. Furthermore, in teaching
and learning situations, the students and the teachers are not interested in
simply acquiring and applying a set of techniques. They want to talk about
them, and therefore they develop a specific language. Then, they can use
this language to question the consistency and the limits of the techniques.
In this way they reach a theoretical understanding of a topic.
A break from teaching based exclusively on training in algorithmic
skills is certainly interesting. However, teachers’ and researchers’ views
of the support of symbolic computation tend to hide the need for a set of
techniques.
So, I emphasised above the role of schemes in the process of concep-
tualisation, and now I stress the need for techniques in the teaching of
concepts. But what is the relationship between schemes and techniques? I
said above that schemes, being internal adaptive constructions of a person,
cannot be taught directly. In contrast, techniques are rational elaborations
used in teaching. Techniques are official means of achieving a task but,
in facing the task, a person doesn’t ‘follow’ a technique, especially when
the task is new or more complex or more problematic than usual. When
knowledge is requested a person acts through schemes.
So, in an educational context, techniques can be seen as official,
rational objects for communicating whereas schemes are structures actu-
ally produced in students’ mind.12 Drilling on a single technique for a
given task without reflection is only able to produce manipulative schemes
and poor knowledge. Many innovators, particularly in the field of the use of
computer, argue this to diminish the role of techniques and try to promote
‘conceptual mathematics’. I observed in the DERIVE experiment that
diminishing the role of techniques encouraged teachers to avoid devoting
time for discussion on these. In contrast, talking of techniques in the
64 JEAN-BAPTISTE LAGRANGE

Figure 2. A task in an experimental exam.

classroom might help students to develop suitable schemes. Furthermore,

in this communication a specific language and theoretical reflection is able
to appear and students can enhance the reflective part of their schemes.

Techniques in the Use of a Complex Calculator

Returning the use of symbolic computation, graphical, numerical and
symbolic facilities make traditional techniques less relevant. In addition,
the role of those techniques is often undervalued because teachers see
them as the routine part of their activity. New techniques should be
taught to help the development of utilisation schemes but teachers often
believe that these techniques are obvious or linked too closely to the
calculator to be relevant. The necessity and relevance of new techniques
may be made clear by considering the task in Figure 2. It was given
in a French experimental exam designed to test the adequacy of a set
of questions when students are allowed to use calculators. The text is
written to avoid giving advantage to students with a symbolic calculator:
a factorised expression of the derivative is given, so students without
symbolic facilities are able to do the subsequent question (variations of
f) in a similar manner to students who obtain this expression from their
symbolic calculator. But, when I consider the TI-92 answer for the deriva-
tive, I see that the task of the user with a symbolic calculator is not
straightforward. The TI-92 answer is neither the expression of the text
nor the raw form obtained when applying the rules of differentiation.
Recognising the expression of the text as a factorised form the user may
apply the factor command. Again, the expression is not the same as in
the text. Therefore the user has first to show how the TI-92 answer can
be obtained from the raw differentiation and then reflect on the two TI-
 COMPLEX CALCULATORS IN THE CLASSROOM 65

92 expressions to show their equivalence with the expression of the text.

Techniques exist to do that (for instance, differentiating sub expressions
helps to obtain the raw form, reflecting on the desired form helps to choose
the right application), and, although linked to the calculator, these tech-
niques might be a topic for teaching. For instance, reflecting on the desired
expression on theTI-92 may help students to focus on the forms of the
expressions.
TI-92 techniques are specific because they rationalise schemes of use
of an instrument, and, according to Rabardel and Vérillon, these schemes
develop in an instrumental genesis. A consequence is that the organisation
of the tasks and associated techniques must comply with the constraints
of that genesis and direct it in a productive way: schemes cannot develop
arbitrary and not all combinations of schemes are productive for mathe-
matical meaning. Below, I look more closely at these constraints and their
implications in terms of tasks and techniques, from the experience of the
TI-92 project.

