Unit 1: Philosophy and History of Mathematics and Mathematics Education

1.1.1 philosophy of Mathematics

Activity 1.1.2: (Brian Storming) (take 8 minutes)

Discuss in groups of five and write a brief agreed up on response on your exercise book
for further reference.
1. What do we mean by philosophy of Mathematics?
2. Could it make sense to doubt 2+2=4? To doubt there is a thing as the empty set?
3. How do students learn mathematics?
4. How do teachers teach mathematics?

The philosophy of mathematics is the branch of philosophy whose task is to reflect on, and
account for the nature of mathematics. It studies the philosophical assumptions, foundations, and
implications of mathematics. The philosophy of mathematics addresses questions such as: What
is the basis for mathematical knowledge? What is the nature of mathematical truth? What
characterizes the truths of mathematics? What is the justification for their assertion? Why are the
truths of mathematics necessary truths?

Controversy in the Philosophy of Mathematics

The fundamental problem of the philosophy of mathematics is that of the status and foundation
of mathematical knowledge. What is the basis of mathematical knowledge? What gives it its
seeming certainty, and is this certainty justified? Two main schools in the philosophy of
mathematics can be distinguished: the absolutist and conceptual change (fallibility) philosophies
of mathematics. Absolutist philosophies of mathematics assert that mathematics is a body of
certain and absolute/unchangeable knowledge. According to this view, mathematical knowledge
is made up of absolute truths, and represents the unique realm of certain knowledge, apart from
logic and statements true by virtue of the meanings of terms, such as ‘All bachelors are
unmarried’.

In contrast, conceptual change (fallibility) philosophies assert that mathematics is corrigible,

subjective, incomplete, fallible and an ever changing social product.

Many philosophers, both modern and traditional, hold absolutist views of mathematical
knowledge. An absolutist mathematics teacher may well teach by emphasizing routine
mathematical tasks and give the expectation that there is one correct answer. The fallibilist
mathematics teacher may well emphasize the collaborative, problem solving, investigational
approach to mathematics. How we view mathematics affects how we teach – of course, this is
not the only influence upon teaching styles, a mathematics absolutist may well teach differently
according to the pedagogical beliefs he/she may hold. Two plus two may equal four, but the
issue is whose concept of ‘two’ and whose concept of ‘plus’ and ‘equals’ are we using? In the
following we will examine different ways in which philosophies view mathematics.

1.1.1.1 Absolutist Views of Mathematics

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Absolutism is a blanket term for several distinct views, which can include: Logicism,
Formalism, Intuitionism, Platonism and Empiricism. The one thing they all have in common is
that at heart all mathematics rests upon a firm foundation. They all, however, violently disagree
over what is that foundation.

i) Logicism

Logicism is the school of thought that mathematics is reducible to laws of pure thought, or
logical principles, and hence nothing but a part of logic. Logicists hold that mathematics can be
known a priori, but suggest that our knowledge of mathematics is just part of our knowledge of
logic in general, and is thus analytic, not requiring any special faculty of mathematical intuition.
In this view, logic is the proper foundation of mathematics, and all mathematical statements are
necessary logical truths.

The major proponents of this view are G.Leibniz, G.Frege, B.Russell, A.N.Whitehead and
R.Carnap. At the hands of Bertrand Russell the claims of logicism received the clearest and most
explicit formulation.

There are two claims:

1. All the concepts of mathematics can ultimately be reduced to logical concepts, provided that
these are taken to include the concepts of set theory or some system of similar power, such as
Russell’s Theory of Types.
2. All mathematical truths can be proved from the axioms and rules of inference of logic alone.

The purpose of these claims is clear. If all of mathematics can be expressed in purely logical
terms and proved from logical principles alone, then the certainty of mathematical knowledge
can be reduced to that of logic. Logic was considered to provide a certain foundation for truth,
apart from over-ambitious attempts to extend logic.

ii) Formalism

Formalism holds that mathematics consists simply of the manipulation of finite configurations
of symbols according to prescribed rules; that is a game independent of any physical
interpretation of the symbols. So that mathematical statements may be thought of as statements
about the consequences of certain string manipulation rules. For example, in the "game" of
Euclidean geometry (which is seen as consisting of some strings called "axioms", and some
"rules of inference" to generate new strings from given ones), one can prove that the Pythagorean
theorem holds (that is, you can generate the string corresponding to the Pythagorean theorem).
According to Formalism, mathematical truths are not about numbers and sets and triangles and
the like — in fact, they aren't "about" anything at all. That is mathematics is not, or need not be,
about anything, or anything beyond typographical characters and rules for manipulating them.
A major early proponent of formalism was David Hilbert, whose program was intended to be a
complete and consistent axiomatization of all of mathematics.

iii) Intuitionism

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In mathematics, intuitionism is a program of methodological reform whose motto is that "there

are no non-experienced mathematical truths" (L.E.J. Brouwer).

Intuitionism is characterized by its rejection of any knowledge- or evidence-transcendent notion

of truth. Hence, only objects that can be constructed (constructivism) in a finite number of steps
are admitted.

Intuitionism in or about mathematics is a revisionary form of mathematical constructivism which

considers the essence of mathematics to be exhausted by our constructive mental activity, i.e.,
mental constructions governed by self-evident laws, rather than be constituted by the analytical
reasoning involving the formal manipulation of linguistic characters, perhaps revealing the
existence and the constitution of independently existing mathematical objects and enabling and
governing their application.
The natural numbers are mental constructions, the real numbers are mental constructions, proofs
and theorems are mental constructions, mathematical meaning is a mental construction…

iv) Platonism

Platonism is the view that the objects of mathematics have a real, objective existence in some
ideal realm; hence mathematics is discovered not invented. It originates with Plato, and can be
discerned in the writings of the logicists . Frege and Russell, and includes Cantor, Bernays,
Hardy and Godel among its distinguished supporters. In this sense Pythagoras and Leibniz were
Platonists. Platonists maintain that the objects and structures of mathematics have a real
existence independent of humanity, and that doing mathematics is the process of discovering
their pre-existing relationships. Thus humans do not invent mathematics, but rather discover it,
and any other intelligent beings in the universe would presumably do the same. According to
platonism mathematical knowledge consists of descriptions of these objects and the relationships
and structures connecting them.

The major problem of mathematical platonism is this: precisely where and how do the
mathematical entities exist, and how do we know about them? Is there a world, completely
separate from our physical one that is occupied by the mathematical entities? How can we gain
access to this separate world and discover truths about the entities? One answer might be
Ultimate ensemble, which is a theory that postulates all structures that exist mathematically also
exist physically in their own universe.

1.1.2.2 Falliblism

The absolutist view of mathematical knowledge has been subject to a severe, and in my view,
irrefutable criticism. Its rejection leads to the acceptance of the opposing fallibilist view of
mathematical knowledge. This is the view that mathematical truth is fallible and corrigible, and
can never be regarded as beyond revision and correction. The fallibilist thesis thus has two
equivalent forms, one positive and one negative. The negative form concerns the rejection of
absolutism: mathematical knowledge is not absolute truth, and does not have absolute validity.
The positive form is that mathematical knowledge is corrigible and perpetually open to revision.

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Self Assessment Exercise

1. Discuss the different schools of thought under absolutist views of mathematics.

2. Explain the Fallibilist Views of Mathematics with examples.
3. What is Ultimate Ensemble Theory? Explain.
4. Write a paragraph that describes the difference between Logicism, Formalism,
Intuitionism, Platonism and Fallibilism.

1.1.2 Constructivism

Activity
Knowledge is not received from outside, but we construct knowledge in our
brain. Do you agree on this idea? Discuss.

Like intuitionism, constructivism involves the regulative principle that only mathematical
entities which can be explicitly constructed in a certain sense should be admitted to mathematical
discourse. In this view, mathematics is an exercise of the human intuition, not a game played
with meaningless symbols. Instead, it is about entities that we can create directly through mental
activity. In addition, some adherents of these schools reject non-constructive proofs, such as a
proof by contradiction.The best known constructivists are the intuitionists L.E.J.Brouwer and A.
Heyting.

Constructivists claim that both mathematical truths and the existence of mathematical objects
must be established by constructive methods. This means that mathematical constructions are
needed to establish truth or existence, as opposed to methods relying on proof by contradiction.
For constructivists knowledge must be established through constructive proofs, based on
restricted constructivist logic, and the meaning of mathematical terms/objects consists of the
formal procedures by which they are constructed.

Within constructivism some now nearly independent strands have developed: social
constructivism; and radical constructivism.

Social constructivism or social realism theories see mathematics primarily as a social construct,
as a product of culture, subject to correction and change. Like the other sciences, mathematics is
viewed as empirical endeavors whose results are constantly evaluated and may be discarded.
Cobb claims that children’s mathematical constructions are ‘profoundly influenced’ by social
and cultural conditions.

For radical constructivists, the first principle is that the teacher recognizes that he/she is not
teaching students about mathematics, he/she is ‘teaching them how to develop their cognition’,
and that he/she is ‘a learner in the activity of teaching’.

Constructivist Teaching and Learning Models

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Constructivism is an approach to teaching and learning based on the premise that cognition
(learning) is the result of "mental construction." In other words, students learn by fitting new
information together with what they already know. Constructivists believe that learning is
affected by the context in which an idea is taught as well as by students' beliefs and attitudes.

Activity 1.1.5 (take 5 minutes)

From the constructivist point of view, there are no direct connections between
teaching and learning. What does this statement mean? Discuss it with your
partner.

Feedback for Activity 1.1.5

From the constructivist point of view, there are no direct connections between
teaching and learning, since the teacher’s knowledge cannot be conveyed to the
students, the teacher’s mind is inaccessible to the students and vice versa. This
supports the notion that pupils actively construct their own learning through
assimilation and accommodation of cognitive structures, a process which is
influenced by the experiences of the pupil, but is dependent upon whether the
existing nature of structures is such as to allow the concepts to be acquired.

"Students need to construct their own understanding of each mathematical concept, so that the
primary role of teaching is not to lecture, explain, or otherwise attempt to 'transfer' mathematical
knowledge, but to create situations for students that will foster their making the necessary mental
constructions. A critical aspect of the approach is a decomposition of each mathematical concept
into developmental steps following a Piagetian theory of knowledge based on observation of, and
interviews with, students as they attempt to learn a concept."

As far as instruction is concerned, the instructor should try and encourage students to discover
principles by themselves. The instructor and student should engage in an active dialog (i.e.,
socratic learning). The task of the instructor is to translate information to be learned into a format
appropriate to the learner's current state of understanding. Curriculum should be organized in a
spiral manner so that the student continually builds upon what they have already learned.

Example

"The concept of prime numbers appears to be more readily grasped when the child, through
construction, discovers that certain handfuls of beans cannot be laid out in completed rows and
columns. Such quantities have either to be laid out in a single file or in an incomplete row-
column design in which there is always one extra or one too few to fill the pattern. These

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patterns, the child learns, happen to be called prime. It is easy for the child to go from this step to
the recognition that a multiple table, so called, is a record sheet of quantities in completed
multiple rows and columns. Here is factoring, multiplication and primes in a construction that
can be visualized."

Self Assessment Exercise

1. What is constructivism and social constructivism? Give supportive examples

for your explanation?
2. Explain the role of constructivism to teaching and learning of mathematics.
3. Mathematics can be seen primarily as a social construct. What does it mean?
Explain.

1.1.3 Debates about the Constructivist View

Activity 1.1.6 (Brain Storming) (take 6 minutes)

Read the statements below and debate with your colleague.
1. External rewards such as “smiley faces” on homework or praises such as “Good
work, Samuel!” are damaging to the goal of having the student become
motivated to learn for the sheer rewards inherent in the learning, itself.
2. Students determine their own knowledge based on their own way of processing
information and according to his / her own beliefs and attitudes towards
learning.

The Great Pedagogical Debate: Behaviorism vs Constructivism

For many years, there has been a debate in education on the advantages and disadvantages of
socio-constructivism and behaviourism. These two philosophies on teaching and learning
mathematics can be depicted as two contrasting views and both have influenced the way
mathematics is being taught in schools.

Socio-constructivism, or constructivism in shorter terms, as opposed to behaviorist models of

teaching and learning, claims that knowledge should not be transferred from one individual to
another in educational environments. For constructivist educationalist, knowledge must be
actively constructed as the learner is an entity with previous experiences that must considered as
a “knowing being”. Learning is therefore seen as an adaptive and experiential process rather than
a knowledge transference activity. As learners encounter new situations, they look for
similarities and differences against their own cognitive representation.

In constructivist terms, learning depends on the way each individual learner looks at a particular
situation and draws his/her own conclusions. People therefore determine their own knowledge
based on their own way of processing information and according to his/her own beliefs and
attitudes towards learning. Constructivist learning strategies include more reflective oriented
learning activities in mathematics education such as exploratory and generative learning. More

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specifically, these activities include problem solving, group learning, discussions and situated
learning.

Behaviourism focuses on the manipulation of external conditions to the learner to modify

behaviours that eventually lead to learning. In a behaviourist oriented environment completion of
tasks is seen as ideal learning behaviour and mastering basic skills require students to move from
basic tasks to more advanced tasks. In addition, learning is considered as a function of rewarding
and reinforcing student learning. Likewise, the emphasis is on correct answers rather than of
partially correct answers. Inspired by linear programming theories developed particularly during
the Second World War, learning and teaching in behaviourist terms is a matter of optimizing and
manipulating the instructional environment towards the fulfilment of rigidly and specifically
designed educational objectives.

It has been said that behaviorism emphasizes a process-product and teacher-centeredness model
of instructions that have been prevalent in classroom teaching and in teacher education programs
in the twentieth century. A behaviorist teaching style in mathematics education tends to rely on
practices that emphasize rote learning and memorization of formulas, one-way to solve
problems, and adherence to procedures and drill. Repetition is seen as one of the greatest means
to skill acquisition. Teaching is therefore a matter of expressing objectives and providing the
means to reach those objectives and situated learning is given little value in instruction. This over
emphasis on procedures and formulas resembles traditional formalist and logicist ideas.

The Implications for Developing Teaching Methodologies

Constructivism is based on a set of assumptions about what goes on inside the learner’s head.
Piaget’s constructivism assumes that genetically controlled brain development governs an
assumed time-table of when a child is capable of learning. This idea asserts that our brain
constructs its own meanings from the social environment when it is ready according to our
genetic abilities and that teachers can have only a minimal effect on learning. Fortunately or
unfortunately for constructivists none of these assumptions can be, or have been, proven. They
can only be inferred to be correct. We cannot pry a subject’s skull open to see what’s going on
inside.

Behaviorism avoids all discussion about what goes on inside the head because we cannot
directly measure or observe it. Likewise, the genetic issue is immaterial to the behaviorist. The
behaviorist focuses on:

 the present environment of a subject (antecedent conditions = A)

 what behavior is exhibited (behavior = B) in that environment and
 what consequences follow (consequences = C).

All factors are observable and subject to experimental verification or refutation. The A-B-C
sequences can be experimentally observed with differing antecedent conditions, A, and differing
consequences, C, that are under control of the experimenter. Thus, one may answer a question
such as, “In a given situation, A, what types of consequences, C, are more effective for
producing a desired behavior, B?”

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Constructivists: Constructivists champion practices that emphasize learning through natural

peer group social interactions. These practices include such concepts as “brain-based learning,”
multi-sensory learning styles, discovery learning, inquiry methods, whole language reading,
balanced literacy, authentic learning environments, and many more.

Constructivists may also argue that external rewards such as “smiley faces” on homework or
praises such as “Good work, Samuel!” are damaging to the goal of having the student become
intrinsically motivated to learn for the sheer rewards inherent in the learning, itself.

Behaviorists: Behaviorists point to decades of data from highly controlled studies of matched
class rooms that show superior performance when rewards – positive reinforcers – are liberally
given for good work.

Self Assessment Exercise

Write a brief summary of the debates on the two major theories of Education:
Behaviorism vs. Constructivism. Add your personal opinion about the two theories.

1.1.4 Basic Theories of Learning and their Implication in Learning and Teaching
of Mathematics

Educational psychology is generally concerned with the study of human behavior. For class
teachers, educational psychology will enable them cope with problem of how children learn and
under what conditions maximum learning can take place. In this subsection you will study the
cognitive aspect of learning mathematics and the contributions of three psychologists-Piaget,
Bruner and Gagne to the learning of mathematics.

i) Contributions of Piaget to the Learning of Mathematics

Jean Piaget was a French-Swiss psychologist who was originally trained as a biologist. For more
than fifty years he studied and analyzed the growth and development of children’s thinking. His
school in Geneva is noted for the study of psychological problems underlying the learning of
mathematics. His work has the greatest significance for teachers of mathematics especially at the
primary level.

According to Piaget, there are four stages of intellectual thinking and development. The stages
are sequential.

1. The Sensory-motor stage: Age (0-2) years:

At this stage, the child relates to his environment through its senses only. By the end of the
second year of life, children have a rudimentary understanding of space and are aware that
objects exist apart from their experience of it.

2. Pre-Operational stage: Age (2-7 years):

This generally covers the cognitive development of children during pre-school years, normally
referred to as a pre-nursery and nursery (kindergarten) years. At this stage, children are able to

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deal with reality in symbolic ways. Their thinking at this stage are, however, limited by centering
inability to consider just one characteristic of an object at a time. At this stage most of them are
not able to understand Reversibility:- the ability to think back to the causes of events. Because of
these inabilities, they cannot conserve – retain important features of objects and events. They
cannot therefore engage in logical thinking in any coherent sense. The child is said not to possess
the concept of conservation of number, volume, quantity or space.

The implication of children not understanding conservation at the pre- operational stage is for the
mathematics teacher not to waste his/her time and not to harm the children by telling the pupils
what they cannot experience through their senses. That is through seeing, feeling as well as
hearing. Abstract mathematical ideas should therefore not be introduced at this stage. Children at
this stage should be permitted to manipulate objects and symbols so as to be able to appreciate
reality. Mathematically-oriented recreational activities such as mathematics games, plays, use of
counters, blocks, stones and marbles etc are important materials for learning mathematics at this
stage.

3. The Concrete Operational Stage (7-12 years):

This stage is very important to every primary school teacher since most of their pupils are in this
stage of development. This stage is the beginning of what is called the logico-mathematical
aspect of experience. At this stage, pupils understand the conservation of objects, counting a set
of objects from front to back, back to front or from the middle give the same answer. This is also
part of logico-mathematical. This logico-mathematical also underlines the physical act of
grouping and classifying in the algebra of sets. Conservation of invariant is usually illustrated by
the pouring of equal amount of liquid to two equal jars of cups. One of the two cups is then
emptied into a thinner cup. When asked which cup contains more liquid, he/she says the new
cup, because the height of the liquid in the thinner cup is higher even though he/she saw that the
liquid poured is the same as in the first case.

4. Formal Operational Stage (12+ years):

This is Piaget’s last stage of intellectual development. At this stage, children can think abstractly
if they are not affected by the limitations of the concrete operational stage. As shown by
research, less than half of adults ever function at the formal operational level. At this stage, the
child now reasons or hypothesizes with ideas or symbols rather than needing objects in the
physical world as a basis of his/her thought. He/she can think scientifically and as a logician.
He/she can reason hypothetically. He/she is no more tied by his/her thoughts to existing reality.
He/she can construct new operations. The ages separating the stages are approximate and they
differ slightly according to cultures.

ii) Contributions of Jerome Bruner

Jerome Burner worked on the process of thought in general. Later he applied this to the process
of learning mathematics. He devised experiments to help him observe how mathematical
thinking in children develops. The investigation concerned the individual strategies by which a
child tries to discover a given logical relationship. The procedure in most of the experiments was
to present a number of cards to the child. Each card has its diagrams of triangle, circle or square
separately or a combination of these. Each card was red or green or blue. So there were three

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variable- number, shape and color-each with three values. A concept such as red triangles was
thought of by the experimenter and the subject chose cards to which the experimenter answered
either Yes or No: if the card was red and had triangles on it, and No if not. Subjects were asked
to find the concept, which the experimenter had in mind in the least number of trials. Sometimes
more variables were used; sometimes the numbers of choice were restricted.
From this single procedure, Bruner was able to claim that learning in general depended on four
factors.
(i) the structure of the concept that is to be learnt,
(ii) the nature of the learner’s intuition,
(iii) the desire of the learner to learn,
(iv) the readiness for learning- (biological readiness ).

Thus Bruner considered adequacy of both the subject matter and the learner him/herself
necessary for the leaning of Mathematics. By this he meant that the learner must be intuitively
ready to learn and the materials to be learnt must be presented in a form (or structure ) that
matches the learners “readiness stage” This led to his controversial, but yet popular, assertion
that ” any concept can be taught effectively in some honest form to any child at any age provided
such a concept is introduced at the child’s language level”. This sort of reasoning let him to
attempt a classification of these levels or stages. The following are the three stages through
which Brunner says a child goes through in cognitive learning.

(i) Enactive Stage

The child thinks only in terms of action. He/she enjoys touching and manipulating objects as
teaching proceeds. Specifically no learning occurs at this stage. Topics can however be
introduced to the child using concrete materials.

(ii) Iconic Stage

The child manipulates images. He/she builds up mental images of things already expressed.
Learning at this stage is usually in terms of seeing and picturing in the mind any objects which
transform learning using pictorial presentations.

(iii) The Symbolic Stage

At this stage the child possesses the ability to evaluate learning. Logic, language and
mathematical symbols are used to discuss what has been learnt. Acquired experiences are
translated into symbolic form.

Activity 1.1.7 (take 7 minutes)

1. Illustrate Brunner’s three stages of cognitive learning using the concept of
addition of positive whole numbers.
2. What are the implications of Bruner’s work to teaching and learning of
mathematics?

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Feedback for Activity 1.1.7

1. Consider the problem: 3 + 2 = 5. First the child must work with block,
marbles, counter or other real objects. Take the three first, take another two,
mix them up and then count the mixture (or union). At the second stage
he/she will be able to work with worksheets containing pictures of objects
(images). Instead of the physical objects he/she is now able to recognize
their image and can solve something like it while not necessarily requiring
the production of the ducks physically. At the final stage he/she can solve
the real problem 3 + 2 =5 using symbols 3 and 2 and 5.

2. Like Piaget, Bruner believes that mathematics can be learnt by discovery

approach by starting early in life using concrete materials relevant to
concepts which are to be learnt at a higher stage. That learning mathematics
should start from known to unknown. It should not be learnt in abstract. It
should be learn first with concrete material, then pictorial, symbols and
then abstract.
The home and school environment help the mathematics teacher to see
them as important in mathematics education. A child exposed to a rich
environment will do well in mathematics. Teachers of mathematics must
make their lessons child-centered. The use of teaching – learning materials
is emphasized. There should not be rote learning and the learning must be
practicable.

iii) The Contributions of Robert Gagne

As a behaviorist psychologist, Gagne devoted his time to study the conditions of learning. He
believes that learning occurs as a result of interaction between the learner and the environment.
Learning is known to have taken place when we notice (observe). Gagne maintains that the
stages described by Piaget are not necessarily the inevitable result of an inborn “timetable” but
are instead a consequence of children having learned sets of rules that are progressively more
complex. How do children acquire these sets of rules? According to Gagne children are “taught”
the rules by their physical and social environment. If we follow Piaget’s (and Burner’s) assertion
we will assume that children will develop complex concepts, understanding and problem solving
skills when they are ready. That is when their nervous systems have matured sufficiently and
they had enough experiences with simpler more elementary problems.

Mathematics teachers who see learning as a process of discovery are likely to borrow heavily
from Piaget and Burner. Others who will see learning as produced primarily by children
environment are likely to take their cues from Gagne. The following illustrates Gagne
contribution to the learning of mathematics. We have already mentioned that he emphasized the
idea of pre-requisite knowledge in learning mathematics. That is the idea that one cannot master
complex concepts without mastering the fundamental concepts necessary for such complex

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concepts. For instance a child cannot successfully add fractions without the knowledge of finding
1 2
common denominator of fraction. That is, a child cannot do  unless he/she has been lead to
3 5
1 5 2 6 1 2 5 6 11
learn that  and  ; therefore     .
3 15 5 15 3 5 15 15 15
Besides there is an intermediate step which we have ignored in this sequence: That is the fact
that fractions with same denominator could be added as like terms.

Self Assessment Exercise

1. Explain Contributions of Piaget to the Learning of Mathematics.
2. Write on any two of the following psychologists, stating their theories and the
implications of their theories to the teaching and learning of mathematics at the
secondary school level.
(a) Jean Piaget (b) Jerome Bruner (c) Robert Gagne
3. Your philosophy of education is a personal statement of your beliefs and emphases
concerning secondary school teaching and learning mathematics. Write a clear and
concise ( 1-2 pages) of your philosophy of mathematics education. You should be
able to explain to an administrator, another teacher, or a parent why you feel the
way you do.

1.2 Some Insight in to the History of Mathematics and Mathematics

Education(8 hrs.)

Introduction

Before the modern age and the worldwide spread of knowledge, written examples of new
mathematical developments have come to light only in a few locales. The most ancient

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mathematical texts available are Plimpton 322 (Babylonian mathematics ca. 1900 BC), the
Moscow Mathematical Papyrus (Egyptian mathematics ca. 1850 BC), the Rhind Mathematical
Papyrus (Egyptian mathematics ca. 1650 BC), and the Shulba Sutras (Indian mathematics ca.
800 BC). All of these texts concern the so-called Pythagorean theorem, which seems to be the
most ancient and widespread mathematical development after basic arithmetic and geometry.

Egyptian and Babylonian mathematics were then further developed in Greek and Hellenistic
mathematics, which is generally considered to be one of the most important for greatly
expanding both the method and the subject matter of mathematics. The mathematics developed
in these ancient civilizations were then further developed and greatly expanded in Islamic
mathematics. Many Greek and Arabic texts on mathematics were then translated into Latin in
medieval Europe and further developed there.

One striking feature of the history of ancient and medieval mathematics is that bursts of
mathematical development were often followed by centuries of stagnation. Beginning in
Renaissance Italy in the 16th century, new mathematical developments, interacting with new
scientific discoveries, were made at an ever increasing pace, and this continues to the present
day.

Why we study history of mathematics?

On the surface, mathematics seems to be a logical theory, complete in itself, not bound to place
or time. But many of its concepts, practices and symbols are clarified and become more
understandable if we know something of the historical reasons behind them. Also, we are not so
much astonished by the difficulty of learning mathematics, if we realize that the invention and
polishing of many of the things now taught in elementary courses has required even centuries of
efforts, trials and errors by some of the sharpest brains of humankind. The student often
encounters these results in a streamlined form. On the other hand, when the student meets an
application problem or tries to do independent research at some level, the problems often seem to
be overwhelming. History teaches us that even the streamlined parts have been put up with much
effort and trouble. The unity of mathematics is dimmed by the fact that as a science it splits into
hundreds of different specialties. The history of mathematics helps us to appreciate mathematics
as a fundamentally unified science.

We need not form a firm opinion about the origins of mathematics –whether it exists
independently of man or whether it is man’s creation – to observe that results of mathematics are
associated to people and their achievements. One reason to learn about the history of
mathematics is to know about the people behind the names and their mutual interaction. One aim
of a course in the history of mathematics is arousing interest in mathematics as an essential part
of human culture. To remind the general public of this often forgotten fact can only be the task of
mathematicians and teachers of mathematics. In any case they hold the key position in attempts
to break the surprisingly widespread prejudices of mathematics as something inhuman, a field of
knowledge beyond the understanding of an otherwise civilized person and also quite unnecessary
to understand. A common feature in very many popular books of mathematics is the reliance on
history of mathematics as a means to convey the meanings of mathematics to the general public.
But this bias itself also recommends that professionals of mathematics ought to have a somewhat

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balanced and extensive picture of the history of mathematics. And the history of mathematics is a
fascinating field of study!

1.2.1 Primitive Counting

Activity

Discuss in groups of three and write a brief response on your exercise book
for further reference.
1. What is number and numeral?
2. How did primitive man count?
3. Why are numerals important?

The origins of mathematical thought lie in the concepts of number, magnitude, and form.
Modern studies of animal cognition have shown that these concepts are not unique to humans.
The number sense is not the ability to count, but the ability to recognize that something has
changes in a small collection. Some animal species are capable of this.

The number of young that the mother animal has, if changed, will be noticed by all mammals
and most birds. Mammals have more developed brains and raise fewer young than other species,
but take better care of their young for a much longer period of time. Many birds have a good
number sense. If a nest contains four eggs, one can safely be taken, but when two are removed
the bird generally deserts. The bird can distinguish two from three.

The Farmer and the Crow is also taken as another example: A farmer wanted to shoot a crow that
was eating his crop. He went out by himself and hid behind a tree waiting for the crow to come
out. Unfortunately, the crow saw him hide behind the tree, and wouldn't return until he'd seen the
farmer (and his rifle) leave. The farmer then figured he would trick the crow; he and his wife
went out and both hid behind the tree. The farmer then sent his wife back inside, in clear view of
the crow. Again, the crow wouldn't come out until the farmer left as well-- the crow could
apparently tell the difference between one and two people. The farmer then repeated his attempts
with three, four, five, six, and seven of his friends. Not until he brought a seventh friend, and sent
all seven friends away from the tree did the crow come out of hiding, at which point the farmer
promptly shot the crow.

Although the crow could distinguish between one through seven, it seemed as though it couldn't
tell the difference between seven and eight individuals. It could "count" as high as seven, since it
has six talons. It could remember the number of people by equating each person to a talon, and
thinking one, two, three, four, five, six, more. Since both seven and eight people equate to
"more" they were indistinguishable to the crow.

One might think people would have a very good number sense, but as it turns out, people do not.
Experiments have shown that the average person has a number sense that is around four.People

So what separates people from the rest~14of

~ the animal kingdom?
 Unit 1: Philosophy and History of Mathematics and Mathematics Education

groups in the world today that have not developed finger counting have a hard time discerning
the quantity four. They tend to use the quantities one, two and many-which would include four.

It may include many things, but the ability to count is one of them. Counting, which usually
begins at the end of our own hands or fingers, is usually taught by another person or possibly by
circumstance. It is something that we should never take lightly for it has helped advance the
human race in countless ways.

The number sense is something many creatures in this world have as well as we do. Although, as
we can see, our human ability is not much better than the common crow’s ability, we are born
with the number sense, but we get to learn how to count.

Even a prehistoric tribe would have needed the notion of counting to make sure that all the cattle
came home, or that the tribe outnumbered its enemies. The primitive form of counting involved
setting up a correspondence between the objects to be counted and some other convenient set of
objects like, for example, the fingers in one’s hand. Even today, some primitive African hunters
keep count of the number of wild boars they kill by collecting the tusks of each animal, and
young girls in the Masai tribe – who live on the slopes of Mt.Kilimanjaro – wear brass rings
around their necks in numbers equal to their ages. This process of counting soon evolved into a
more sophisticated process of keeping ‘tally marks’ on a bone or a stone.

Evidence exists that early counting involved women who kept records of their monthly
biological cycles; twenty-eight, twenty-nine, or thirty scratches on bone or stone, followed by a
distinctive scratching on the bone or stone, for example. Moreover, hunters had the concepts of
one, two, and many, as well as the idea of none or zero, when considering herds of animals.

The oldest known possibly mathematical object is the Lebombo bone, discovered in the
Lebombo mountains of Swaziland and dated to approximately 35,000 BC. It consists of 29
distinct notches cut into a baboon's fibula. Another example (the Ishango bone), found near the
headwaters of the Nile river (northeastern Congo), may be as much as 20,000 years old and
consists of a series of tally marks carved in three columns running the length of the bone.

The Ishango bone dating to perhaps 18,000 BC to 20,000 BC

From such humble beginnings, mathematics progressed rather rapidly. In counting, one realize
that ‘one cow and one cow make two cows’ just as ‘one spear and one spear make two spears’.
To extract from such a concrete experience the abstract idea that ‘one and one make two’ was the
first major breakthrough in mathematical thought. This idea, which appears so obvious to us
today, involves thinking of ‘one’ and ‘two’ as independent abstract entities existing on their own.
(This abstraction has not been achieved by some tribal societies even today. For example,
Fijitribals distinguish ten boats (called bole) from ten coconuts (koro) and call thousand coconuts
by a separate name, saloro!) In practical terms, this idea requires words in a language to describe

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these numbers as separate nouns. All ancient civilizations– Egyptian, Chinese, Babylonian and
Indian developed such verbal descriptions for numbers at some stage in their development. In the
early stages, words for numbers often originated from parts of the body, names of fingers, etc.,
and covered only small numbers. In fact, words originally existed only for 1, 2, rarely for 3 and
for ‘many’. The written forms of these ‘number words’ formed the earliest of mathematical
notations. Very soon, they were condensed into forms, which were more compact and useful.
The most primitive and complicated among them, which have surprisingly enough survived till
today, are the Roman numerals: I, V, X, L, C, D, M etc. The most useful, of course, are the
Arabic numerals or the Hindu Arabic numerals, which we will discuss fully later.

Activity

Discuss in groups of three and write a brief response on your exercise book.
1. When did man first use numerals?
2. How did ancient man write?

Feedback

1. The earliest written numbers so far discovered were used in ancient Egypt
and Mesopotamia about 3000 BC. These people, living many miles apart,
each independently developed a set of numerals. Their simple numerals, 1, 2,
3, were copies of the cave man's sticks or notches. It is interesting to note that
in many of the numeral systems found throughout the world, 1 was written as
a single stroke (like a stick) or as a dot (like a pabble).
2. The ancient Egyptians wrote their numerals on papyrus, a special paper made
from reeds, painted them on pottery and carved them into the walls of their
temples and pyramids. The Sumerians taught the Babylonians how to cut
their numerals into soft clay tablets. The ancients Chinese did their number
writing with ink and a bamboo brush or pen on cloth.
In the Western Hemisphere, with no contact with the rest of the world, the
Mayas of Central America developed one of the most remarkable of the early
number systems. They made their numerals using only three symbols: a dot
(•), a straight line (---- ), an oval .

