CHAPTER FIRST

INTRODUCTION

1.0 INTRODUCTION

Human being needs minimum knowledge of Mathematics for their survival. It

is needed daily by everybody. In this complex world passing through scientific and

technological age, the practical value of Mathematics is going to be increasingly felt

and recognised. Counting notation, addition, subtraction, multiplication, division,

weighing, measuring, buying, and many more are simple and fundamental process of

Mathematics which have got an enormous practical value in life.

The knowledge and skill in these processes can be provided effectively by

adopting innovative practices in teaching Mathematics in schools. It is our common

observation that the Mathematics subject is mainly taught by lectures. The teachers

rarely use other methods of instruction. Self developed materials are also used rarely

in teaching learning of the Mathematics subject, which results in memorization for

examination purpose and it is also observed that the students forget the content and

rarely apply it in their day-to-day. Self developed materials can help to increase the

interest of students and teachers.

Mathematics Experiment Notebooks (MEN) is new Self developed material

that can be used at upper primary classes. Development of Mathematics Experiment

Notebooks (MEN) has provided new opportunities for delivering instruction in

innovative ways. The researcher is a teacher educator working in the field since last

17 years and related to teaching “Mathematics”. He felt unsatisfied about strategy of

instruction and hence decided to undergo a research on use of Mathematics

Experiment Notebooks (MEN) to teaching Mathematics for upper primary classes.

The Mathematics Experiment Notebooks (MEN) can be more effective in comparison

with the traditional.

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In this chapter the researcher has discussed about the history of Mathematics

teaching, importance of Mathematics in human life, methods of teaching

Mathematics, innovative practices in Mathematics teaching, present status of

Mathematics at upper primary level, statement of the study, definitions of the terms,

significance , scope , delimitations , objectives , assumptions , hypotheses and

organisation of the study.

1.1 HISTORICAL REVIEW OF DEVELOPMENTS IN MATHEMATICS

Mathematics is a man made and progressive science. Sudhir Kumar (2000)

and Sidhu, Kulbir Singh (1985) explained the some remarkable development, ideas

and processes in Mathematics in his books. They are considered to be landmarks in

the history of Mathematics in the following way.

The Shulba Sutra‟s (800 BC–200 AD), which gave simple rules for constructing altars

of various shapes, such as squares, rectangles, parallelograms, and others. The Shulba

Sutras gave methods for constructing a circle with approximately the same area as a

given square, which imply several different approximations of the value of π. The

Shulba Sutras compute the square root of 2 and gave a statement of the Pythagorean

Theorem.

In the 5th century AD, Aryabhata wrote the Aryabhatiya, it is related to rules

of calculation used in astronomy and mathematical mensuration.

In the 7th century, Brahmagupta identified the Brahmagupta theorem,

Brahmagupta's identity and Brahmagupta's formula, and for the first time, he clearly

explained the use of zero as both a place holder and decimal digit, and explained the

Hindu-Arabic numeral system.

In the 12th century, Bhaskara II lived in southern India and wrote extensively

on all then known branches of mathematic. His work contains mathematical objects

equivalent or approximately equivalent to infinitesimals, derivatives, the mean value

theorem and the derivative of the sine function. To what extent he anticipated the

invention of calculus is a controversial subject among historians of mathematics

In the history of mathematics there have been some very remarkable

developments in the form of discovery and evolution of certain ideas and processes.

These ideas and processes claim special status and significance in the overall progress

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of mathematical knowledge. They are considered to be landmarks in the history of the

subject. At this moment we shall discuss only few of them.

A) Notation System

Number is one of the roots of mathematics, but it has taken roots and flowered

into a full-fledged system over the period of centuries. The origin of notation system

is as old as the man himself. It is claimed that birds and insects also have some

number sense. This elementary concept took centuries to take any concrete shape.

The primitive man used various ways to count. He used fingers, cuts in the

tree, lines on the ground, pebbles etc. for the purpose. The notation system originated

and developed differently in different countries.

a) Babylonians

The Babylonians used wedge-shaped symbols. One was represented by V, ten

was represented by <, hundred by V<.

b) Hebrews and Greeks

They used letters of the alphabet to denote numbers. For example X (alpha)

stood for one, B (beta) stood for two, and i (iota) stood for ten.

c) Roman System

The roman system is based on the idea of counting by fingers or lines. Thus I,

II, III represented one, two, and three respectively. V stands for the whole hand or the

gap between the thumb and the four fingers. To avoid the clumsy IIII for four, they

wrote I before V. This symbol also gave rise to the idea of positional value. Similarly

symbols VI, VII etc., were developed. The symbol X for ten seems to be the

combination of two fives.

d) Hindu-Arabic System

Our present system in the form of numerals 1, 2, 3, etc., was originated by

Hindus. It was transmitted to the West through the Arabs and hence the name Hindu-

Arabic System.