TEACHING PRE-CALCULUS WITH COMPUTER-LIKE

CALCULATORS

Our team choose a level where the legitimacy of an unusual and relatively
expensive calculator might be accepted.13 In the French general upper
secondary level, students are in three main branches: literature, economy
and science. In this latter branch students’ use of calculators with sophis-
ticated numerical and graphical capabilities is now well established. Thus,
we expected that students accept the TI-92 in spite of its unusual aspect,
as an ‘enhanced’ substitute for their familiar calculator. We chose the
first year (eleventh grade), because the ‘baccalaureat’, at the end of the
second and last year of this course brings students much anxiety, with
possible negative effects on the experiment. The curriculum of this first
year is an introduction to calculus concepts (functions, limits and deriva-
tives) and to their application, based on problem solving and on experi-
menting. This curriculum suited our approach well, because it focuses on
the development of abilities in algebra and calculus and on the under-
standing of functional concepts, an interesting frame for an instrumental
genesis.
We worked with two teachers in two distinct regions of France. Thus,
although the teachers collaborated, we might observe two distinct experi-
ences of the integration of the TI-92. In the first year, our work was mainly
observing classroom sessions and students. The teachers had been working
with us in the DERIVE experiment, and we asked them to adapt the many
66 JEAN-BAPTISTE LAGRANGE

sessions that teachers built in this experiment, in order to use the calculator
in every suitable classroom situation.
The observation of the students was done by way of three attitudinal
questionnaires and three individual interviews of a sample of students.
From the observation in the first year and from the analysis of classroom
sessions in the same year, we built our project, a series of lessons and
classroom activities that the French Ministry of Education will publish as
a guideline for teachers. We experimented this project in the second year:
the teachers taught the lessons and we did an observation like in the first
year.
The aim of this paper is not to report this whole experimentation, but
to emphasise the role of teaching. Lagrange (to appear) will focus on the
observation of students. It will show how, in the first year, the acquisi-
tion of utilisation schemes was a long and complex process, effective for
some students and more problematic for others, with significant differ-
ences between individual students and between the two classes. It will
also discuss the improvement that the project that we experimented the
second year brought in students’ attitudes and abilities. Here, in this paper,
from the lessons that we experimented in the second year, I offer a view on
how teaching might help the development of schemes productive to mathe-
matical meaning. The observation of students’ genesis in the first year will
be used to show the necessity of this view,14 and classroom observations
in the second year will help to discuss its effectiveness.

Tasks and Techniques to Develop an Appropriate Instrumental Genesis

for Algebra and Functions
Obviously, at the beginning of an instrumental genesis, a user exercises
the schemes s/he built for other familiar instruments. For instance, when
a beginner uses his/her new TI-92 to do a division, like 34 divided into
14, s/he keys in 3 4 ÷ 1 4 ENTER, like on an ordinary numerical
calculator and s/he is very surprised when the TI-92 answers 17/7. Tasks
and techniques are to be organised to help him/her learn that, in the default
mode, the TI-92 simplifies radicals and rational numbers symbolically and
that decimal approximations must be specifically requested. Moreover,
the user has to consider that the graph window handles functions in an
approximate mode. Meanwhile, s/he has to consider, more acutely than
usual, the difference between the mathematical treatment of numbers and
the approximations of everyday practice.
Then schemes of use of the algebraic capabilities are essential.
Symbolic applications like DERIVE or the main module of the TI-92 are
basically algebraic, even when they include facilities in calculus: their core
 COMPLEX CALCULATORS IN THE CLASSROOM 67

Figure 3. Algebraic tasks.

is the treatment of expressions that a student has to understand before

s/he is able to use them for problems on functions. A key point is the
notion of equivalence of expressions and the need for awareness of the
different equivalent forms of an expression. A TI-92 user meets auto-
matic simplification as soon as an expression is entered. The following
screen displays an example of a puzzling phenomenon occurring with
an automatic simplification. Two obviously equivalent expressions are
‘simplified’ in two radically different forms.