1.2.2 Babylonians’ Contribution in Mathematics

Babylonian mathematics refers to any mathematics of the peoples of Mesopotamia (modern Iraq)
from the days of the early Sumerians until the beginning of the Hellenistic period. It is named
Babylonian mathematics due to the central role of Babylon as a place of study, which ceased to
exist during the Hellenistic period. From this point, Babylonian mathematics merged with Greek
and Egyptian mathematics to give rise to Hellenistic mathematics. Later under the Arab Empire,

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Iraq/Mesopotamia, especially Baghdad, once again became an important centre of study for
Islamic mathematics.

In contrast to the scarcity of sources in Egyptian mathematics, our knowledge of Babylonian

mathematics is derived from more than 400 clay tablets unearthed since the 1850s. Written in
Cuneiform script, tablets were inscribed whilst the clay was moist, and baked hard in an oven or
by the heat of the sun. Some of these appear to be graded homework.

The Babylonian mathematical tablet Plimpton 322, dated to 1800 BC.

The earliest evidence of written mathematics dates back to the ancient Sumerians, who built the
earliest civilization in Mesopotamia. They developed a complex system of metrology around
3000 BC. From around 2500 BC onwards, the Sumerians wrote multiplication tables on clay
tablets and dealt with geometrical exercises and division problems. The earliest traces of the
Babylonian numerals also date back to this period.

The majority of recovered clay tablets dates from 1800 BC to 1600 BC, and cover topics which
include fractions, algebra, quadratic and cubic equations, and the calculation of Pythagorean
triples. The tablets also include multiplication tables, trigonometry tables and methods for
solving linear and quadratic equations. The Babylonian tablet YBC 7289 gives an approximation
to √2 accurate to five decimal places.

Babylonian mathematics was written using a sexagesimal (base-60) numeral system. This system
has its influence in our own world today. From this we derive the modern day usage of 60
seconds in a minute, 60 minutes in an hour, and 360 (60 x 6) degrees in a circle. Babylonian
advances in mathematics were facilitated by the fact that 60 has many divisors (1, 2, 3, 4, 5, 6,
10, 12, 15, 20, 30 and 60 - in fact, 60 is the smallest integer divisible by all integers from 1 to 6).
Also, unlike the Egyptians, Greeks, and Romans, the Babylonians had a true place-value system,
where digits written in the left column represented larger values, much as in the decimal system.
They lacked, however, an equivalent of the decimal point, and so the place value of a symbol
often had to be inferred from the context. However, there is no evidence that they used a number
for zero, and they did not use fractions.

The Babylonians also developed another revolutionary mathematical concept, something else
that the Egyptians, Greeks and Romans did not have, a circle character for zero, although its
symbol was really still more of a placeholder than a number in its own right.

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 Unit 1: Philosophy and History of Mathematics and Mathematics Education

The famous and controversial Plimpton 322 clay tablet, suggests that the Babylonians may well
have known the secret of right-angled triangles (that the square of the hypotenuse equals the sum
of the square of the other two sides) many centuries before the Greek Pythagoras. The tablet
appears to list 15 perfect Pythagorean triangles with whole number sides, although some claim
that they were merely academic exercises, and not deliberate manifestations of Pythagorean
triples.

Ultimately, their knowledge passed to the Greeks and formed the basis of pure mathematics as
the master manipulators of numbers, the Greeks, took this knowledge and began to explore the
relationships between numbers.

Activity

Discuss in groups of four and write a brief response on your exercise book.
1. What are the unique Babylonians’ contributions to mathematics?

Feedback
Some of the unique Babylonians’ contributions to mathematics are:

 The Babylonians developed a revolutionary mathematical concept, a circle

character for zero, as a placeholder.
 The Babylonians’ sexagesimal (base-60) numeral system used to derive the
modern day usage of 60 seconds in a minute, 60 minutes in an hour, and 360
(60 x 6) degrees in a circle. And 60 has many divisors (1, 2, 3, 4, 5, 6, 10, 12,
15, 20, 30 and 60 - in fact, 60 is the smallest integer divisible by all integers
from 1 to 6).

1.2.3 Egyptians’ Contribution in Mathematics

Egyptian mathematics refers to mathematics written in the Egyptian language. From the
Hellenistic period, Greek replaced Egyptian as the written language of Egyptian scholars, and
from this point Egyptian mathematics merged with Greek and Babylonian mathematics to give
rise to Hellenistic mathematics. Mathematical study in Egypt later continued under the Arab
Empire as part of Islamic mathematics, when Arabic became the written language of Egyptian
scholars.

The oldest mathematical text discovered so far is the Moscow papyrus, which is an Egyptian
Middle Kingdom papyrus dated ca. 2000 BC —1800 BC. Like many ancient mathematical texts,
it consists of what are today called "word problems" or "story problems", which were apparently
intended as entertainment.

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 Unit 1: Philosophy and History of Mathematics and Mathematics Education

The Rhind papyrus (ca. 1650 BC) is another major Egyptian mathematical text which was an
instruction manual in arithmetic and geometry. In addition to giving area formulas and methods
for multiplication, division and working with unit fractions, it also contains evidence of other
mathematical knowledge, including composite and prime numbers; arithmetic, geometric and
harmonic means; and simplistic understandings of both the Sieve of Eratosthenes and perfect
number theory (namely, that of the number 6) . It also shows how to solve first order linear
equations as well as arithmetic and geometric series.

Also, three geometric elements contained in the Rhind papyrus suggest the simplest of
underpinnings to analytical geometry:

(1) first and foremost, how to obtain an approximation of π accurate to within less than one
percent;
(2) second, an ancient attempt at squaring the circle; and
(3) third, the earliest known use of a kind of cotangent.

Finally, the Berlin papyrus (ca. 1300 BC ) shows that ancient Egyptians could solve a second-
order algebraic equation.

The Egyptian method for recording quantities is based on 10 with a symbol for one, ten, and
each successive power of ten. A distinct hieroglyphic was used for each power of 10. There was
no symbol for zero; therefore a particular symbol was omitted in a numeral when that multiple of
ten was not part of the number.

Ancient Egyptians had an understanding of fractions, however they did not write simple fractions
3 4
as or because of restrictions in notation. The Egyptian scribe wrote fractions with the
5 9
numerator of 1. They used the hieroglyph "an open mouth" above the number to indicate its
1
reciprocal. The number 5, written , as a fraction would be written . There are some
5
2 3
exceptions. There was a special hieroglyph for , , and some evidence that also had a
3 4
3
special hieroglyph. All other fractions were written as the sum of unit fractions. For example
8
1 1
was written as + .
4 8
The Egyptians used a written numeration that was changed into hieroglyphic writing, which
enabled them to note whole numbers to 1,000,000. It had a decimal base and allowed for the
additive principle. In this notation there was a special sign for every power of ten. For 1, a
vertical line; for 10, a sign with the shape of an upside down U; for 100, a spiral rope; for 1000, a
lotus blossom; for 10,000, a raised finger, slightly bent; for 100,000, a tadpole; and for
1,000,000, a kneeling genie with upraised arms.

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 Unit 1: Philosophy and History of Mathematics and Mathematics Education

Activity

Discuss in groups of three and write a brief response on your exercise book.
1. What are the unique Egyptian contributions to mathematics?

Feedback

 The first step towards written numbers was taken in ancient Egypt when
tally marks came into use probably 4000-5000 years ago.
 The Rhind papyrus is a rich primary source of ancient Egyptian
mathematics, containing 84 worked problems and describing the Egyptian
methods of adding - subtracting - multiplying and dividing with whole
numbers and fractions, the solution of linear equations and the measure of
simple areas and volumes.
 Tables (calculation charts) were invented in Babylon, but Egyptian
mathematics developed and perfected them in forms that were used and
unchanged for millennia. The Rhind papyrus gives clear evidence of a two
times table for addition, which could also be used for its complement,
subtraction.
 Perhaps the most dazzling mathematical insight of the Egyptians was that
the four arithmetical processes are closely related.
 Egyptian unit fractions continued in use for thousands of years, up to the

modern period, except for , which was accorded a special symbol. The

development of unit fractions in ancient Egypt was a big advance, a way to

refine measurement, and a way to express the results of division.

1.2.4 The Greeks’ Contribution in Mathematics

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The history of mathematics is not complete without Greeks. Greek mathematics refers to
mathematics written in Greek between about 600 BC and 450 AD. Greek mathematicians lived
in cities spread over the entire Eastern Mediterranean, from Italy to North Africa, but were
united by culture and language. Greek mathematics of the period following Alexander the Great
is sometimes called Hellenistic mathematics.

Greek mathematics was much more sophisticated than the mathematics that had been developed
by earlier cultures. All surviving records of pre-Greek mathematics show the use of inductive
reasoning, that is, repeated observations used to establish rules of thumb. By contrast, the Greeks
revolutionized the study of mathematics by introducing the deductive method in their study of
math. The best example here is the Pythagorean Theorem. The proof of the theorem was the first
to use deductive method and logical reasoning.

Greek mathematics is thought to have begun with Thales of Miletus (ca. 624 BC —ca.546 BC)
and Pythagoras of Samos (ca. 582 BC —ca. 507 BC). Although the extent of the influence is
disputed, they were probably inspired by Egyptian and Babylonian mathematics. According to
legend, Pythagoras travelled to Egypt to learn mathematics, geometry, and astronomy from
Egyptian priests.

Thales used geometry to solve problems such as calculating the height of pyramids and the
distance of ships from the shore. Pythagoras is credited with the first proof of the Pythagorean
theorem, though the statement of the theorem has a long history. In his commentary on Euclid,
Proclus states that Pythagoras expressed the theorem that bears his name and constructed
Pythagorean triples algebraically rather than geometrically.

The Pythagoreans discovered the existence of irrational numbers. Eudoxus (ca.408 BC —ca.355
BC) invented the method of exhaustion, a precursor of modern integration. Aristotle (ca.384 BC
—ca.322 BC) first wrote down the laws of logic. Euclid (ca. 300 BC) is the earliest example of
the format still used in mathematics today, definition, axiom, theorem, and proof. He also studied
conics. His book, Elements, was known to all educated people in the West until the middle of the
20th century. In addition to the familiar theorems of geometry, such as the Pythagorean theorem,
Elements includes a proof that the square root of two is irrational and that there are infinitely
many prime numbers. The Sieve of Eratosthenes (ca. 230 BC) was used to discover prime
numbers.

Plato (428/427 BC – 348/347 BC) is important in the history of mathematics for inspiring and
guiding others. His Platonic Academy, in Athens, became the mathematical center of the world
in the 4th century BC, and it was from this school that the leading mathematicians of the day,
such as Eudoxus of Cnidus, came from. The Academy of Plato had the motto "let none unversed
in geometry enter here". Plato also discussed the foundations of mathematics, clarified some of
the definitions (e.g. that of a line as "breadthless length"), and reorganized the assumptions. The
analytic method is ascribed to Plato, while a formula for obtaining Pythagorean triples bears his
name.

Archimedes (ca.287 BC – ca.212 BC) of Syracuse, widely considered the greatest mathematician
of antiquity, used the method of exhaustion to calculate the area under the arc of a parabola with

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 Unit 1: Philosophy and History of Mathematics and Mathematics Education

the summation of an infinite series, in a manner not too dissimilar from modern calculus. He also
showed one could use the method of exhaustion to calculate the value of π with as much
precision as desired, and obtained the most accurate value of π then known, 310⁄71 < π < 310⁄70.

Apollonius of Perga (ca. 262 BC – ca.190 BC) made significant advances to the study of conic
sections, showing that one can obtain all three varieties of conic section by varying the angle of
the plane that cuts a double-napped cone.

The 3rd century BC is generally regarded as the "Golden Age" of Greek mathematics, with
advances in pure mathematics henceforth in relative decline. Nevertheless, in the centuries that
followed significant advances were made in applied mathematics, most notably trigonometry,
largely to address the needs of astronomers.

Following a period of stagnation after Ptolemy, the period between 250 AD and 350 AD is
sometimes referred to as the "Silver Age" of Greek mathematics. During this period, Diophantus
made significant advances in algebra, particularly indeterminate analysis, which is also known as
"Diophantine analysis".

The ancient Greek numeral system, known as Attic or Herodianic numerals, was fully developed
by about 450 BC, and in regular use possibly as early as the 7th Century BC. It was a base 10
system similar to the earlier Egyptian one (and even more similar to the later Roman system),
with symbols for 1, 5, 10, 50, 100, 500 and 1,000 repeated as many times needed to represent the
desired number. Addition was done by totaling separately the symbols (1s, 10s, 100s, etc) in the
numbers to be added, and multiplication was a laborious process based on successive doublings
(division was based on the inverse of this process).

Activity
Discuss with your colleagues and write a brief response on your exercise book.
1. What are the unique Greek contributions to mathematics?

Feedback

We can identify three elements.

 The first is an insistence that all mathematical results must be established by

deductive reasoning.
 Secondly the Greeks made mathematics abstract.
 And the third notable feature of Greek mathematics was their emphasis on
geometry and the use of geometrical methods for solving problems.

1.2.5 The Hindu Mathematics System

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 Unit 1: Philosophy and History of Mathematics and Mathematics Education

The earliest civilization on the Indian subcontinent is the Indus Valley Civilization that
flourished between 2600 BC and 1900 BC in the Indus river basin. Their cities were laid out with
geometric regularity, but no known mathematical documents survive from this civilization.

Golden Age Indian mathematicians made fundamental advances in the theory of trigonometry, a
method of linking geometry and numbers first developed by the Greeks. They used ideas like the
sine, cosine and tangent functions (which relate the angles of a triangle to the relative lengths of
its sides) to survey the land around them, navigate the seas and even chart the heavens. For
instance, Indian astronomers used trigonometry to calculate the relative distances between the
Earth and the Moon and the Earth and the Sun. They realized that, when the Moon is half full
and directly opposite the Sun, then the Sun, Moon and Earth form a right angled triangle, and
were able to accurately measure the angle as 1⁄7°. Their sine tables gave a ratio for the sides of
such a triangle as 400:1, indicating that the Sun is 400 times further away from the Earth than the
Moon.

The oldest extant mathematical records from India begins in the early Iron Age, with the
Shatapatha Brahmana (c. 9th century BC), which approximates the value of π to 2 decimal
places., and the Sulba Sutras (ca. 800 BC- ca.500 BC) were geometry texts that used irrational
numbers, prime numbers, the rule of three and cube roots; computed the square root of 2 to five
decimal places; gave the method for squaring the circle; solved linear equations and quadratic
equations; developed Pythagorean triples algebraically and gave a statement and numerical proof
of the Pythagorean theorem.

The Surya Siddhanta (ca. 400) introduced the trigonometric functions of sine, cosine, and inverse
sine, and laid down rules to determine the true motions of the luminaries, which conforms to
their actual positions in the sky. The cosmological time cycles explained in the text, which was
copied from an earlier work, corresponds to an average sidereal year of 365.2563627 days,
which is only 1.4 seconds longer than the modern value of 365.25636305 days. This work was
translated to Arabic and Latin during the Middle Ages.

In the 5th century AD, Aryabhata wrote the Aryabhatiya, a slim volume, written in verse,
intended to supplement the rules of calculation used in astronomy and mathematical
mensuration, though with no feeling for logic or deductive methodology. Aryabhata of
Kusumapura developed the place-value notation.

Aryabhata in 499 introduced the versine function, produced the first trigonometric tables of sine,
developed techniques and algorithms of algebra, infinitesimals, differential equations, and
obtained whole number solutions to linear equations by a method equivalent to the modern
method, along with accurate astronomical calculations based on a heliocentric system of
gravitation. He also computed the value of π to the fourth decimal place as 3.1416. Madhava
later in the 14th century computed the value of π to the eleventh decimal place as
3.14159265359.

In the 7th century, Brahmagupta identified the Brahmagupta theorem, Brahmagupta's identity
and Brahmagupta's formula, and for the first time, in Brahma-sphuta-siddhanta, he lucidly
explained the use of zero as both a placeholder and decimal digit and explained the Hindu-

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Arabic numeral system. It was from a translation of this Indian text on mathematics (around 770)
that Islamic mathematicians were introduced to this numeral system, which they adapted as
Arabic numerals. Islamic scholars carried knowledge of this number system to Europe by the
12th century, and it has now displaced all older number systems throughout the world. In the
10th century, Halayudha's commentary on Pingala's work contains a study of the Fibonacci
sequence and Pascal's triangle, and describes the formation of a matrix.

In the 12th century, Bhaskara first conceived differential calculus, along with the concepts of the
derivative, differential coefficient and differentiation. He also stated Rolle'stheorem (a special
case of the mean value theorem), studied Pell's equation, and investigated the derivative of the
sine function.

Activity
Discuss in groups of three and write a brief response on your exercise book.
1. What are the unique Indian contributions to the development of mathematics?

Feedback

Some of the contribution to mathematics by India can be stated as follows:

1. Zero and the place-value notation for numbers:

 The number zero is the subtle gift of the Hindus of antiquity to mankind.
 The Hindu ancients were using a decimal positional system, that is, a system in
which numerals in different positions represent different numbers and in which
one of the ten symbols used was a fully functional zero.
2. Vedic Mathematics and arithmetical operations:
 Vedic Mathematics recognizes that any algebraic polynomial may be
expressed in terms of a positional notation without specifying the base.
3. Geometry of the Sulba Sutras:
 The Bodhayana Sutras contains a general statement of the Pythagorean
Theorem, an approximation procedure for obtaining the square root of two
correct to five decimal places and a number of geometric constructions.
4. Astronomy:
 The contribution to Astronomy by ancient Indians is so great.
5. Indian Trigonometry
 Though Trigonometry goes back to the Greek period, the character of the
subject started to resemble modern form only after the time of Aryabhata.
From here it went to Europe through the Arabs and went into several
modifications to reach its present form.

1.2.6 The Arabs’ contribution in Mathematics

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 Unit 1: Philosophy and History of Mathematics and Mathematics Education

Arab contributions to human civilization are noteworthy. In arithmetic the style of writing digits
from right to left is an evidence of its Arab origin.
Another invention that revolutionized mathematics was the introduction of the number zero by
Muhammad Bin Ahmad in 967 AD. Zero was introduced in the West as late as the beginning of
the thirteenth century. Modern society takes the invention of the zero for granted, yet the Zero is
a non-trivial concept, that allowed major mathematical breakthroughs. Arab civilizations also
made a great contribution to fractions and to the principle of errors, which is employed to solve
algebra problems arithmetically.

Concerning Algebra, al-Khawarzmi is credited with the first treatise (writing). He solved
Algebra equations of the first and second degree (known as quadratic equations, and are
prevalent in science and engineering) and also introduced the geometrical method of solving
these equations.

He also recognized that quadratic equations have two roots. His method was continued by
Thabet Bin Qura, the translator of Ptolemy's works who developed Algebra and first
realized(recogonized, understood) application in geometry. By the 11th century the Arabs had
founded, developed and perfected geometrical algebra and could solve equations of the third and
fourth degree.

Another outstanding Arab mathematician is AbulWafa who created and successfully developed a
branch of geometry which consists of problems leading to equations in Algebra of a higher
degree than the second. He made a number of valuable contributions to polyhedral theory.

Al-Karaki, of the 11th century is considered to be one of the greatest Arab mathematicians. He
composed one arithmetic book and another on Algebra. In the two books, he developed an
approximate method of finding square roots, a theory of indices, a theory of mathematical
induction and a theory of intermediate quadratic equations.

Arabs have excelled in geometry, starting with the transition of Euclid and conic section of
Apolonios and they preserved the genuine works of these two Greek masters for the modern
world, by the 9th century AD and then started making new discoveries in this domain.

It is also at the hand of the Arabs that the geometry of conic sections was developed to a great
extent. However, Arab achievements in this field were crowned by the discovery made by Abu
Jafar Muhammad ,Ibn Muhammad, Ibn al-Hassan, known as Nassereddine al-Tusi. Al-Tusi
separated trigonometry from astronomy. This contribution recognizes and explains weakness in
Euclid's theory of parallels, and thereby may thus be credited as founder of non-Euclidian
geometry.

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 Unit 1: Philosophy and History of Mathematics and Mathematics Education

Activity
Discuss in groups of three and write a brief response on your exercise book for
further reference.
1. What are the unique Arab contributions to mathematics?

Feedback

 In arithmetic the style of writing digits from right to left is an evidence of

its Arab origin.
 Another invention that revolutionized mathematics was the introduction of
the number zero by Muhammad Bin Ahmad in 967 AD.
 Arab civilizations also made a great contribution to fractions and to the
principle of errors, which is employed to solve Algebra problems
arithmetically.
 The numbers we use are called Arabic numbers (numerals) which is a
system of tens, with place values, and a zero to show an empty place.
 Algebra was first fully developed by Al Khwarism, the "father of algebra".
 Nassereddine al-Tusi separated trigonometry from astronomy. This
contribution recognizes and explains weakness in Euclid's theory of
parallels, and thereby may thus be credited as founder of non-Euclidian
geometry.

1.2.7 The Hindu-Arabic Numeration system

The Hindu-Arabic numeration system evolved around 800AD. It is basically the numeration
system that is widely used today. The following lists four main attributes of this numeration
system.

 First, it uses 10 digits or symbols that can be used in combination to represent all possible
numbers. The digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
 Second, it groups by tens, probably because we have 10 digits on our two hands. Interestingly
enough, the word digit literally means finger or toes.
In the Hindu-Arabic numeration system, ten ones are replaced by one ten, ten tens are
replaced by one hundred, ten hundreds are replaced by one thousand, 10 one thousand are
replaced by 10 thousands, and so forth.
 Third, it uses a place value. Starting from right to left, the first number represents how many
ones there are, the second number represents how many tens there are, the third number
represents how many hundreds there are, the fourth number represents how many thousands
there are, and so on.

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 Unit 1: Philosophy and History of Mathematics and Mathematics Education

 Finally, the system is additive and multiplicative. The value of a numeral is found by
multiplying each place value by its corresponding digit and then adding the resulting products.
Place values: thousand hundred ten one
Digits 4 6 8 7
Numeral value is equal to 4 × 1000 + 6 × 100 + 8 × 10 + 7 × 1 = 4000 + 600 + 80 + 7 = 4687
Notice that the Hindu-Arabic numeration system requires fewer symbols to represent numbers as
opposed to other numeration system. Each Hindu-Arabic numeral has a word name. Numbers
from 1 through 12 have unique names. Numbers from 13 through 19 have "teens" as ending and
the ending is blended with names for numbers from 4 through 9. For numbers from 20 through
99, the tens place is named first followed by a number from 1 through 10. Numbers from 100
through 999 are combinations of hundreds and previous names.

Self Assessment Exercise

1. Write the important characteristics of the Hindu – Arabic system of numeration.

2. Discuss why the Hindu-Arabic system is the international system of
numeration?
3. Compare the Hindu-Arabic Numeration system with that of Saban Numeration
System?

1.2.8 Great Mathematicians and their contributions

Often called the language of the universe, mathematics is fundamental to our understanding of
the world and, as such, is vitally important in a modern society such as ours. Everywhere you
look it is likely mathematics has made an impact, from the faucet in your kitchen to the satellite
that beams your television programs to your home. As such, great mathematicians are
undoubtedly going to rise above the rest and have their name embedded within history. There
were numerous mathematicians throughout history all of whom contributed to our modern day
mathematics. All these famous mathematicians played the most significant role in our lives much
more than we realize. This subsection includes only some such people. I also suggest one looks
deeper into the lives of these men, as they are truly fascinating people and their discoveries are
astonishing – too much to include here.

Thales of Miletus

He was born in 640 BC and lived in Miletus. He was a merchant politician. He visited Egypt and
Babylon to buy and sell wares. So, in Babylon, he came in contact with its people and got their
ideas of Astronomy and Earth Measurement from Egypt. After retiring from merchandise, he

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 Unit 1: Philosophy and History of Mathematics and Mathematics Education

devoted his time to the study of Astronomy and Mathematics. He started Deductive Geometry.
He successfully predicted an eclipse of the sun from May 28th in 585 BC.
Pythagoras of Samos

He was born in 580 BC on the Island of Samos. He later moved to Crotona in Southern Italy,
where he did most of his mathematics. He studied under Thales. He founded a school in Crotona
and his students lived like a brotherhood or cult, (the Pythagoreans). Some of their knowledge
were treasured orally but later became written. Their specific contributions to Mathematics
included:
1) discovery of the harmonic progression;
2) invention (innovation)of the terms odd and even numbers;
3) Pythagoras theorem;
4) they were the first to use the word parabolas, ellipse, hyperbole, Appotonus borrowed these
words in conics .
5) he was the first to discover that the world was a sphere.

Plato

He lived 400 BC in a place near Athens. He founded a school called the Academy. His
philosophy was that anyone who would become a leader of men should learn and know
Mathematics. This philosophy influenced the great American leader Abraham Lincoln to learn
the thirteen (13) books of Euclid called “Elements”. He believed that mathematics was the best
discipline for the human mind. His ideal was that mathematics should be taught with amusement
and pleasure and made very interesting. He wrote at the entrance of his school “Let no man
destitute of Mathematics, enter my door”.

Euclid

His name was first met in the records around 300 BC. Before him, mathematical knowledge was
in fragments and pieces. He collected all these knowledge and wrote them in 13 volumes known
as “Euclid Elements” He taught mathematics in the Royal School of Alexandria. He was the
mastermind that collected all the muddled, confused pieces of mathematical jigsaw, puzzle and
put them together in such a way that a clear and beautiful picture suddenly emerged. All the
proofs in the Elements were based on deductive reasoning.

Archimedes

He was born in Syracuse in 287 BC. He was perhaps the world’s greatest Mathematician. He too
studied in the Royal School of Alexandria. His father was a Mathematician and Astronomer. He
was so much a man of ability, energy and power of application that he brought the mathematics
of his time to such a height that not much further progress were made until new mathematical
tools were invented. He was said to have remarked, “Give me a place to stand andI will move the
earth”. His achievement included:
1) Calculated an approximate value of π;
2) He invented a method for finding square roots;
3) Discovered how to find area of an ellipse.

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 Unit 1: Philosophy and History of Mathematics and Mathematics Education

He wrote a number of books on spheres, cylinder and cones. With his death in 212 BC, came the
end of the Golden Age of Greek Mathematicians.

Fermat Prierre

Fermat was a lawyer by profession but by 1629 AD, he began to make discoveries of capital
importance in mathematics. He discovered many theorems in analytical geometry. One of which
was in 1636.The extremity of one of these describing a straight line not curved. Most of his
works were however published after his death. Fermat used not only a method of finding the
tangent to curves of the form y = mx but he also in 1629 came up with a theorem on the areas
under these curves. He eventually discovered differential calculus by studying “rate of change”.

So what did all great mathematicians have in common? Are they born geniuses or
did they realize the existence of theorems and other mathematical discoveries in
their journey of life?

Feedback

There are two schools of thoughts that answer this question. Plato believed that
these theorems and concepts are out there in the universe just waiting to be
revealed whereas the ‘realists’ have a different philosophy; according to them
mathematical concepts and ideas arise from our physical presence and we solely
are responsible for their being. Whichever theory is true the fact remains that
mathematics is creative boldness and all these famous mathematicians found that
creativity and boldness within them that encouraged them to discover the
mathematical principles we now know and study in our books.

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 Unit 1: Philosophy and History of Mathematics and Mathematics Education

Self Assessment Exercise

1. Does history of mathematics have a role in mathematics education? Explain.

2. What impact did the writing materials (papyrus, clay, bamboo, paper) have on the
development of mathematics and our knowledge of it? How did the different
writing materials influence the mathematical notation?
3. Summarize what we have learned from the following papyri:
(a) The Moscow papyrus (b) The Rhind papyrus
4. Write short notes on the life and contributions of any three of the following
mathematicians.
(i) Blaise Pascal
(ii) René Descartes
(iii) Isaac Newton and Wilhelm Leibniz
(iv) Carl Friedrich Gauss
(v) Leonhard Euler
(vi) G. F. Bernhard Riemann
5. Discuss the concept of number as abstract versus concrete. Is ”number" a
philosophical concept or a concrete object with physical existence?
6. Where Did Numbers Originate?

Unit Summary
There are different forms of knowledge, each of which has a different source. Some knowledge
does enter our heads through our eyes and ears and fingertips, but the most critical type of
knowledge (which Piaget called “logico-mathematical knowledge”) is built within the brain.

Piaget identified three types of knowledge: Physical knowledge, Social knowledge, and Logico-
mathematical or metaphysical knowledge.

Two main schools in the philosophy of mathematics can be distinguished: the absolutist and
conceptual change (fallibilists) philosophies of mathematics.

Absolutist philosophies of mathematics assert that mathematics is a body of certain and

absolute/unchangeable knowledge.

In contrast, conceptual change (fallibilist) philosophies assert that mathematics is corrigible,

subjective, incomplete, fallible and an ever changing social product.

Absolutism is a blanket term for several distinct views, which can include: logicism, formalism,
intuitionism, Platonism and empiricism.

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In the view of constructivism, mathematics is an exercise of the human intuition, not a game
played with meaningless symbols. Instead, it is about entities that we can create directly through
mental activity.

Constructivism is an approach to teaching and learning based on the premise that cognition
(learning) is the result of "mental construction." In other words, students learn by fitting new
information together with what they already know.

Students need to construct their own understanding of each mathematical concept, so that the
primary role of teaching is not to lecture, explain, or otherwise attempt to 'transfer' mathematical
knowledge, but to create situations for students that will foster their making the necessary mental
constructions.

For many years, there has been a debate in education on the advantages and disadvantages of
socio-constructivism and behaviourism. These two philosophies on teaching and learning
mathematics can be depicted as two contrasting views and both have influenced the way
mathematics is being taught in schools.

No single theory provide a complete model of either teaching, in spite of the limitation of the
theories, each has applications for teaching and learning secondary school mathematics.

Piaget’s theory of intellectual development postulated four stages beginning from childhood to
adulthood.

The theories of Bruner and Gagne were also treated with the implications of their theories for the
teaching and learning of Mathematics.

The Egypt, Babylon, and Greek mathematicians made wonderful contributions in different
mathematical fields.

Further Reading Materials for the Unit:

 Ernest, Paul(1990): The Philosophy of Mathematics Education;Falmer Press.

 Burton (2007): The History of Mathematics: An Introduction, 6th Edition; the
McGraw−Hill Companies.
 Roger Cooke (2005): The History of Mathematics: A Brief Course, 2nd Edition;
University of Vermont, Wiley.
 Sudhir Kumar, D.N. Ratnalikar (2003): Teaching of Mathematics; Anmol Publications
Pvt. Ltd. New Delhi.

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 Unit 2: Education Values of Mathematics and Mathematics Curriculum

UNIT 2:Educational Values of Mathematics and Mathematics

Curriculum (7 hrs.)

Introduction
Learning to teach mathematics is beyond learning mathematics. Mathematics teachers who know
little about the importance and purpose of learning mathematics will have little influence to
motivate and assist students particularly with reduced motivation and / or interest in doing
mathematics. Mathematics teachers are expected to be curriculum developers, reviewers, as well
as councilor of their students regarding mathematical issue and student-related problems. The
purpose and aims of mathematics, under any educational system, are the foundation for selection
of methods and techniques to teach it. Accordingly, you are encouraged to appreciate the aims of
mathematics vis-à-vis the international as well as national contexts, develop and evaluate
mathematics syllabi fit for purpose. Accordingly, this unit will provide you with the theoretical
as well as practical knowledge and skills of syllabus development and evaluation in mathematics.
This unit has two sections. Section one deals with educational values of mathematics while the
second section is about mathematics curriculum. As this section is central to your future career,
your independent as well as collaborative learning is mandatory. Hence, you are advised to use
and / or refer to as many materials and resources as possible including websites in the internet in
the areas of this unit. It is also mandatory to complete the activities and self assessment exercises
mentioned in this unit as well as those which will be given by your instructor in classroom.

Unit Learning Outcomes

Upon completion of this unit, students will be able to:

 Appreciate the values of mathematics education in relation to other subjects

and in real life;
 Design lesson capitalizing the values and applications of mathematics to a day
to day social life;
 Identify the basic elements of mathematics curriculum;
 Evaluate and critique secondary mathematics programs in Ethiopia;
 Examine the aims of mathematics in Ethiopia;
 Write appropriate instructional objectives for mathematics programs and
lessons.

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 Unit 2: Education Values of Mathematics and Mathematics Curriculum

2.1 Educational Values of Mathematics (3 hrs.)

Activity

1. What is the importance of mathematics in life?

2. What is the place of mathematics in any scheme of education?
3. What shall be the advantage of devoting so much effort, time, and money to the
teaching of mathematics?

There are a number of questions including the following which need to be answered at this stage.
Why should everybody learn mathematics? What are the purposes and aims of teaching
mathematics? How does it make any contribution in the development of an individual? It will be
seen that these questions pertain to the same aspect viz., the educational values of mathematics.

A genuine teacher of mathematics will be interested in finding out the answers to the above
questions. He/she needs to be informed and convinced about the educational values of his/her
subject. Something in the personality of the child will certainly remain unrealized in the absence
of the study of mathematics. The knowledge of its values and aims will stimulate and guide the
teacher to adopt effective methods, devices and illustrative materials. His/her own conviction
about its soundness enables him/her to convince the students, parents and the society.

Mathematics is a subject of great educational values and makes a major contribution in achieving
the aims of education. It has got many educational values which determine the need of teaching
the subject in schools.