The early Hindus had no symbol for zero and they had different symbol for

10, 20, and 30 etc. Later on the symbol 0 was evolved by the Hindu mind, meaning

„Sunya‟ i.e. nothingness. One interpretation connects the concept of zero with Hindu

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concept of Nirvana, the ideal state of nothingness. Arabs also made certain

modifications in the Hindu numerals. During the 13th

century the Hindu Arabic

numerals became popular in the West and virtually spread all over the world.

By and by, certain modifications and improvements were effected and the

Hindu Arabic system assumed the present shape.

B) Metric System of Weights and Measurements

Before the evolution of metric system of weights and measurements, there was

a wide variety of measuring units prevalent in different parts of the world. Due to lack

of means of communication and exchange of experiences, different regions even in

the same country used different units of measurement.

For measurement of length, man used various limbs of the body e.g. foot,

span, cubit, etc. yard came from the word yard which means a stick. In England a yard

was fixed as the distance from nose to the thumb of King Henry I. Ancient people in

India also measured the yard similarly. The unit of weight „Pound‟ was used by

Romans. At some place, the pound was considered to be of 13 ounces, at another of

13 ounces.

So there was a lot of confusion in the system of weights and measures. In

1970, the king of France decided to have uniformity in the system. He appointed a

committee of scientists for the job and also invited other nations to send their

scientists.

These scientists took the distance from the North Pole to the equator on a line

running through Paris and the one ten-millionth part of this distance was named as

meter. Now the meter is defined as the length of standard bar kept in Paris. For shorter

measurements, the meter was divided by tens.

Area

Similarly unit of area were developed on the same principle, and we have

square meters, square decimeters‟ and square centimeters as the units of area.

Weight

The weight of one cubic centimeter of water at 40C was taken as a gram. The

gram was further divided into decigrams, centigrams and milligrams and multiplied

into decagrams, hectograms, and kilograms.

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The metric system of weights and measurements is definitely superior to non-

metric system on various grounds.

C) Logarithms

The word Logarithm is the combination of Greek words „logos‟ and

„arithmos‟. Logos means to reason, and to calculate. Arithmos means a number. Thus

Logarithm stands for calculations involving numbers.

Logarithms were invented by a Scotman , John Nepier after many years of

hard labor. The problems before him were to find out some way to cut short lengthy

calculations. Napier spent about twenty years on the theory and finally Logarithms

were evolved.

Napier began to multiply, divide, square and extracts square roots of various

numbers. Finally 1614 the first table of Logarithm of numbers was prepared.

The terms Mantissa and Characteristic were introduced by Briggs in 1624.The

first table of Logarithm of trigonometric functions to the base 10 was made by Gunter

in 1620.Gauss suggested using it for the fractional parts of all decimals.

D) Computer Mathematics

This is the age of automation. Computer have their origin in mans attempts to

find ways and means to facilitate calculations.

Probably the earliest attempt was a table of dust or sand on which calculations

were carried out with the help of stick. This was known as abacus. Next attempt

probably ruled table with small stick, pebbles or counters arranged in lines. Using the

principle of position a single bead on a line would represent 1, in next line would

represent 10 and so on.

The next development was given by Napier in1617. He arranged system of

rods to represent the method of calculation. These rods were called „Napier‟s Bones.

In 1642, Pascal developed first mechanical computer. He attached cylinders with the

notched wheels of rocks. Each wheel was divided into ten small divisions. The system

was so arranged that one complete rotation of unit wheel would turn one division on

the tens wheels and so on. It was an adding machine. Leibnitz introduced in this, a

device to calculate multiplications and divisions.

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Towards the beginning of 19th

century, Charles Baves, a professor of

Mathematics in the Oxford University invented computer consisting of 18 wheels. In

1944, world‟s first electronic computer was invented. It was called electronic brain of

the world. It consisted of 3500 parts. It would take 300,000 micro seconds in addition

of two numbers. Then came more and more sophisticated computers.

Sudhir Kumar: (2000) pp171 to 175. Sidhu, Kulbir Singh: (1985) pp 44 to 47.

1.2 IMPORTANCE OF MATHEMATICS IN HUMAN LIFE

Right from human existence on this earth, the man wanted to answer the

questions like; How many? How much? How big? How long? etc. For answering

these questions man invented arithmetic, algebra, geometry. This knowledge was born

in the need of human.

There is need of Mathematics in anybody‟s day-to-day and lifelong planning,

evaluation, assessment, judgment, guidance and direction for future.

The prices, rates, discounts, commissions, rebates, interest, taxes, production,

distribution, profit, loss, bank, post, shares trading etc. are the issues concerned to

everybody. There is no escape from Mathematical concepts of life and livelihood.