So a student cannot rely on automatic simplification to obtain the form

s/he needs for an expression. S/he must consciously learn to use the items
of the algebra menu (Factor, Expand, ComDenom), to decide whether
expressions are equivalent as well as anticipate the output of a given
transformation on a given expression.
The above screens (Figure 3) give a hint of the tasks involved in our
scheme, to develop students’ algebraic instrumental schemes.
In a first task students had to enter the expressions on the left of the
screen A and then observe the TI-92 simplification on the right of the
screen A. They then identified the mathematical treatment: expanding,
factoring, ordering, partial fractional expanding. In another task, an expres-
sion G was given, with three other apparently similar expressions H, I, J.
68 JEAN-BAPTISTE LAGRANGE

The student had to find a TI-92 command to decide what expression H,

2 −6x+2
I, J is equivalent to G. In this task, G was x 2x−1 , H was −11x+4
2x−1
+ x2 ,
√ √
3
I was 4(2x−1) − x2 + 114
and J was (x+ 7−3)(x−
2x−1
7−3)
. By expanding G,
students had no difficulty seeing that I was the opposite of G (screen B). In
contrast, showing that G and H are equivalent, is not straightforward: in the
screen B, the function for the reduction of a sum of rational expressions has
been used to reduce a sub-expression. Factor was the appropriate function
to obtain J from G.
Work on the equivalence of expressions proved necessary not only
at the beginning of the TI-92 use. For example, towards the end of the
academic year of student TI-92 use the teacher asked
them to differ-
entiate the trigonometric function x → cos 3x − π6 by hand and
with the TI-92, and to explain why resulting expressions are equivalent.

The application of rules of differentiation

gives −3 sin 3x − π
6
when
the TI-92 gives 3 cos 3x + 3 . We expected that students could give a
π



reason like cos a + π3 = − sin a + π6 , because they knew the property


cos a + π2 = − sin(a) but only 8 in a class of 26 were able to do that.
Others expressed general reasons like “the calculator doesn’t work like we
do”. So the work on the equivalence of expressions had to be continued
when new expressions were introduced to help students to build suitable
utilisation schemes.
As stated before, conscious use of the algebraic capabilities of the TI-92
may help students to focus on the most suitable form for a given task,
whereas paper and pencil schemes focus on the rules of transformation.
One may reasonably think that the joint development of the TI-92 and
paper/pencil schemes is able to give an understanding of the equivalence of
expression. This is an example of how paper and pencil and TI-92 practices
are to be thought complementary in teaching, rather than opposed.
Like many calculators, the TI-92 offers a graphical window and
a numerical table with a wide range of capabilities. Therefore, it
may enhance early functional thinking because graphical and numerical
schemes are essential for the growth of the function concept. As seen
above, in Figure 1, notions like the variations of a function implies the
co-ordination of algebraic and graphico-numerical utilisation schemes.
Able users adjust settings of the graph window to make visible proper-
ties that they see algebraically, and use algebraic transformations to prove
properties that they read from a graph.
A relevant task for developing these schemes is the study of functions
whose properties are not obvious in a standard graph (see Figure 1 for
an example, and Guin, Trouche, 1999 for others). From a task like this,
teaching may focus on techniques for useful zooming (identifying values
 COMPLEX CALCULATORS IN THE CLASSROOM 69

of interest, specifying the graph window to show those values . . . ) and

for relevant algebraic transformation (Expand for finding the asymptotic
behaviour, Solve for the intersection with axis . . . ). From this, students can
get a better view of the treatment of functions in the graph window.