2.1.1 Intellectual Values

The study of mathematics helps us in the development of many intellectual traits like power of
thinking and reasoning, inductive, analysis, synthesis, originality, generalization, discovery, etc.
Every mathematical problem poses an intellectual challenge and is a unique mental exercise. The
subject is taught for the development of power rather than knowledge.

Its problem solving is helpful in the development of one’s mental faculties. The statement of
every problem is studied and analyzed to know what is given and what is to be found out. All the
possible approaches are carefully analyzed and sorted out for choosing the most suitable one.
With the help of the chosen techniques and facts, its student tries to reach a conclusion. The
obtained conclusions are again verified for confirmation and acceptance. The problems provide
enough of opportunities for the training of the thought process.
The proving of statements requires argumentation, discrimination, cancellation, comparison, etc.,
which are experiences of intellectual value. Mathematics develops our powers of acquiring
knowledge, thinking, reasoning, judgment and generalization.

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 Unit 2: Education Values of Mathematics and Mathematics Curriculum

2.1.2 Utilitarian/Practical Values

We find a large number of practical applications of mathematics in life. The knowledge of

mathematics is needed at every step or stage of life. Everyone uses some mathematics in every
form of life.

A common man sometimes can do without reading or writing but he cannot do without counting
and calculating. Any person who is ignorant of mathematics can be easily cheated. He/she will
always be at the mercy of others. We have to make purchases daily. We buy cloth, food items,
fruit, vegetables, grocery etc. We have to calculate how much we have to pay for everything. A
house-wife also needs mathematics for looking after her house, preparing family budgets and
estimates, writing various expenses and noting down various household transactions.

Mathematics is needed by all of us whether rich or poor, high or low. Not to speak of engineers,
bankers, accountants, businessmen, planners etc., even petty shopkeepers, humble coolies,
carpenters and laborers need mathematics not only for earning their livelihood but also to spend
wisely and save for future. Whoever earns and spends uses mathematics.

We are living in a world of measurements. We have to measure lengths, areas, volumes and
weights. We have to fix timings, prices, wages, rates, percentages, targets, exchanges etc. In the
absence of these fixations, the life in the present complex society will come to a standstill. There
will be utter confusion and chaos. Just think if a fairy descends on earth and removes all
mathematics. There will be no calendar, no maps, no accounts, no fixations or measurements, no
industrial activity, no plans or projects.

Thus we see that mathematics has tremendous value or application in our daily life. It is essential
for leading a successful social life.

2.1.3 Disciplinary Values

Mathematics trains or disciplines the mind to function in a particular manner. It develops

thinking and reasoning power. According to Locke, "Mathematics is a way to settle in the mind a
habit of reasoning." Mathematics is 'an exact and definite science'. Every student of mathematics
has to reason properly without any prejudices or unnecessary biases.

Reasoning in mathematics possesses certain characteristics which are suitable for the training of
learner’s mind. These are:-

(i) Simplicity—It teaches that definite facts are always expressed in simple language which
are always easily understand.
(ii) Accuracy—Accurate reasoning thinking and judgment are essential for its study.
Accuracy, exactness and precision compose the beauty of mathematics.
(iii) Certainty of Results—The answer is either right or wrong. Subjectivity or difference of
opinion between the teacher and the taught is missing. The student can verify his/her result
by reverse process. It is possible for the student to remove his/her difficulty by self-effort

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 Unit 2: Education Values of Mathematics and Mathematics Curriculum

and to be sure of the removal. He/she develops faith in self-effort which is the secret of
success in life.
(iv) Originality—Most work in mathematics demands original thinking. Reproduction and
cramming of ideas of others is not very much appreciated. When the child has a new or a
different mathematical problem, it is only his/her originality which keeps him/her going.
The discovery or establishment (derivation) of a formula or conversion of formula in one
form to another is also his/her original work. This practice in originality enables the child
to face new and challenging problems with confidence.
(v) Similarity to the reasoning of life—Clear and exact thinking is as important in daily life
as in mathematics. Before starting with the solution of the problem, the student has to
grasp the whole meaning. Similarly in daily life, while undertaking a task, one must have
a firm grip over the situation.
(vi) Verification of results—Results can be easily verified. This gives a sense of
achievement, confidence and pleasure. It inculcates the habit of self-criticism and self
evaluation.
(vii) Power not knowledge—In this ever advancing society the important thing is not only to
learn facts, but also to know how to learn facts. The main thing is not the acquisition of
knowledge but the acquirement of the power of acquiring knowledge.
(viii) Application of knowledge—Knowledge becomes real and useful only when the mind is
able to apply it to the new situations. Ability to apply knowledge to new situations is
inculcated in students. They acquire the power to think effectively. It generates the
otherwise latent powers of thinking, reasoning, discovery and judgment of the child.

2.1.4 Cultural Values

Study of an individual's behavior without a study of the world in which he/she lives in is
incomplete. For this, one has to know about the civilization as well as cultural heritage of the
concerned society. Mathematics is associated closely with the cultural heritage of each country.
It has got a great cultural value which is steadily increasing day to day. Mathematics has made a
major contribution to our cultural advancement. The progress of our civilization has been mainly
due to the progress of various occupations such as agriculture, engineering, industry, medicine,
navigation, rail road building etc. These occupations build up culture. Mathematics makes direct
or indirect contribution to the development of all occupations.

The history of mathematics shows how mathematics has influenced civilization and culture at a
particular time. Progress in mathematics, of Greeks and Egyptians in the past led to their cultural
advancement and the progress of their civilization. That is mathematics is the mirror of
civilization.

Mathematics is a pivot for cultural arts such as music, fine arts, poetry and painting. Perhaps that
is why the Greeks, who were the greatest geometers of their times, were quite adept in fine arts.
Therefore mathematics shapes culture as a play back pioneer. Some of the important aspects of
cultural heritage have been preserved in the form of mathematical knowledge only and learning
of mathematics is the only medium to pass on this heritage to the coming generations.

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 Unit 2: Education Values of Mathematics and Mathematics Curriculum

Activity

1. Describe the major contributions mathematics has made to our cultural

advancement.
2. Reasoning in mathematics possesses certain characteristics which are suitable
for the training of learner’s mind. Describe them.

Feedback

1. Mathematics is associated closely with the cultural heritage of each

country.
 Mathematics makes direct or indirect contribution to the development
of all occupations;
 Mathematics is a pivot for cultural arts such as music, fine arts, poetry
and painting;
 Some of the important aspects of cultural heritage have been preserved
in the form of mathematical knowledge only and learning of
mathematics is the only medium to pass on this heritage to the coming
generations; etc.
2. Characteristics of reasoning in mathematics are:
 Simplicity:- definite facts are always expressed in simple language
 Accuracy:- Accuracy, exactness and precision compose the beauty of
mathematics.
 Certainty of Results:- The answer to mathematics questions is either
right or wrong.
 Originality:- Most work in mathematics demands original thinking.
 Similarity to the reasoning of life:-Clear and exact thinking is as
important in daily life as in mathematics.
 Verification of results:- Mathematical Results can be easily verified.
 Power not knowledge:- The main thing is not the acquisition of
knowledge but the acquirement of the power of acquiring knowledge.

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2.1.5 Moral Values

The study of mathematics helps in moral development and character formation. It helps in
developing proper moral attitudes as there is no place for prejudiced feelings, biased outlook,
doubts and half truths, discrimination, maldistribution of resources, unreasonableness, and
irrationality in the learning of this subject. It is the only subject which helps in objective analysis,
correct reasoning, valid conclusions and impartial judgment.

The qualities like honesty, truthfulness, justice, dutifulness, punctuality, self-confidence,

discrimination between good and evil, observation of rules and beliefs in systematic organization
and arrangement are indirectly inculcated through the teaching of mathematics. These qualities
go towards developing a morality and sound character.

The Greek philosopher Dutton has rightly remarked that… “gossip, flattery, slander, deceit- all
speak for a slovenly mind that has not been trained by mathematics.”

2.1.6 Social Values

Mathematics plays an important role in the organization and maintenance of our social structure.
Society is the result of the inter-relations of individuals. It consists of big and small groups and
there are sub-groups within each group. Mathematics enables us to understand the inter-relations
of individuals and the possibilities of various groups. Society is phenomena of balancing and
counter-balancing of various social forces. Mathematics helps in creating a social order in these
phenomena. It regulates the functioning of society in many ways. Social conditions like justice,
fair play, healthy competition, symmetry, harmony, etc. have often to be described in
mathematical terms for the purpose of clarity.

For smooth transactions, exchange, trade, business and bargaining, mathematics becomes a
useful tool. It has its own role to play in the development of means of communication. It has
helped in knitting the vast society into a family. When the dealings between individuals are given
a mathematical touch, it leads to social progress, prosperity and welfare.

History of mathematics reveals that whenever a society gave due weight age to the knowledge of
mathematics, it made a tremendous progress. When mathematics makes its contribution in the
advancement of science and technology, society draws huge benefits. Its contribution is evident
in the fields of atomic energy, space research, space travel, and man-made satellites.

The harnessing of social resources can best be done by mathematical and scientific approach.
Mathematics helps in the formation of social norms and their implementation. The dominance of
materialistic outlook in our society is one of the chief attributes of mathematics. Our monetary
dealings are a major domain of our social dealings and relations. We earn a social status by
virtue of our economic status and behavior. The social status is governed by our property,
income, bank balance and economic potential. Social security is bound to imply economic
security.

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The ideas like manpower planning have originated partly due to the influence of mathematics.
The statistical data and the census provide bases for short-term and long range planning for the
welfare of the society.

2.1.7 Artistic/Aesthetic Values

Mathematics possesses immense aesthetic value. People wrongly consider mathematics as

unartistic and non-aesthetic. But for a true student of mathematics, the subject is all beauty,
symmetry, balance, harmony, fitness, art and music. There is a great pleasure in successfully
solving a mathematical problem. It was the reason why Pythagoras sacrificed one hundred oxen
to the goddess for celebrating the discovery of the theorem that goes by his name. In the same
way, Archimedes forgot his nakedness while announcing to the people the discovery of his
principle. Similarly a young student is overwhelmed with satisfaction and delight when the
answer obtained by him/her tallies with that given in the text. What we enjoy and appreciate in
the arts like drawing, painting, architecture, music, dance, etc., bear appearance of mathematics.

Apart from giving pleasure through its application to various arts, it also provides entertainment
through its own riddles, games and puzzles. While developing the subject, its dedicated students
have been playing with its numbers, figures, shapes and problems.

There is no exaggeration in saying that mathematics is the creator as well as the nourisher and
savior of all the arts. What we enjoy in the arts like Drawing, Painting, Architecture, Music or
Dance etc. is all due to mathematics. Mathematical regularity, symmetry, order and arrangement
play a leading part in beautifying and organizing the work of these arts. Even the poetry is not
enjoyable without mathematics. Music is nothing but the mathematically organized sound. All
the musical instruments Harmonium, Drum, Table, Flute, Guitar, Sitar, Violin etc. are played on
the set of rules of mathematics. Therefore, Leibnitz is right when he says “Music is a modern
hidden exercise in Arithmetic of a mind unconscious of dealing with numbers.”

In dancing too one has to take care of mathematics in taking steps and responding to the tunes.
Moreover, the secret of the beauty of a garden, an ornament or a flowering pot lies in the hands
of the arrangement made with the help of mathematics. In brief we can say that secret of the
beauty no matter whatever, lies in regularities, precision symmetry, order and arrangement and it
is only mathematics that is itself capable of decorating a thing with so many characteristics.

2.1.8 Vocational Values

The main aim of education is to help the children to earn their living and to make them self
dependent. Mathematics has a great role in achieving that aim of education. The study of
mathematics prepares us for various occupations like engineering, accountancy, auditing,
taxation, banking, surveying, trade, designing, teaching, agriculture, planning, financing, weights
and measures inspection, quality control, budgeting, construction, computer applications, etc.
These occupations have immensely benefited from mathematics in their development. The
knowledge of mathematics is helpful in achieving vocational efficiency in many spheres.

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 Unit 2: Education Values of Mathematics and Mathematics Curriculum

Almost every vocation involves investment, loan, interest, profit and loss, percentage, purchases,
planning of financial and other resources, accounts, shares, etc., which can be better managed
with the help of mathematical knowledge and understanding.

A sound and productive vocational life demands a sound mathematical background. Thus the
study of mathematics prepares the students for a wide variety of vocations.

2.1.9 International Values

Mathematics is a universal subject and it helps in creating international understanding. Its history
presents a very good picture of the overall development of our civilization. What we possess in
the form of mathematical knowledge today is the fruit of the combined efforts of all human
beings, the inhabitants of all the corners of the world, the scholars of all ages, the followers of all
religions and members of all the races. Mathematics is the common heritage of mankind and it is
not an exclusive property of any particular nation, race or country. All mathematicians
irrespective of their casts, colors or creed worked devotedly towards a common cause.
Mathematics is a symbol of agreement all over the world. It is the same everywhere. It is an
intellectual bond between various nations and countries of the world. The man-made barriers of
boundaries cannot restrict or check the free flow of mathematical knowledge and cooperation
among the mathematicians belonging to various places. Any new idea discovered in the field of
mathematics does not take much time to become an international asset. The countries with
divergent political views have at least some common platforms in the sphere of scientific and
mathematical knowledge. We may differ from one another in many ways but in the field of
mathematics we find our differences removed. Mathematical measurements and their latest
modifications towards decimal system are bringing the regions of the world closer and
establishing universal agreement.

There is a continuous flow of teachers, mathematicians and researchers from one nation to
another nation for the exchange of mathematical and scientific ideas. The mathematics books and
research journals are also exchanged and circulated among almost all the nations of the world.
All these things add to the feeling of international understanding and are helpful in bringing
international peace.

Mathematics has international value in the sense that it is helpful in creating international
understanding and brotherhood.

Self Assessment Exercise

1. Debate and discuss in small groups and then reflect in whole class session on the
values of mathematics?
2. What are the Artistic/Aesthetic values of mathematics?
3. There are a number of misunderstandings in the minds of some parents about the
utility of mathematics as a school subject for their wards. How will you try to
convince them?
4. Form a group of three and discuss & write a brief summary of Utilitarian/Practical,
disciplinary and cultural values of mathematics.

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 Unit 2: Education Values of Mathematics and Mathematics Curriculum

2.2 Mathematics Curriculum (4 hrs.)

What do you understand by the term “curriculum”? What are the roles a
curriculum can play in instruction?

A curriculum is a central state (governmental) document that contains the general objectives of
education and the most important directions for planning, shaping and performing the
instructional processes in practice. In the curriculum the objectives of all subjects are given after
deriving them from the general objectives of education of a country. It also determines the
volume of the subject matter of all subjects, the time necessary for imparting as well as for
acquiring the planned objectives with respect to the development of students’ personality such as
knowledge, capability, convictions, habits, etc.
As curriculum contains the most important elements of instruction, it can be used as sources of
information for teachers in the teaching learning process. That means the curriculum can play a
great role in the preparation of objectives, in the selection of contents, in the selection of
methodology and assessment tools while teachers are preparing their daily, unit, semester or
annual plans.
A curriculum is more than a collection of activities: it must be coherent, focused on important
mathematics, and well articulated across the grades. An effective mathematics curriculum
focuses on important mathematics--mathematics that will prepare students for continued study
and for solving problems in a variety of settings.

In setting a mathematics curriculum the following important questions are to be answered:

(i) What aspects of mathematics need to be utilized and emphasized in order to satisfy the
needs of the society and the needs of the individual?
(ii) To what depths of mathematics content should we teach to reach the goals?
(iii) What teaching techniques and teaching aides are available for mathematics?
(iv) What human and material resources are available to handle the necessary aspects of
mathematics necessary for the achievement of the goals?
(v) If these resources are not available, do we need some trainings or some fundamental
foundation works to be done to make them available?

Mathematics cannot be taught in isolation of the societal needs or the needs of the individuals. A
teacher of mathematics must be convinced that what he/she is teaching will be useful to the
students and to the society. With this conviction, he/she can then convince the students. In our
modern society of attachment of high values to material living, every student will love
mathematics more if he/she understands in advance what mathematics can do for him/her, for
his/her future success as an individual and as a good citizen. This leads us to the utilitarian
aspects of mathematics.

Curriculum Framework

The curriculum framework for Mathematics embodies the key knowledge, skills, values and
attitudes that students are to develop at the senior secondary level. It forms the basis on which

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schools and teachers can plan their school-based curricula, and design appropriate learning,
teaching and assessment activities.

Design Principles
The following principles are used in designing the curriculum:

(a) Building on knowledge developed at the basic education level

To ensure that the curricula at different levels of schooling are coherent, the development of the
Mathematics Curriculum (Senior Secondary mathematics) is built on the knowledge, skills,
values and attitude acquired through the mathematics curriculum for basic education from
Primary to Junior Secondary.

(b) Providing a balanced, flexible and diversified curriculum

The curriculum thus provides flexibility for teachers to:

 offer a choice of courses within the curriculum to meet students’ individual needs,
 organize the teaching sequence to meet individual situations; and
 make adaptations to the content.

(c) Catering/supplying for learner diversity

The curriculum provides opportunities for organizing a variety of student activities to cater for
learner diversity.

(d) Achieving a balance between breadth and depth

The curriculum covers the important and relevant contents for senior secondary students, based
on the views of mathematicians, professionals in mathematics education and overseas
mathematics curricula at the same level. The breadth and depth of treatment in the secondary
mathematics are intended to provide more opportunities for intellectually rigorous study in the
subject.

(e) Achieving a balance between theoretical and applied learning

An equal emphasis is given on theories and applications in both real-life and mathematical
contexts to help students construct their knowledge and skills in mathematics. The historical
development of selected mathematical topics is also included to promote students’ understanding
of how mathematics knowledge has evolved and been refined in the past.

(f) Fostering lifelong learning skills

Knowledge is expanding at an ever faster pace and new challenges are continually posed by
rapid developments in technology. It is important for our students to learn how to learn, think
critically, analyze and solve problems, and communicate with others effectively so that they can

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confront current and future challenges. The curriculum provides a suitable context for
developing such abilities.

(g) Promoting positive values and attitudes to learning

Positive values and attitudes to learning, which are important in learning mathematics, permeate
the secondary Mathematics Curriculum. Some particular topics in the curriculum would help to
develop in students an interest in learning mathematics, keenness to participate in mathematics
activities, and sensitivity and confidence in applying mathematics in daily life. It also helps to
foster open-mindedness and independent thinking.

Self Assessment Exercise

1. Discuss about the design principles of mathematics curriculum.
2. Identify the basic elements of mathematics curriculum.

2.2.1 Curriculum Development in Mathematics as a Subject in Ethiopia

Curriculum change is essential if the existing curriculum shows high people dissatisfaction and
controversies in its effectiveness that can be supported by evidences. The mathematics
curriculum in Ethiopia has been changing from time to time. But such changes were not based on
evidences or research rather it was based on foreign political influences and other benefits that
can be obtained from foreign countries. For instance, as soon as the Italian invasion was over,
Ethiopia started to have trade relationship with Britain. At that time, Ethiopia was not importing
only goods but educational curriculum too. Latter Ethiopia developed a better relationship with
America and as a result the Ethiopian educational system became more oriented to the American
educational system. The American educational experts have initiated and developed the
mathematics program in African countries. Thus the mathematics program was introduced in
African countries such as Ethiopia, Ghana, Kenya, Liberia, Malawi, Nigeria, Siera-Leone,
Zambia, Tanzania and Uganda in 1967,which was known as “The Entebbe Mathematics
Program” As a result of the influence of worldwide changes in the mathematics curriculum,
Ethiopia also introduced another mathematics program in high schools ( ninth grade) in the year
1967 which was again based on the changes and influences of foreign educational system. Thus
it is questionable whether such changes have satisfied the needs of the broad masses or not
throughout the time.

The New Mathematics Program in Ethiopia

In 2003 the Ethiopian education system experienced wide-ranging reform that touches every
aspect of the educational system. This reform is called TESO (Teacher Education System
Overhaul). Designed to address educational problems in Ethiopia, TESO introduced significant
structural changes and promised to bring a ‘paradigm shift’ in the Ethiopian educational system
by engaging teacher education in changing society and promoting democratic, practical, and
problem-solving education. As a result of the influence worldwide changes in the mathematics
curriculum, the mathematics program of Ethiopia has also undergone a change or reform. This
change was a change in goals, contents, approaches and assessment methods as other subjects.

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Hence, training has given to the teachers of mathematics to handle the revised syllabus and the
new curriculum easily, so that to improve the quality of education in Ethiopia that is why the
TESO program is designed.

The new mathematics curriculum has brought many changes and has many useful features in its
reformation. Of the many features, some of the main characteristics of the new mathematics
program are:
 Developing problem solving and reasoning skills is the main goal of the program,
 At all levels of study, its emphasis is on the understanding of the underlying principles
based on reasoning,
 Terms are carefully defined and considerable attention is given to their precise use,
 The method of encouraging the students to make discoveries for him/herself is used
whenever it is appropriate,
 Computational skills are based on understanding of theory rather than solely on drill or rote
memorization,
 The need for some undefined terms and assumptions ( unproved statements, postulates ) is
recognized,
 The recognition and use of patterns is emphasized at all levels, etc.

A curriculum in general or a mathematics curriculum in particular expresses the learning goals of

the students. In setting a mathematics curriculum the first important question is “What are the
educational goals and aims of the mathematics curriculum?” Accordingly the objectives of the
mathematics program are:
 To build logically coherent, unified courses,
 To introduce at any early level, powerful but elementary ideas first presented by concrete
examples, later abstracted and placed in an axiomatic system,
 To place great emphasis on the meaning of mathematics rather than on memorization and
manipulation procedures,
 To develop in the students the capacity for original mathematical thought,
 To prepare the students for scientific study at a higher level.

Self Assessment Exercise

1. Discuss the trend of curriculum reform in mathematics in Ethiopia.
2. Why do we require a change in the mathematics curriculum?
3. Mention some of the main characteristic of the new mathematics program.

2.2.2 Selection of Goals and Objectives of Mathematics

What should be the appropriate goals for mathematics education?

Before we start the teaching of a subject, it is important for us to know as to why we are going to
teach it. The process of teaching can be kept on right lines only with the help of clear-cut aims.
Aimlessness in teaching would result in the wastage of time, energy and other resources.

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 Unit 2: Education Values of Mathematics and Mathematics Curriculum

What should be the goals of teaching mathematics in our schools? The answer requires the
knowledge of all the advantages that can be drawn from the teaching of mathematics. Goals help
in the realization of the values possessed by a subject.

Selection of goals and objectives for a curriculum design is never done without considering the
needs, problems and aspiration of the people or the society for whom it is being designed. With
respect to mathematics the question asked are which of the problems or needs can mathematics
be used to address or solve and in what ways can mathematics be taught to address those needs
and problems?

While some educators/teachers prefer to use the two terms interchangeably, it is useful to
distinguish between goals and objectives with a view to facilitating effective teaching and
learning in the classroom. Goals without objectives can never be accomplished while objectives
without goals will never get you to where you want to be. The two concepts are separate but
related and will help you to be who you want to be.

Goals of Teaching Mathematics

The term goals of teaching mathematics stands for the targets or broader purposes that may be
fulfilled by the teaching of mathematics in the general scheme of education. Goals are like
ideals. Their attainment needs a long-term planning. Their realization is not an easy task.
Goals are usually broad and abstract, and they are of little immediate value to the classroom
teacher in planning a particular lesson in that they cannot assist him/her in his/her decision on
content, teaching strategy and/or evaluation. However, the objectives of the teachers set for the
student in the classroom should be consistent with the overall goals of education.

We may mention the goals of teaching mathematics as under:

1. To enable the students to solve mathematical problems of daily life. We have to select the
content and methods of teaching so that the students are able to make use of their learning of
mathematics in daily life;
2. To enable the students to understand the contribution of mathematics to the development of
culture and civilization;
3. To develop thinking and reasoning power of the students;
4. To prepare a sound foundation needed for various vocations. Mathematics is needed in
various professions such as those of engineers, bankers, accountants, statisticians etc.;
5. To prepare the child for further learning in mathematics and the related fields. School
mathematics should also aim at preparing him/her for higher learning in mathematics;
6. To develop in the child desirable habits and attitudes like habit of hard work, self-reliance,
concentration and discovery;
7. To give the child an insight into the relationship of different topics and branches of the
subject;
8. To enable the child to understand popular literature. He should be so prepared that he finds
no handicap in understanding mathematical terms and concepts used in various journals,
magazines, newspapers etc.;
9. To teach the child the art of economic and creative living;
10. To develop in the child rational and scientific attitude towards life.

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 Unit 2: Education Values of Mathematics and Mathematics Curriculum

Objectives of Teaching Mathematics

Goals are divided into some definite, functional and workable units named as objectives. The
objectives are those short-term, immediate goals or purposes that may be achieved within the
specified class situation. They help in bringing about behavioral changes in the learners for the
ultimate realization of the goals of teaching mathematics. The goals are broken into specified
objectives to provide definite learning experiences for bringing about desirable behavioral
changes. The objectives should always be specified in terms of student’s behavior.

The major objectives of teaching mathematics can be classified in the following manner:

 Knowledge and understanding objectives,

 Skill objectives,
 Attitude objectives,
 Appreciation and Interest objectives.

1) Knowledge and Understanding Objectives

Through mathematics, a student acquires the knowledge and Understanding of the following:

(i) The language of mathematics in terms of mathematical symbols, signs, formulae, figures,
diagrams, technical terms, definitions etc.
(ii) The various mathematical concepts like concept of area, volume, number, measurement
and direction etc.
(iii) Fundamental mathematical ideas, facts, principles, processes, rules and relationships.
(iv) Historical background of various topics and contribution of mathematicians.
(v) The basic nature of the subject mathematics.

2) Skill Objectives

Mathematics helps a student in developing the following skills:

(i) To express thoughts clearly and accurately;

(ii) To develop speed, precision, brevity, accuracy and neatness in the computation and
calculation work;
(iii) To perform calculations orally or mentally;
(iv) To develop the ability to estimate and check results;
(v) To reach accurate conclusions by accurate and logical reasoning;
(vi) To analyze problems and discover fundamental relationships;
(vii) To develop the techniques of solving problems;
(viii) To develop the skill to draw accurate geometrical figures;
(ix) To develop the ability to use mathematical apparatuses and tools skillfully;
(x) To develop essential skills in surveying or measuring and weighing processes;
(xi) To develop essential skills in drawing, reading, organizing and interpretation of
graphs, statistical tables and data.

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 Unit 2: Education Values of Mathematics and Mathematics Curriculum

3) Attitude Objectives

Mathematics helps in the development of the correct attitudes in students in the following
manner:

(i) Develops the attitude of systematically pursuing a task to completion;

(ii) Develops the habit of systematic thinking and objective reasoning;
(iii) Develops heuristic attitude and tries to make independent discoveries;
(iv) Develops the habit of logical reasoning and verification;
(v) Tries to collect enough valid evidences for drawing inferences, conclusion or
generalization;
(vi) Tries to express his/her opinions precisely, systematically and logically without any bias
prejudices;
(vii) Recognizes the adequacy or inadequacy of given data in relation to the problem;
(viii) Develops power of concentration and independent thinking;
(ix) Develops habit of self-reliance for solving mathematical problems;
(x) Develops personal qualities, such as regularity, punctuality, honesty and truthfulness;
(xi) Develops mathematical prospective and outlook for observing the realm of nature and
social world;
(xii) Showed originality and creativity.

4) Appreciation and Interest Objectives

Mathematics helps a student in acquiring the power of appreciation and interest in the following
manner:

(i) appreciates the contribution of mathematics to the development of various subjects and
occupations;
(ii) appreciates the contribution of mathematics to the development of culture and
civilization;
(iii) appreciates mathematics as the science of all arts;
(iv) appreciates the role of mathematics in everyday life;
(v) appreciates the role of mathematics in understanding his/her environment;
(vi) appreciates the role played by mathematics in modern life;
(vii) appreciates the aesthetic value of mathematics by observing symmetry, similarity, order
and arrangement in mathematical facts, principles and processes;
(viii) appreciates the recreational value of mathematics and learn to utilize his/her leisure time
properly;
(ix) appreciates the vocational and cultural values of mathematics;
(x) appreciates the power of computation.

We have discussed the goals and objectives of teaching mathematics in general. The teacher
should carefully choose the objectives regarding a particular topic. The nature of students will
also be kept in view.

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 Unit 2: Education Values of Mathematics and Mathematics Curriculum

Activity 2.2.1: (take 6 minutes)

1. What is the difference between goals and objectives?

Feedback for Activity 2.2.1

 Goals are long-term aims that you want to accomplish. Objectives are
concrete attainments that can be achieved by following a certain number of
steps.
 Goals and objectives are often used interchangeably, but the main difference
comes in their level of concreteness. Objectives are very concrete, whereas
goals are less structured.
 Here is an easy way to remember how they differ:
Goals has the word ‘go’ in it. Your goals should go forward in a specific
direction. However, goals are more about everything you accomplish on your
journey, rather than getting to that distant point. Goals will often go into
undiscovered territory and you therefore can’t even know where the end will
be.
 Objective has the word ‘object’ in it. Objects are concrete. They are
something that you can hold in your hand. Because of this, your objectives
can be clearly outlined with timelines, budgets, and personnel needs. Every

Self Assessment Exercise

1. What are the factors that influence the selection of Educational objectives?
2. What guiding principle could be used in formulating appropriate
objectives?
3. What is the importance of goals in curriculum development in
Mathematics?
4. List any six major goals of teaching mathematics.
5. Review and present, in group of not more than five, goals of secondary
mathematics in Ethiopia in particular and their implication for selection of
teaching approach and methods.

2.2.3 Mathematics Syllabus

A syllabus represents the application of general principles given in the curriculum. It is, like a
curriculum, a central state (governmental) planning document for the training and education of
students in instruction. Thus a syllabus is defined as: a concise and condensed written outline of
a course of study; a list of items which have been written down to be taught and learnt from year

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 Unit 2: Education Values of Mathematics and Mathematics Curriculum

to year, a blueprint enabling teachers to carry out a particular part usually a formal part of the
curriculum. The mathematics syllabus contains directions and information on objectives,
contents, methods and evaluation of the mathematics instruction.
Teachers have certain roles to play in the implementation of the proposals contained in the
syllabus. The professional roles/duties include the following:

1. The teacher should make a thorough study of the syllabus so as to have an adequate
knowledge of what is to be taught;
2. The teacher should acquaint himself or herself with what the students have done in the
previous class and the work the students will do when they move on to the next class. This
is necessary because in order for the current syllabus to be well understood, the teacher
must build upon the foundation of knowledge gained in the previous year and work
towards preparing a good foundation for the students’ work in the next class;
3. The instructional and learning materials that would be required to achieve the objectives of
a given syllabus should be sought for by the teacher and used at the appropriate time;
4. There should be modification of the syllabus whenever occasion calls for such, say
adjustment of the syllabus to community resources. In other words, the syllabus should not
be regarded as a rigid handout to be studied and obeyed indiscriminately;
5. The teacher should make a clear and logical breakdown of the syllabus yearly, termly
and/or weekly topics.
6. Evaluation of each stage of the syllabus is the responsibility of the teacher in order to
ascertain two basic elements:
(a) achievement of the stated objectives; and
(b) the effectiveness of the teacher’s instruction.

A given syllabus is only an indication of content, and does not prescribe specifically the order in
which topics should be taught. It is from the syllabus that the teacher will work a detailed plan
indicating what he/she will do in each term, and then break it down even further and show what
ground he/she will cover in each week of the term.

Activity 2.2.2: (take 5 minutes)

What is the difference between curriculum and syllabus?

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 Unit 2: Education Values of Mathematics and Mathematics Curriculum

Feedback for Activity 2.2.2

 The terms Curriculum and School Syllabus have erroneously been taken to
mean the same thing and the two have been used interchangeably.
 Curriculum is not equated to the school syllabus.
 Curriculum is perceived to include all the various activities and learning
experiences available in school situations.
 Mathematics syllabus is described as sequential arrangement of Mathematical
topics at the Primary, Secondary and Teacher Training Schools.
 It is therefore seen that curriculum subsumes syllabus and not verse versa.

Self Assessment Exercise

1. Prepare hypothetical mathematics syllabus that fulfils all qualities and
components of real syllabus, in group of not more than five students and
present in whole class.
2. Discuss the syllabus of preparatory program with respect to objectives,
contents, methods and teaching aids.

2.2.4 Motivating Students towards Learning of Mathematics

Negative attitude and lack of motivation are among the factors given as being responsible for
poor performance of learners in mathematics. It follows then by hypothesis that well motivated
learners will improve in their performance in mathematics and consequently will develop
positive attitude toward the subject. For one thing, the poor performance of students in
mathematics may be the cause of negative attitudes as inability to do mathematics and not liking
it seen to go together. Motivating learners towards learning of mathematics will directly or
indirectly lead to development of positive attitude towards mathematical learning.

There is therefore a great need to motivate learners towards learning mathematics. Motivation
will serve as an inner force that will propel students’ interest towards mathematical learning. So
the major task facing every mathematics teacher is how to arouse his/her learner’s interest
towards the learning of mathematics through appropriate motivation. This will go a long way
towards improving learner’s attitude towards the subject.