If we shut off Mathematics from daily life and all civilization comes to a

standstill. In this world of today nobody can live without mathematics for a single

day. Human being needs minimum knowledge of Mathematics for their survival. It is

need daily by everybody consciously or unconsciously out of his /her necessity. Any

person ignorant of Mathematics will be easily cheated. Mathematics has a constant

power on our everyday lives, and contributes to the wealth of the country. Thus

knowledge of Mathematics fundamental processes in Mathematics and the skill to use

them are the preliminary requirements of a human being these days.

The main reason for studying mathematics is that it is interesting and

enjoyable. People like its challenge, its clarity, and the fact that you know when you

are right.

Sidhu, Kulbir Singh (1985) explained the importance of Mathematics and it is

helpful to create the values, which is useful to meaningful human life.

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These are as under:-

A) Practical value

He cannot do without learning how to count and calculate. Any person

ignorant of it, he is easily cheated in counting, notation, addition, subtraction,

multiplication, division, weighing, measuring, selling. These fundamental processes

of Mathematics have many practical values.

B) Disciplinary value

It trains and disciplines the mind. Mathematics creates discipline in the mind

because it is exact time and knowledge. It develops reasoning and thinking powers.

Reasoning in Mathematics possesses certain characteristics. These are:-

1. Characteristic of Accuracy

Accurate reasoning thinking and judgment are essential for its study.

Accuracy, exactness and precision compose the beauty of mathematics

2. Characteristic of 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

result by reverse process. It is possible for the child to remove his difficulty by self-

effort and to be sure of the removal. He develops faith in self-effort which is the

secret of success in life.

3. Characteristic of 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 originality which keeps him

going. The discovery or establishment (derivation) of a formula or conversion of

formula in one form to another is also his original work. This practice in originality

enables the child to face new and challenging problems with confidence.

4. Characteristic of 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

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meaning. Similarly in daily life, while undertaking a task, one must have a firm grip

over the situation.

5. Characteristic of 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.

6. 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.

7. 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.

In addition to these major and fundamental disciplinary values, there are few

other values, which is of equal importance. These are: -

1. Development of concentration.

2. Power of expression.

3. Self-reliance.

4. Attitude of Discovery.

5. Quality of hard work.

C) Cultural Value

It is said, “Mathematics is the mirror of civilization”. It helped man to

overcome difficulties in the way of his progress. The prosperity of man and his

cultural advancement have depended considerably upon the advancement of

mathematics. The modern civilization owes its advancements to the progress of

various occupations such as agriculture; engineering, surveying, medicine, industry,

navigation; road-rail building etc. and contribution of mathematics in their

advancement cannot be undermined.

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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. Mathematics is also a spin for cultural arts,

such as music, sculpture, poetry and paining.

Sidhu, Kulbir Singh: (1985) pp 9 to 14

1.3 HISTORY OF TEACHING METHODS

According to the Encyclopedia of Educational Research, (Fourth Edition) , Dr. J.V.

Naralikar , C.V. Bhimasankaram and a project of the American Educational Research

Association (1969), it was possible to saw many precursors of contemporary research

issues in the history of teaching methods.

1. The Rhetoricians used the method of systemized limitation to teach young men of

ancient Athens to speak effectively.

2. Socrates used exhortation to induce the learner to become concerned about

achieving the good, true and beautiful and engages the learner in dialectical self-

examination. He also used a kind of conditioning to discipline appetites. His

method culminated in intellectual training in dialectics by means of the Socratic

dialogue, whereby deductive relations were discovered to exist in reality,

definitions were clarified, and the relations between them were tested.

3. In the 8th

century A.D. Alcuin made use not only of riddles, poems, puzzles and

witty exchanges but also emphasized drill and repetition.

4. Abelard, in the 12th

century used the lecture method, structured materials

logically, and argued theological and philosophical issues by the method of

reconciliation between affirmative and negative sides of question.

5. In the 16th

century, Ascharm, taught the classics by the tutorial method and also

used the method of double translation in teaching languages. Individual

differences were also noticed during the teaching process.

6. The Jesults, in the 16th

century, systemized materials, methods and teacher

training, in order to standardize procedures for motivation, presentation, practice,

and testing. Individual and team rivalry, corporal punishment, reviews, lectures,

written assignments, correction work, etc. were used as techniques.

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7. Comenius, in the 17th

century, advocated the principle of proceeding from the

universal to the particular, understanding rather than memorization, learning

through teaching others, correlation etc.

8. Pestalozzi (1746-1827) advocated the principle of proceeding from concrete to

abstract, the particular to universal, etc; full freedom was given to the physical

movements of the pupils; real object were discussed in a progression from sense

impression to formal definitions i.e. inductive process of concept formation.