Helping Students to Develop Flexible Links between Calculus Concept

Representations
Work and schemes in algebra and functions is not itself calculus, but it is
the first part of a genesis in which limits and differentiation can develop.
The use of the TI-92 in pre-calculus is quite simple: a menu entry for the
symbolic calculation of limits and a key for the calculation of derivatives.
So, as a difference with algebra and functions, no specific instrumental
learning will be necessary. However, we saw above, with limits, that this
use tends to produce symbolic manipulative schemes, likely to generate
a narrow understanding of these concepts if they are alone. This implies
a deep reflection on how teaching can help the development of other
schemes, instrumental or not, that students could associate to the idea of
limit and derivative.
Tall (1996) stresses that there is not a single way, but ‘a spectrum of
possible approaches . . . from real-word calculus . . . through the numeric,
symbolic and graphic representations in elementary calculus, and on the to
the formal . . . approach of analysis’. He emphasises the need for helping
students to move flexibly from one representation to another. He notes,
moreover, that in teaching the balance between various approaches is diffi-
cult to establish because an approach well suited for one student will
not necessary suit another, due to differing cognitive profiles. Traditional
balances exist but technology tends to toss them about.
For instance, in France, every student in the secondary level now has a
graphico-numerical calculator. This situation clearly changes the balance
of numeric and graphical representations and of the symbolic view of the
concepts of calculus. For instance, in a traditional approach, searching
for the variations of a function began with symbolic study. The result of
this study was a set of particular values of the variable (turning points,
asymptotic branches . . . ). Values of the function were calculated for those
particular values: they were few because of the lengthy calculations by
hand. These values helped to scale, then plot the graph of the function,
which, in turn, helped to check the consistency of the symbolic study.
So, in this traditional approach, the symbolic study commanded the study
of variations. In contrast, with numerico-graphical calculators, graphing,
zooming, calculating tables of values is very easy and a student will use
these facilities extensively to get graphical or numerical evidence of limits
70 JEAN-BAPTISTE LAGRANGE

or variations. Symbolic calculations are relatively difficult, as they neces-

sitate paper and pencil work, and a different reflection. Many authors
emphasise the potential of graphic calculators for developing new under-
standings and skills (see for instance Shoaf-Grubbs, 1995). It is, neverthe-
less, true that these calculators make the traditional balance of symbolic
and graphico-numerical representations redundant, but a new balance is
not yet clearly established. A cautious approach would be to encourage
students to use a symbolic approach to control their calculator work,
but students often prefer experimenting rather than analysing. Trouche
(1996) noticed this behaviour and emphasises the schemes that students
should develop to control the graphs and numbers they obtain on their
calculators: these schemes are theoretical (through algebraic or analytic
considerations), enactive (the student acts on the picture, for instance
s/he traces a graph) or reflective (producing other graphs/pictures).
He suggests specific teaching strategies to help students to get these
schemes.
Now, with the TI-92,15 calculators are graphico-numeric and symbolic.
Little is known of how this new feature will affect the balance between
representations. In most studies, students do not have enough practice of
computer algebra to effect a change in the links between representations,
or authors do not consider this question.16
A first difficulty is the higher complexity of this instrument. With
common graphing calculators, graphical and numerical schemes are
instrumental when analytic schemes are associated with pencil and paper
practices. In contrast, with the TI-92, co-ordinating analytic and graphico-
numerical schemes implies controlled switches between windows. In
those switches, a flexible view of the organisation of functions in the
calculator is essential. An example of the difficulties occurring because of
a limited view was observed with one student. She had limited knowledge
of the capabilities of her TI-92 and one of her peers entered the following
into her calculator:

So, the second function, y2, was systematically defined as the derivative
of the first function y1. When she had to study a function, she introduced
this function as y1, and then, without doing anything, y2 was its deriva-
tive. She did know that, but not the way it worked. So in ordinary tasks
 COMPLEX CALCULATORS IN THE CLASSROOM 71

with her calculator she was comfortable. But when she had to use another
calculator, or in a task involving two functions, she was totally confused.
Whatever the difficulties, the design of teaching modules for pre-
calculus courses forced our team to make assumptions on how, with easier
symbolic calculation, teaching may obtain an adequate balance between
representations. We assumed first that easier symbolic calculation enlarges
the possibility of linking enactive representations and theoretical calculus,
and also that teaching must avoid the danger of too close an association
between concepts and symbolic manipulative schemes, wiping out other
representations.