Methods of motivating learners towards learning of mathematics

There are a number of methods that can be used to motivate learners towards learning of
mathematics. They include the following:

a) Learner’s participation in the learning process

When learners are actively involved in the learning process in mathematics lessons through
being given the opportunity to find out by themselves, it arouse their curiosity to know or to find

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 Unit 2: Education Values of Mathematics and Mathematics Curriculum

out. Therefore, teachers should discourage the use of chalk-talk approach of teaching
mathematics. Rather, they should embrace the investigative approach of teaching mathematics
where learners are given the opportunity to find out through the use of concrete or relevant
illustrative instructional materials. This may require the teacher in using deductive approach of
teaching in which students are led through step by step activities to arrive at generalization or the
rules behind the concept being taught.
b) Teaching applications of mathematics
Although mathematics has been widely acclaimed as the queen and servant of all subjects; that
no other subject has greater application than mathematics; yet it is equally true that learners are
more often not aware of the various applications of mathematics. They do not see its usefulness
in everyday life. In fact, some students have attributed their hatred of mathematics as a subject to
the fact that they cannot see its relevance to everyday life. It is therefore important that learners
are taught and made to understand the importance of mathematics and its relevance to everyday
life.

Learners need to know for example that mathematics was developed as a result of human’s effort
to solve a major problem of his/her day. For example, numeral was invented to help human have
a way of counting and keeping record of his/her properties such as sheep, goats and other
personal belongings. As per its applicability, mathematics has applications in all areas of human
endeavor, for example, it can be used to forecast population growth, business growth; it is used
to predict rainfall and agricultural yields; administrators use mathematics for budgetary planning
and to make educational projections through statistical generalizations and conclusions etc.
c) Home background and parental encouragement
The home background of a learner has a strong influence on his/her educational achievement say
in mathematics. This is so because the child understands about the values his/her home (family
background) places on mathematics and successful performance in it serves as a motivation and
consequently affects his/her actual performance in it. From research it has been found out that
what the child learns at home and how his/her family motivates him/her towards education
contribute to the child’s success or failure in school. Furthermore, the interest and attitude
parents show towards mathematical achievement of their children are also source of strong
motivational factors to the child.This is done through:
(i) Continuous encouragement given to the child by the parent towards learning
mathematics;
(ii) High expectation which parents have about their child’s performance; and
(iii) The attitude of parents themselves towards mathematics as a subject.

Therefore the home and particularly the parents can do a lot in motivating their children towards
learning mathematics and getting them to develop interest in the subject

Teachers’ Role in Motivating Learners

The role that teachers can play in motivating their learners towards learning mathematics cannot
be brushed aside. It is to be noted that teachers’ attitudes towards mathematics will significantly
affect his/her learners’ attitudes towards mathematics. Hence a well-motivated teacher will, to a

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 Unit 2: Education Values of Mathematics and Mathematics Curriculum

large extent, be able to motivate his/her learners. However an unmotivated mathematics teacher,
who him/herself feels insecure in teaching it, who fears and dislikes it, and who him/herself does
not have a proper understanding of the basic concepts he/she is called upon to teach, cannot help
transferring these negative traits to his/her learners. In a similar vein, a teacher who is confident,
and has a good grasp of the subject matter, who displays interest in the subject, will affect his/her
learners with these positive traits. This is so because his/her positive disposition towards the
subject has provided a conducive learning environment for his/her learners which is a basic
requirements for learning.

In addition, other things a teacher can do to motivate his/her learners include the following:

1. Be passionate about what you're teaching. If you have a favorite mathematics concept or if
one problem really challenged you, point that out to students. Let your own enthusiasm for
the subject show in your attitude towards teaching.
2. Find out what most interests your students. If your students are interested in a specific sport or
to current events, tie in any applicable mathematics concepts to that sport or to events in the
newspaper.
3. Discuss the history behind the mathematics. Often, explaining when a problem was solved
and who solved it can help students relate to the people behind the mathematics process, as
well as giving them mathematical role models that they can look up to.
4. Explain how the information they learn in class can help them in real life. For example, you
may want to introduce them to different interesting fields in which they will need this
information, such as astronomy, nuclear physics or even cooking (when you are teaching
fractions and proportions).
5. Teach through discovery learning. Instead of teaching a concept and having students apply it
to several problems, give students a problem and challenge them to solve it. When they are
engaged in the problem-solving process, they will be more interested in the concept.
6. Give students the freedom to choose as much as possible. If students can choose which
concept they will learn next, they'll be more invested in understanding it well. If students can
choose the project they would like to do to illustrate a concept, they will feel as if they have
more control over their learning.
7. Praise your students when they succeed in challenging problems or projects.
8. Do not overemphasize testing or grades. Doing so can cause students to lose interest in the
concepts they are learning and encourage them to focus only on their scores.

Other ways of motivating learners

Other ways of motivating learners include:

(i) Use of games and mathematical recreations; mathematical games can be used to sparkle
interest and bring about a lively atmosphere in a mathematics classroom. In this way it can
become a medium for motivating the learning of mathematics. The use of mathematical
recreations can also be used to relieve boredom, this can be in form of creating patterns,
tessellations, paper folding, giving mathematical puzzles.
(ii) Mathematical clubs formation and running a mathematical clubs can provide a good
forum for motivating learners towards learning of mathematics. The club can organize
programs of mathematical interest, like symposium, debates that are mathematically

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 Unit 2: Education Values of Mathematics and Mathematics Curriculum

oriented. For example, in such program topics that make students see importance of
mathematics as a discipline; careers that needs mathematics; how mathematics enters
other discipline etc. will go a long way, to both motivate and enlist other learners’ interest
in mathematics.

Self Assessment Exercise

1. Discuss methods of motivating learners towards learning of
mathematics.
2. Describe teachers’ role in motivating learners.
3. What is the role of using games, mathematical recreations and
mathematical clubs towards motivating learners?

Unit Summary
The study of mathematics helps us in the development of many intellectual traits like power of
thinking and reasoning, inductive, analysis, synthesis, originality, generalization, discovery, etc.

Mathematics is needed by all of us whether rich or poor, high or low. Not to speak of engineers,
bankers, accountants, businessmen, planners etc., even petty shopkeepers, humble coolies,
carpenters and labourers need mathematics not only for earning their livelihood but also to spend
wisely and save for future.

Mathematics trains or disciplines the mind to function in a particular manner. It develops

thinking and reasoning power.

Some of the important aspects of cultural heritage have been preserved in the form of
mathematical knowledge only and learning of mathematics is the only medium to pass on this
heritage to the coming generations.

The study of mathematics helps in moral development and character formation. The qualities like
honesty, truthfulness, justice, dutifulness, punctuality, self-confidence, discrimination between
good and evil, observation of rules and beliefs in systematic organization and arrangement are
indirectly inculcated through the teaching of mathematics.

Mathematics helps in the formation of social norms and their implementation. The dominance of
materialistic outlook in our society is one of the chief attributes of mathematics.

What we enjoy in the arts like Drawing, Painting, Architecture, Music or Dance etc. is all due to
mathematics. Mathematical regularity, symmetry, order and arrangement play a leading part in
beautifying and organizing the work of these arts.

A sound and productive vocational life demands a sound mathematical background.

Thus the study of mathematics prepares the students for a wide variety of vocations.

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 Unit 2: Education Values of Mathematics and Mathematics Curriculum

Mathematics has international value in the sense that it is helpful in creating international
understanding and brotherhood.

A curriculum is more than a collection of activities: it must be coherent, focused on important

mathematics, and well articulated across the grades.

The following principles are used in designing the curriculum:

 Building on knowledge developed at the basic education level.

 Providing a balanced, flexible and diversified curriculum.
 Catering for learner diversity.
 Achieving a balance between breadth and depth.
 Achieving a balance between theoretical and applied learning.
 Fostering lifelong learning skills.
 Promoting positive values and attitudes to learning.

Thus a syllabus is defined as: a concise and condensed written outline of a course of study; a list
of items which have been written down to be taught and learnt from year to year, a blueprint
enabling teachers to carry out a particular part usually a formal part of the curriculum. The
mathematics syllabus contains directions and information on objectives, contents, methods and
evaluation of the mathematics instruction.

There are a number of methods that can be used to motivate learners towards learning of
mathematics. They include the following:
 Learner’s participation in the learning process
 Teaching applications of mathematics
 Home background and parental encouragement

Use of games and mathematical recreations and mathematical clubs formation and running a
mathematical clubs can provide a good forum for motivating learners towards learning of
mathematics.

Further Reading Materials for the Unit:

To complete your study of this unit, you will need to refer to:

 Pamela COWAN(2006): Teaching MATHEMATICS A HANDBOOK FOR PRIMARY AND

SECONDARY SCHOOL TEACHERS, LONDON AND NEW YORK.
 MOE. (2002): Mathematics Syllabi for Grades 9- 12; Mathematics Panel , ICDR.
 Bloom, B.S. (ed) (1956) Taxonomy of Educational Objectives Handbook I Cognitive
Domain. David Mckay Co Inc.
 TGE. (1994a). Education and Training Policy EEP 86. Addis Ababa, Transitional
Government of Ethiopia, April.

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 Unit 3: Instructional Methods and Instructional Resources in the Teaching and Learning of Mathematics

UNIT 3:Instructional Methods and Instructional Resources in

the Teaching and Learning of Mathematics(14 hrs.)

Introduction
This unit has two sections. Section one deals with Instructional methods of mathematics teaching
and learning while the second section is about instructional resources for teaching in
mathematics classrooms.
Instructional method is nothing but a scientific way of presenting the subject, keeping in mind
the psychological and physical requirements of the pupil. Communication of ideas and
development of concepts in a precise manner based on a logical development of subject is the
most important prerequisite in teaching a subject like mathematics. For effective learning of
mathematics, the method has to be as good as the content. It is through method that is possible
to make a subject interesting and useful.
In this unit, you will be exposed to some methods of teaching, which can also be applied in the
teaching of mathematics whenever a particular type is relevant.
In addition, the importance of using instructional aides in the teaching and learning of
mathematics can never be over – emphasized. It can both enhance understanding of
mathematical concepts and also stimulate students’ interest in mathematical learning.
As this unit is central to your future career and that is more of practical exercise, your
independent as well as collaborative learning is mandatory. Hence, you are advised to use and /
or refer to as many materials and resources as possible including websites in the internet in the
areas of this unit. It is also mandatory to complete the assignments and activities mentioned in
this unit as well as those which will be given by your instructor in classroom.

Unit Learning Outcomes

Upon completion of this unit, students will be able to:

 describe teacher-centered andstudent-centered approaches in the teaching and learning

of mathematics;
 Justify how learning styles are important for selection of appropriate methods
of teaching mathematics;
 explain different active learning methods of teaching, as well as the advantages
and disadvantages of using them;
 explain what lecture method of teaching is all about, as well as its advantages
and disadvantages;
 appreciate the use of mathematics laboratory and recreational activities for
teaching mathematics;
 identify instructional materials necessary for teaching and learning
mathematics.

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 Unit 3: Instructional Methods and Instructional Resources in the Teaching and Learning of Mathematics

3.1 Teaching - Learning Methods in Mathematics(7 hrs.)

What is the best method to teach a certain topic? How can we enable children
to learn mathematics?

These are some of the questions for which every teacher wants to find a solution. Teaching
and Learning is not an easy task, it is a complex process. Each pupil is an individual with a
unique personality. Pupils acquire knowledge, skills and attitudes at different times, rates and
ways.
Different methods of teaching mathematics have been proposed by different
educators and the knowledge of these methods may help in working out a better
teaching strategy. It is not appropriate for a teacher to commit to one particular method. A
teacher should adopt a teaching approach after considering the nature of the
children, their interests and maturity and the resources availabl e. Every method has
certain merits and few demerits and it is the work of a teacher to decide which
method is best for the students. Some of the methods of teaching mathematics are as follows:
• Lecture Method • Project Method • Co-operative learning
• Laboratory Method •Discovery Learning • Group Discussion
• Inductive-Deductive Method • Heuristic Method (Inquiry Method)
• Analytical-Synthetic Method • Problem Solving Method • Project Method etc.
All the above mentioned methods may not be equally appropriate and suitable for all levels of
mathematics teaching. The teacher, after knowing about all these methods, their
merits and demerits, should be able to make his/her own method by taking the good qualities of
all the methods. The method finally adopted by the teacher must:
• ensure maximum participation of the child,
• proceed from concrete to abstraction, and
• provide knowledge at the understanding level.

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Activity 3.1.1: (Brian Storming)(take 5 minutes)

Read the following and discuss in groups of two about the finding.
What I hear I forget;
What I hear and see I remember a little;
What I hear, see and discuss I begin to understand;
What I hear, see, discuss and do, I acquire knowledge and skill;
What I teach to another, I master.
In support of this are the findings that we remember:

 20% of what we hear,

 40% of what we see,
 90% of what we see, hear, say and do or what we discover for ourselves.

3.1.1 Learning Styles

How do you learn? Are you the type who needs to arrange all information
clearly in the form of notes? Or Are you the type who needs to make a mental
picture or drawing to help you to learn?

Your learning style is the way you prefer to learn. It doesn't have anything to do with how
intelligent you are or what skills you have learned. It has to do with how your brain works most
efficiently to learn new information. Your learning style has been with you since you were born.

There's no such thing as a "good" learning style or a "bad" learning style. Success comes with
many different learning styles. There is no "right" approach to learning. We all have our own
particular way of learning new information. The important thing is to be aware of the nature of
your learning style. If you are aware of how your brain best learns, you have a better chance of
studying in a way that will pay off when it's time to take that dreaded exam.

A learning style is the way in which students respond to and use different types of stimuli in their
learning. The three most common learning styles are visual, auditory, and kinesthetic.VAK
(Visual, Auditory, and Kinaesthetic) learning style is a simple model and helps the teacher to
assess how important different types of stimuli are to the students. This uses the three main
sensory receivers: Vision, Auditory, Movement (Kinaesthetic) to determine the dominant
learning styles. Students use all the three to receive information, but the dominant style defines
the best way to learn new information.

i. Visual Learners

You learn by seeing and looking. Visual Learners:

 take numerous detailed notes;

 tend to sit in front;
 are usually neat and clean;

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 often close their eyes to visualize or remember something;

 find something to watch if they are bored;
 like to see what they are learning;
 benefit from illustrations and presentations that use color;
 are attracted to written or spoken language rich in imagery;
 prefer stimuli to be isolated from auditory and kinesthetic distraction;
 find passive surroundings ideal.

Learning Strategies for the Visual Learner

To aid recall, make use of "color coding" when studying new information in your textbook or
notes. Using highlighter pens, highlight different kinds of information in contrasting colors.
Write out sentences and phrases that summarize key information obtained from your textbook
and lecture.

Make flashcards of vocabulary words and concepts that need to be memorized. Use highlighter
pens to emphasize key points on the cards. Limit the amount of information per card so your
mind can take a mental "picture" of the information. When learning information presented in
diagrams or illustrations, write out explanations for the information. When learning
mathematical or technical information, write out in sentences and key phrases your
understanding of the material.

When a problem involves a sequence of steps, write out in detail how to do each step. Make use
of computer word processing and copy key information from your notes and textbooks into a
computer. Use the print-outs for visual review. Before an exam, make yourself visual reminders
of information that must be memorized. Make "stick it" notes containing key words and concepts
and place them in highly visible places --on your mirror, notebook, etc.

ii. Auditory Learners

You learn by hearing and listening. Auditory Learners:

 sit where they can hear but needn't pay attention to what is happening in front;
 may not coordinate colors or clothes, but can explain why they are wearing what they are
wearing and why;
 hum or talk to themselves or others when bored;
 acquire knowledge by reading aloud;
 remember by verbalizing lessons to themselves (if they don't have difficulty reading maps
or diagrams or handling conceptual assignments like mathematics).

Strategies for the Auditory Learner:

You learn best when information is presented auditory in an oral language format. In a classroom
setting, you benefit from listening to lecture and participating in group discussions. You also
benefit from obtaining information from audio tape. When trying to remember something, you

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can often "hear" the way someone told you the information, or the way you previously repeated
it out loud. You learn best when interacting with others in a listening/speaking exchange.

Join a study group to assist you in learning course material. Or, work with a "study partners" on
an ongoing basis to review key information and prepare for exams.

When studying by yourself, talk out loud to aid recall. Get yourself in a room where you won't
be bothering anyone and read your notes and textbook out loud.

Use audio tapes such as commercial books on tape to aid recall. Or, create your own audio tapes
by reading notes and textbook information into a tape recorder. When preparing for an exam,
review the tapes on your tape player or on a "Walkman" player whenever you can.

When learning mathematical or technical information, "talk your way" through the new
information. State the problem in your own words. To learn a sequence of steps, write them out
in sentence form and read them out loud.

iii. Kinesthetic Learners

You learn by touching and doing. Kinesthetic Learners

 need to be active and take frequent breaks.

 speak with their hands and with gestures.
 remember what was done, but have difficulty recalling what was said or seen.
 rely on what they can directly experience or perform.
 activities such as cooking, construction, engineering and art help them perceive and learn.
 enjoy field trips and tasks that involve manipulating materials.
 sit near the door or someplace else where they can easily get up and move around.
 are uncomfortable in classrooms where they lack opportunities for hands-on experience.
 communicate by touching and appreciate physically expressed encouragement, such as a
pat on the back.

Strategies for the Kinesthetic Learner:

You learn best when physically engaged in a "hands on" activity. In the classroom, you benefit
from a laboratory setting where you can manipulate materials to learn new information. You
learn best when you can be physically active in the learning environment. You benefit from
instructors who encourage in-class demonstrations, "hands on" student learning experiences, and
field work outside the classroom.

To help you stay focused on class lecture, sit near the front of the room and take notes
throughout the class period. Don't worry about correct spelling or writing in complete sentences.
Jot down key words and draw pictures or make charts to help you remember the information you
are hearing.

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When studying, walk back and forth with textbook, notes, or flashcards in hand and read the
information out loud.

Think of ways to make your learning tangible, i.e. something you can put your hands on. For
example, make a model that illustrates a key concept. Spend extra time in a lab setting to learn an
important procedure. Spend time in the field (e.g. a museum, historical site, or job site) to gain
first-hand experience of your subject matter.

When reviewing new information, copy key points onto a chalkboard, easel board, or other large
writing surface.

Make use of the computer to reinforce learning through the sense of touch. Use word processing
software to write essential information from your notes and textbooks. Use graphics, tables, and
spreadsheets to further organize material that must be learned.

Listen to audio tapes on a Walkman tape player while exercising. Make your own tapes
containing important course information.

Activity 3.1.2: (take 4 minutes)

How can you help students who are Visual, Auditory and Kinesthetic
learners?

Feedback for Activity 3.1.2

1. Visual Learners: are helped by charts, diagrams, demonstrations and other
visual aides and like to learn through reading and writing tasks.
2. Auditory Learners: are helped by a brief explanation of what is coming, and
conclude with a summary of what has been covered.
3. Kinesthetic Learners: do best while touching and moving and helped by
varying activity and keeping them actively involved.

Self Assessment Exercise

1. Review your teaching style in your school. Is it suitable for your Students’
learning styles? Also, is your teaching style appropriate for their level of
thinking? Write a reflective essay.
2. Explain the importance of identifying learning Styles of students in the
teaching learning process.
3. What type of learning style do you have?

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3.1.2 Teacher-Centered versus Student-Centered Approach

Teacher-Centered Approach
This approach includes instruction where the teacher’s role is to present the information that is to
be learned and to direct the learning process of students. In this method the teacher identifies the
lesson objectives and takes the primary responsibility for guiding the instruction by explanation,
demonstration of the information, and modeling and then followed by students practice.
In the teacher - centered instruction methods teachers follow the following procedures:
a) Pre – instructional task: Preparing objectives and performing task analysis.
b) Task during the lesson: Providing objectives, conducting presentations, demonstrations,
providing guided discovery in the form of question and answer interaction and checking
understanding and providing feedback.
c) Post instructional task: Providing independent and extended practice and testing.

The most commonly used teacher-centered methods of teaching mathematics are:

 Lecture methods
 Demonstration methods
 Lecture–discussion (Questioning) Method
 Deductive Methods
 Inductive Methods
During expository methods, the students should be thinking about the spoken information, and it
is difficult for them to do this if they are also making notes about the information. This method
of teaching needs the following aspects:

 Any necessary notes (modules) before or at the start of a lesson;

 Reference to certain text book pages which contain the information;
 Pause every now and again during the exposition so that the students can make notes.

Under which situation the teacher – centered approach is more preferable?

Learner-Centered Approach
You must accept that nobody can learn for anyone else. The person who has to learn must
become aware of what he/she has to learn, how to learn etc. In other words, learners should be
guided to gain knowledge themselves about what they are going to achieve through the teaching
- learning sessions, and how to achieve the target that they have fixed in their mind.
In learner-centered approach of teaching-learning process, the ‘learner’ or ‘child’ and not the
‘teacher’ is the main focus of the educational program. It emphasizes ‘learning’ rather than
‘teaching’.

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The overall goal of education, according to this approach is all round development of the child
and not only that of acquiring knowledge. It is based on the constructivist model in which
students construct rather than receive or assimilate knowledge. Curriculum, according to this
approach, should be based upon needs, interests, aptitudes and abilities of learners at different
levels so that it enables them to acquire the necessary skills, knowledge, attitudes and values for
realizing their full potential.

Do we follow this approach in our classrooms?

Learner -centered approach means planning and transacting curriculum in the classroom in a way
that allows for flexibility in pace and style of learning keeping in mind that children in a
classroom are widely different. It therefore suggests an approach of teaching different from the
existing practice of having uniform curriculum, uniform learning material and activities, uniform
time and uniform instructional and evaluation strategies for the whole class.
Learner-centered approach advocates:

 flexible curriculum;
 varied methods of teaching;
 varied learning experiences/tasks/activities;
 varied learning time;
 varied methods of assessing children’s progress.

To cater to children in a class with varied ability and intelligence levels, varied interests and
attitudes, varied age levels and varied socio-economic background, this approach emphasizes
‘learning by doing’, ‘experiencing’ and ‘active participation’ on the part of the child. The
methodology employed in this approach is largely based on creating a learning environment for
the child through planned activities, which are joyful and involve active thinking/learning from
the child. The learner-centered approach is thus largely guided by the following two basic
principles or assumptions:

 Children create their own knowledge from their experiences and interactions with the world
around them.
 Teachers foster children’s learning and development best by building on the existing
knowledge, abilities, interests, needs, styles of learning and strengths of the children in the
class.

You can see that the learner-centered approach is more effective than the teacher-centered
approach in ensuring effective learning. Let us now discuss in what ways the learner-centered
approach is effective:

 Learner-centered approach encourages individualized learning. It means that each learner

gets an opportunity to participate actively in the learning process.
 Learner-centered approach emphasizes independence in learning. Teacher creates a situation
where most of the learners feel free to react and give responses.

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 Learner-centered approach minimizes discipline problems among learners. In this case,

learners create interest within themselves to learn new things.

Activity 3.1.3: (take 5 minutes)

1. What is the role of teachers in learner-centered approach?

Feedback for Activity 3.1.3 What is the role of teachers in learner-centered

approach?
In the learner-centered approach, your role as a teacher is very crucial and
demanding. You have to motivate the learners and help them develop a habit
of self learning through various planned activities. The emphasis is not on
“teaching” but on “learning”. Hence, the entire teaching-learning process has
to be activity-based. If you are facilitating learning among the learners, then
your task is to increase learners’ participation in the teaching-learning process.
It is found that all learners do not understand a particular concept or idea in a
specific time or in some particular way. A few learners learn easily, once you
teach them a concept, and some learners take more time to learn. This is
because of the different abilities of the learners. But, as a teacher, you can take
certain steps to increase the learners’ participation.
Some such steps can be:

 encourage learners to respond to the questions;

 ask as many questions as possible;
 encourage those learners who hesitate to respond to questions;
 allow learners to repeat the response given by a particular pupil;
 get the response of a pupil corrected by another pupil;
 allow learners to cooperate with each other;
 encourage group activities among the learners; and
 ask learners some thought-provoking questions.

Self Assessment Exercise

1. Write in five lines the concept of ‘learner-centered approach’.

2. Write some of the characteristics of teacher – centered and student –
centered methods of teaching.
3. Describe five main characteristics of the learner-centered approach.
4. Can you conclude that the learner-centered approach is more effective
than the teacher-centered approach? Why?

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3.1.3Some Specific Active Learning Methods

3.1.3.1 Discovery Learning

Discovery learning refers to those situations in which the learner achieves the instructional
objectives with little or no guidance from the teacher. There are two varieties of this method
namely: (i) the guided discovery and (ii) the open discovery

(i) Guided Discovery

Under this approach, the teacher assists the learners either through the art of questioning or by
explaining to them what they are to do or in the recall and/or the application of relevant
principles; the teacher then allows them to work on their own, carrying out the activities
designed for the period. With the appropriate guidance provided by the teacher, the learners
usually discover the concept to be learnt; or the generalization to arrive at. We should be clear
here that the teacher guides the students and does not simply provide the solution to them.
Not all pupils find it easy to ‘discover’ under all circumstances and this may lead to frustration
and lack of interest in the activity. To avoid this, it may be necessary to have cards available with
additional clues. These clues will assist the pupils, through guidance, to discover the rule or
generalization.

(ii) Open Discovery

It is an unguided discovery learning situation in which the learners are given free hand to interact
with the learning materials and then come up with whatever discovery they can make. The
teacher gives the learner neither the guidance nor the solution to a problem. By implication, the
learners themselves arrive at the various possibilities independently, by applying their skills, and
then find out the solution. Using this method, learners can come up with discoveries that may be
new to the teacher him/herself, owing to the fact that it waste a lot of time.

Advantages of Guided Discovery

The application of the discovery technique as a method of learning leads to individual
capabilities of various kinds.
(i) It increases intellectual capability (the way a learner acquires knowledge helps him/her
solve any kind of problem in a teaching learning situation);
(ii) It increases learners' intrinsic motivation (the learner carries out the learning activities
independently);
(iii) It increases the learner's skills of observation, investigation, problem solving and
independent learning (The learner's competency of inquiry, is thus, sharpened
tremendously);
(iv) It increases the retention of learning among learners (Because discovery learning involves
active participation on the part of the learner it helps him/her develop capacity of his/her
storage system of information);
(v) It makes learners to participate actively in the lesson – they are the major actors and
actresses, the teachers merely guide;

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(vi) Learners will also be able to transfer the knowledge gained;

(vii) Since learners work on their own to discover knowledge, it will help them to develop
ability and courage to solve other life problems.

Disadvantages of Guided Discovery

The main limitations of guided discovery method are as follows:
(i) Generally discovery method of teaching wastes time, for this reason the teacher should be
well prepared to give hints and direct students when appropriate in order to reduce the
time spent;
(ii) Under this method, the classroom tends to be noisy as learners interact with materials and
themselves – the teacher should use his/her expertise to check this;
(iii) Even with guidance offered by the teacher, not every learner succeeds in making the
expected discovery. However, when majority of the learners have succeeded in their
discovery, the teacher can use their own discovery to bring others to the results;
(iv) It demands a fair amount of expertise from the teacher. Requires technical expertise (i.e.
how best to organize or present the subject) and a good knowledge of the pupils (i.e. how
much help/guidance should be given.)
(v) It demands a high level of intellectual maturity in the learner in terms of ability to inquire,
analyze, and apply concepts and to generalize.

3.1.3.2 Questioning and Answering Method

This is one of the most common methods employed at the secondary school level. Pattern
typically begins with the teacher asking a question, and then recognizing one student who
answers. Next the teacher reacts verbally in some way to the student’s response and asks a
question to other students, who then responds, and so forth.
Many teachers believe questioning and answering is a method that enables the teacher to find out
who knows what. It is extremely valuable as a way to guide development thinking, to stimulate
creative problem solving, to initiate discussions and to stimulate quick recall of requisites needed
for the day’s lesson. The questioning and answering method can be used effectively in
combination with every other method. The kind of question posed and the variety of ways used
to encourage and accept responses are all skills that make the differences between thoughtful
interaction and dull sequences.
There are three major components of questioning and answering that needs special attention:

A. The Question:
1. Write down the major question in a developmental sequence and analyze the possible
responses ahead of time;
2. Precede a question and answer sequence by a brief lecture, or demonstration designed to set
the stage for the sequence;
3. Do not ask frequent yes-no questions or fill in the blank question;
4. Increase the number of question requiring a phrase or a sentence in response;

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5. Do not try to elicit development thinking by the all-encompassing “what about” question,
such as “what about the circle?”
6. Use a variety of opening question- phrases, such as “How?” “What” “why?”

B. Getting Responses from Students:

1. Pose the question before you call on someone;
2. Do not call on students in only one area of the room for all answers;
3. Ask slower students low-level question;
4. Save high-level questions for brighter students;
5. Do not direct a series of quick questions to students row by row (or in any clear pattern);
6. Wait at least five seconds prior to accepting responses to high level question inform the
students you are going to do this;
7. Tell the students that there is no penalty for incorrect or partially correct answer. Tell them
it is not a quiz, but a learning experience.

C. Handling Student Responses:

1. Ask another student to agree or disagree and give his/her reasons for doing so;
2. Take “straw vote” and follow up with a request for justification;
3. Frown a bit and ask, “Are you sure?”
4. Ask another student to add to the answer of the first student;
5. Ask the student to explain how he/she arrived at the solution;
6. Ask if there is another way to solve the problem;
7. Do not accept mixed chorus responses;
8. If a student cannot answer a difficult question, ask a contingency backup question on a
lower level;
9. Refuse to accept responses that are not audible to all students;
10. Give praise for partially correct responses to complicated questions.

The physical location of the teacher greatly influences the nature of verbal interaction in the
class.
3.1.3.3 Group Discussion Method

In group discussion method, Students work in small groups (4-6) and teachers encourage them to
discuss and solve problems in groups. Each group is accountable for management of time and
resources both as individuals and as a group. The teacher moves from group to group giving
assistance and encouragement, ask thoughts provoking questions as the need arises. Teacher’s
role is changed from leader to facilitator and initiator. Group work is visually reported to the
entire class and further discussion ensues.
This method allows pupils to work together as a team fostering co-operation rather than
competition. It also provides students discussion, social interaction and problem solving abilities
for students.

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Advantages and Disadvantages of Group Discussion Method

Listed below are some advantages and disadvantages of group discussion method.

(a) Advantages of Group Discussion Method

(i) As learners work together, their interaction and inter communication within the group
fosters social relationship which in itself is an achievement of major educational objectives;
(ii) There is a development of healthy competition as the different groups work together and
share their interests;
(iii) In some kind of class work where learners work together to discover a formula or
generalization, each group produces a pool of thinking and this can lead to making more
discoveries in mathematical learning;
(iv) The capable learners that lead the groups also have additional benefits. As they lead the
groups, they learn mathematics, learn how to communicate and also gain experience in
leadership roles;
(v) Pupils learn to accept responsibility for their own learning (autonomy);
(vi) Reinforces understanding –each pupil can explain to other group members;
(vii) Implies change in teachers role from leader to facilitator and initiator.

(b) Disadvantages of Group Discussion Method

(i) As learners work in group, their discussions and sharing of ideas may make the
classroom more noisy than usual;
(ii) Also because of the nature of work in the groups, class control may not be as easy as
during formal full-class lessons;
(iii) Requires more careful organization and management skills from the teacher;
(iv) Demands careful pre-planning and investment of time and resources in preparing
materials.

Activity 3.1.4: (take 5 minutes)

What do you suggest for effective use of group work?

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Feedback for Activity 3.1.4

(i) As far as possible, the group should be homogeneous in level of

intelligence and level of achievement;
(ii) The group should neither be too large nor be too small. It should have
members from 3 to 7 but there cannot be any rigidity about it;
(iii) A single student should not be encouraged to monopolize the group work.
Teacher should ensure that every member of the group share the activity;
(iv) The purpose of group work should be adequately clear;
(v) The teacher should encourage healthy group competition but should guard
against mutual jealousies or differences.

3.1.3.4 Laboratory Method

It is based on the principles “learning by doing”, “learning by observation” and “proceed

from concrete to abstract”. It is a more elaborated and practical form of the Inductive method.
Pupils do not only listen for information, but do something practical also. Principles have to be
discovered, generalized and established. The method, if properly used, should help in the
removal of the abstract nature of mathematics. It makes the subject interesting as it combines
play and activity.
Kinesthetic learners excel in environments where they can physically manipulate objects. We
incorporate kinesthetic learning into mathematics instruction by holding mathematics laboratory.
Within the laboratory setting, we give students tangible objects to work through mathematics
equations and test theories. Teacher creativity is vital to teaching mathematics via the laboratory
method, as you will be required to develop projects for the students to take on. Some examples
may include using tiles for basic addition or using toothpicks for examining the principles of
geometry.
The construction work in geometry is on the whole a laboratory work. The drawing of a line,
construction of an angle, construction of a triangle or a quadrilateral or parallelogram etc., all
involve the use of some equipment and therefore their nature is that of practical or laboratory
work. There can be many more illustrations to explain the procedure. For example, suppose
Pythagoras theorem is required to be proved.
AB2 = AC2 + BC2
In the laboratory we can prove by cutting squares from cardboard and then weight by finding
actual size with ruler or by measuring tape.

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Merits of Laboratory Method

1. It is interesting and joyful for the learner. A successful laboratory work is a source of joy
and encouragement to the learner;
2. It is very useful for children because they love to do work with their hands;
3. It produces results more lasting and potent than produced by lecture method;
4. It is a natural way of making discoveries;
5. The learner acquires a clear understanding of the subject. He/she finds or discovers facts
with his/her own effort;
6. It gives students attain better mastery of concepts and principles by deriving them from
concrete experiences and these become more functional and meaningful when they are
seen in relation to actual applications;
7. It provides great scope for independent work and individual development;
8. It helps in the growth of self-reliance;
9. It inculcates the spirit of cooperation and exchange of ideas when the students are required
to perform laboratory work in groups;
10. Shyness of hands is removed, as the learner has to handle apparatus and materials;
11. The application of mathematics becomes increasingly evident to the learner. Thus the
subject becomes functional and meaningful to the learner;
12. To a certain extent, laboratory work imitates conditions of real life.

Laboratory method on the whole is highly useful and plays a formidable role in teaching
mathematics effectively.