9. Froebel (1782-1852) invented the kindergarten method of teaching.

10. Herbart (1876-1941) formulated ideas to identify steps in teaching, preparation,

presentation, association, systemization, and application.

11. Kilpatric (1871-1965) John Dewey‟s foremost disciple advocated the project

method.

In these sketches of teaching methods advocated by historical figures,

anticipations of following contemporary ideas issues may be discerned; the learners

developmental stages, theories of cognitive development; categories of educational

objectives; types of motivation, social arrangements, ranging from independent study

through tutoring and dialogues to competitions and lectures, issues in cognitive

structuring , such as the inductive verses the deductive arrangement of learning

experiences and the use of drill and games ; the interaction of teaching methods with

learner characteristics , so that optimal outcomes are attainable only through some

individualization , the standardization of materials and the systemization of

experience.

All these concerns of the historical examples have counter parts in the research

and developmental work of the 1960‟s.

The post-war years really witnessed remarkable and breath-taking advances in

the realm of mathematical learning. A brief review of such movements, together with

a study of factors that gave rise to them, would not therefore be out of place here.

Encyclopedia of Educational Research, (Fourth Edition) A project of the American

Educational Research Association: (1969)

Dr. J.V. Naralikar and C.V. Bhimasankaram: pp 36 to 38.

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1.4 METHODS OF TEACHING MATHEMATICS

Students have different intellectual capacities and learning styles that favor to

knowledge growth. As a result, teachers are interested in ways to effectively cause

students to understand better and learn. Teachers want to bring about better

understanding of the matter. It is the responsibility of the educational institutions and

teachers to seek more effective ways of teaching in order to meet individual‟s and

society‟s expectations from education. Improving teaching methods may help an

institution meet its goal of achieving improved learning outcomes. Sudhir Kumar

(2000) explained the methods of teaching Mathematics in his book in the following

way.

Lecture Method:

In this method teacher prepared his talk at home and pours it out in the class. The

student listens silently, attentively and tries to grasp the point. The listeners remain

passive. He may not even write anything on the black board simultaneously or may

not even argue a point with the listeners by cross questioning.

Inductive Method:

It is the method of constructing formula with the help of a sufficient number of

concrete examples. The learner proceeds from particular to general, concrete to

abstract and examples to formula. The student gets generalizations after the numbers

of concrete cases have been understood.

This method proceeds in the following way.

Observation

Comparison

Findings

Generalisation

Deductive Method:

This method is the opposite of inductive method. The learner proceeds from

general to particular, abstract to concrete and formula to examples.

The teacher told and explains the pre-constructed formula to the students and

applies the formula to solve the examples.

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Heuristic Method:

There are no typical steps for this method. The teacher told the problem, creates a

learning situation and ask the students to solve the problem with own effort. The

teacher in the role of looker and students tries to move independently. Self

confidence, self reliant and thinking power is to be developed in the individual to

make him successful pupil.

Problem Solving Method:

The student is involved in the finding the answer to a given problem. This

method proceeds in the following way.

1. Recognising the problem

2. Defining the problem

3. Collecting relevant data

4. Organising the data

5. Formulating the tentative solution

6. Find out the correct solution

7. Verifying the result

Project Method:

Students learn better through association, cooperation and activity is the principle

of project method.This method involves the following steps

1. Providing a situation

2. Choosing

3. Planning

4. Executing

5. Evaluating

6. Recording

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Analytical Method:

Teacher split up the unknown problem into simpler parts then these can be

recombined to find the solution or to know hidden aspects. Thorndike says that all the

highest intellectual performance of the mind is analysis.

Synthetic Method:

We proceed from known to unknown. It is the opposite of analytical method. The

teacher starts with the known data and connects to the unknown part. It is the process

of putting together of known parts of information to reach the point where unknown

information becomes obvious and true. Sudhir Kumar : (2000) pp54-82

Laboratory Method or Self-Discovery Method:

It is more elaborated and practical form of the inductive method. It makes the

subject interesting as it combines play and activating.

The construction work in geometry is on the whole a laboratory work e.g., the

drawing of a line; construction of an angle; construction of a triangle, quadrilateral,

parallelogram etc. all involve the use of some equipment and so their nature is that of

laboratory work.

This method helps to improve students‟ hand skills, makes them more

productive and increases their active involvement in learning. Students can create a

relationship between theory and practice by using experimental teaching method

.They applying theory into their real life problems through experiments.

The most challenging task of a teacher is to ensure self-learning of

Mathematics among the students. This can be best realized through “self-discovery”

method of teaching of Mathematics. The experiments envisaged precisely promote

this spirit of teaching and learning.

According to the Dr.N.M.Rao (2009) the major steps to be followed in this

approach are the following

The students predict the result experimentally by repeating the experiment

several times.

1. Prove or disprove the prediction by the deductive method of Mathematics.

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2. Try to answer the open-ended questions at the end of the experiment and

generalize wherever possible.