Linking Enactive Representations and Theoretical Calculus

Enactive representations (Tall, 1996) exist in the prior differential know-
ledge of students. For instance, most students have a sense of the tangential
behaviour of curves from their geometrical experience. It seems important
to use this knowledge as a basis for the theoretical concept of derivative,
because differentiation is an analytic answer to the question of the tangent
line for a curve defined by a function. However, in the ordinary context of
paper and pencil calculations, students cannot really question their enactive
differential notions because they would have to consider, and give sense to,
expressions which are beyond their experience.17 Using symbolic compu-
tation potentially helps students to work with these expressions and to
understand their meaning.
As an example, let us consider the introduction of the concept of
derivative we experimented. It starts from the following problem: Let (G)
be the graph of the function f defined by f (x) = x 2 − 1 and A be the
point (1, 0). For every straight line passing through A, a number m exists
such that an equation is y = m(x − 1). What straight line gives the best fit
of the graph (G)? Through geometrical experience students were able to
guess that the line of best fit is when m = 2. But the teacher stressed that,
for m = 1.9 and m = 2.1, the lines also fit well, and students recognised
that zooming does not help to distinguish the ‘fitness’ of the three lines.
72 JEAN-BAPTISTE LAGRANGE

From this reflection it appeared necessary to consider the distances

between an arbitrary point on the curve, near A, and the points of same
abscissa (1 + h) on the lines. The first three lines of the TI-92 main screen
below display the expressions of this distance for the three lines.

The simplified forms (on the right of the screen) gave a hint of why
the line fits better for m = 2. In ordinary paper and pencil practice, the
expressions and simplification would have been complex for students and
would have hidden the sense of the expressions. A feature of symbolic
computation is that students were able to focus on the symbolic forms
without being disturbed by the complexity of hand calculations.18 Students
were, moreover, able to address the more general question of what line fits
the best among all lines passing through A. Calculating with a parameter
is never easy for students at this level, but the symbolic computation (last
line of the screen) made it similar to the preceding calculations.
Furthermore, given the aim to develop students’ links between their
enactive conceptions of tangent lines and the theoretical notion of
derivative, one example is clearly not enough. Symbolic computation
again may make a contribution because students were able to address
the same question, first for other points of the same graph (G), then for
other functions. The example below shows a screen for the same function
defined by f (x) = x 2 − 1 at the point x = 2, then for a cubic function at
the point x = 1.
 COMPLEX CALCULATORS IN THE CLASSROOM 73

In this process, students operated by themselves progressive modi-

fications of the above expression, and in doing so, concentrated on the
expression and grasped its sense.19 The symbolic capabilities of the TI-
92 were essential in helping students to focus on the algebraic forms.
However, they needed good algebraic schemes of use of the TI-92 to give
sense to the transformations.

Symbolic Aspects of Calculus Concepts

I expressed my concerns above that students may use the symbolic capa-
bilities for very simple limits or derivatives and see nothing more in
those concepts than the manipulative aspects. In the paper and pencil
context the tendency for students to consider mathematics as meaningless
symbolic manipulation exists, but these manipulations are often tedious
and better replaced by a reflection which involves other representations
of the concepts. As Monaghan et al. (1994) argue, symbolic computation
may make manipulations effortless but tends to obscure other represen-
tations linked with infinitesimal approaches. For that reason, we preferred
to introduce the TI-92 capabilities for limits and derivative only after
students did considerable work on the concepts, linking enactive views
with graphico-numerical approaches and symbolic forms. This is clearly
distinct from approaches like Watkins (1992) which introduce calculus
concepts as outputs of symbolic computations. Watkins’ concern, however,
is with vocational courses where little time is devoted to conceptual devel-
opments. At the secondary general level the balance of representations
should be a major goal.
So, in our proposition, the limit concept was introduced from an intui-
tive view that a function ‘tends toward zero as x tends toward zero’.
Students did a lot of graphic and numeric work, passing from ‘f(x) is small
when x is small’ to ‘f(x) can be arbitrarily small provided that x is small
enough’. Then students had to study by the same means standard, as well
as non regular, limits before the limit function of the TI-92 was introduced.
The concept of derivative was introduced from a geometrical represen-
tation, as outlined above, and another two sessions focused on activities
where students had first to build a formal definition of the derivative
and then as for the limits, search for derivatives of standard as well as
nonregular functions. In this work, students used the symbolic and graphic
facilities of the TI-92, but not the key for the symbolic differentiation (see
screens).
74 JEAN-BAPTISTE LAGRANGE