Drawbacks of Laboratory Method

1. It is very expensive. Every school cannot afford to spend a large amount of money on
laboratory equipment;
2. It acquaints the students with facts and not with mathematical reasoning;
3. All the topics of mathematics cannot exclusively be taught by this method;
4. It needs thorough planning and supervision, otherwise students may just play with
instruments without deriving any substantial gain. Since teacher will be required to play
individual attention, it may not be practicable in large classes;
5. It is an exceedingly laborious and slow method;
6. It is not at all easy to make the students discover mathematical facts experimentally,
especially on lower classes;
7. The tendency of cooking up results or coping may develop among them, as it is not easy
to check;
8. It requires a good supply of materials and suitable designed classrooms;
9. Demands a fair amount of teacher preparation and creativeness.

3.1.3.5 Demonstration Method

Demonstration is a method used mainly to develop the psychomotor and manipulation skills of
the students. Unlike most of the lecture sessions it is possible that the students get chances to
actually participate in the practical sessions.

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It is essential, therefore, that the teacher does not only treat the elements of the theme at the
theoretical level, but also designs practical work. It is always helpful to stipulate the objectives of
such demonstrations in advance for informing both the students and the teacher-demonstrator of
the expected outcomes.
Though the demonstrations activities may be well planned, they often fail to achieve educational
objectives beyond the psychomotor domain. And so it is essential to evolve means by which we
can make the practical sessions more individualized and student centered. What is obvious here
is that the teacher plays a dominant role in the teaching-learning process. Demonstration is used
in combination with lecture, questioning and answering, and laboratory methods.
To focus students’ attention on a demonstration, you must first be sure that the object can be
seen, the specimen must be large enough to be visible from the rear, or the teacher must move
about the room with a smaller object, White cardboard sheets or the over-head screen can
provide an effective backdrop.

Requirements of good Demonstration:

The success of any demonstration depends on the following points.
1. It should be planned and rehearsed by the teacher before hand;
2. The apparatus used for demonstration should be big enough to be seen by the whole class.
If the class is disciplined, the teacher may allow them to sit on the benches to enable them a
better view;
3. Adequate lighting arrangements be made on demonstration table and a proper background
table need to be provided;
4. All the pieces of apparatus are placed in order before starting the demonstration. The
apparatus likely to be used should be placed in the left hand side of the table and it should
be arranged in the same order in which it is likely to be used;
5. Before actually starting the demonstration a clear statement about the purpose of
demonstration be made to the students;
6. The teacher makes sure that the demonstration lecture method leads to active participation
of the students in the process of teaching;
7. The demonstration should be quick and slick and should not appear to remain on
unnecessarily;
8. The demonstration should be interesting so that it captures the attention of the students;
9. It would be better if the teacher demonstrates with materials or things the children handle in
everyday life;
10. For active participation of students the teacher may call individual students in turn to help
him/her in demonstration;
11. The teacher should write the summary of the principles arrived at because of demonstration
on the blackboard. The black board can also be used for drawing the necessary diagrams.

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Self Assessment Exercise

1. Explain the difference between the laboratory method in mathematics teaching and
the one in the basic sciences.
2. What are the advantages and disadvantages of the guided discovery method of
teaching?
3. State the advantages and disadvantages of the teaching methods listed below?
(a) Demonstration method (b) Laboratory method(c) Group Discussion method

3.1.4 Lecture Method

Lecturing has been classified as the most conventional teacher centered method of teaching. It is
the most widely used form of presentation.
In lecturing, the teacher prepares his/her talk at home and pours it out in the class. The students
sit silently, listen attentively and try to catch the point. The teacher may not even write anything
on the black board simultaneously or may not even argue a point with the listeners by cross
questioning. So a lecture is an oral presentation given to a class by the teacher, and its main
purpose is to present a large amount of information in a short period of time.
The lecture method has three phases:
1. Definitions of the terms, expressions or symbols which form the subject of the lecture.
2. Explanations of the definitions just mentioned. Each individual definition must be broken
down into simple components so that the students can understood them.
3. A summary which brings these components together so that the whole subject is
understood by the students.
This method of teaching does not require the opinions of learners. After the teacher has delivered
the lectures, he/she may allow a few questions from learners for him/her to clarify – depending
on the remaining available time.
Lecturing method heavily relies on the teacher as the only source or transmitter of knowledge,
ignoring the role and contributions learners can make.
In the less formal lectures, the class is invited to ask a few questions but these are largely for the
sake of clarification, not of discussion. The essence of this kind of teaching and its purpose are
for a steady transmission of information from the teacher to the students.
Since the study of mathematics should ultimately encourage thinking and reasoning and should
not end with memorizing bits of information through repeated practice or drill, this method is not
suitable. In this method, child’s participation, experimentation and scope for enabling him/her to
discover/explore are totally neglected. This method should be followed with care and for limited
objectives only and when the teacher finds it unavoidable.

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3.1.4.1 Advantages of Lecture Method

(i) A large number of facts can be presented in a short period of time.

(ii) Lectures are used to introduce new topics, summarizing ideas, showing
relationships between theory and practice, reemphasizing main points, etc;
(iii) When the number of students in a class is very large. The teacher’s voice is heard
clearly even in farthest corner of the class room. All the students are provided with an
equal opportunity to listen and learn;
(iv) When heavy syllabus is to be covered in a short time. The teacher can teach the topic at
his/her own speed. He/she need not adjust his/her speed to the learning speed of the
students;
(v) It can be an effective means of providing new information and clarifying existing
information to a large heterogeneous group in a short period of time;
(vi) It can be a good means to set the stage and lay the necessary groundwork
and parameters for a subsequent activity;
(vii) It may be recorded for future use;
(viii) It gives students the information not elsewhere available. This is especially true when
the lectures are based on the unpublished research projects and on the crystallized
wisdom out of the life-long academic pursuits of the instructor;
(ix) It affords opportunities for an instructor to explain a particularly equivocal ambiguous
point of idea, or a complicated, difficult, abstract process or operation ; thus
unnecessary obstacles to learning are removed;

3.1.4.2 Disadvantages of Lecture Method

(i) Learners are exposed to only the teacher’s interpretation of content to be learnt;
(ii) Sometimes, learners do not have opportunity to ask questions, especially during a rush
to cover the content to be learnt;
(iii) It is a difficult method to use when individual differences need to be considered;
(iv) It requires longer interest span than most children possess unless the lecture is kept
short;
(v) It does not help most pupils remember much of what they have heard unless they take
good notes, which most do not;
(vi) It gives the students no opportunity to express their reactions and is therefore less
"democratic" than other procedures in teaching. This lack of class participation
dampens the learner's motivation to learn and impedes learning progress;
(vii) It promotes the authoritarian role of instruction and minimizes the importance of
student's spirit of curiosity and scientific inquisitiveness. It discourages critical thinking
and initiative. The result might turn the learner into a passive, apathetic individual;
being satisfied to do minimal work necessary for passing the course;
(viii) It tends to widen the gap between the instructor and the students by setting them apart
and on different levels in the classroom;
(ix) It bores the students, especially when the instructor has a hypnotic, monotonous voice
which lulls the class into sleep;
(x) Requires the teacher to have effective speaking skills.

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Activity 3.1.5: (take 8 minutes)

1. Which points are to be considered while preparing a good lecture?

2. When should we use the lecture method?

Feedback for Activity 3.1.5

1. In order to prepare a good lecture, many points should be taken into

consideration like:
 Lecture must be well organized and short,
 It should proceed from the simple to the complex,
 Good grammar should be used along with words that pupils can
understand,
 Capture the attention of the class right at the beginning,
 Every general statement or fact used must be accompanied by an
example,
 In order to add variety to a lecture, a teacher can use the chalkboard,
illustrations, or many of the visual aides,
 Do not read word-for-word from notes,
 Duplicated copies can be helpful to give to students,
 Talk to the entire class, not just to one or two students.

Thus lectures are often interspersed by brief discussion periods or followed by

such innovations as buzz groups or brainstorming sessions.

2. i) To introduce a new topic, summarizing ideas, showing

relationships between theory and practice, reemphasizing main points,
etc.;
ii) when the lectures are based on the unpublished research projects;

Self Assessment Exercise

1. Discuss with your partner about the three phases of the lecture method of
teaching mathematics.
2. Which of the following teaching methods will you recommend for the
teaching of mathematics in secondary schools? Why?
(a) Lecture method (b) Demonstration method
(c) Questioning and Answering method(d) Discovery method
(e) Group Discussion method (f) Laboratory method

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3.2 Instructional Resources in the Teaching and Learning of

Mathematics(7 hrs.)
What are instructional resources? Where do we get instructional resources?
How do we use in the teaching-learning of mathematics?

Teaching is a complex activity in which an attempt is made to ensure learning among pupils.
Pupils learn things through their senses. Teachers can also find means to have effective
interaction with pupils’ senses in order to help them learn better. The materials which
supplement teachers’ efforts and facilitate teaching-learning are known as teaching materials.
Teaching aides are the materials used for effective teaching and enhancing the learning of
students. It can be anything ready-made or made by the teacher or made by students. Different
teaching aides should be used in teaching mathematics like charts, models, computers, audio,
visual and audio-visual materials. Such materials, if presented properly at the appropriate time,
may prove to be of great help in clarifying difficult concepts, making abstract things
understandable and arousing and maintaining pupils interest in mathematical learning. It is not
always desirable to employ very costly things as teaching aides. There are many materials
available in our immediate environment which may be either used as such or in a modified form
to generate low-cost or no-cost teaching materials.
In the present section an attempt has been made to clarify the role and importance of teaching
materials. Various illustrations are provided with a view to giving you an idea of how to develop
no-cost or low-cost teaching aides and their judicious use in classrooms for enhancing learners’
competence in mathematics.
3.2.1 Advantages of using Instructional Aides in the Mathematics Classroom

Instructional aides are learning resources used by the teacher or learners or both in classroom
teaching to enhance understanding. They therefore help to make teaching and learning more
effective. Using appropriate instructional aide will go a long way to helping learners understand
the abstraction of mathematics and help improve their attitude toward learning the subject.
The importance of using instructional aides in the teaching and learning of mathematics can
never be over – emphasized. It can both enhance understanding of mathematical concepts and
also stimulate students’ interest in mathematical learning.
Mathematics teachers should see the use of instructional aides as a matter of necessity in their
teaching, especially at the primary and secondary levels.
Mathematics has to do with development of concepts and skills; these concepts are abstraction
from concrete real life situations. When learners are given the opportunity of going through the
process of such abstraction through the use of instructional aide, it will help them to both
appreciate and understand the concepts. Piaget, a psychologist, who worked on the cognitive
development of children proposes that in the course of learning, learners should be allowed to
proceed from concrete through semi concrete and finally to the abstract stage. This proposition is
particularly relevant in the mathematics classroom. In the teaching of any concept, the teacher
should provide opportunity for learners to interact with the concrete situation or material,

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investigations or experiment, from this stage to the semi – concrete and finally to the abstraction
stage. This procedure can be followed only through the use of instructional aides.
Among other things, instructional aides serve the following purposes in a mathematics
classroom.
(i) They make lessons clear and practical to learners;
(ii) They help learners to retain the knowledge gained from the lesson;
(iii) Curiosity and interest to learn are evoked in learners as they interact with the teaching
aides;
(iv) Learners gain firsthand experience which is not possible with ordinary chalk and talk
approach;
(v) Learners are helped to discover mathematical principles by themselves;
(vi) Mathematical lessons become more interesting and more meaningful to learners;
(vii) Learners find it easier to understand concept taught to them especially when concrete
apparatus are used;
(viii) They provide opportunity for activity –learning, i.e. learning by doing;
(ix) They help learners to recall what they have been taught more easily;
(x) Teachers are also relieved of the problem of making long explanations;
(xi) As learners work in groups as they use the apparatus, the spirit of cooperation is
developed among them.

Self Assessment Exercise

Mention and discuss the advantages and disadvantages (if any) of using
instructional aides in the teaching and learning of mathematics.

3.2.2 Some Specific Types of Instructional Aides

Mathematics instructional aides that can be used in mathematics classroom are many and varied
in types. Some of them include – real or concrete objects, models, pictorial aides, charts, chalk
board, flannel boards, calculator or computer, audio-visual aides, mathematical games and
mathematics laboratory. You will become more intimate with them as we examine some of them.
3.2.2.1 Real Life Objects

These are real objects or concrete objects that can be used in the course of teaching or learning.
They are the most useful and most effective means of providing direct experiences to the pupils.
For example, for primary mathematics topics like counting, and basic number works like
addition, subtraction multiplication, division etc. real materials like seeds, bottle tops, square
blocks, set of sticks, beads etc. may be used as counters. In teaching money arithmetic, actual
currency may be used at the initial stages to illustrate buying and selling at the classroom corner
shops. When teaching weight and measures, actual objects like tables, length & breath of
classroom etc. can be measured; books and other real life objects can be weighed. All such
objects that are used are instructional aides in real or concrete forms.

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3.2.2.2 Models

Mathematical models are concrete materials which are representations of the objects to be learnt
about and are utilized by teachers and students to demonstrate mathematical concepts. They are
usually three dimensional i.e.having length, breadth, and height /depth. Examples of such models
include cube, cuboids, dieses block etc.
Some models are made of opaque materials such as cones, cylinders, cubes etc. while others are
made of transparent materials such as glass, water proof etc. Models are very useful aides
because they can be easily handled and manipulated. The pupils can be encouraged to make
certain models themselves.
So models are used to:
(i) provide a concrete visualization of the real object;
(ii) aide, simplify and clarify the description of mathematical ideas;
(iii) make the abstracted ideas clear and facilitate creative thinking.

3.2.2.3 Charts, Graphs, Maps, Pictures, and Diagrams

Charts
A chart as an instructional aide is a sheet of paper containing information in the form of curves,
diagrams, maps, graphs, equations etc., which guides and trains students for accurate work. It
can be used to display formulae, symbols, mathematical and geometrical figures. Charts can be
used for making students familiar to the symbols and for memorization of basic formulae. Even
it can be used to bring to the students two-dimension geometry and the graphical representation
in a better way.
In teaching of mathematics charts are an inevitable material aide, charts can be used almost in
every topic. Charts help in saving time because instead of drawing them on the black-board, the
teacher can depend upon the pre-drawn diagrams. More over it is not always possible to draw a
diagram on black-board with accuracy. The charts, as far as possible, should be accurate,
interesting and good looking. If the charts do not fulfill these requirements, they shall not be
useful for the class-room teaching.

Flow charts
A flow chart is a diagram showing a series of step-by-step operations which make up a particular
process. The main elements of the process are shown in picture form and are linked by arrows to
indicate how one operation leads to the next. A flow chart can, for example, be used to show
stages in the flow of money or probability.

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Diagrams
Diagrams are employed by teachers in a variety of situations. They may be used to illustrate
properties of lines and shapes. They can show something complex like a scale drawing. The best
diagrams are clear, with all the necessary details, and labels to identify features and explain
processes.

Activity 3.2.1: (take 5 minutes)

What do you suggest for effective use of charts?

Feedback for Activity 3.2.1

Following points should be kept in mind while using charts as teaching aides.
i) The chart should be neatly prepared. Important results or figures should be
systematically drawn;
ii) The size of a diagram, figure or shape should be appropriate. It should
neither be too big nor too small;
iii) To draw attention to some specific points different colors can be used while
making a chart;
iv) Charts should be accurate;
v) It is desirable that the students are encouraged to prepare simple charts.

3.2.2.4 Instructional Games and Simulations

Instructional games and simulations are not purely recreational activities that provide only
exercise or just fill time. They are designed to help students to learn, to achieve specific goals or
objectives, in an active rather than a passive climate. There are many games which are
educational. However, we will concentrate on those games, simulations and related activities that
are especially suited to classroom and other in school use.

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An Instructional Game is a structural activity with set of rules for play in which two or more
students interact to reach clearly designated instructional objectives. Competition and chance are
generally factors in the instruction and usually there is a winner. Although games are valuable as
instructional activities, they don’t necessarily attempt to imitate real life situations.
A simulation is in many respects a model of the real world. Simulation participants are assigned
specific roles; they make decision and solve problems according to specified conditions. As with
instructional games, simulations also have instructional objectives. But a simulation is usually
more loosely structured than a game, with a simulation there is no winner as such but merely a
changed condition or situation to be achieved by participants.
Playing games is considered to be one of the natural ways through which children acquire
knowledge. Therefore, a mathematics teacher can also use games so as to facilitate the teaching
learning process. Whenever the teaching is associated with games, the learning process becomes
more interesting and enjoyable to the students.
Games stimulate students’ interests in mathematics and they encourage thinking and creativity.
Through playing them, students develop positive attitude toward mathematics and they are
intrinsically motivated. Games help students stay alert and active in the mathematics classroom
while connecting the material with a fun experience.

Activity 3.2.2: (take 6 minutes)

What do you suggest for effective use of Games and Simulations in Class rooms?

Feedback for Activity 3.2.2

If a teacher plans to use games and simulations for instruction, he/she may need
to consider the following suggestions:

 Decide whether a particular game will suit his/her purposes and help
his/her students.
 Find ways to adapt games and simulations to the capabilities and interest of
his/her students as well as to the physical environment in which he/she uses
them.
 Determine, in advance, whether to involve all or only part of the class in a
game.
 Begin the use of games or simulations with simple but interesting
activities.
 Reserve time for post game discussions and evaluations of game

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3.2.2.5 Textbooks and Workbooks

The text book is an instructional aide that is most frequently used by teachers.
A good mathematics text book will offer tremendous help to mathematics teachers whose
background and methodology is not adequate. Most textbooks provide most of the content for a
course, they also provide exercises for practice, and it can be used for independent study by
learners too. In selecting the mathematics text to use, it must be checked to be adequate in much
respect, some of which include the following:
(i) The topics must conform to the current curriculum being emphasized by the school;
(ii) It must be appropriate in terms of interest, correctness, difficulty as well as its usefulness
for learners offering the course;
(iii) The mathematical computation of worked examples must be correct straight forward and
easy to understand;
(iv) The language must be simple and comprehensible.

Self Assessment Exercise

Mention and describe five types of instructional aides that can be used in the
teaching and learning of mathematics.

3.2.3 Creation of Games and Simulations

Should a teacher and his/her students create their own games and simulations?

The answer to this question is positive but creating games and simulations may not be an easy
task. We may start to use commercially prepared games and then create simple games or
simulations. Students should involve in such activities, so that students will learn not only
through playing games but also in developing them.
Following step-by-step sequence of events should help the teacher and/or students with the
creation process:
 Define learning objectives.
 What will students be able to do after having played the game that they were unable to do
before?
 Set Parameters.
 What is the time scheme within the game itself?
 For how long does he/she want his/her students to be occupied in playing it?
 Identify players and their goals.
 What role will each participant play?
 What will each seek to accomplish?
 Identify and specify resources to be used.
 With what will each player work?
 Will he/she use dice, cards, spinner, or some other means to control the action?

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 Identify the principal rules of play and determine how players will interact.
 What will determine the sequence of events?
 What does each player do or not do to reach specified goals?
 Decide the direction in which players will move. Will players meet obstacles in route? If
so, what will determine their actions?
 How would bonuses or penalties for chance occurrences be distributed?
 Will specific incentives be provided, either for progress during the game or for the
winners?
 Establish how and when the game is to be won.
 Will it be won by a single player or by a team?
 Does winning mean reaching a fixed goal first, or do quality and quantity in meeting
goals determine who wins?
 While problems to be explored through games and simulations may be simplified to suit
the capabilities of groups who use them, they should retain as much reality as possible.
 List and describe materials and arrangements required for play.
 Are required materials readily available, or obtainable at reasonable cost?
 Must materials be constructed?
 Are the appropriate facilities available to play the game?
 Undertake a trial run.
 Try the game with a typical group of players; workout any problems. This step may save
time, inconvenience, and later frustration for the teacher and his/her students.
 Develop a post-game evaluation plan.
 These will assist players in assessing what they have learnt from the experience and how
their future performance in playing it, and even the game itself, might be improved.

Self Assessment Exercise

Mention the sequence of events that can help the teacher in the game creation
process and discuss with your partner.

3.2.4 Criteria for Choosing and Using Instructional Materials

Choosing Instructional Materials

If instructional aides are to serve the purposes for which they are engaged; they must be chosen
carefully.
(i) The aides should serve some useful purpose. Aides should not be used just for the sake
of using an aide. Rather it should help in teaching a particular lesson. The teacher should
be clear about the purpose for which he/she is using the aide.
(ii) The aide should be selected according to the general interests, abilities of the pupils.
(iii) The size of the aide should be neither too large nor too small. It should be clearly visible
to the students.
(iv) Instructional aides should be relevant to the lessons they are used to illustrate.

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(v) They should be easy to understand, and must be suitable for the age and experience of
the learners.
(vi) They should not be too costly or elaborate but they must be good enough to worth the
time expense and efforts sacrificed to obtain them.
(vii) Whenever possible, local materials should be used in making them to reduce their cost.
(viii) They should be neat, brightly colored and attractive especially when they are to be used
for learners in lower classes.
(ix) If the aides are charts, they should be well labeled, with clear bold lettering. It must have
a heading, written clearly so that pupils at the back can see it without problems.
(x) An instructional aide should not be too complex.
(xi) In mathematics, the diagrams, concepts, figures, formulae etc. depicted through the aides
must be accurate. Accuracy is very important.

Making use of Instructional Materials

Teaching aides serve their purposes best only when they are highly used. Following suggestions
be kept in mind while making use of these aides:
(i) Aides should be used only at the right moment. If it is meant for introduction of a lesson, it
should be shown at the proper time. Aide used at the wrong time may prove harmful.
(ii) If a number of aides are to be used, then every aide must be used at the proper time and not
in a haphazard manner. The systematic display gives good results.
(iii) The aide should be kept before the students as long as it serves some purpose. It should be
removed when it has served its purpose.
(iv) While using aides the teacher should always bear in mind that teaching aides are only
means and not an end.
(v) If the actual objects needed, are not available, then the aides should be a true picture of
what they represent.
(vi) Whenever possible, each learner or groups of learner should be given the specimen, if this
is not possible, their sizes must be adequate for learners to see from a distance.
(vii) The most important function of instructional aide is to contribute to the understanding of
the lessons they are used to teach; the teacher must ensure that this purpose is achieved.
(viii) If the instructional aide is a gadget that requires operation, the teacher must posses the
ability to operate, use and explain such a gadget; otherwise he/she must bring someone who
can operate to assist.
(ix) The facilities or materials needed to operate or use the aide must be available. For example,
if electricity is required for its operation, then light must be available in the premises; or if
a video or CD is to be watched, then a television set or CD player must be available.

Factors to be considered in the use of instructional aides

In considering the type of instructional aide to be used for a particular lesson the following need
to be born in mind.

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(a) The instructional aide should not replace the teacher. It is supposed to aide the teacher in
his/her effort to help learners understand the lesson. Therefore, Instructional aide must not
be made to play the role of a teacher.
(b) In all teaching endeavors, there are always objectives to be achieved; the teacher should be
guided by the objective of the mathematical lesson in selecting the Instructional aides to
use.
(c) Instructional aides selected for use must help both the fast and slow learners i.e. it must
cater for individual differences.
(d) The teacher should endeavor to vary the instructional aides he/she uses; it should not be
the same for every lesson, otherwise it will no longer interest the learner.

Self Assessment Exercise

1. Describe the criteria for choosing and using instructional materials.

2. What four factors will you take into consideration in using instructional aides?

3.2.5 Production of Instructional Materials Locally

In order to make the concepts comprehensible to pupils and making teaching learning interesting,
teachers require some materials. It is not advisable to depend on the commercially available
materials. It is the teacher who can develop certain teaching-learning materials as per his/her
needs. Also no foolproof prescription or advice may be offered to the teachers. It is the training,
experience, foresight and creativity of teachers which will help them innovate, redesign or
develop new types of materials for use in their own classrooms.
Whenever possible, teachers should endeavor to produce some of the instructional aides needed
in the mathematics classrooms locally. To do this, the learners too may be involved in the
construction of such materials. Some special period say a free period can be arranged during
which the whole class works on production of particular instructional aides to be used in some
next classes.
Sometimes interdisciplinary arrangement can be made such that learners work on production of
specified instructional aides during the art classes. It can also be given to learners as take home
assignment such that learners produce such from home after all the technicalities involved has
been shown to them at school.
nother major way to produce instructional aides locally is to make all final year learners produce
a specific instructional aide as their project and leave such for the school use; this is particularly
suitable for learners in Colleges of Education.

Activity 3.2.3: (take 6 minutes)

Give three reasons why a mathematics teacher should consider producing

instructional aides locally.

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Feedback for Activity 3.2.3

The effort to produce instructional aides locally is important for a number of

reasons.
(i) It may not be feasible to expect the school authority or the Government
agency concerned to buy all instructional aides that will be needed in the
classroom, the cost will be formidable and mathematical instructional aides
are lacking in most schools in Ethiopia. Hence local production of such
instructional materials may be the way out. This will drastically reduce cost
of procurement of such aides, or may even eliminate money altogether in
some cases.
(ii) Even if funds to buy the materials are available, some important/basic and
relevant aides may not be available in the market.
(iii) Another good reason for local production is that the process of production
itself is very educative. The fact that learners have been part of the
production may help them to understand the concepts faster.

Guidelines in Production Process

This subsection is not about showing how to construct different type of instructional aides, but to
give some general guidelines.
Generally some basic materials that can be used as tools should be available; such includes
cardboards, rulers, colored pencils, letter stencils, blade, gum, knife, scissors, nail, plank or
plywood, etc. Generally too, the type of instructional aide to be produced will determine the
specific type of materials to use. For example, in production of charts, three types of materials
are basically needed, which includes:
(i) Fabrics, i.e. the basic structure that will hold the drawing – this may be cardboard,
newsprint or wrapping paper.
(ii) Lettering materials – this can be ink of various colors, markers.
(iii) Choice of suitable colors.

After production, if the edges are to be protected, it can be binded or if the surface is to be
protected, it can be laminated. In the production of models, the type or make will also determine
the materials to use. For models in which the internal structures are required to be seen,
transparent materials like glass should be used. However, if only the external are required to be
examined, opaque materials may be used to make the models.
Generally however, for any type of instructional aides, it is advisable to use durable materials to
avoid quick damaging so that the items can be used for long. It is also a good idea to consider a
good storage facility where all the instructional aides can be safely kept, and from where they
could be brought out when needed.

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 Unit 3: Instructional Methods and Instructional Resources in the Teaching and Learning of Mathematics

3.2.6 Mathematics Laboratory

Mathematics Laboratory is a place where students can learn and explore mathematical concepts and
verify mathematical facts and theorems through a variety of activities using different materials. These
activities may be carried out by the teacher or the student to explore, to learn, to stimulate interest and
develop favorable attitude towards mathematics.

So a mathematics laboratory is a place where we find a collection of games, puzzles, teaching

aides and other materials for carrying out activities. Although mathematics is not an
experimental science in the way in which physics, chemistry and biology are, a mathematics
laboratory can contribute greatly to the learning of mathematical concepts and skills.

3.2.6.1 Importance of Mathematics Laboratory

The main aim of teaching mathematics is the development of power of abstract thinking and
reasoning. When students handle concrete objects their learning is quicker and understanding is
better. Use of geometrical models helps students in grasping various geometrical facts. If
mathematical facts are verified physically then they could be understood more easily and can be
easily applied in new situations. Furthermore any practical work in mathematics makes the
subject interesting to the students.

Here are some ways we think a mathematics laboratory could contribute to learning
mathematics:

(i) A mathematics laboratory provides an opportunity for the students to discover through
doing. In many of the activities, students learn to deal with problems while doing
concrete activity, which lays down a base for more abstract thinking.
(ii) It gives more scope for individual participation. It encourages students to become
autonomous learners and allows a student to learn at his or her own space.
(iii) It widens the experiential base, and prepares the ground for later learning of new areas in
mathematics and of making appropriate connections.
(iv) In various puzzles and games, the students learn the use of rules and constraints and have
an opportunity to change these rules and constraints. In this process they become aware
of the role that rules and constraints play in mathematical problems.
(v) Because of the larger time available individually to the student and opportunity to repeat
an activity several times, students can revise, resee and rethink the problem and solution.
This helps to develop meta cognitive abilities.
(vi) It builds up interest and confidence in the students in learning and doing mathematics.
(vii) Importantly, it allows variety in school mathematics learning.

In the lower classes, laboratory work helps the students in learning elementary mathematics.
Even the mentally deficiency children can profit from practical work. Moreover, mathematics
laboratory is a rich repository of audio-visual aides that can be used in day to day teaching.

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 Unit 3: Instructional Methods and Instructional Resources in the Teaching and Learning of Mathematics

3.2.6.2 Equipment for Mathematics Laboratory

Mathematics laboratory serves not only as a laboratory but also as mathematics room and
mathematics museum. Keeping in view the various purposes, it should have the following
equipments:
(i) Concrete Materials:
It should be provided with concrete material connected with simple arithmetical topics such as
beads, sticks, pebbles, ball frames, number of cards, seeds, balances, coins, weights, measuring
tapes, didactic, apparatus, scissors, pins, card-board, chart paper, graphs, nails, hammers, ropes
etc. It may also be provided with the following materials.
(a) A number kit: consisting of a set of wooden blocks of different colors, labeling the digits
from one to ten. Each block has the numeral and word representing the numeral.
(b) Place value packets: is a box having three or four portion labeled as Units, Hundreds,
and Thousands. This is used to teach place value of numbers, idea of carrying processes
of addition and subtraction.
(c) Fractional parts: is a series that is divided into halves, thirds, fourth, fifths etc. These are
quite useful to illustrate the concept of fraction, addition, multiplication etc.
(d) Charts: A number of mathematical charts should be kept in the mathematics laboratory.
Charts can be used to explain certain points which otherwise would be difficult to
explain. These charts are drawn on paper with the help of markers as well as pencils.
These charts may be got prepared by the students on different topics such as percentage,
average, fractions, circles, cubes etc. Some set of charts are also available from the
market.
(ii) Pictures and Photographs
The picture and photographs of various mathematicians be prominently displayed in the
mathematics room. It would be much useful if the contributions of these mathematicians are also
indicated on such charts.

(iii) Models
Various mathematical models such as those of triangles, squares, solids etc. be stored in a
mathematics laboratory.

(iv) Black-board and Geometrical instruments

The size of black-board provided in mathematics room should be larger than ordinary size.
Colored chalks, a set of geometrical instruments such as protractors, compass, rulers etc. be kept
in a mathematics laboratory.
In general Bulletin Boards or display boards, proportional dividers, slide rules, calculating
machines, surveying instruments, projective aides, collections of mathematical data, etc. are
materials in mathematics laboratory.

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 Unit 3: Instructional Methods and Instructional Resources in the Teaching and Learning of Mathematics

Self Assessment Exercise

1. What is mathematics Laboratory?

2. Give the list of equipments which you would like to have for the mathematics
laboratory for secondary classes of your school?
3. How will you take steps to organize a laboratory of mathematics in your own
school?
4. Justify the need of a mathematics laboratory for effective teaching of
mathematics?

3.2.7 Mathematics Club and Recreational Activities in Mathematics

For supplementing the teaching of mathematics in the class-room and to widen the knowledge of
students, a good mathematics teacher can involve his/her students in a number of co-curricular
activities such as mathematics club, visits and excursion etc. There is no limit to such
extracurricular activities and teacher is free to undertake one or more of such activities in the
school for the benefit of his/her students.

3.2.7.1 Mathematics Club

As in some other subjects, the students can be encouraged to organize themselves into
mathematics clubs and mathematics associations. Under the auspices of these organizations,
discussions and lectures etc. are arranged. Such functions are quite helpful in creating interest in
mathematics. Under the auspices of such organization certain games based on some concepts of
mathematics and mathematical problems can also be arranged. This is likely to help the students
in having an idea of the practical utility of mathematics in addition to creating their interest in
mathematics.

The values of mathematics clubs may be summarized as under:

(i) It is useful in arousing and maintaining interest in mathematics.
(ii) It stimulates the active participation of the students.
(iii) It develops in the students a habit of selective study. This helps them to make a
distinction between a relevant and irrelevant material.
(iv) The knowledge gained by students in various functions of such club activities
supplements the class teaching.
(v) It provides the students an opportunity of free discussions and they are benefited from
one another’s view.
(vi) Gifted students get an opportunity to satisfy their needs and interests by actively
participating in the activities of such clubs and associations.
(vii) Students get an opportunity to listen to some well known and distinguished
mathematicians.
(viii) It is helpful in making proper utilization of leisure time.
(ix) Through participation in such clubs students get acquainted with the contribution of great
mathematicians in their fields.

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 Unit 3: Instructional Methods and Instructional Resources in the Teaching and Learning of Mathematics

3.2.7.2 Recreational Activities in Mathematics

Mathematics is considered as dry subject and students do not find anything interesting in it. This
impression about mathematics can be reversed with the help of recreational activities. It is also
possible to present a lively and interesting picture of mathematics with the help of these
activities. Recreational activities bring variety and also develop taste for mathematics.

In ancient times, mathematics was mainly studied for its recreational value and as a leisure time
activity. Playing with numbers and other mathematical problems was enjoyed by people. Riddles
in mathematics provided fun and were quite witty. Some examples of such mathematics riddles
are as under:

(i) A clock strikes six in five seconds. How long does it take to strike twelve?
The answer is not 10 sec., but 11 seconds.
(ii) An oil merchant has a tank of oil and he has two measures, one of 3 liters and another of 5
liters. He is to take 7 liters from the tank. How will he do this by using these two measures?

Aesthetic Enjoyment in Mathematics

In mathematics we come across many number games and patterns which give aesthetic
enjoyment.