Broadly, the following steps are suggested for every experiment.

1. Title of the experiment

2. Objective of the experiment

3. Description/Analysis of the experiment, with figure

4. The procedure of the experiment- How the experiment is to be conducted.

5. The prediction obtained after doing the experiment several times. The prediction

may be a correct result or may not be correct.

6. Mathematical proof/disproof of the above prediction.

7. The result (Thermos/Generalisation).

8. The open-ended questions relating to this experiment for the students to pursue the

Discovery method.

Learning by Doing:

There are several ways of learning, viz., Learning by seeing and Learning by

doing. “Learning by Doing” is one of the most effective ways of learning. It is in this

context that mathematics experiments play a vital role in not only arousing the interest

of the learner but also in making the learning of mathematics more meaningful. In the

process of “Learning by Doing”, students also learn some applications of

Mathematics in real life situations, Hence the use mathematics experiments ensures

better understanding among the students.

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 students to explore, to learn, to stimulate interest and develop favorable

attitude towards mathematics.

Laboratory method is based on the principles of “learning by doing” and

“Learning by observation” and proceeding from concrete to abstract. Students do not

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just listen to the information given but do something practically also. Principles have

to be discovered, generalized and established by the students in this method. Students

learn through hands on experience. This method leads the student to discover

mathematical facts. After discovering something by own efforts, the student starts

taking pride in his achievement, it gives him happiness, mental satisfaction and

encourages him towards further achievement.

Need and purpose of Mathematics Laboratory:

Dr.N.M.Rao (2009) explained some of the ways in which a Mathematics

Laboratory can contribute to the learning of the subject are:

1. It provides an opportunity to students to understand and internalize the basic

mathematical concepts through concrete objects and situations.

2. It enables the students to verify or discover several geometrical properties and facts

using models or by paper cutting and folding techniques.

3. It helps the students to build interest and confidence in learning the subject.

4. The laboratory provides opportunity to exhibit the relatedness of mathematical

concepts with everyday life.

5. It provides greater scope for individual participation in the process of learning and

becoming autonomous learners.

6. The laboratory allows and encourages the students to think, discuss with each other

and the teacher and assimilate the concepts in a more effective manner.

7. It enables the teacher to demonstrate, explain and reinforce abstract mathematical

ideas by using concrete objects, models, charts, graphs, pictures, posters, etc.

Dr.N.M.Rao: (2009) p.p.11-12

1.5 AIMS OF TEACHING OF MATHEMATICS

Education is imparted for achieving certain ends and goals. Various subjects of

the school curriculum are different means to achieve these goals so with each subject

some goals are attached which are to be achieved through teaching of that subject.

According to Sidhu, K.S. (1995) the goals of teaching mathematics are as below:

1. To develop the mathematical skills like speed, accuracy, neatness, brevity, estimation,

etc.

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2. To develop logical thinking, reasoning power, analytical thinking, and critical-

thinking.

3. To develop power of decision-making.

4. To develop the technique of problem solving.

5. To recognize the adequacy or inadequacy of given data in relation to any problem.

6. To develop scientific attitude i.e. to estimate, find and verify results.

7. To develop ability to analyze, to draw inferences and to generalize from the collected

data and evidences.

Aims of teaching Mathematics at the elementary level.

1. To provide a good start to the pupils in learning Mathematics.

2. To provide pupils to clarify about fundamental concepts and processes of

Mathematics.

3. To create in pupils an enduring interest on the subject and to develop a love for it.

4. To introduce pupil to mathematical games puzzles, recreations, hobbies and to unfold

before them the mysteries of the subject.

5. To develop in pupils a taste for and confidence in Mathematics.

6. To develop in pupils accuracy and efficiency in fundamental processes.

7. To acquaint pupils with mathematical language and symbolism.

8. To develop in them the habits such as regularity, practice, patience, self reliance and

hard work.

9. To prepare pupils for the learning those subjects, which are intimately related with

Mathematics.

10. To acquaint pupils with mathematical language and symbolism.

11. To prepare pupils for the learning of Mathematics of higher classes.

12. To intimate and develop required discipline in the learner‟s mind.

Sidhu Kulbir Singh: (1992)