After this it was time to consider the symbolic aspects of the concepts,
namely the algebraic rules by which a person or a machine is able to obtain
derivatives or limits of expressions. The question was how to use the TI-92
to teach students algebraic rules that their calculator uses, and to give those
rules a meaning.
Authors introduced the “black-box-white-box process” which could be
an answer to the above question.
Using the CAS as a black box enables students to discover mathematical theories, concepts
or algorithms (. . . ) to the point where the students say ‘we are able to do what the CAS
can do’ (Heugl, 1997, p. 34).

This is, in my opinion, too simplistic a view of the support computer

algebra can give: using a CAS as a black box, students will only discover
symbolic entities. Learning theories and concepts implies wider strategies,
as we have seen above. However, as the black-box-white-box process
focuses on symbolic aspects of concepts, it could be useful for teaching
symbolic rules.
For instance, students could consider several examples of how the TI-
92 computes limits and derivatives and then learn to do those calculations
by themselves. In this process the student is likely to have more self-
reflection than in a formal approach where the teacher demonstrates the
rules. However, from experience, we consider that implementing such
processes is not so simple. The first problem is that in this process students
are prone to see only the manipulative aspect of the rules, even when
previous teaching focused on other representations of the concepts.
Another difficulty occurs when students have insufficient algebraic
maturity to give a suitable meaning to the feedback from the computer.
Pozzi (1994) gives an example where students had to understand the rule
for the differentiation of a product of functions. They considered the differ-
entiation of x → cos(x)(7x 3 + 2x), and, using DERIVE, they obtained
(21x 2 + 2) cos(x) − x(7x 2 + 2) sin x. Then, they deeply reflected on the
central part of the expression cos(x) − x(7x 2 + 2) and found it to be
very similar to the original function. They tried unsuccessfully to derive
a general rule from this example. Clearly, their reflection was misplaced
 COMPLEX CALCULATORS IN THE CLASSROOM 75

because they did not see that the central part is not a sub-expression of the
derivative. Once more, we see how good algebraic schemes are essential
to be able to make sense of computer algebra output.
Pozzi further stresses that ‘computer algebra systems can support
students to make sense of their algebraic generalisation’ but he maintains
that ‘this is only likely to be achieved if (students) use the computer to
explore and verify their conclusions and not simply as a symbolic calcu-
lator’. So students should be encouraged to make conjectures about general
properties and produce examples to test these conjectures.
As an example of this we designed a session to enable students to
discover the algebraic properties of limits and to learn how to use them.
This algebra of limits is summarised in the following TI-92 table.

In this table, the four indefinite limits appear ‘undef’. The actual aim
of the lesson was that the students bear in mind these four cases as well
discovering the means to solve these limits.

For instance, they should be
able to recognise that limx→0 x14 − x12 is indefinite and to find that this
expression has actually a value (+∞).
We asked students to experiment on an example of explicit func-
tions and not on symbolic notations like in the above table. The data-
matrix editor21 of the TI-92 was used to support this investigation. The
teacher introduced the first examples of limits of sums to give students
a method, and to introduce the problem of indefinite cases. Students
were then requested to produce others examples of possible values for
limx→0 (f (x)+g(x)), when limx→0 f (x) = +∞ and limx→0 g(x) = −∞.
Then they had to make conjectures for products and quotients and produce
examples to illustrate these. The TI-92 gave the values of the limits but the
teacher asked the students to explain the values by qualitative reasons or
by calculation.
At this time students’ knowledge about limits was new and they
encountered many difficulties recalling even simple limits. We were aware
that this lack of mathematical maturity might cause them too much to
rely on the TI-92 for the calculation of limits and thus to use try and
76 JEAN-BAPTISTE LAGRANGE