Examples are:
(i) Magic Square

In this square the digit 1 to 9 are arranged in such a manner that the sum of digits in all directions
is 15.
4 9 2
3 5 7
8 1 6

(ii) Some interesting number patterns.

0*9 + 1 = 1 9*9 + 7 = 88
1*9 + 2 = 11 98*9 + 6 = 888
12*9 + 3 = 111 987*9 + 5 = 8888
123*9 + 4= 111 etc. 9876*9 + 4 = 88888 etc.

The importance of recreational activities can be summarized as under:

(i) Recreational activities bring healthy change in the class atmosphere.

(ii) Recreational activities develop taste for mathematics.
(iii) The students learn to appreciate the power and beauty of mathematics.
(iv) Recreational activities prepare the students for leisure time.
(v) These develop many abstract relationships which would otherwise remain vague.
(vi) They sharpen intelligence and stimulate quick thinking.

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 Unit 3: Instructional Methods and Instructional Resources in the Teaching and Learning of Mathematics

Self Assessment Exercise

1. How can you make teaching of mathematics more interesting and joyful?
2. How can you take care of fast learners, average learners and slow learners in
the classroom?
3. What type of teaching material a teacher can develop for teaching of
mathematics?
4. Highlight eight purposes of Instructional aides in the mathematics classroom.

Unit Summary
All the methods may not be equally appropriate and suitable for all levels of mathematics
teaching. The teacher, after knowing about all the methods of teaching, their merits
and demerits, should be able to make his/her own method by taking the good qualities of all the
methods.

A learning style is the way in which students respond to and use different types of stimuli in their
learning. The three most common learning styles are visual, auditory, and kinesthetic.

In teacher-centered method the teacher identifies the lesson objectives and takes the primary
responsibility for guiding the instruction by explanation, demonstration of the information, and modeling
and then followed by students practice.

In learner-centered approach of teaching-learning process, the ‘learner’ or ‘child’ and not the
‘teacher’ is the main focus of the educational program. It emphasizes ‘learning’ rather than
‘teaching’.

Discovery learning refers to those situations in which the learner achieves the instructional
objectives with little or no guidance from the teacher. There are two varieties of this method
namely: (i) the guided discovery and (ii) the open discovery.
The questions and answer method is extremely valuable as a way to guide development thinking,
to stimulate creative problem solving, to initiate discussions and to stimulate quick recall of
requisites needed for the days lesson.

Group discussion method allows pupils to work together as a team fostering co-operation rather
than competition. It also provides students discussion, social interaction and problem solving
abilities for students.

The Laboratory method, if properly used, should help in the removal of the abstract nature of
mathematics. It makes the subject interesting as it combines play and activity.

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 Unit 3: Instructional Methods and Instructional Resources in the Teaching and Learning of Mathematics

Demonstration is a method used mainly to develop the psychomotor and manipulation skills of
the students. Unlike most of the lecture sessions it is possible that the students get chances to
actually participate in the practical sessions.

Lecturing method heavily relies on the teacher as the only source or transmitter of knowledge,
ignoring the role and contributions learners can make. In the less formal lectures, the class is
invited to ask a few questions but these are largely for the sake of clarification, not of discussion.
The essence of this kind of teaching and its purpose are for a steady transmission of information
from the teacher to the students.
Teaching materials, if presented properly at the appropriate time, may prove to be of great help
in clarifying difficult concepts, making abstract things understandable and arousing and
maintaining pupils interest in mathematical learning. It is not always desirable to employ very
costly things as teaching aides. There are many materials available in our immediate
environment which may be either used as such or in a modified form to generate low-cost or no-
cost teaching materials.
If Instructional aides are to serve the purposes for which they are engaged, they must be chosen
carefully and used highly.
Whenever possible, teachers should endeavor to produce some of the instructional aides needed
in the mathematics classrooms locally. To do this, the learners too may be involved in the
construction of such materials.
For supplementing the teaching of mathematics in the classroom and to widen the knowledge of
students a good mathematics teacher can involve his/her students in a number of co-curricular
activities such as mathematics club, visits and excursion etc.

Further Reading Materials for the Unit:

To complete your study of this unit, you will need to refer to:
 Pamela COWAN(2006): Teaching MATHEMATICS A HANDBOOK FOR
PRIMARY AND SECONDARY SCHOOL TEACHERS, LONDON AND NEW
YORK
 Callahan, F. Joseph and Clard, L.H. (1988): Teaching in the Middle and Secondary
Schools, New York: Macmillan publishing Company.
 Van de Walle, John A. (2004) Elementary and Middle School Mathematics: Teaching
Developmentally (fifth edition). New York, Pearson
 Peter G. Dean., Teaching and learning mathematics, University of London Institution
of Education.

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 UNIT 4:Assessment Methods and Planning in the Teaching and Learning of Mathematics

UNIT 4: Assessment Methods and Planning in the Teaching

and Learning of Mathematics (15 hrs.)
Introduction
This unit has two sections. Section one deals with assessment of mathematics learning while the
second section is about planning for teaching in mathematics classrooms.

Evaluation can be regarded as an integral part of an instructional program of activity regardless

of the length of time or period the instruction lasted. Evaluation is an ongoing process throughout
the period of instruction. It is also the last activity that completes the teaching exercise whether
the teaching lasted for a year, a term or whether it is a teaching unit or single period of
instruction.
In this unit, you will be exposed to evaluation as an integral part of mathematical teaching and
learning, you will learn about its purpose, its importance and the techniques of evaluation in the
mathematics classroom.

In addition, this unit will also provide you with the theoretical as well as practical knowledge and
skills of planning. Furthermore, teaching always takes place after planning for teaching.
Although planning is dynamic, formal teaching is always based on systematic planning and
implementation contextually. You will also be acquainted with basic skills of designing annual
plan, unit plan, and daily lesson plan.

As this unit is central to your future career and that is more of practical exercise, your
independent as well as collaborative learning is mandatory. Hence, you are advised to use and /
or refer to as many materials and resources as possible including websites in the internet in the
areas of this unit. It is also mandatory to complete the assignments and activities mentioned in
this unit as well as those which will be given by your instructor in classroom.

Unit Learning Outcomes

Upon completion of this unit, students will be able to:

 explain in their own words the meaning of evaluation;
 state or describe the purposes of assessment;
 list the techniques of assessment;
 explain the components of annual plan, unit plan , and daily lesson plan for
teaching mathematics;
 relate the meaning and importance of annual plan, unit plan and daily lesson
plan;
 list the main features of daily lesson plan, and write a good lesson plan.

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 UNIT 4:Assessment Methods and Planning in the Teaching and Learning of Mathematics

4.1 Assessment in the Teaching and Learning of Mathematics(7 hrs.)

What is assessment, measurement and evaluation? Why we assess students learning?

Assessment can be regarded as an integral part of an instructional program of activity regardless

of the length of time or period the instruction lasted. Assessment is an ongoing process
throughout the period of instruction. It is also the last activity that completes the teaching
exercise whether the teaching lasted for a year, a term or whether it is teaching a unit or single
period of instruction.

In this section you will be equipped with some knowledge and understanding of this important
educational process, evaluation, as well as you will be made conversant with its indispensable
purposes for educational thought and practices.
As a teacher at any level, this knowledge is basic for you to be able to assess your students’
performance effectively.

Evaluation is a continuous process and can be represented through a triangular diagram.

Objective

Learning Experiences Evaluation Procedure

Teachers assess their students in a variety of ways for a variety of reasons. In today’s
mathematics class, assessment is a key component of daily practice. Not all assessment is in
search of a letter or number grade for the sole purpose of reporting student achievement.
Teachers use formative assessment to determine what their students know already and what they
need to know. Students can also use assessment to become aware of their own progress and areas
of strength. They have access to rubrics and scoring guides that help to shape learning. Teachers
use information about their students’ prior knowledge to help plan relevant lessons. Information
gained from assessment is used to monitor student progress, inform parents of student
achievement, and plan instruction.

4.1.1 Purpose of Assessment in Mathematics

One of the first things to consider when planning for assessment is its purpose. Who will use the
results? For what will they use them? Evaluation should be seen as an essential ingredient for
effective mathematical teaching and learning. Its main purpose is to provide feedback to all
stakeholders in this enterprise. This group includes the teacher, the learner, the parents, and the
school as a system, etc. and also to motivate learning.

1. Decision Making

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 UNIT 4:Assessment Methods and Planning in the Teaching and Learning of Mathematics

At the end of a lesson, course, or unit, the teachers, school administrators and other school
personnel make many decision about the students progress and in addition lead the students’ to
make many decisions for themselves. But it is necessary to realize that decisions are not made on
scanty or irrelevant information about a students’ progress. Hence the school should keep or
have cumulative and considerable record of information about each student.

2. Predictive Measurement

Moreover, evaluation helps for the purpose of prediction. For example questions like: “In what
mathematics class is this student likely to make the best progress” etc. This question for example
demands predictions. Yes he/she is likely to do well in such a class; he/she is not likely to do
well in that class. He/she will not cope with the demand. These predictions now serve as
prerequisites for individual or institutional decisions. But it should be emphasized that the
accuracy of the judgments and inferences you make will only be increased by relevant data.

3. Placement and Promotions

Another important purpose of evaluation is that it helps for placement and promotion. So,
evaluation tests the readiness of a student or child to be placed in a new school or promoted to a
new class as the case may be. Common Entrance Examinations are all evaluative strategies for
placement of pupils or students to new and higher institutions relative to the former. Sessional
examinations in schools are evaluative strategies for the purpose of promotion of students from
one class to a higher class. But better decisions of placement and promotion can only be made
with cumulative considerable and relevant data and not just by one or two single achievement
tests.

4. Guidance and Counseling

Evaluation also helps in guidance and counseling and career choice.

Choosing a career is a type of decision-making which a student has to make by him/herself and
or by parents, based on the predictions they can make or the school helps him/her to make by
making available his/her cumulative records.

5. Assessment of Instructional Strategies

Evaluation helps in the assessment of the teaching methods and materials on how effective they
were in the course of an instruction. Therefore, it helps in remediation purposes. This implies
that if the feedback obtained from evaluation is unsatisfactory there will be the decision of
making up (remedying the situation) what was lost. Remediation will therefore involve re-
teaching and re-planning on the part of the teacher as well as relearning on the part of the learner.

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 UNIT 4:Assessment Methods and Planning in the Teaching and Learning of Mathematics

Self Assessment Exercise

1. Discuss about the following purposes of Assessment.

Assessment provides feedback to:
(i) the teacher and the learners;
(ii) the school and administration;
(iii) parents and guardians.
2. How does evaluation help in decision making? Give three examples of such
possible decisions.

4.1.2 Types of Assessment

There are diagnostic, formative and summative types of assessment.

Diagnostic Assessment: involves the assessment of learners’ previous knowledge before any
unit of instruction. This type of test is used to locate areas not well understood by learners in the
course of teaching, such that when dictated, suitable remedial teaching can be planned and given
by the teacher. Although not frequently used, every mathematics teacher should plan and
administer this type of evaluation.
Formative Assessment: Formative evaluations involve the assessment of learners’ achievement
after a brief unit of instruction. It is the on-going type, which is the type given when the
instruction is on-going; that is, when it is given at the intervals of the on-going course, or the
currently taught course. It also serves the purpose of diagnosing errors in learning that is areas
not understood by learner etc. Unit tests, quizes and assignments are the essential components of
formative assessment.

Summative Assessment: Summative evaluation aims at much more general or large scale
assessment of the extent or degree to which the larger stated aims of instruction have been
attained over the entire course or some substantial aspect of it. Sessional examinations are
examples of summative types.

Summative evaluation is used to determine the end-of-course of a semester or a term, end of a

session or year achievement. The result of this is used to assign grades and determine whether
the objectives identified at the beginning have been achieved. Term tests, annual tests and
external examination conducted by school or public agency are the essential part of summative
evaluation.

Self Assessment Exercise

1. Describe the three types of assessment and give examples for each type.
2. Discuss the difference between formative and summative assessments
with your partner.

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 UNIT 4:Assessment Methods and Planning in the Teaching and Learning of Mathematics

4.1.3 Techniques of Assessing Students’ Learning in Mathematics

The techniques of obtaining information on mathematical attainment of a learner are many and
varied. For assessing mathematical behavior at the intellectual level, or cognitive level, the
following techniques can be used.

(i) Written Test

A test is a tool or measuring instrument for evaluation. It is an examination of mathematical

knowledge of the learner; it can be in form of questions to answer or activities to carry out. The
response of the learner will be assigned a numerical score based on his/her performance in the
test. The numerical score assigned becomes an indicator of how well he/she possesses the
characteristics or quality being measured.

Broadly speaking, there are two types of tests which are the essay and the objective tests; these
will be given detailed treatment in the next subsection.

(ii) Assignment

This is an allocation of a piece of work given to the learner to do within a specified period of
time. The learner is expected to do the pieces of work using the experiences to which he/she has
been exposed by the teacher. Such work is usually individual based and may require efforts and
use of initiative of the learner. It may be paper and pencil work, may involve drawing, sketches
and diagrams, construction of models, use of library, etc. On completion, such assignment will
be graded and is useful in obtaining information on the mathematical achievement of the learner
in the examined area.

(iii) Oral Questions

Another technique of assessment is in the use of oral questions. Under this technique the
mathematics teacher through questioning tries to determine how much his/her learners have
learnt, or to recapitulate or consolidate what has been learnt; to detect areas not well understood
by students in what he/she has taught, etc. Through learners’ responses, the teacher is able to
evaluate the extent of their understanding of the mathematical concepts taught, or if any aspect is
not well understood.

(iv) Learners Self-Evaluation

This technique gives each learner the opportunity to evaluate him/herself by stating how well
he/she has understood the lesson or the course. It is assumed that every learner can evaluate
him/herself to some extent. Therefore, after the mathematics teacher finishes teaching a unit or a
topic in mathematics, learners may be giving the opportunity to evaluate him/her self.

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 UNIT 4:Assessment Methods and Planning in the Teaching and Learning of Mathematics

Self Assessment Exercise

Describe the techniques of assessing students’ learning in mathematics. Which one

is most used in your teaching? Why?

4.1.4 Factors Influencing Assessment Quality

(i) Community, district and school factors:

It includes geographic location, community and school population, socio-economic profile and
race/ethnicity, stability of community, political climate, community support for education, and
other environmental factors.

(ii) Classroom factors:

It includes physical features, availability of technology equipment and resources and the extent
of parental involvement. Some other relevant factors include classroom rules and routines,
grouping patterns, scheduling and classroom arrangement.

(iii) Student characteristics:

It includes age, gender, race/ethnicity, special needs, achievement/developmental levels, culture,

language, interests, learning styles/modalities or students’ skill levels. In addition, it includes
student’s skills and prior learning that may influence the development of your learning goals,
instruction and assessment.

(iv) Instructional implications:

It shows how contextual characteristics of the community, classroom and students have
implications for instructional planning and assessment. Include specific instructional
implications for at least two characteristics and any other factors that will influence how you
plan and implement your unit.

Factors to improve the Validity of Assessments

(i) Motivation:
 Devote to help students learning.
(ii) Trust:
 The teacher is encouraging, constructive, and sensitive to student's feelings.
 Class/peer relationships and attitudes support student's learning.
 Students feel safe to admit difficulties and uncertainties.

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 UNIT 4:Assessment Methods and Planning in the Teaching and Learning of Mathematics

(iii) Criteria:
 Identify and explain well the qualities sought.
 Students understand clearly what is needed.
(iv) Standards:
 Set standards appropriate to students.
 Through descriptions and examples, the standards are explained.
 Students understand the standards and accept them as appropriate.
(v) Self-assessment:
 Help students to develop self-assessment skills.
 Student takes increasing responsibility for his/her own learning.
(vi) Peer involvement:
 Encourage collaboration among students to improve work.
 Peers learn to be constructive and generous in offering feedback.
(vii) Monitoring:
 Monitor student's work to track both process and progress.
(viii) Insight:
 Detect misunderstandings or other obstacles to success.
 Detect exciting possibilities in student's work.

(ix) Timing:
 Feedback is given at times when a student is most receptive to it.
(x) Balance:
 Feedback gives attention to strengths as well as weaknesses.
(xi) Selectivity:
 Feedback addresses mainly the aspects likely to have biggest benefit.
 Feedback is convincing, appreciated, and useful to students.

Self Assessment Exercise

Discuss the factors influencing assessment quality with your partner.

4.1.5 Test Construction and Types of Tests

A test refers to a set of questions or items designed and to be responded to by one or a group of
individuals within a specified period of time. Tests are used mainly as measures of achievement,
to show how much an individual has learnt or not learnt in an aspect of instruction to which
he/she has been exposed. It is therefore a major instrument in the process of evaluation.

Criteria of a Good Test construction

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 UNIT 4:Assessment Methods and Planning in the Teaching and Learning of Mathematics

In test construction, there are certain criteria to be satisfied. There are a number of criteria that
should be met when designing a test.
They include the following:

(1) Validity
This means that the test should measure what it is intended to measure.
For example, if a test is meant to measure mathematical achievement of learners let it measure
achievement. If it is a diagnostic test, then it should be able to reveal weaknesses in mathematics
that should be put right. A single test cannot fulfill all purposes; one should ensure that the right
type of test is used for the right purpose.

(2) Reliability
This means that results obtained in the tests should be consistent all other things being equal; i.e.
a candidate should score approximately the same mark if he/she took the test repeatedly
assuming that he/she does not gain new knowledge from his/her previous attempt of the test.
While this ideal is not obtainable, efforts can be made to curb the factors that spoil the perfect
testing situation. Such factors that can reduce reliability include:

(i) Mental and physical condition of the learner e.g. lack of sleep may affect a learner’s mental
effectiveness.
(ii) the condition under which the test is given, e.g. a test written in an area where civil fight
breaks out and people are under tension will negatively affect performance of candidates.
(iii) Inconsistency in the standard of marking adopted by different examiners (say in a large
scale examination) or by the same examiner on different occasions.
(3) Fairness
The test should be fair even from the point of view of the candidates or outsiders e.g. questions
should not be set outside the syllabus, time allotted for the work should not be too much or too
little etc.
(4) Discrimination

A test should discriminate between the abilities of candidates. A test in which every candidate
score the same mark has not discriminated abilities and would not serve any useful purpose. At
the same time, too wide a spread in marks can be misleading.

(5) Comprehensiveness
A test should cover every aspect of the contents being examined; for example an end-of-course
test should cover every area of the objectives highlighted at the beginning of the course.

(6) Ease of Administration and Scoring

The test or examination should be easy to administer and score. This is particularly important
when the number of candidates is very large. It may not be a problem when candidates are few.

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 UNIT 4:Assessment Methods and Planning in the Teaching and Learning of Mathematics

Classification of Tests

Broadly speaking, there are two types of tests namely the essay type and the objective type. Each
of this will be examined below;
(a) Essay Type Tests
The essay type tests examine the learners’
(i) understanding of the subjects;
(ii) ability to organize his thoughts and demonstrate logical argument;
(iii) ability to think critically;
(iv) ability to apply knowledge to the solution of problems.
In essay type tests, the learner is required to provide fully written answer in response to
questions. Examples are the traditional end-of-course or end-of- session tests or examinations.

Advantages and Limitations of Essay Tests

Essay tests have both advantages and limitations. Its advantages include the following:
(i) Questions are easy to compose.
(ii) They test learners’ ability to recall information rather than merely recognize factual
information.
(iii) They appraise higher-level intellectual abilities like ability to reason and think in
abstraction etc.
(iv) They test learners’ ability to apply knowledge to solution of problems.

Its limitations include the following among others:

(i) Inability to sample representatively;
(ii) The scoring can be subjective; and
(iii) Scoring takes much time.
However, the limitations can be controlled by appropriate measures e.g.
(i) Combining the use of essay and objective Tests for better sampling;
(ii) Preparing detailed marking scheme which should strictly be adhered to;
(iii) Using essay test for only problems that cannot be easily adapted to objective type. For
example, in mathematics, some areas of the subject matter that are better examined by
essay type questions include: Geometrical constructions, Solving word or verbal
problems and Proving Theorems.

(b) Objective Type Tests

These are type of tests in which questions (usually many) have been prepared with a set of
possible solution, and the examinee is required to select the correct one.

Types of Objective Tests

Objective tests can be classified as follows:

(i) The completion (or short answer) items

Here, a question is followed by a blank space in which the examinee is to supply correct answer
that will meaningfully complete the idea or sentence.

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(ii) True/False or alternate choice items

In this type, the question is in form of a statement of facts or explanation and the examinee is
expected to confirm by selecting either true or false; right or wrong or Yes or No.

(iii) The Multiple Choice items

In this type, a question or an incomplete sentence is given – usually called a stem. Then a set of
suggested answers (known as options or alternatives) are given in which only one of them is
correct, the others are distracters.
The examinee is expected to pick the correct answer either by ticking or circling or as may be
contained in the instructions.

(iv) The Matching items

In this type of test a number of items and a number of responses are listed in different orders.
The examinee is expected to match the items correctly.

Activity 4.1.1: (take 4 minutes)

What are the purposes of tests?

Feedback for Activity 4.1.1

Four purposes of tests are listed below:-

(i) It provides feedback to teachers and learners

(ii) It motivates learners to hard work
(iii) It motivates teachers to hard work
(iv) It provides means of measuring teachers’ effectiveness.

Self Assessment Exercise

1. List and discuss the criteria of a good test construction.

2. What are the classifications of Tests? Describe their appropriateness in
Mathematics.
3. Consider the different forms of assessment discussed in section 4.1 to answer the
following questions.
 Write a list of advantages and disadvantages for each form of assessment.
 Identify the most appropriate purpose for which this form of assessment could be
used.

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4.2 Planning for Teaching in Mathematics Classroom (8 hrs.)

What is instructional planning? What is the advantage of instructional planning?

You would agree that success of any activity depends largely upon its planning. Proper planning
of activities leads to fruitful results. The same is true with teaching as well. As a teacher you are
given charge of a certain class to teach mathematics. So you have to cover the given course in
the available time span and also ensure effective learning amongst children. Now the challenge is
to do it successfully. This very idea may generate thinking with regard to sequencing, ordering,
arranging and grouping the items of the curriculum, matching these with the available time slot
and identifying suitable activities to be performed with children etc. This is nothing but planning
for teaching.

You may plan teaching in various ways.

 Firstly you may develop a rough layout for the whole year in which you may decide before
and how much time you would devote to various topics or units in the curriculum.
 Secondly you may like to develop a detailed planning of the separate units of work wherein
you may decide the number of lessons for each segment of work along with the method or
approach to deal with them.
 Thirdly you may like to go into details of activities pertaining to each lesson. So the planning
for teaching involves the process of making decisions about why, how and what to teach
which may range from one lesson to the whole curriculum for the year. In this section we
will discuss the various aspects of unit and lesson planning in mathematics.

4.2.1 The Need to Plan

Teachers plan in order to modify the curriculum to fit the unique characteristics of their students
and resources. The plan helps teachers allocate instructional time, select appropriate activities,
link individual lessons to the overall unit or curriculum, sequence activities to be presented to
students, set the pace of instruction, select the homework to be assigned, and identify techniques
to assess student learning.

Planning helps teachers in five basic ways:

1. By helping them feel comfortable about instruction and giving them a sense of
understanding and ownership over the teaching they plan.
2. By establishing a sense of purpose and subject matter focus.
3. By affording the chance to review and become familiar with the subject matter before
actually beginning to teach it.
4. By ensuring that there are ways in place to get instruction started, activities to pursue, and a
framework to follow during the actual delivery of instruction.
5. By linking daily lessons to broader integrative goals, units, or curriculum topics.

Classrooms are complex environments that are informal rather than formal, ad hoc rather than
linear, ambiguous rather than certain, process oriented rather than product-oriented, and people-
dominated rather than concept-dominated. The realities and strains of the classroom call for
order and direction, especially when teachers are carrying out formal instruction. In such a

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world, some form of planning and organization is needed. Planning instruction is a context-
dependent activity that includes consideration of students, teachers, and instructional materials.
A lesson that fails to take into account the needs and prior knowledge of the students or that
poorly matches lesson aims to lesson instruction is doomed to failure. Similarly, a lesson that
does not take into account the context in which it will be taught can also lead to difficulty.

Teachers have a great deal of control over many classroom features associated with instructional
planning. For example, most teachers have control over the physical arrangement of the
classroom, the rules and routines students must follow, the interactions with students, the kind of
instruction planned and the nature of its delivery, and the methods used to assess and grade
students. However, there are important features that teachers do not control. For example, most
teachers have little control over the number and characteristics of the students in their classes, the
size of their classroom, the quality of their instructional resources, and the Ministry/Department
curriculum guidelines. In planning, teachers must arrange the factors they do control to
compensate for the factors they do not.

Self Assessment Exercise

Discuss about the benefits of planning for teachers.

4.2.2 Types of Instructional Planning

There are different types and stages of academic or instructional planning. The first stage is that
of curriculum development, this is usually done by a team of curriculum experts for a particular
program or course. When this is handed down, further work of breaking it down into learning
units is done by the various authorities concerned. When a curriculum is broken down into list of
topics that students should study in a particular subject in the particular course or program, this is
called the syllabus. This may further be broken down into yearly plan. This is usually done by
the various educational controlling bodies for schools. Following this, the next stage is to break
the topics further and organize them into Scheme of Work, Unit Plans, and Daily lesson plans.
This is essentially the work of the teachers, and it is a stage where their professional expertise is
challenged.

In delivering specific grade mathematics, instructional plans will be divided into three types;
namely:
 Annual(yearly) plan
 Unit Plan
 Daily Lesson Plan,

4.2.2.1 Annual(yearly) Plan

This is a rough layout for the whole year. The teacher has to decide before hand how much time
is devoted to various topics or units of the year. The main purpose of planning the annual plan is
the coordination of the demands of the syllabus and the actual conditions to be found in the

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school and the classroom. In order to prepare the annual plan, the teacher has to consider the
following basic points:

1. The function of the course with in the whole course of teaching mathematics.
 The teacher has to recognize the aims and the contents of the course regarding the
development of the students’ personality.
 The teacher must know the whole course of teaching mathematics exactly.
2. The subject matter related to the time available.
 The teacher has to make sure that all topics being included in the syllabus can be
treated in an appropriate time. It must not happen that there is no time to deal with
last chapters of the course uncovered.
3. The knowledge of the students gained in previous courses of teaching mathematics or in
other subjects.
 The teacher should check whether there was a loss of certain lessons within the last
academic year.
4. The production and repair of teaching aides.
 The teacher should check whether there is appropriate teaching aides or not for all
topics.
5. The school time table.
 The teacher should identify number of working days and holidays.

4.2.2.2 Unit Plan

The unit plan is the mid-way stage between the scheme of work and daily lesson plan. It is made
up of a set of related learning experiences, obtained from the breakdown of a mathematical
theme, concept or topic into units of different lessons.

Meaning of Unit Planning

Let us first understand what do we mean by a unit in mathematics. A unit in mathematics

comprises of a chunk of interlinked competencies/concepts/content which have some common
basis or characteristics. So, within any area of mathematical learning several units can be
formed. It is the nature of competencies content and the experience of the teacher about teaching
mathematics and his/her perception of learning styles of children which will enable him/her to
decide about formulating the units.

Now you will appreciate that teacher has to organize the given set of competencies/content
prescribed for the given class in a meaningful manner which will make his/her teaching and
evaluation systematic and convenient. A unit in mathematics may be covered in one day, several
days or even several weeks. You will have to decide the number of lessons to be delivered under
one unit.

Having arranged the mathematical competencies in a graded manner and divided them into units
for purposes of classroom transaction, you would like to think of the ways of communicating the
same to the children. This will obviously make you think of the sequence of lessons within a

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unit, the method of teaching, instructional aides, students’ activities and the evaluation
procedures. This decision if presented in an organized manner would result into a unit plan.

Need and Importance of Unit Planning

Why we need a unit plan? Can’t we do without a unit plan?

The answer is simple that unit planning may bring about significant changes in the quality of
teaching-learning. The following points highlight the advantages of unit planning and thus
clarify how unit planning makes teachers talk easier and effective:
 It helps teachers to have a holistic view of teaching-learning, which may help in organizing
time and resources available at his/her disposal.
 It helps in designing a systematic, sequential and graded arrangement of course content
which may give insight to develop teaching activities in the best possible manner.
 It helps in giving a balanced emphasis to various aspects of course content or competency
under reference.
 It provides an opportunity to correlate textual content with the competencies to be dealt with
in the class.
 It may help thinking about alternative approaches to teaching-learning and adapt to
individual differences.
 It may help unit-wise evaluation of children and in organizing remedial teaching and
undertaking enrichment measures as per the requirements.
Steps Involved in Unit Planning

Some steps to be followed in unit planning are suggested below:

(a) Estimate the whole course content/set of competencies for the grade during the year.
(b) Estimate the teaching time available to the teachers.
(c) Arrange the given course content /set of competencies in a teaching-learning sequence.
(d) Identify inter-linked aspects of course content /competencies.
(e) Distribute the whole course content/competencies into units. Hence you may like to
consider the following:
i) A unit should not be too small or too lengthy.
ii) It should have some element of commonness within its components.
iii) It should be such that it should not require more than a month in any case to complete
in the class, and
iv) It should be such that its completion develops a sense of accomplishment to both the
teacher and the students.
(f) For each listed unit, further breaking up of teaching lessons would be required.
(g) For each lesson within the unit, decide about the appropriate teaching methods, teaching
aides, students activities and the evaluation procedure.
(h) Present these decisions and the break-up in a tabular form which may be considered to be
unit plan.

Limitations of Unit Planning

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While unit planning is of much value to teachers it may suffer from the limitations mentioned
below:
 It is sometimes difficult to clearly anticipate the teaching-learning approach in advance.
 The division of content / competencies is artificial.
 It gives a piecemeal view of the competencies developed during a year.
 It requires a conscious effort on the part of a teacher. A less experienced teacher sometimes
find it difficult to plan units.
 It puts a check to the flexibility of the teacher when followed rigidly.

Format of a Unit Plan

Essentially, a Unit Plan must contain the following:

(1) The subject,
(2) The topic of the unit,
(3) The grade for which the unit is being planned,
(4) Objectives of the unit,
(5) Entry behavior: It is the pre-requisite behavior necessary or essential for the new concept.
(6) Contents of the Unit,
(7) Methods and Activity,
(8) Instructional Materials,
(9) Evaluation Mechanism.

Frame Work of a Unit plan

Preliminary part
Name of the school: _______________ Academic year: _______
Teacher’s Name: _________________ Semester: ____________
Grade: __________________________ Number of periods: _____
Subject: _________________________
Title of the unit: ___________________

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Main part
Objectives of the unit:
Begin with: By the end of the unit you should be able to … and then include your objectives
for the specified unit
1._______________________ 3. __________________________
2.________________________ 4. __________________________
No of Periods Topics Subject Main Evaluation and Teaching
Matter instructional Assessment Aides
Methods
Indicate the Include the Include the Include the Identify the Include
number of main topics main contents suitable methods methods you have the
periods to be of the unit or subject or techniques you chosen to assess teaching
spent in each matter of the have chosen to knowledge, skills aides that
unit unit in details teach each content and understanding you will
in the unit of the students to use in
each content each
content

Teacher’s sign.__________ Comments and approval of the department

head: __________________

Self Assessment Exercise

1. What is the meaning of a unit plan?

2. State briefly how unit planning is helpful to teachers.
3. Mention two limitations of unit planning.
4. What is meant by planning is a continuous process?
5. List the main elements (parts) of a unit plan format.
6. Discuss about the advantages of annual plan for instruction.
7. Develop a Sample unit plan for the unit solving quadratic equations.

4.2.2.3 Daily Lesson Plan

A daily lesson plan outlines in detail the various steps which the teacher proposes to undertake in
his/ her class. As such, a daily lesson plan concerns itself with the teaching of one period. This is
the most specific part of planning. Planning for a lesson means identification of the sequence and

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style of presentation and evaluation procedure to be adopted for classroom teaching of a lesson.
Hence it is a proposition in advance which establishes a linkage between the why, what and how
of teaching in one period. While attempting to do this the teacher may foresee likely problems in
classroom communication and may arrange certain materials and decide about techniques to be
adopted to ensure a smooth and effective teaching-learning situation. Thus a lesson plan is a
means of taking advance decisions about the selection, sequencing and execution of various
activities to be performed in a classroom with a view to ensuring learning of children.

On the basis of the unit plan, from which the function, the position, the topic of the lesson, etc.
can be taken, the teacher starts preparing the daily lesson plan.

While planning the daily lesson, the teacher might question him/herself the following and answer
with regard to the respective conditions in his locality:

 What are the major ideas of the daily topic? How are these ideas related to previous work?
 How can the period be subdivided with regard to didactic elements and time?
 What pre-requisite skills and understanding should the student know? What prerequisites
are necessary with the students for a successful treatment of the topic?
 Do I need to review certain materials before introducing the new topic? Which ideas
should review major emphasis? How the new subject matter will be introduced? How
motivation of the students and aim-orientation will be taken for each part of the lesson?
 Do I need to supplement the text?
 Do I need teaching aides? Which teaching aides have to be kept ready for the support of
the presentation of the new subject matter and the consolidation process as well as for the
motivation?
 How will particularly the textbook, numerical tables, calculation aides and drawing
instruments (if available) be used for effective implementation of the periods and for the
development of certain abilities of the students?
 Which methods, strategy or technique should I use? How can the process of all round
active and creative learning of the students be organized by a variety of methods?
 At what time of the term and in which form you should plan tests or tasks to evaluate
students' attainments?
 Which activities will be planned for the teacher, for the class or for a single student?
 Which objectives should be attained after teaching this topic? In what way will the
checking of the teaching learning outcomes be done?
 In what way will the summaries be given? How will a systematization of the contents be
implemented?
 What has to be given as notes? In which manner are they to be carried out? How will the
black-board work be prepared?
 What has to be assigned for class work, homework, assignment? Which hints must be
given to the students?