1.6 INNOVATIVE PRACTICES IN MATHEMATICS TEACHING

Mathematics is a subject which has to be learnt by doing rather than by

reading. Looking to the aims of teaching mathematics it can be seen that more focus is

laid to the higher level of objectives underlying the mathematics subject, like critical

thinking, analytical thinking, logical reasoning, and decision-making, problem-

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solving. Such objectives are difficult to be achieved only through verbal and

mechanical methods that are usually used in the class of mathematics. The verbal

methods of instruction give all importance to speech and texts, to the book and to the

teacher. From an historical point of view this method was majorly used until the end

of the nineteenth century. In one of these verbal methods teachers are simply satisfied

with giving the mathematical rules to pupils and having them memorize it. They

justify this method by saying pupils would not understand explanations. Their task is

to transmit to their pupils the knowledge which has accumulated over the centuries, to

stuff their memory while asking them to work exercises, e.g. the rule of signs and

formulas in algebra; students memorize this and remember it! Another verbal method

involves explanation. Teachers who use this method assume that the mental structure

of the child is same as the adult‟s. But a developmental stage according to Piaget is a

period of years or months during which certain developments take place. Teachers

think teaching must imply logic, and logic being linked to language, or at least to

verbal thought, verbal teaching is supposed to be sufficient to constitute this logic.

This method leads to series of explanations and students at the initial steps of logical

explanations trying to understand and grasp but slowly the gap is created between the

explanations transmitted by teacher and received by students which lead to the poor

understanding on part of students and they develop a fear of the subject: Math‟s

phobia. The Education Commission (1964-66) points out that “In the teaching of

Mathematics emphasis should be more on the understanding of basic principles than

on the mechanical teaching of mathematical computations”. Commenting on the

prevailing situation in schools, it is observed that in the average school today

instruction still confirms to a mechanical routine, continues to be dominated by the

old besetting evil of verbalism and therefore remains dull and uninspiring.

1.7 PRESENT STATUS OF MATHEMATICS AT UPPER PRIMARY LEVEL

National Curriculum Framework 2005 proposes five guiding principles for

curriculum development:

1. Connecting knowledge to life outside the school;

2. Ensuring that learning shifts away from rote methods;

3. Enriching the curriculum so that it goes beyond textbooks;

4. Making examinations more flexible and integrating them with classroom life;

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5. Nurturing an overriding identity informed by caring concerns within the

democratic policy of the country.

The development of the upper primary syllabus has attempted to emphasis the

development of mathematical understanding and thinking in the child. It emphasises

the need to look at the upper primary stage as the stage of transition towards greater

abstraction, where the child will move from using concrete materials and experiences

to deal with abstract notions. It has been recognized as the stage wherein the child will

learn to use and understand mathematical language including symbols. The syllabus

aims to help the learner realise that mathematics as a discipline relates to our

experiences and is used in daily life, and also has an abstract basis. All concrete

devices that are used in the classroom are scaffolds and props which are an

intermediate stage of learning. There is an emphasis in taking the child through the

process of learning to generalize, and also checking the generalization. Helping the

child to develop a better understanding of logic and appreciating the notion of proof is

also stressed. The syllabus emphasises the need to go from concrete to abstract,

consolidating and expanding the experiences of the child, helping her generalise and

learn to identify patterns. It would also make an effort to give the child many

problems to solve, puzzles and small challenges that would help her engage with

underlying concepts and ideas. The emphasis in the syllabus is not on teaching how to

use known appropriate algorithms, but on helping the child develop an understanding

of mathematics and appreciate the need for and develop different strategies for

solving and posing problems. This is in addition to giving the child ample exposure to

the standard procedures which are efficient. Children would also be expected to

formulate problems and solve them with their own group and would try to make an

effort to make mathematics a part of the outside classroom activity of the children.

The effort is to take mathematics home as a hobby as well. The syllabus believes that

language is a very important part of developing mathematical understanding. It is

expected that there would be an opportunity for the child to understand the language

of mathematics and the structure of logic underlying a problem or a description. It is

not sufficient for the ideas to be explained to the child, but the effort should be to help

her evolve her own understanding through engagement with the concepts. Children

are expected to evolve their own definitions and measure them against newer data and

information. This does not mean that no definitions or clear ideas will be presented to

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them, but it is to suggest that sufficient scope for their own thinking would be

provided. Thus, the course would de-emphasise algorithms and remembering of facts,

and would emphasis the ability to follow logical steps, develop and understand

arguments as well. Also, an overload of concepts and ideas is being avoided. We want

to emphasis at this stage fractions, negative numbers, spatial understanding, data

handling and variables as important corner stone‟s that would formulate the ability of

the child to understand abstract mathematics. There is also an emphasis on developing

an understanding of spatial concepts. This portion would include symmetry as well as

representations of 3-D in 2-D. The syllabus brings in data handling also, as an

important component of mathematical learning. It also includes representations of

data and its simple analysis along with the idea of chance and probability.

The underlying philosophy of the course is to develop the child as being

confident and competent in doing mathematics, having the foundations to learn more

and developing an interest in doing mathematics. The focus is not on giving

complicated arithmetic and numerical calculations, but to develop a sense of

estimation and an understanding of mathematical ideas.