error strategies rather than anticipating. After a first experimentation of

the lesson we decided to make the limit point zero for all the limits.
With these settings students produced a lot of examples and convinced
themselves that if limx→0 f (x) = +∞ and limx→0 g(x) = −∞,
limx→0 (f (x) + g(x)) may ‘give everything’. The other indefinite cases
did not appear immediately. For example, limx→0 (f (x) × g(x)) presented
difficulties when limx→0 f (x) = +∞ and limx→0 g(x) = +∞, because
many students produced examples where this limit was zero, and thought it
was a general rule. But other students produced examples where this limit
was different, and convinced the class that it is again an indefinite case.
The need for controlled anticipation induced students to think of limits
on the basis of their prior infinitesimal knowledge and the emphasis of
the teacher that their reasoning corroborate values obtained of the TI-92
helped here. The problem of indefinite cases clearly appeared to students,
and they could easily recall them from the examples that they produced.
This is an example on how the symbolic calculator might help the students
to conduct a mathematical activity in symbolic aspects of a concept in
calculus without forgetting previously constructed representations.
A number of sessions of more open research followed these intro-
ductory lessons. This paper is a first look into how a teaching of pre-
calculus might help the development of productive calculator use schemes.
Thus, it leaves the students at a first stage of their genesis. Analysis of the
sessions would help to see how in these sessions students put their schemes
at work, questioned and enhanced them.

CONCLUSION

To conclude, I will outline the issues discussed in this paper, and point out
questions for further analysis.
The table in Figure 4 summarises the key points which arise when I
tried to conceptualise changes in the mathematical activity in a classroom
when every student uses a ‘computer-like’ calculator and to see how
teaching can take this use into account in a subject like pre-calculus. I
saw the role of mediation of these calculators from the many new poten-
tialities and constraints that they bring: when a student has one of these
for everyday work in mathematics, his/her action depends strongly on
these. Using it along a year s/he develops schemes specific to the calcu-
lator, together with other schemes. This instrumental genesis has its own
constraints deriving from the specificity of the calculator as well as of the
mathematical topic. As a student understands a mathematical topic from
the schemes s/he builds to do tasks in this topic, teaching has to be attentive
 COMPLEX CALCULATORS IN THE CLASSROOM 77

Figure 4. Key points in teaching pre-calculus with complex calculators.

to this genesis. The teaching experiment I did with the DIDIREM team is
a practical example of how a reflection on the instrumental genesis helps
to design lessons, developing students’ suitable schemes and connecting
various representations of concepts.
The role of schemes in the understanding of mathematics is not a new
idea. In contrast, the need for conceptualising the development of specific
schemes of use recently appeared in the research studies when students
uses of technology moved from occasional to regular. The context of long
term everyday use of technology forced researchers to look at this instru-
mental genesis. Techniques are now seen by mathematics educators as an
important level between the tasks and the theoretical reflection. However,
this role has rarely been considered in the use of computerised tools. This
paper offers to look at the techniques as official, rational objects in the
classroom and to schemes as more ‘private’ entities making up a frame for
the learners’ knowledge. Highlighting various techniques and encouraging
discussion on them, teaching influences students’ development of utilisa-
tion schemes and is thus able to direct it in a mathematically productive
way.
In this paper, my approach of the changes induced by a complex calcu-
lator in the learner’s action was a broad one and issues would deserve
further analysis. Particularly, I had just a short look at the effect of calcu-
lator language use on the students’ work. A reasonable assumption would
be that this language gives students an ‘expressing power’ that they could
use when working with the calculator, and also in classroom interactions
as observed by Hoyles and Noss (ibid., p. 153). These authors, however,
demonstrated that this potential is not a general property of the use of
technology, but a consequence of particularities of the microworlds that
they analysed. Thus a more precise analysis of students’ uses of calculator
expressions to handle objects is to be done. This analyse should search for
the possible schemes and technologies which would give sense to these
uses. More generally, with or without calculator, we have to consider
the instrumental dimension in students’ work. A deeper look into this
78 JEAN-BAPTISTE LAGRANGE

dimension would help to appreciate the respective contributions of the

paper/pencil work and of calculator use.