The answers of such questions would enable the teacher to plan and implement class room
activities in a very organized manner. In answering these questions the teacher must proceed

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always from the fundamental positions of pedagogic and methodical principles of the teaching-
learning process in schools.

Activity 4.2.1: (take 7 minutes)

Describe the importance of daily lesson plan for a mathematics teacher.

Feedback for Activity 4.2.1

Preparing a lesson plan ahead of time gives you a sense of security and
confidence in the teaching-learning process. If you are going to teach a class
successfully, you must have a clear idea about the topic you will be teaching
about and you must make the necessary preparations for the lesson.

In general lesson planning helps the teacher in the following ways:

 It makes teaching systematic and well organized.
 It helps teachers in identifying adequate content and its proper sequencing
for teaching a lesson.
 It helps teachers to learn to foresee and tackle learning difficulties of
children.
 It enables teachers to utilize the available time properly.
 It helps in developing insights about learning needs and abilities of
children.
 It helps teachers to develop the habit of undertaking immediate corrective
measures.
 It gives confidence to teachers during teaching.
 It motivates the learners and ensure their attentiveness.
 It ensures that the classroom surroundings are conducive to learning.

Self Assessment Exercise

Discuss about the questions a teacher must ask him/herself during planning
the daily lesson with your partner.

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4.2.3 Structure of a Daily Lesson

4.2.3.1 The Basic Characteristics of a Lesson

Although there are many types of lessons, they have a number of characteristics in common.
These should be borne in mind when preparing a lesson, as well as when teaching it. The
characteristics address issues which are important to the lesson as a unit of work.
Included in the basic characteristics are:

 A lesson is a self-contained unit of work. Regardless of its length, each lesson should address
only one topic. This is a basic property. It gives lesson integrity.
A lesson should be able to stand on its own feet. At the same time, it is also part of a larger
whole or program. Unless it stands on its own feet, it loses a sense of theme. If a lesson is too
long, it loses touch with its objectives and audience.

 A lesson should be tied to a particular standard of performance. This relates to the objectives
of the lesson. It also relates to the expectations that something concrete will come from the
experience. This does not mean that the results always have to be measurable and observable.
It does mean that something worthwhile must be achieved.
 A lesson should be adapted to the needs of the learners.
The objectives of a lesson, and the learning activities necessary to realize the objectives, must
be within the capabilities of the trainees and students. They must be reasonable, yet
challenging.
A lesson must be cast at the level of the people involved. Otherwise it is irrelevant to their
needs. This includes making sure that the language used is at their level of comprehension.

 A lesson should have a definite structure.

People in a learning situation should understand what is going on. Just as an audience
watching a play needs to know which act it is watching, so trainees need to know where they
are in the lesson.
Every lesson needs a framework. The role of instructor and trainees in one part of the lesson
is different to their role in other parts of the lesson.
These four characteristics insure that the lesson is a reasonable learning experience. They should
not be taken for granted.

The Events of Instruction

Whenever instruction takes place, a number of things have to be attended to while the teaching is
underway. It does not matter what instructional strategy is employed. Effectiveness, as well as
efficiency, depends upon them. Since every lesson should have a beginning, a middle, and an
end, the “events” are best though of under these headings.

The events of instruction are described as follows:

 Events in the beginning part of the lesson;

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 Gain the attention of the learners.

 Tell them what is expected.
 Events in the body of the lesson;
 Remind them of what they already know, relevant to the new task.
 Present them with the new knowledge, skill, or attitudes they are to learn.
 Provide them with encouragement and guidance.
 Insure that they practice what they have learned.
 Tell them how they are doing.
 Events in the last part of the lesson:
 Consolidate what they have learned.
 Assess their level of mastery.
 Assist them to apply what they have learned to new situations.
This is a lot to think about and remember. The main purpose of the lesson structure is to insure
that it all gets done. And at the right time, too!

4.2.3.2 Constructing a Daily Lesson Plan

Every lesson should have three basic parts. Each part, of course, should lead naturally to the
next. However, each part also needs to have a form of its own. It should be definite and clear cut,
with its own beginning, middle, and end.
The three parts of the basic lesson are:
I. The Introduction
This should be relatively short in duration. It must lead naturally to the main body of the
lesson. As a rule of thumb, it should account for less than 10 percent of the instructional time.
However, everything depends upon the specific circumstances of the lesson.
II. The Development
This is the main body of the lesson. As a rule of thumb, the development will probably
account for up to 65 percent of the instructional time in a knowledge lesson. When a skill is
involved, this part of the lesson will account for only about 25 percent of the training. This is
when the skill is demonstrated.

III. The Consolidation

This is the end part of the lesson. In a knowledge lesson, it is likely to be relatively short. It
will probably account for something like 25 percent of the training. In a skill lesson,
however, the consolidation part of the lesson is used to imitate and practice the skill. For this
reason, a large chunk of time is normally involved. Sometimes it accounts for up to 60
percent or more of the training.
As we have seen, each of the three parts has a different task to accomplish. Each involves a
different set of the “events” of instruction.

I. The Introduction to the Lesson

Despite the relatively short duration of the introduction, much needs to be done. Two events of
instruction, in particular, have to be taken care of:

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 Gain the attention of the learners.

 Tell them what is expected.
The former involves attracting their interest. The latter emphasizes the importance of giving
them clear objectives to realize.
The introduction to a lesson is one of the more difficult things to arrange. It is hard to think of a
suitable introduction, and hard to put it into effect. To some extent, it demands a degree of
showmanship or a sense of theater. For this reason, the quality of an introduction can be a good
indicator of an instructor’s skill. But like all skills, introductions improve with practice.

Activity 4.2.2: (take 6 minutes)

Describe the purposes of an introduction to a lesson.

Feedback for Activity 4.2.2

An introduction has a number of jobs to do. It should:

 Concentrate attention and arouse interest.

 Set the scene for the lesson.
 Set the climate or atmosphere for what is to follow.
 Establish links with past lessons and future ones.
 State the aim of the lesson and the objectives to be realized.
 Point out the importance of the task.
This is a lot to do in short amount of time. But unless it is done, trainees simply
will not be motivated. In effect, the six functions serve as a checklist for effective
introductions.

II. The Development of the Lesson

The development forms the main body of the lesson. As its name suggests, it involves a gradual
unfolding of the topic. Inevitably, sequence is a key concern. What should be taught first? What
should be taught last? What is logical, and what is not? These are the kinds of decisions that have
to be made.
The development portion of the basic lesson accounts for something like 65 percent of the
instructional time. During this time there is a lot to do and a great deal to think about. Planning is
essential so that nothing will be overlooked.
Five events of instruction have to be taken care of in the development. The training must:

 Remind trainees what they already know, relevant to the new task.
 Present them with the new knowledge, skill, or attitudes they are to learn.

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 Provide them with encouragement and guidance.

 Insure that they practice what they have learned.
 Tell them how they are doing.

The first event will insure that the prerequisites for learning are known. The last event
emphasizes the importance of trainees and students receiving appropriate feedback in a timely
manner.

Activity 4.2.3: (take 6 minutes)

Describe the purposes of the development of a lesson.

Feedback for Activity 4.2.3

The purposes of the development are to:

 Develop or unfold the topic in a gradual manner.

 Insure smooth continuity and logical order in the material.
 Provide sufficient periods of class involvement on a group as well as
individual basis.
 Insure that there is variety and vividness in the instructional approach.
 Manage time and resources in an efficient and effective manner.
Realizing these objectives casts teachers and instructors in a manager rather than
operator role.

The development part of the lesson includes a great deal of telling and asking. It also includes
showing and doing. Trainee and student participation is vital. For this reason, instructors must
resist the temptation of lecturing. They must not shoulder the whole burden; otherwise, they
become teacher-operators. Group activity is the key.

III. The Consolidation of the Lesson

The consolidation is the concluding part of the lesson. It involves strengthening what has been
learned. For this reason, it should be planned as carefully as the other parts. It is too important to
be done hastily.
Unfortunately, due to poor time management, far too many teachers and instructors fail to
consolidate. They run out of time. Yet there is little point increasing the amount of time spent on
the development if there is going to be little or no opportunity for trainees to consolidate what
they have learned.
The consolidation part of the basic lesson accounts for something like 25 percent of the
instructional time. During this time, three events of instruction have to be taken care of
somehow. All three are important.

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Instruction, during the consolidation part of the lesson, must do the following:

 Consolidate what trainees have learned.

 Assess their level of mastery.
 Assist them to apply what they have learned to new situations connected with the job or
task.
In this way, the knowledge, skill and attitudes that have been acquired during the learning
module will be nurtured and strengthened.

Activity 4.2.4: (take 6 minutes)

Describe the purposes of the consolidation of a lesson.

Feedback for Activity 4.2.4

A period of review and the consolidation period of a lesson are not the same
thing. A comparison of the purposes or objectives of the two will highlight the
difference. The two thrusts have very little in common.
The main purposes of the consolidation period are to provide opportunities for
trainees and students to:

 Rehearse their learning until mastery has been obtained.

 Assimilate and retain their learning.
 Apply their knowledge and understanding to real world situations such as
they might meet in their jobs.
 Obtain an assessment of the quality of their performance in real world or
simulated situations.

Frame work of a daily lesson plan

As discussed earlier, you are the best person to prepare a blueprint of the lesson plan that you are
going to execute. However, for your convenience, the specimen of a lesson plan has been given
below. You may follow it and/or may make changes, if necessary.

Name of the school: _____________________ Grade: ________________

Teacher’s Name: ________________________ Date: _________________
Subject: _______________________________ Duration of period: ___________
Topic of the lesson: ______________________

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Specific Learning Objectives:

Begin with: By the end of the lesson students will be able to… and then list your SMART --
specific, measureable, achievable, realistic, and time bound objectives for the given lesson.

Time Contents Teacher’s Activity Student’s Activity

Indicate how Include the Include details of what the teacher Include the details of
much time is to main contents will be doing: the activities that the
be spent on each or subject  Introducing the lesson students will do and
activity by the matter of the  Directing whole class or group list the expected
teacher and the lesson teaching, giving instructions outcomes of the task.
students  Organizing active learning
situations
 Consolidation
 Assessing students learning
 Summarizing key points.
Include the key questions the
teacher will ask students in order
to check understanding.
Resources:
Include the material and teaching aides that you will use in your lesson
Assessment methods to be used:
Identify the methods you have chosen to assess the knowledge, skills and understanding of the
students. The methods may include:
 Listening to pairs or group discussion,
 observation of an activity,
 students written work,
 problem solving, presentation, quiz, etc.

Teacher’s Sign. __________________ Comments and approval of the department head:

_______________________________________

Self Assessment Exercise

1. What type of knowledge is needed on the part of the teacher before

lesson planning?
2. Describe the four characteristics of lesson plan.
3. List the important steps of lesson planning.
4. Design a lesson plan to teach congruency of triangles.

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 UNIT 4:Assessment Methods and Planning in the Teaching and Learning of Mathematics

Unit Summary
Instructional planning is putting and organizing the necessary information, objectives, contents
and resource materials into some sort of order for use in the teaching learning process.

The main elements of instructional planning are instructional objectives, contents and learning
experiences, methods of teaching, resources and assessment of learning outcomes.

Since the purpose of education is to develop knowledge, skills and attitudes, the objectives have
fall under the three domains of learning (Cognitive, affective and psychomotor domains).

Good teaching can’t be defined because the criteria differ for every instructional situation and
every teacher. But different scholars said about observable indicators in effective teaching.
According to Ryan (1960) and his colleagues stated that teachers rated to the positive poles of
the three factors associated with effective teaching are considered more effective than teachers
rated to the negative poles. According to Flanders (1970) and his associates said that teachers
following “ Indirect Teaching” style are more effective than those following “ Direct Teaching “
styles.

Observation of the teacher’s pupils is also a method of perceiving quality of teaching. Some of
the observable indications of effective teaching indicated by pupil’s behavior are studied
“Academic engaged time.” is an important factor in school achievement.

Instructional plans may be objectives for a daily lesson prepared at the level of curriculum
development, Syllabus preparation, annual plan, unit plan or lesson plan. The action verbs used
in the construction of objectives of unit and annual plans are more general compared to those of
lesson plans.

Efficient and effective lesson planning is undoubtedly one of the keys to successful instruction.
The measure of a good instructor or teacher is that person’s ability to get the right things done.
This requires intelligence, imagination, and knowledge. But these are not enough. Steps must be
taken to convert them into results.

Further Reading Materials for the Unit:

To complete your study of this unit, you will need to refer to
 Peter G. Dean., Teaching and learning mathematics, University of London Institution of
Education.
 Elizabeth Perrott, Effective Teaching: a practical guide to improve your Teaching,New
York; (1982).
 Gray L. Musser, William F. Burger, Mathematics for Elementary Teachers, a Contemporary
Approach.
 Douglas K. Brumbaugh, Teaching Secondary Mathematics, 1997 Printed in the United
States of America.

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 UNIT 5: Some Specific Methods to Teach Mathematics

UNIT 5: Some Specific Methods to Teach Mathematics(13 hrs.)

Introduction
Teaching mathematics is helping students to learn mathematics. It is not merely telling
something to a group of listeners, nor explaining some topic, nor demonstrating your mastery of
mathematics. Of course, when you are helping students to learn you may be engaged in telling,
explaining, or demonstrating, but you do these only as a means of helping your students to learn.
In the final analysis your students learn. In this ways your success as a mathematics teacher will
be determined by how well your students have learned and changed their behavior, attitudes,
performance, and achievements on mathematics. Therefore, it is very crucial for you to analysis
and synthesizes different methods of teaching in general and mathematics in particular in order
to help students learn. In this unit you will look at inductive and deductive approaches of
teaching, synthetic and analytic approaches of teaching, and problem solving approaches for
teaching and learning mathematics. The unit also deals about teaching concepts and
misconceptions. You are advised to review and examine the appropriateness of different methods
of teaching and determine methods suitable for teaching mathematics at secondary school level.
But bear in mind that a teaching method may serve for teaching various subjects; therefore , it is
your task and skill to select, adopt and use appropriate teaching methods for your students to
learn mathematics. However, problem solving approach, when implemented contextually
considering the needs and learning styles of your students, is the most appropriate strategy of
teaching and learning mathematics. This approach is also the one that is recommended by the
national education and training policy of the country. Problem solving is a learnable skill that
you can easily but systematically develop your students’ problem solving capacity. Through
problem solving approach students not only learn mathematics but also learn how to learn
mathematics. Teaching and learning mathematics concepts play a vital role. Misconceptions of
concepts are also serious factor affecting students’ development of higher order thinking. The
term "concept" has been defined somehow differently by different scholars in the area.
However, there is no disagreement among them about the importance and necessity of concepts
development in learning mathematics. The basic and crucial stage in learning mathematics is the
development of one concept up on the previous concepts.

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Unit Learning Outcomes

Upon completion of this unit, the students will be able to:

 Explain concepts and misconceptions of mathematics;
 Identify sources of misconceptions in secondary mathematics;
 compare and contrast inductive and deductive methods of teaching;
 compare and contrast synthetic and analytic methods of teaching;
 apply the synthetic and analytic methods to solve mathematical problems ;
 explain the meaning of problem solving as a method of teaching;
 identify the characteristics of problem solving which distinguish it from other
methods of teaching;
 prepare and teach mathematical lessons using problem solving method.

5.1. Teaching and Learning of Mathematics Concepts(2 hrs.)

5.1.1Meaning of ‘Concept’ and ‘misconception’

What are mathematical concepts?

Concept is a form of thought which reflect important characteristics of the objects studied.
The process of formulating a concept is a gradual process. We can roughly describe the process
in the following way:
 The initial and most simple step of being aware of the concept is observation and
introduction to concrete objects and their concrete characteristics related to the concept
and sensory awareness – observation.
 The second step is observing something general and common to elements in the observed
group of objects – having an idea about the concept.
 The third step is pointing out the important characteristic of such objects – formulation
and acquisition of the concept.

It is not difficult to recognize some important scientific procedures in the described process:
analysis, synthesis, abstraction and generalization. That means that any concept, including
mathematical concepts, after careful analysis develop through abstracting characteristics of
objects which exist in nature and through generalization. In that way mathematical concepts,
although abstract concepts, reflect some characteristics of the real world and in that way
contribute to their awareness. According to that, in teaching mathematical concepts, the teacher
realizes the science principle if the process of formulating concepts is appropriately implemented
(observation, the idea about the concept, formulating the concept) and if he/she adheres to the
rules which must satisfy the definition of a concept (appropriateness, content minimum,
conciseness, naturalists, applicability, and contemporariness).

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One of the characteristics of a concept as a form of thought is that formulating a concept as part
of human awareness is inseparable from expressing words or recording or using symbols. This
characteristic is especially emphasized in mathematics.

The issue of language in teaching mathematics is very sensitive. There can be vagueness and
violation of the science principle in this area. As an example we can look at several formulations
from mathematics books:

 Parallelogram is a quadrilateral whose opposite sides are parallel.

 Parallelogram is a quadrilateral whose opposite sides are parallel and congruent, opposite
angles is congruent and the angles on the same side are supplement.
 The bisector of a length is the set of all points of a plane which are of the same distance
from the end points of a length.
 An equation in the form ax2+bx+c = 0, where a, b, c are real numbers and a≠ 0, is called
equation of the second degree or quadratic equation.

The first sentence is a concrete definition of a parallelogram; however it would be even better
and more precise in the following form: A quadrilateral whose opposite sides are parallel is
called a parallelogram.
The second statement is not a definition since it has redundant words and concepts and it is
unlikely that all sixth grade students would know how to use it. It actually consists of the first
definition and three theorems.
The third sentence causes ambiguity. It can be a definition of the symmetric length of a line;
however, since in teaching the usual definition is the symmetry of the length as a line which
passes through the midpoint of the length and is perpendicular to it, the mentioned theorem needs
to be proven.

The fourth sentence is a concrete, abstract-deductive definition of a quadratic equation.

What is mathematics Misconception?

Student thinking consists of many things. Formulas, relevance, boredom, and enjoyment are part
of their attitudes and thinking about mathematics. One problem that leads to very serious
learning difficulties in mathematics is the set of misconceptions students may have from
previous inadequate teaching, informal thinking, or poor remembrance. We begin with a
definition.

Certain conceptual relations that are acquired may be inappropriate within a certain context. We
term such relations as "misconceptions." A misconception does not exist independently, but is
contingent upon a certain existing conceptual framework. Misconceptions can change or
disappear with the framework changes.

Changing students' misconceptions is one of the key goals in repairing mathematics and science
misconceptions. It is not usually successful to merely inform (e.g. lecture) the student about a
misconception. The misconception must be changed partly through the students' belief systems
and partly through their own cognition. Many students do not come to the classroom as "blank

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slates. Rather, they come with informal theories constructed from everyday experiences. These
theories have been actively constructed. They provide an everyday functionality to make sense of
the world but are often incomplete half-truths. In this topic, student misconceptions about
mathematics are considered, particularly those that impact algebra and algebraic thinking. Yet,
misconceptions are one facet of faulty, inaccurate, or incorrect thinking. All of these are
intertwined and all are sources of students' trouble in grasping with mathematics from the most
elementary concepts through calculus. In turn, student misconceptions cause teachers immense
frustration about why their teaching isn't "getting through."

First thoughts on misconceptions might be that, once rooted in the student's memory, they are not
easy to erase. But the situation is very complex, and the answer to the question of eliminating
misconceptions is neither well understood nor researched. Mostly, what is known are simple tips
on what and what not to do. Even the problem of properly identifying misconceptions is not easy
without intense teacher-to-student interaction.

Research also suggests that repeating a lesson or making it clearer will not help students who
base their reasoning on strongly held misconceptions. Students tend to be emotionally and
intellectually attached to their misconceptions, partly because they have actively constructed
their misconceptions and partly because they provide ready methodologies for solving various
problems. Misconceptions definitely interfere with learning when students use them to interpret
new experiences.

Examples of Misconceptions in Algebra

Many misconceptions apparent in algebra are rooted in misconceptions of arithmetic. In fact, the
arithmetic misconceptions below are all have an algebraic counterpart. The purpose is not to
explain how to teach students away from the misconception as there is so very little research on
specific misconceptions. Besides, teaching students to resolve a misconception is very much
dependent on the nature of the misconception within the specific student’s mind. The purpose is
to delineate them, although this list is by no means exhaustive.

1. Number sense: Students do not understand the difference between rational and irrational
3
numbers. Some think that is irrational, and many think that repeating decimals such as
5
23.45454545… are irrational.
1
2. Exact vs approximate: If you compute on your calculator = 0.142857142857, many
7
students will assume this answer is exact. This misunderstanding may well arise from over-
reliance on calculators or improper teaching of the meaning of numbers generated by the
calculator.
3. Fractions: For most of mathematics through calculus, misconceptions about fractions
provide the root source of many student difficulties. Many of these problems come from
fractions not being properly understood.
ab  c
a. Incorrect cancelling of to obtain a + c.
b

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b. Working with large numerators and denominators rather than reducing when multiplying
13 14 13  14 182 2
fractions such as  to get   , instead of cancelling common
7 65 7  65 455 5
13 14 1 2 2
factors as     .
7 65 1 5 5
4. Addition of fractions generates a number of errors, most particularly with treating different
denominators incorrectly. There is also the notational conflict between adding and
multiplying fractions. In multiplying fractions the numerators and denominators of each
fraction, respectively, are multiplied. It is natural for students to want to do the same for
addition, without understanding the absolute need to determine a common denominator at
the first step.
5. Magnitudes for negative numbers: For example, which number is larger -8 or -5? The
problem is how the student perceives "large". Some teachers mistakenly use expressions
such as the "larger negative value". This causes confusion to many students.

6. Order and rule of operations: This is perhaps the most common misconception.
a. Many students will compute 4 + 3x² as 7x², effectively interpreting 4 + 3x as 7x and
then multiplying by x to obtain the incorrect result.
b. Students often misuse the distributive rule. We may see x²-2(x-3) written as x²-2x-6.
The -2 is not correctly distributed.
7. Powers: Students have trouble with precedence of operations.
a. In computing -4² = -16, many will incorrectly compute -4² as (-4)² to obtain 16.
1
( )
b. In computing 16 any number of answers may be reported including -2 and 2 rather
4

1
than the correct answer, .
2
c. Is a x a x a the same as 3a? Similarly is a3 the same as a x 3? Some students are not
secure in the correct mathematical meaning of various notations. This is partly a fault of
the existence of multiple representations of the same thing. These are aspects of not
fully understanding notation and not being functionally fluent in making computations.
As misconceptions, these are examples of not recognizing and correctly interpreting
mathematical notation.
8. Using the definition of the absolute value function, particularly for negative numbers.
The definition causes many students to pause. They understand the idea of the absolute
value. However, it is the actual mathematics notation that causes trouble.
9. Square roots - with sums and differences: Typical errors many students make are to
write a  b  a  b and a b  a  b .
10. Simplification/factorization of algebraic expressions: Students typically abandon the
rules or misinterpret them in many types of simplification problems.
a. In expanding perfect square binomials:Many students compute (x+3)² as x²+9 even
worse x²+6.
b. Many students do not see a common factor in an expression such as 2 · 3 · 4 · 25 +
5. This could be why they have difficulty seeing the common factor in 2x²y³ + 3x.

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11. Inequalities: Students often have trouble in solving various types of inequalities. The
worst offenders are inequalities involving quadratic terms and inequalities with an
absolute value.
12. Logarithm:
a. In the study of logarithmic properties students mix up the exponential and logarithmic
rules.
b. When solving a logarithmic equation, students forget to check if the answer is in the
domain.
13. Functions: Finding the domain of a rational function when a common factor is present in
the numerator and denominator.

Self Assessment Exercise

1. Discuss about mathematical concepts and misconceptions? Support

your discussion with examples.
2. Describe the process of formulating a concept.

5.1.2 Importance of Concept Learning

In teaching mathematical concepts and definitions, it is important to present some basic

theoretical aspects about them. Mathematical concepts are the most learnable part of subject
matter of mathematics and are building blocks of the subject mathematics. This is because
concepts help students to learn other concepts and other kinds of subject matter. By means of
concepts other kind of subject matter are learned. Thus grasping concepts and definitions are
fundamental for:
 Understanding mathematical coherencies
 Discriminating and classifying mathematical objects and relations
 Drawing deductions and conclusions
 Communications
 Generalizations
 Discovering new knowledge
 Applying the knowledge creatively
 Developing students’ mental abilities.

The set of mathematical concepts are subdivided into three categories. These are the following:
1. Concepts about objects: These concepts characterize the classes of real or thought things
that can be given by representatives. For instance, point, line, rectangle etc
2. Concepts about operations: These concepts characterize actions carried out by means of
d
objects. For example, , , , , , dx
, reflection, rotations, etc.

3. Concepts about relations: These concepts reflect relations existing between the objects.
For example, , , , , , , , ,  , etc.

The teaching of mathematical concepts and definitions are carried out by using various methods.
Among these methods, the most common ones are the inductive and deductive approach. With

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these and other methods, teaching mathematical concepts may take place in several steps in a
single lesson. Therefore, in dealing with mathematical concepts we will follow the following two
partial processes.

 The formation of the concepts

 The acquisition of the concepts

1. The formation of concepts: This partial process takes place in order to introduce the
concept. This process is closely related to the objectives of enabling students to define the
new concept. At this stage the teacher should secure the starting level by
(i) Revision of previous concepts: Revise all the main concepts that are necessary for the
concept of the new one.
(ii) Aim and topic orientation: Telling the task expected and aim to the students
(iii) Motivation: Making the students ready for the new concept, capturing their attention.
(iv) Attributes of the concept: Finding out the common and non common characteristics
or properties of the concepts of the representatives (examples) up to definition or
explanation of that concept.

2. The acquisition of the concept: After the starting level is secured and after the motivation
and aim orientation have already takes place, the next step will be acquiring or imparting the
new concept. It is a process which contains stabilizing the concept in the students’
consciousness. Identifying, realizing and applying concepts are the basic actions that could
be carried out in acquiring a concept by the students. Expected generalization, definition and
further strengthening of concept are part of this process.

Self Assessment Exercise

1. Identify the concepts that you know in mathematics and then classify them into
objects, operations and relations.
2. What teaching method(s) is/are suitable to teach mathematical concepts? Why?

5.2 Inductive and Deductive Methods in Teaching Mathematics(3 hrs.)

5.2.1 Inductive Reasoning and Method of Teaching

What is Inductive Reasoning? How do you apply inductive reasoning as a

method of teaching mathematics?

Induction is the form of reasoning in which a general law is derived from a study of particular
objects or specific processes. The child can use measurement, manipulator or constructive
activities, patterns etc. to discover a relationship which he/she shall him/herself, later, formulate
in symbolic form as a law or rule. The law, the rule or definition formulated by the child is the
summation of all the particular or individual instances.

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Induction leads from concrete to abstract, particular to general and from examples to the rule.
Here we take a few examples and arrive at the rule i.e. it is a method of constructing a formula
from a number of concrete examples.

Schematically, this is represented as follows:

Induction means; “to provide the universal truth by showing that if it is true for a particular case,
it is true for all such cases.” In this method we arrive at a formula or generalization through
reasoning and solving problems.

Inductive reasoning is common in science, where data is collected and tentative models are
developed to describe and predict future behavior, until the appearance of the anomalous data
forces the model to be revised.

Example: To arrive at the generalization that; “The sum of all the angles of a triangle is 180o”
the teacher draws different types of triangles and asks students to measure all the angles of the
triangles and state the sum of the three angles in each case. Every time students realize that the
three angles measure up to 180o. The triangles may be as shown below.

The teacher can ask students to cut the three corners of the triangles and put them at a point so
that they form a linear.

They observe the results and can make the generalization that “The sum of all the angles of a
triangle is 180o.”

Merits of Inductive Method of Teaching:

 It is a psychological method and the interest of the student is sustained throughout.
 Helps in the development of students understanding.
 It is a natural method of making discoveries.

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 It is a logical method and thus suits the purpose of mathematics.

 Encourages active pupil participation.
 Discourages cramming and reduces homework.
 It bases on actual observation, thinking and experimentation.
 Found most suitable in the initial stage of the lesson.
 Helps in increasing pupil-teacher contacts.

De-Merits of Inductive Method of Teaching:

 Limited in range and not suitable for all topics.
 All formulae cannot be generalized in this method.
 The formulae may be more valid by increasing the number of cases. So that it is likely to
be more laborious and time-consuming.
 Inductive reasoning is not absolutely conclusive; this process establishes a certain degree
of probability.
 A complete topic cannot be covered by this method.

5.2.2 Deductive Reasoning and Method of Teaching

What is Deductive method of teaching mathematics? How can you compare it

with Inductive method of teaching?

Deductive Method is the exact opposite of the inductive method. Here we proceed from general
to particular, abstract to concrete. In this method the rule or generalization or formula is given at
the very beginning. The formula or rule is accepted as the universal truth and the student uses
them to solve problems.

Deductive reasoning is common in mathematics and logic, where elaborate structures of

irrefutable theorems are built up from a small set of basic axioms and rules.

Example: To teach the angle sum theorem of a triangle, the teacher states that; “The sum of all
the angles of triangle is 180o.”The teacher then draws different types of triangles along with the
measures. The teacher asks students to state the sum of the angles in each case. Students then
with the help of this property, solve different types of problems such as; “In triangle PQR,
measure of angle P is 50o, the measure of angle Q is 75o.Find the measure of angle R?

In deduction the law is accepted and then applied to a number of specific examples. The child
does not discover the law but develops skills in applying the same, proceeds from general to
particular or abstract to concrete. In actual practice, the combination of induction and deduction
is practiced. The laws discovered by pupils inductively are further verified deductively through
applications to new situations.

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Merits of Deductive Methods of Teaching:

 It is short and time-saving. The solution of problems by pre-established formulae takes little
time.
 It encourages memory as students have to memorize a considerable number of formulae.
 It is advantageous at the "practice and revision" stage.
 It removes the incompleteness and inadequacy of inductive method.
 The method suits all types of students.
 It is suitable for all topics.
 It enhances speed and efficiency in solving problems.

De-Merits of Deductive Methods of Teaching:

 It is not a psychological method.
 Students are passive learners and thus lose interest.
 It puts more emphasis on memory.It demands memorization of large number of formulae
and this will cause an unnecessary and heavy burden on the brain of children.
 The beginners find it very difficult to understand abstract formulae, if they are not
acquainted with a number of concrete instances.
 In this method, memory becomes more important than understanding and intelligence and
that is educationally unsound.
 Blind cramming leads very often to forgetting the formulae and the children are at a loss to
recollect. This ultimately leads to no learning.
 This method is not suitable for development of thinking, reasoning and discovery.

5.2.3 Combination of Inductive and Deductive Methods

Induction and deduction are not opposite modes of thought. There can be no induction without
deduction and no deduction without induction. Inductive approach is a method for establishing
rules and generalization and deriving formulae, whereas deductive approach is a method of
applying the deduced results and for improving skill and efficiency in solving problems. Hence a
combination of both inductive and deductive approach is known as “inducto-deductive
approach” is most effective for realizing the desired goals.

In classroom usually the instructions directly start with the abstract concepts and are being taught
in a way that does not bring understanding on the part of majority of the students. Formulae,
theorems, examples, results are derived, proved and used. But the teacher needs to start with
specific examples and concrete things and then move to generalizations and abstract things. Then
the teacher again needs to show how generalization can be derived and it holds true through
specific examples. This method will help students for better understanding; students don’t have
to cram the things and will have long lasting effect.

Thus we have seen that both the inductive as well as deductive methods by themselves have
merits and demerits. These demerits can be negated when both methods are used together in the
form of “INDUCTODEDUCTVE METHOD”. Induction and deduction are complementary to
one another. So in the beginning inductive method must be used to elicit the rule or
generalization and then this must be followed by sufficient practice using the deductive method.

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This will promote better understanding of mathematics and speed, accuracy and command over
the subject increases.

Activity 5.2.1: (take 10 minutes)

Discuss in groups of three to compare Inductive and Deductive methods of teaching

Mathematics and write a summary of your response in tabular form.

INDUCTIVE APPROACH DEDUCTIVE APPROACH

Base: Base:
Inductive reasoning Deductive reasoning
Proceeds from: Proceeds from:
 Particular to general  General to particular
 Concrete to abstract  Abstract to concrete
Method: Method:
 A psychological method  An unpsychological method
 A method of discovery and  A method of presentation and does
stimulates intellectual powers not develop originality and creativity.
Learning: Learning:
 Emphasis is on reasoning.  Emphasis is on memory
 Encourages meaningful learning  Encourages rote learning.
Level: Level:
Most suitable for initial stages of Suitable for practice and application
learning
Class: Class:
Suitable for lower classes Most suitable for higher classes
Participation: Participation:
Enhances active participation of the Makes the student passive recipient of
students knowledge
Time: Time:
Lengthy, time consuming and Short, concise and elegant
laborious
Facilitates discovery of rules and Enhances speed, skill and efficiency in
generalizations solving problems

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5.2.4 Designing Lessons to be Taught Deductively and/or Inductively

Lesson Title: Discovering Pi and circumference of a circle

Teacher Name: Mrs. Hana Kassu

Grade: Grade 9
Subject: Mathematics

Topic: Geometry: Discovering pi and how it was established.