Taking into consideration the national aspirations and expectations reflected in

the recommendations of the National Curriculum Framework developed by NCERT,

the Central Board of Secondary Education had initiated a number of steps to make

teaching and learning of mathematics at school stage activity-based and

experimentation oriented. In addition to issuing directions to its affiliated schools to

take necessary action in this regard, a document on „Mathematics Laboratory in

Schools – towards joyful learning’ was brought out by the Board and made

available to all the schools. The document primarily aimed at sensitizing the schools

and teachers to the concept of Mathematics Laboratory and creating awareness among

schools as to how the introduction of Mathematics Laboratory will help in enhancing

teaching- learning process in the subject from the very beginning of school education.

The document also included a number of suggested hands-on activities.

The researcher has been working as lecturer in Education with Mathematics

method at the different Colleges of Education since 1993. The researcher has also

worked as an assistant teacher of Mathematics at Secondary School for four years.

The general observation of the researcher is that Mathematics has been the result

making subject at the S.S.C. and H.S.C. examination. The failure percentage of

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students in Mathematics subject remains a point to think over. To fulfill the objectives

of the course book, there should be three main principles.

i. „Learning by doing‟,

ii. „Learning by observation‟, and

iii. Proceeding from concrete to abstract.

As per the nature of the course book, wording and language of the content is

based on practical method. NCERT published New Curriculum framework in 2001

focusing on Mathematics laboratory. This framework indicates that Mathematics

laboratory should have a place in the science laboratory.

Keeping in mind all such things, the researcher has been encouraged to

develop such a Mathematics Experiment Notebook that would enhance the pupils to

learn Mathematics in a better way which ultimately affect their achievement in

Mathematics positively.

1.8 STATEMENT OF THE PROBLEM

The statement of the problem for research is

“DEVELOPMENT OF MATHEMATICS EXPERIMENT NOTEBOOK FOR

TEACHING UPPER PRIMARY SCHOOL PUPILS’ –A STUDY.”

1.9 DEFINITIONS OF THE TERMS

The conceptual and operational definitions of the terms used in the statement

of the problem are defined below for the sake of clarity and also for delimiting the

scope of study as follows:

A) CONCEPTUAL DEFINITIONS:

1. Development:

1. The systematic use of scientific and technical knowledge to meet specific

objectives or requirements.

2. An extension of the theoretical or practical aspects of a concept, design,

discovery, or invention. (Oxford English Dictionary: 1933)

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2. Mathematics

1. The abstract science which investigates deductively the conclusions implicit in the

elementary conceptions of spatial and numerical relations, and which includes as

its main divisions geometry, arithmetic, and algebra. (Oxford

English Dictionary: 1933).

2. The study of the measurement, properties, and relationships of quantities and sets,

using numbers and symbols. (American Heritage Dictionary:2000)

3. Mathematics is the science of structure, order, and relation that has evolved from

elemental practices of counting, measuring, and describing the shapes of objects.

(Encyclopedia Britannica:2010)

3. Achievement

1. A standardized test used to assess knowledge and skill in any subjects.

2. Performance appraisal test that measures the extent to which a trainee has

acquired certain information, or has mastered the required skills.

(Encyclopedia Britannica: 2010)

B) OPERATIONAL DEFINITIONS:

1. Development: The term development includes planning, designing, constructing and

the testing of an instructional system.

2. Mathematics Experiment Notebook (MEN): It is a Mathematics Experiment

Notebook consisting of teaching learning material includes aim of the experiment,

procedure , observation table, calculation ,diagram and conclusion i.e. Mathematical

principles or properties etc. developed for the learners in order to assist them in the

comprehension of certain mathematical concepts.

3. Mathematics: One of the essential subjects at Upper Primary School Level.

4. Achievement: Marks obtained by the pupils in Upper Primary classes in

Achievement Test.

5. Upper Primary School: The Middle Stage of education comprises Classes VI-VIII.

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1.10 SIGNIFICANCE OF THE STUDY

Mathematics is a science. As like other science subjects, Mathematics also

provides activities. According to these things the researcher will develop the

Mathematics Experiment Notebook.

It is important that Mathematics Experiment Notebooks were useful to

teachers who are teaching Mathematics at upper primary classes. The present study

has great significance because it is helpful to Board of studies to introduce the

Mathematics Experiment Notebook (MEN).

It is useful to teachers to teaching Mathematics with activity based and gave

the hands on experience.

It is helpful to pupils to develop the interest and confidence in learning.

The results of this study will be useful to develop a Mathematics laboratory (as per

NCERT curriculum framework 2001).

Mathematics Experiment Notebook gives new approach to upper primary

school teachers. This approach embraces the pupils and learning in a more complete

way than traditional method.

For researchers in the field of Education and Psychology, this study makes

available the Mathematics Experiment Notebook for their research purposes and also

to develop these types of Mathematics Experiment Notebook for other classes,

medium.