ACKNOWLEDGEMENT

Special thanks are extended to John Monaghan. He made many helpful

suggestions on the content as well as helping to overcome language
difficulties.

NOTES
1 This paper is based on a project where every student had a relatively expensive Texas
Instrument TI-92. Manufacturers now offer complex calculators pricing like ordinary
graphing calculators. For instance, the new TI-89 has the capabilities of the TI-92, except
geometry, that we did not use in the project. Looking at the interface, the TI-92 is like
a small computer (high resolution screen, alphabetical keyboard) and the TI-89 is like a
graphing calculator. This difference is not very consequential for the discussion in this
paper. Casio offers also a graphing calculator with symbolic capabilities (the GRAPH 80).
2 Michele Artigue was the leader of the team. Badre Defouad and the author participated
with the teachers, Michele Duperrier and Guy Juge, to the definition of the sessions and
did classroom observations and interviews. A report on the project can be obtained from
DIDIREM Université Paris VII 75251 Paris Cedex 05, France. The project was founded
by the French ministry of Education (DISTEN B 2). In the paper, ‘we’ and ‘our’ will refer
to the team. ‘I’ and ‘me’ will be used to express my own ideas and work.
3 Legitimacy of a technological tool is a complex question, which do not limit to improved
efficiency. Use by professional mathematicians, acceptance by parents, allowance at exams
are other important factors of legitimacy.
4 Papert (1980) is an influential example of a Piagetian approach in a computer environ-
ment.
5 “Mediation is a trick of the mind” Hegel quoted by Moro, Scheuwly (1997, p. 2).
6 Quoted by Moro, Scheuwly (ibid, p. 3).
7 Artigue (1995) named “pseudo-transparency” this phenomenon: in the mediation, the
instrument is transparent for the teacher, but not for students.
8 In the paper, TI-92 may be replaced by TI-89 or other complex calculator with the same
symbolic, graphic and numeric capabilitites. See note 1.
9 See also, in French, Rabardel (1995, p. 37).
10 See also Rabardel (ibid, p. 93).
11 Trouche (1996, p. 303) produces a comprehensive analysis and classification of
schemes in a search for a limit of a function.
12 In this paragraph, I look briefly at the relationship between schemes and techniques,
to emphasise their respective functions. Schemes and techniques may be viewed in a more
dialectical relation. There is a wide range, from personal hidden elementary schemes to
social global schemes. The latter are more easily rationalised. Teaching can act more
directly on these schemes, very similar to techniques. See again Trouche (ibid.)
13 See footnote 3.
 COMPLEX CALCULATORS IN THE CLASSROOM 79
14 Defouad (1999) analyses more comprehensively the varied individual genesis of
students.
15 And other calculators: see note 1.
16 Ruthven (1997) reviewed a number of researches into CAS in mathematics education.
The prevalent topic appearing in this review was the comparison of student performances
between CAS and non CAS students. Research reports on the impact of CAS in the
everyday teaching are very recently available. One of them is Guin, Trouche (1999)’s study.
17 Motion and velocity are other enactive differential notions that could be considered.
Questioning this notion seemed even more difficult for students. So we did not consider
this notion in the introduction of the concept of derivative. It seemed however interesting
that students establish the link between this notion and the differentiation when the concept
of derivative and associated schemes were steady enough.
For a very stimulating picture of problems arising when students have to build mathe-
matical representations of motion, see Boyd and Rubin (1996). Interestingly, they study
the effect of mediation by a noncomputer technology: the interactive video.
18 Students were however interested for knowing the reason why the expressions simpli-
fied in that form. That is why the teacher asked them to develop and simplify with paper
and pencil one of the expressions. Done separately of the main process of solving, and
for only one expression, this hand calculation gave this process a complementary meaning
without obscuring it.
19 This process has clear links with the formalisation of proportionality in expanding
patterns of Noss and Hoyles: pupils have a perceptive (enactive) idea of patterns in propor-
tion, and the computer helps them to consider a formalised relation between those patterns.
20 Computer Algebra System.
21 This module is like a symbolic simplified spread-sheet software.

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