Content: Students are often just told what pi is, and many are never able to find why pi is pi. This
activity allows students to discover pi and its relationship to the circle and how it can be
used to find the circumference.
Goals: 1. Students will know how pi came to 3.14.
2. Students will find the relationship between circumference and pi.
Objectives: 1. Students will measure the circumference and the diameter of the same object with a
measuring tape to the nearest millimeter.
2. Students will explain through a one-sentence summary how the number 3.14 was
determined.
3. Students will demonstrate dividing the circumference of a shape by its diameter results
in pi or 3.14.
4. Students will find the formula for circumference using pi and demonstrate it.
Materials: round objects, measuring tapes, calculators
Introduction: Students will see the symbol pi on the board. Students will also see the goal posted for the
class period so they know where our learning will be focused. We will work independently
for two minutes and then share ideas with the class. This will allow for each student to
become actively involved.
Development: I will demonstrate for students the procedure during their math lab. I will use a measuring
tape to illustrate how to measure both the circumference and the diameter to the nearest
millimeter. Students will observe how the information is placed in the chart on their math
lab. In the last column of their chart is the section where they will divide the
circumference by the diameter and round that answer to the nearest hundredths. Students
will be given the opportunity to ask questions.
Practice: Students will work in pairs visiting four different stations. Students will be given
approximately two minutes at each station and then will proceed to a total of four stations.
Each group will have a calculator to use for the division of the circumference by the
diameter. Each independent student will need to fill in their own chart to illustrate their
learning. After students have visited the four stations, I will put two groups together to
compare their results. Students will be given three minutes to compare and then we will
meet back together. We as a whole group will discuss their results and post them on the
board. Students will make connections between pi and circumference. We will create a
formula together for finding the circumference of a circle.
Accommodations: As groups are working, I will rotate around helping those who need assistance. In a
routine setting, I would know which students needed accommodations and I would have
prepared an alternate assessment with less problems.
Checking For Students will return to their seat and fill in the last column. Whole class will share some
Understanding: of the pieces that we learned today.
Closure: Students will complete a one-sentence summary of what they learned today. After
completing their summary if there are no questions, students will be given the opportunity
to begin working on their assignment.
Evaluation: Students will attain at least a 70% on their homework assignment tomorrow. Students will
turn in their math lab which will illustrate their learning and participation.

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Self Assessment Exercise

1. In the above lesson plan which method of teaching is used? Why?

2. Can we use other teaching methods to teach the same lesson effectively?
Which one?

5.3 Analytic and Synthetic Methods in Teaching Mathematics (3 hrs.)

5.3.1 Analytic Method of Teaching

What is analytic method? How can you apply it in the teaching of mathematics?
Is there a disadvantage of using it?

The word “analytic” is derived from the word “analysis” which means “breaking up” or
resolving a thing into its constituent elements. The original meaning of the word analysis is to
unloose or to separate things that are together. In this method we break up the unknown
problem into simpler parts and then see how these can be recombined to find the solution. So we
start with what is to be found out and then think of further steps or possibilities that may connect
the unknown, built the known and find out the desired result. It is believed that all the highest
intellectual performance of the mind is analysis.

Analysis also means, “Breaking up of a given problem, so that it connects with what is already
known.” It leads to
 conclusion to hypothesis
 unknown to known
 abstract to concrete

Analytic method is used under the following conditions:

 When we have to prove any theorem.
 Can be used for construction problems.
 To find out solutions of new arithmetical problems.

Analytic method is particularly suitable for teaching of arithmetic, algebra and geometry as it
analyses the problem into sub-parts and various parts are reorganized and the already learnt facts
are used to connect the known with unknown. It puts more stress on reasoning and development
of power of reasoning which is one of the major aims of teaching of mathematics.
Example: If a2 + b2=7ab, then prove that 2log (a+b) = 2log3 + loga + logb.

Proof:
To prove this using analytic method, begin from the unknown. The unknown is 2log (a+b) =
2log3+loga+logb.
Now, 2log (a+b) = 2log 3+ log a + log b is equivalent to the following:

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log (a+b)2 = log 32 + log a + log b, why?

log (a+b)2 = log 9 + log ab, why?
log (a+b)2 = log 9ab, why?
(a+b)2 = 9ab, why?
a2+b2=7ab which is known and true.
Thus if a2+b2= 7ab then 2log (a+b) = 2log3+loga+logb.

5.3.2 Merits and Demerits of Analytic Method

Merits of analytic methods are as follows:
 Logical, leaves no doubt.
 Each step has reason and justification.
 The method suits the learner and the subject.
 It develops the power of thinking and reasoning.
 It develops originality and creativity amongst the students.
 It helps in a clear understanding of the subject because the students have to go through the
whole process themselves.
 There is least home work.
 In analytic method student’s participation is encouraged and maximum.
 It is a psychological method.
 No cramming is required in analytic method.
 It develops self-confidence and self reliant in the pupil.
 Knowledge gained by analytic method is more solid and durable.
Demerits of Analytic Methods are:
 It is time consuming and lengthy method, so it is uneconomical.
 In analytic method, facts are not presented in a neat and systematic order.
 Analytic method is not suitable for all the topics in mathematics.
 It does not find favor with all the students because below average students fail to follow
analytic method.

Self Assessment Exercise

Discus how analytic method of teaching is effective in teaching mathematics?

5.3.3 Synthetic Method of Teaching

What is synthetic method? How can you compare it with analytic method of
teaching?

In synthetic method we proceed from known to unknown. Synthetic is derived from the word
“synthesis”. Synthesis is the complement of analysis. To synthesis is to combine the elements to
produce something new. Actually it is reverse of analytic method. So in it we combine together
a number of facts, perform certain mathematical operations and arrive at a solution.

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It is the method of formulation, recording and presenting concisely the solution without any trial
and errors. It leads to:
 hypothesis to conclusion
 known to unknown
 concrete to abstract

Example:If a2 + b2 = 7ab, then prove that 2log (a + b) = 2log3 + loga + logb.

Proof:
To prove this using synthetic method, begin from the known.The known is a2 + b2 = 7ab.

Adding 2ab on both sides, we get:

a2 + b2 + 2ab = 7ab + 2ab which is equivalent to:
(a + b)2 = 9ab.
Taking logarithm on both sides, we have:
log (a + b)2 = log 9ab, which is equivalent to the following:
2log (a + b) = log 9 + log ab. Why?
2 log (a + b) = log 32 + log a + log b. Why?
2log (a + b) = 2log 3 + log a + log b. Why?

Thus if a2 + b2 =7ab, then 2log (a + b) = 2log3 + loga + logb.

5.3.4 Merits and Demerits of Synthetic Method

Merits of Synthetic Method are as follows:

 It saves time and labour.

 It is short and precise.
 It is a neat method in which we present the facts in a systematic way.
 It suits majority of students.
 It can be applied to a majority of topics in mathematics.
 It glorifies the memory of the child.
 Omits trial and error as in analysis method.
 Accuracy is developed by the method.

Demerits of Synthetic Method are:

 It is an unpsychological method.
 There is a scope for forgetting.
 It makes the students passive listeners and encourages cramming.
 In synthetic method confidence is generally lacking in the student.
 There is no scope of discovery, no opportunity to develop the skills of thinking and
reasoning, as understanding is hampered.
 The recall of each step cannot be possible for every child.

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From the above discussion we can see that both the methods of analysis and synthesis by
themselves have their advantages and disadvantages. In order to ensure the complete
understanding of mathematics in the learners, both the methods can be used together to teach
mathematics. By using a combination of these two methods the teacher can ensure that effective
teaching learning takes place.

Activity 5.3.1: (take 10 minutes)

Discuss in groups of five to compare Analytic and Synthetic methods of teaching

mathematics and write a summary of your response in tabular form.

Feedback for Activity 5.3.1

ANALYTIC METHOD SYNTHETIC METHOD

Meaning: Meaning:
Analysis means breaking up into Synthesis means combining the
components. elements to get something new.
Leads from: Leads from:
 Unknown to known.  Known to unknown.
 Conclusion to hypothesis.  Hypothesis to conclusion.
 Abstract to concrete.  Concrete to abstract.
 Complex to simple.  Simple to complex.
Method: Method:
 A method of discovery and  A method for the presentation of
thought. discovered facts.
 A psychological method.  A logical method.
Time: Time:
Lengthy, laborious and time Short, concise and elegant.
consuming.
Sequence: Sequence:
Valid reasons to justify every step No justification for every step in the
in the sequence. sequence.
Learning: Learning:
Encourages meaningful learning. Encourages rote learning.
Informal and disorganized. Formal, systematic and orderly.
Easy to rediscover. Once forgotten not easy to recall.
Encourages: Encourages:
Encourages originality of thinking Encourages memory work.
and reasoning.
Thinking: Thinking:
Process of thinking Product of thinking
Participation: Participation:
Active participation of the learner Learner is a passive listener

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5.4 Problem Solving Method in Teaching Mathematics (5 hrs.)

5.4.1 Meaning and Definitions of Problem Solving

What is the difference between exercise and problem? What is problem solving?

When most students, and many faculty members, think of problem solving, they imagine a
homework exercise with a single correct answer. Working out homework exercises can be
considered problem solving to the extent that this activity reinforces the use of standard
equations and cultivates specific problem-solving skills such as pattern recognition. However,
open-ended problem solving presents a higher level of challenge because it requires the problem
solver to respond to situations which are completely new to him or her. The following definition
by Woods captures this essence. “The problems that we focus on to solve are ones where there is
no immediately apparent procedure, idea, or routine to follow; if one has an idea how to solve
‘the problem,’ then this problem is simply an exercise. What we call a problem is a real
challenge; it is a situation where we really have to struggle to define it, figure out what it means,
and resolve it”.

A task (question) is said to be a routine problem (exercise) if the students know an algorithm or
solving procedures for answering this questions at the beginning. On the other hand, a question is
said to be a problem if the students do not know an algorithm or procedure for answering this
question at the beginning of the solving process, which means there is a gap between a current
state of the question and a desired state of the question.

Problem-solving is a mental process that involves discovering, analyzing and solving problems.
The ultimate goal of problem-solving is to overcome obstacles and find a solution that best
resolves the issue.

Problem–solving may be a purely mental difficulty or it may be physical and involve

manipulation of data. Problem-solving method aims at presenting the knowledge to be learnt in
the form of a problem. It begins with a problematic situation and consists of continuous,
meaningful, well-integrated activity. The problems are test to the students in a natural way and it
is ensured that the students are genuinely interested to solve them.

Self Assessment Exercise

Explain what mathematical problems and problem solving mean. Give illustrative
examples.

5.4.2 Significance of Problem Solving

Problem solving is an important component of mathematics education because it is the single

vehicle which seems to be able to achieve at school level the functional, logical and aesthetic
values of mathematics. It has already been pointed out that mathematics is an essential discipline
because of its practical role to the individual and society. Through a problem-solving approach,
this aspect of mathematics can be developed. Presenting a problem and developing the skills

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needed to solve that problem is more motivational than teaching the skills without a context.
Such motivation gives problem solving special value as a vehicle for learning new concepts and
skills or the reinforcement of skills already acquired. Approaching mathematics through problem
solving can create a context which simulates real life and therefore justifies the mathematics
rather than treating it as an end in itself.

Problem solving is more than a vehicle for teaching and reinforcing mathematical knowledge
and helping to meet everyday challenges. It is also a skill which can enhance logical reasoning.
Individuals can no longer function optimally in society by just knowing the rules to follow to
obtain a correct answer. They also need to be able to decide through a process of logical
deduction what algorithm, if any, a situation requires, and sometimes need to be able to develop
their own rules in a situation where an algorithm cannot be directly applied. For these reasons
problem solving can be developed as a valuable skill in itself, a way of thinking, rather than just
as the means to an end of finding the correct answer. Many writers have emphasized the
importance of problem solving as a means of developing the logical thinking aspect of
mathematics. 'If education fails to contribute to the development of the intelligence, it is
obviously incomplete. Yet intelligence is essentially the ability to solve problems: everyday
problems, personal problems ... '(Polya, 1980).

Although mathematical problems have traditionally been a part of the mathematics curriculum, it
has been only comparatively recently that problem solving has come to be regarded as an
important medium for teaching and learning mathematics. So that the mathematics curriculum
should be organized around problem solving, focusing on:

(i) developing skills and the ability to apply these skills to unfamiliar situations;
(ii) gathering, organizing, interpreting and communicating information;
(iii) formulating key questions, analyzing and conceptualizing problems, defining problems and
goals, discovering patterns and similarities, seeking out appropriate data, experimenting,
transferring skills and strategies to new situations;
(iv) developing curiosity, confidence and open-mindedness .

Goals of Mathematical Problem-Solving

The specific goals of problem solving in mathematics are to:

i) Improve pupils' willingness to try problems and improve their perseverance when
solving problems.
ii) Improve pupils' self-concepts with respect to the abilities to solve problems.
iii) Make pupils aware of the problem-solving strategies.
iv) Make pupils aware of the value of approaching problems in a systematic manner.
v) Make pupils aware that many problems can be solved in more than one way.
vi) Improve pupils' abilities to select appropriate solution strategies.
vii) Improve pupils' abilities to implement solution strategies accurately.
viii) Improve pupils' abilities to get more correct answers to problems.

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Self Assessment Exercise

1. What is the significance of problem solving?
2. What are the goals of problem solving? Discuss with your group
members.

5.4.3 Steps in Problem Solving

Polya’s Four-Step Process for Problem Solving

Polya’s model is recommended as a guide to be used in teaching problem-solving, first outlined

by George Polya in 1945. Using this model he identifies four steps in the problem-solving
process:

1. UNDERSTANDING THE PROBLEM

You cannot solve a problem if you do not understand what you are asked to find. The problem
must be read and analyzed carefully. You will probably need to read it several times. After you
have done so, ask yourself, “What must I find?” So you need to answer the following questions.

 Can you state the problem in your own words?

 What are you trying to find or do?
 What are the unknowns?
 What information do you obtain from the problem?
 What information, if any, is missing or not needed?

2. DEVISING A PLAN

There are many ways to attack a problem and decide what plan is appropriate for the particular
problem you are solving. The following list of strategies, although not exhaustive, is very useful.

 Look for a pattern.

 Examine related problems, and determine if the same technique can be applied.
 Examine a simpler or special case of the problem to gain insight into the solution of
the original problem.
 Make a table.
 Make a diagram.
 Write an equation.
 Use guess and check.
 Work backward.
 Identify a sub goal.

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3. CARRYING OUT THE PLAN

Once you know how to approach the problem, carry out your plan. You may run into “dead
ends” and unforeseen roadblocks, but be persistent. If you are able to solve a problem without a
struggle, it isn’t much of a problem, is it?

 Implement the strategy or strategies in step 2, and perform any necessary actions or
computations.
 Check each step of the plan as you proceed. This may be intuitive checking or a formal
proof of each step.
 Keep an accurate record of your work.

4. LOOKING BACK
Check your answer to see that it is reasonable. Check and interpret the results in terms of the
original problem. If possible, determine other related or more general problems for which the
techniques will work. In addition you need to answer the following questions.

 Does it satisfy the conditions of the problem?

 Have you answered all the questions the problem asks?
 Does your answer make sense? Is it reasonable?
 Can you solve the problem a different way and come up with the same answer?

Self Assessment Exercise

Explain the steps in the problem-solving process.

5.4.4 Additional Strategies used while Teaching Problem Solving

In Step 2 of Polya’s problem-solving process, we are told to devise a plan. Here are some
strategies that may prove useful.

Make a table or a chart. If a formula applies, use it.

Look for a pattern. Work backward.
Solve a similar simpler problem. Guess and check.
Draw a diagram. Use trial and error.
Use inductive reasoning. Use common sense.
Write an equation and solve it.
Look for a “catch” if an answer seems too obvious or impossible.
Use a variable
Use symmetry

A particular problem solution may involve one or more of the strategies listed here, and you
should try to be creative in your problem-solving techniques.

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The following table shows the different types of strategy in solving problems with some of the
corresponding clues.

Strategies The strategy may be appropriate when

 You have a good ideas of what the answer is
Guess and check  There is a limited number of possible answers to test
 You can systematically try possible answers
 There is no other obvious strategy to try

 A problem suggests an equation

Use a variable  A proof or a general solution is required
 There is unknown quantity related to known quantity
 You are trying to develop a general formula

 A list of data is given

 A sequence of numbers is involved
Look for a pattern  Listing special cases helps you deal with complex problems
 Information can be expressed & viewed in organized manner

 Data can easily be generated

Make a list  Listing the results obtained by using guess and check
 Asked “in how many ways” something can be done, etc

 The problem involves sets, ratios, probabilities

Draw a diagram  A diagram is more efficient to solve the problem
 Representing relationships among quantities

 A direct solution is too complex

Solve a simpler  The problem involves complicated computations
problem  You want to gain a better understanding of the problem, etc

 A proof is required
Use direct reasoning  A statement of the form “ if …, then …”is involved
 You see a statement that you want to imply from a collection of

 A proof is required
 Direct reasoning seems too complex or does not lead to a solution
Use indirect reasoning  Assuming the negation of what you are trying to prove narrows the
scope of the problem

 A problem is related to another problem you have solved previously

Solve equivalent  You can find an equivalent problem that is easier to solve
problem  A problem can be represented in a more familiar setting

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 The final result is clear and the initial portion of a problem is obscure
 A problem proceeds from being complex initially to being simple at the
Work backward end
 A direct approach involves a complicated equation
 A problem involves a sequence of reversible actions

Use cases  A problem can be separated into several distinct cases

 Investigations in specific cases can be generalized
 Solving problems involving in the problem
 A problem suggests a pattern that can be generalized
Look for a formula  Ideas such as area, volume, distance, or other measurable attributes are
involved
 Application problems are involved
 Geometry problems involve transformations
Use symmetry  Interchanging values does not change the representation of the problem
 Symmetry limits the number of cases that need to be considered
 Pictures or algebraic expressions are symmetric
 known conditions

Example: Solving a Problem by Using a Table or a Chart

A man put a pair of rabbits in a cage. During the first month the rabbits produced no offspring
but each month thereafter produced one new pair of rabbits. If each new pair thus reproduces in
the same manner, how many pairs of rabbits will there be at the end of one year?

This problem is a famous one in the history of mathematics and first appeared in Liber Abaci, a
book written by the Italian mathematician Leonardo Pisano (also known as Fibonacci) in the year
1202. Let us apply Polya’s process to solve it.

1. Understand the problem:

After several readings, we can reword the problem as follows: How many pairs of rabbits will
the man have at the end of one year if he starts with one pair, and they reproduce this way:
During the first month of life, each pair produces no new rabbits, but each month thereafter each
pair produces one new pair?

2. Devise a plan:
Since there is a definite pattern to how the rabbits will reproduce, we can construct a table as
shown below. Once the table is completed, the final entry in the final column is our answer.

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3. Carry out the plan:

At the start of the first month there is only one pair of rabbits. No new pairs are produced during
the first month, so there is one pair present at the end of the first month. This pattern continues
throughout the table. We add the number in the first column of numbers to the number in the
second column to get the number in the third.

There will be 233 pairs of rabbits at the end of one year.

4. Look Back and Check:

This problem can be checked by going back and making sure that we have interpreted it
correctly, which we have. Double check the arithmetic. We have answered the question posed by
the problem, so the problem is solved.

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The sequence shown in last column of the table is the Fibonacci sequence, and many of its
interesting properties can be seen in different fields of study.

Self Assessment Exercise

1. Solve the following word problem by “Guessing and Checking”. One-fourth
of a herd of camels was seen in the forest; twice the square root of that herd
had gone to the mountain slopes; and 3 times 5 camels remained on the
riverbank. What is the numerical measure of that herd of camels?
2. Solve the following word problem by “Use a variable”. Find the sum of the
whole numbers from 1 to 1000.
3. Solve the following word problem by “Draw a picture”. Can you cut a pizza
into 11 pieces with four straight cuts?

5.4.5 Designing Problem Solving Lesson

A three-part lesson format

You may be inclined to agree that teachers typically spend a small portion of the allocated time
in explaining or reviewing an idea, followed by learners working through a list of exercises – and
more often than not, rehearsing the procedures already memorized. This approach conditions the
learners to focus on procedures so that they can get through the exercises.
Before, during and after
Teaching through problem solving does not mean simply providing a problem or task, sitting
back and waiting for something to happen. The teacher is responsible for making the atmosphere
and the lesson work. To this end, Van de Walle (2004) sees a lesson as consisting of three main
parts: before, during and after. He proposes the following simple three-part structure for lessons
when teaching through problem solving:

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If you allow time for each of the before, during and after parts of the lesson, it is quite easy to
devote a full period to one seemingly simple problem. In fact, there are times when the ‘during’
and ‘after’ portions extend into the next day or even longer! As long as the problematic feature
of the task is the mathematics you want learners to learn, a lot of good learning will result from
engaging learners in only one problem at a time.

Teacher’s actions in the before phase

What you do in the before phase of a lesson will vary with the task. The actual presentation of
the task or problem may occur at the beginning or at the end of your ‘before actions’. However,
you will have to first engage learners in some form of activity directly related to the problem in
order to get them mentally prepared and to make clear all expectations in solving the problem.
The following strategies may be used in the before-phase of the lesson:
 Begin with a simple version of the task – reduce the task to simpler terms.
 Brainstorm: where the task is not straight forward, have the learners suggest solutions and
strategies – producing a variety of solutions.
 Estimate or use mental computation – for the development of computational procedure,
have the learners do the computation mentally or estimate the answer independently.
 Be sure the task is understood – this action is not optional. You must always be sure that
learners understand the problem before setting them to work. Remember that their
perspective is different from yours.

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 Have them restate the problem in their own words – this will force them to think about
the problem.
 Establish expectations. This action is essential. Learners need to be clearly told what is
expected of them. For example:
 Explain (in writing) why you think your answer is correct.
 When learners are working in groups, only one written explanation should come from
the group.
 Share your ideas with a partner and then select the best approach to be presented.

Teacher’s actions in the during phase

 Once you are comfortable that learners are ready to work on the task, it is time to let go –
your role now shifts to that of a facilitator:
 You must demonstrate confidence and respect for your learners’ abilities.
 Your learners should get into the habit of working in groups – to treat in co-operative
group work.
 Listen actively – find out what your learners know, how they think, and how they are
approaching the task.
 Provide hints and suggestions – when the group is searching for a place to begin, when
they stumble. Suggest that they use a particular manipulative or draw a picture if that
seems appropriate.
 Encourage testing of ideas. Avoid being the source of approval for their results or ideas.
Instead, remind the learners that answers, without testing and without reasons, are not
acceptable.
 Find a second method. This shifts the value system in the classroom from answers to
processes and thinking. It is a good way for learners to make new and different connections.
The second method can also help learners who have made an error to find their own mistake.
 Suggest extensions or generalizations. Many of the good problems are simple on the surface.
It is the extensions that are excellent. The general question at the heart of mathematics as a
science of pattern and order is: What can you find out about that? This question looks at
something interesting to generalize.
The following questions help to suggest different extensions: What if you tried…….? Would the
idea work for …….?

Teacher’s actions in the after phase

This ‘after’ phase is critical – it is often where everyone, learners as well as the teacher learn the
most. It is not a time to check answers, but for the class to share ideas.
In the ‘after phase’ of a lesson, teachers may find that they will engage in the following
activities:

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 Engage the class in discussion.

Rule number one is that the discussion is more important than hearing an answer. Learners must
be encouraged to share and explore the variety of strategies, ideas and solutions – and then to
communicate these ideas in a rich mathematical discourse.
 List the answers of all groups on the board without comment.
Unrelated ideas should be listened to with interest, even if they are incorrect. These can be
written on the board and testing the hypothesis may become the problem for another day – until
additional evidence comes up that either supports or disproves it.
 Give learners space to explain their solutions and processes.
A suggestion here is to begin discussion by calling first on children who are shy, passive or lack
the ability to express themselves – because the more obvious ideas are generally given at the
outset of a discussion. These reticent learners can then more easily participate and thus be
valued.
 Allow learners to defend their answers, and then open the discussion to the class.
Resist the temptation to judge the correctness of an answer.
In place of comments that are judgemental, make comments that encourage learners to extend
their answers, and that show you are genuinely interested. For example: ‘Please tell me how you
worked that out.’
 Make sure that all learners participate, that all listen, and that all understand what
is being said.
 Encourage learners to ask questions, and use praise cautiously.
You should be cautious when using expressions of praise, especially with respect to learners’
products and solutions. ‘Good job’ says ‘Yes, you did that correctly’. However, ‘nice work’ can
create an expectation for others that products must be neat or beautiful in order to have value – it
is not neatness, but good mathematics that is the goal of mathematics teaching.

Self Assessment Exercise

1. Discuss about the three-part lesson formatand write a paragraph about teacher’s
actions in the before, during and after phase.
2. Why are problem-solving skills important to the young child?
3. Give examples of problem-solving situations in mathematics. Give examples
of word problems that are technically not problem-solving situations.

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Unit Summary
Mathematical concepts, although abstract concepts, reflect some characteristics of the real world
and in that way contribute to their awareness.

A misconception does not exist independently, but is contingent upon a certain existing conceptual
framework. Misconceptions can change or disappear with the framework changes.Misconceptions
definitely interfere with learning when students use them to interpret new experiences. Teaching
students to resolve a misconception is very much dependent on the nature of the misconception within
the specific student’s mind.

Mathematical concepts are the most learnable part of subject matter of mathematics and are
building blocks of the subject mathematics.

The set of mathematical concepts are subdivided into three categories. These are Concepts about
objects, Concepts about operations, and Concepts about relations.

The teaching of mathematical concepts and definitions are carried out by using various methods.
Among these methods, the most common ones are the inductive and deductive approach.

Inductive method leads from concrete to abstract, particular to general and from examples to the
rule.

Deductive Method is the exact opposite of the inductive method. Over here we proceed from
general to particular, abstract to concrete.

Induction and deduction are complementary to one another. So in the beginning inductive
method must be used to elicit the rule or generalization and then this must be followed by
sufficient practice using the deductive method. This will promote better understanding of
mathematics and speed, accuracy and command over the subject increases.

In Analysis method we break up the unknown problem into simpler parts and then see how these
can be recombined to find the solution. So we start with what is to be found out and then think of
further steps or possibilities the may connect the unknown built the known and find out the
desired result.

In Synthetic method we proceed from known to unknown. Actually it is reverse of analytic

method. So in it we combine together a number of facts, perform certain mathematical
operations and arrive at a solution.

We can see that both the methods of analysis and synthesis by themselves have their advantages
and disadvantages. In order to ensure the complete understanding of mathematics in the learners,

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both the methods can be used together to teach mathematics. By using a combination of these
two methods the teacher can ensure that effective teaching learning takes place.

A task (question) is said to be a routine problem (exercise) if the students know an algorithm or
solving procedures for answering this questions at the beginning. On the other hand, a question is
said to be a problem if the students do not know an algorithm or procedure for answering this
question at the beginning of the solving process which means there is a gap between a current
state of the question and a desired state of the question.

Problem-solving method aims at presenting the knowledge to be learnt in the form of a problem.
It begins with a problematic situation and consists of continuous, meaningful, well-integrated
activity.

Problem solving is more than a vehicle for teaching and reinforcing mathematical knowledge
and helping to meet everyday challenges. It is also a skill which can enhance logical reasoning.

Polya’s model is recommended as a guide to be used in teaching problem-solving, first outlined

by George Polya in 1945. Using this model he identifies four steps in the problem-solving
process:

1. Understanding the Problem

2. Devising a Plan
3. Carrying out the Plan
4. Looking Back

Further Reading for the Unit:

To complete your study of this unit, you will need to refer to:
 G. Polya (1945) A new Aspect of Mathematics Method, USA
 Sudhir Kumar, D.N. Ratnalikar (2003): Teaching of mathematics (3rd Ed.), Anmol
Publications, New Delhi.
 Gray L. Musser, William F. Burger, Mathematics for Elementary Teachers, a
Contemporary Approach
 Douglas K. Brumbaugh, Teaching Secondary Mathematics, 1997 Printed in the United
States of America.
 Pamela COWAN(2006): Teaching MATHEMATICS A HANDBOOK FOR
PRIMARY AND SECONDARY SCHOOL TEACHERS, LONDON AND NEW
YORK.

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Module Summary
In this first module of mathematics subject area teaching methods, you have covered five units
each containing two sections except the fifth unit, which contains four sections. Each section
again contains several subsections, activities, and self assessment exercises. I hope you pass
through all of the contents of the module and gain enough information which will be helpful in
your future teaching career.

The following are some of the major points discussed in the module.

 The teaching and learning of mathematics cannot be justified without some understanding of
the historical as well as philosophical backgrounds of the subject. This is because the
philosophical perception of an individual teacher could definitely affect his /her methods of
teaching as well as learning.

 Mathematics is a subject of great educational values and makes a major contribution in

achieving the aims of education. It has got many educational values which determine the need
of teaching the subject in schools.

 As curriculum contains the most important elements of instruction, it can be used as sources
of information for teachers in the teaching learning process. That means the curriculum can
play a great role in the preparation of objectives, in the selection of contents, in the selection
of methodology and assessment tools while teachers are preparing their daily, unit, semester
or annual plans.

 A teacher should adopt a teaching approach after considering the nature of the
children, their interests and maturity and the resources available. Every method
has certain merits and few demerits and it is the work of a teacher to decide
which method is best for the students.

 Pupils learn things through their senses. Teachers can also find means to have effective
interaction with pupils’ senses in order to help them learn better. The materials which
supplement teachers’ efforts and facilitate teaching-learning are known as teaching materials.

 Evaluation can be regarded as an integral part of an instructional program of activity

regardless of the length of time or period the instruction lasted.

 Planning helps teachers allocate instructional time, select appropriate activities, link
individual lessons to the overall unit or curriculum, sequence activities to be presented to
students, set the pace of instruction, select the homework to be assigned, and identify
techniques to assess student learning.

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Thus, a successful completion of this module helps you to make a significant contribution to the
teaching profession in the quality of education. So, my hope is that you will take what you learn
in Mathematics Subject Area Teaching Methods I with you and make a difference.

Course Assessment Strategies:

Continuous assessment will be the major assessment strategy to be used in assessing and
evaluating students’ learning and the attainment of the overall objectives of the course in general.
Particularly students’ major projects, portfolios, term papers and tests given at the end of each
unit will serve as tools to assess the attainment of the objectives of the course. Depending on the
need of the trainees a final paper and pencil exam that may not count more than 30% of the
overall assessment will be used.

Course requirement:

 Each candidate must fully attend all lecture classes (at least 80%) and presentation sessions
regularly for each units within the course
 Each candidate is expected to demonstrate active participation in the classroom
 Each candidate must complete all assignments and activities within each unit of the course to
the satisfaction of the instructor(s)
 Each candidate is expected to meet deadlines for submission of assignments as well as
presentations
 Each candidate must sit for tests at the end of each unit and perform to the satisfaction of the
course instructor (s)
 Each candidate must sit if there will be a final paper and pencil exam

Grading Schemes:

Grading scale will be criterion referenced

 Portfolios 20%
 Tests at the end of each unit 20%
 Pear teachings 10%
 Major Term papers and presentations 20%
 Final paper and pencil exam 30%

The following fixed scale will be used for determining students’ letter grade:
90 – 100 % ……………………… A
70 – 89 % …………………………B
50 – 69 % ………………………....C
30 – 49 % …………………………D
0 – 29 % …………………………F

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References:
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Vikaspublishing House Pvt. Ltd.

Anthony Orton and Wain, G. (eds). (1994). Issues in Teaching Mathematics.

London: Cassell.
Arnold, A., Shiu, C. &Ellerton, N. (1996) Critical issues in the distance teaching
ofmathematics and mathematics education.In A. J. Bishop K.Clements, C, Keitel,
J.
Asturias, H. (1994). Implementing the assessment standards for school
mathematics:using students’ portfolios to assess mathematical
understanding.Mathematics Teacher, 87(9), 698-701.
Borich, G.D. (1988). Effective Teaching Methods.New Macmillan PublishingCompany
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Bloom, B.S. (Ed.) (1956).Taxonomy of educational objectives: The classification ofeducational
goals. Handbook 1: Cognitive domain. New York: McKay.
Boyer, C. B. (1991).A history of mathematics (2nd ed.). New York: Wiley.
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Callahan, s. g. 1966.Successful Teaching in Secondary Schools,New York: Macmillan
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Columba, L., and Dolgos, K. A. (1995).Portfolio assessment in mathematics.Reading

Improvement, 32 (3), 174-176.
Davis, I.K. 1981. Instructional Techniques. New York: MC Graw Hill BookCompany.
David, P., &Eric, L. (eds). (1991). Teaching and Learning School Mathematics.London: Open
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Learning.New York: Macmillan.
EB Rutikam, P. (2002).Philosophies, Historical Dimensions of Mathematics, Mathematics
Education and Logic.University of South Africa, Pretoria
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Computing Teacher.v21 n6 p22-23.
Kilpatrick, & C. Laborde (Eds.) ().International handbook of mathematics education(pp.701-
754).Dordrecht, the Netherlands: Kluwer.
Kochhar, S.K. (1985). Methods and Techniques of Teaching.New Delhi: SteringPublishers
private Limited.
Laborde (Eds.) ().International handbook of mathematics education (pp. 327-
370).Dordrecht, TheNetherlands: Kluwer.
Larcombe A. (1985). Mathematical Learning Difficulties in the Secondary
School.Philadelphia: Open University Press.

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Midkiff, R. B.,&Thomasson, R. D. (1993).A practical approach to using learning styles in

math instruction. Springfield, IL: ThomasBooks.
Paul, E. (1990). The Philosophy of Mathematics Education.Falmer Press.
Perrott, E.(1982. Effective Teaching: A practical guide to improving your teaching. New
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Struik, J. (1948). A Consinse History of Mathematics.Dover publication Inc; U.S.A
Sudhir Kumar, D.N. Ratnalikar (2003): Teaching of mathematics (3rd Ed.), Anmol Publications,
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