1.11 SCOPE AND DELIMITATIONS OF THE STUDY

The scope of this study was covering Mathematics subjects at Upper Primary

classes in Marathi language to teaching pupils in Maharashtra. The Mathematics

Experiment Notebooks were useful not only the teachers but also students of Upper

Primary classes in Marathi medium.

However, the present study has the following delimitations.

1. The present study is limited only to the students of Upper Primary classes.

2. The present study is related only to the urban area that is Satara city.

3. The research is limited for the subject Mathematics of Upper Primary classes.

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4. The study is limited to the students who are studying in Upper Primary classes

in the Maharashtra State Education Pattern 10+2+3.

5. The study is delimited to specific units i.e. arithmetic, area, volume, similarity,

congruency etc. of Upper Primary classes for Mathematics subject.

1.12 OBJECTIVES OF THE STUDY

The general and the specific objectives of the study were as follows –

a. General Objectives

1. To develop a Mathematics experiment notebooks to teach Mathematics to Upper

Primary classes.

2. To study the efficacy of the Mathematics Experiment Notebooks developed by

the researcher in terms of pupils achievement.

b. Specific Objectives

1. To analyse the textbook of upper primary classes in order to locate areas

suitable for developing Mathematics Experiment Notebooks.

2. To determine the teaching strategy in accordance with developed Mathematics

Experiment Notebooks.

3. To compare the effectiveness of developed Mathematics Experiment

Notebooks with the conventional system of instruction.

4. To study the comparative effect of developed Mathematics Experiment

Notebooks in terms of the pupils‟ achievement in knowledge, comprehension,

application and skill.

1.13 ASSUMPTIONS OF THE STUDY

The study was based on the following assumptions.

1. Students learn better by „Learning by doing‟ method.

2. Teachers know the different methods of teaching Mathematics but they don‟t

use most of them while teaching.

1.14 HYPOTHESES OF THE STUDY

Research Hypothesis

Teaching Mathematics with MEN affects pupils‟ achievement in Mathematics.

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Null Hypotheses

1. There is no significant difference between the achievement score of the students from

control and experimental group in knowledge, taught by using MEN and that with

conventional method.

2. There is no significant difference between the achievement score of the students from

control and experimental group in comprehension taught by using MEN and that with

conventional method.

3. There is no significant difference between the achievement score of the students from

control and experimental group in application taught by using MEN and that with

conventional method.

4. There is no significant difference between the achievement score of the students from

control and experimental group in skill taught by using MEN and that with

conventional method.

5. There is no significant difference between the performances of the students from

control group in knowledge, comprehension, application and skill in pre over post-

testing.

6. There is no significant difference between the performances of the students from

experimental group in knowledge, comprehension, application and skill in pre over

post-testing.

7. There is no significant difference in the mean achievement of the students by using

MEN and conventional instructional system.

8. There is no significant difference between the performances of the students in upper

primary classes by using MEN.

9. There is no significant difference between the performances of the students in

knowledge, comprehension, application and skill from upper primary classes by using

MEN.

1.15 ORGANISATION OF THE STUDY

Chapter I : Introduction

The chapter includes background of the research, statement of the

problem, objectives of the study, hypotheses of the study, significance of the study,

scope and limitations.

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Chapter II : Review of Related Literature and Researches

The chapter was devoted to the theoretical aspects of Mathematics

Experiment Notebook, Upper Primary Level, review of related Literature and

Researches and rationale of the present study.

Chapter III : Plan and Procedure of the Research

In this chapter the researcher explained the procedure used in the

development of a Mathematics Experiment Notebook, improvements in the prototype,

experimentation procedure, variables in the experiment, control of experiment,

research and null hypotheses of the study, experimental design, sampling design ,

conduct of Experiment , pilot study , data collection and description of the tools used

in the study.

Chapter IV : Analysis and Interpretation of the Data

The data obtained through pilot and main study of experimentation was

analysed and interpreted accordingly. The analysis and interpretation of the collected

data in the form of tables, graphs and figures and statistical measures was elaborated

in this chapter.

Chapter V : Summary, Conclusion, Suggestions and Recommendations

The chapter is a brief summary of the study. The chapter includes

discussion of procedure and results of the experiment also the conclusions based there

upon, which is followed by recommendations based on conclusions and topics for

further research were suggested at last.

1.16 CONCLUDING REMARK

Thus this chapter discussed with historical review of Mathematics and

teaching method. The researcher discussed that the importance, aims and innovative

practices of Mathematics teaching also explained the various methods of Mathematics

teaching. The researcher further explained the statement of the problem, objectives of

the study, hypotheses of the study, significance of the study, scope and limitations.

It is essential to know about review of researches related to the development

of Mathematics Experiment Notebook for upper primary level. The further chapter

deals with brief review of the related literature and researches.