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CHAPTER-I

Background of the study

Introduction

The main aim of teaching mathematics in schools is to develop

scientific attitude towards Mathematics. Now-a-days Mathematics

is being a compulsory subject of primary and secondary school

students. Mathematics is considered by many learners as a dry

subject. Every child’s right is to get quality mathematics education.

So it is the duty of the teachers to give mathematics education to

be easy, enjoyable and also affordable to every child.

The mother of all sciences is mathematics. It is very important in

everyone’s life. Without the use of mathematics, it is very difficult

to survive in life. Everyone uses mathematics in one or other way

in his / her daily life. We cannot imagine a life without

mathematics. From beggar to businessman, everyone uses

mathematics in their life.

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Most of the problems in Mathematics have magic and mysteries.

Our ancients understood all these mysteries and developed some

simple ways / techniques to solve mathematical problems. Many

years ago our Indians used some techniques in various fields like

construction of temples, medicine, science, astrology, etc., due to

which, we can proudly say that India developed as the richest

country in the world.

1.1 Mathematics – Meaning and Definitions

The term ‘Mathematics’ may be defined in a number of ways. The

dictionary meaning of mathematics is that “it is either the science

of number and space or the science of measurement, quantity and

magnitude. Bacon said “Mathematics is the gateway and key to all

sciences”.

All the above definitions emphasize mathematics as a tool

especially suited for dealing with scientific concepts. According to

Lindsay, ‘Mathematics is the language of physical sciences and

certainly no more marvelous language with its signs, symbols,

terms and operations, which can handle ideas with a precision and

conciseness that is unknown to other languages.

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The National Policy of Education (NPE) (1986) stated

“Mathematics could be considered as a medium to train a child to

develop his thinking capacity, to develop his reasoning power, and

to coherent logically”. Mathematics should be shown as a way of

thinking, an art or form of beauty, and as human achievement.

1.2 Nature of Mathematics

1.2.1 Mathematics – A science of Discovery: The expression of

mathematics relationships are in symbolic form-in words, in

letters, by diagrams or by graphs (E.E.Biggs, 1963). Initially a

child’s discoveries may be observational. But, later, when its

power of abstraction is adequately developed, it will be able to

appreciate the certitude of the mathematical conclusions that it has

drawn. This will give it the joy of discovering mathematical truths

and concepts. Mathematics gives an easy and early opportunity to

make independent discoveries.

1.2.2 Mathematics – An intellectual Game: Mathematics can be

treated as an intellectual game with its own rules and without any

relation to external criteria. From this viewpoint, mathematics is

mainly a matter of puzzles, paradoxes and problem solving – a sort

of healthy mental exercise.

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1.2.3 Mathematics- The Art of Drawing Conclusions: One of

the important functions of the school is to familiarise children with

a mode of thought which helps them in drawing right conclusions

and inferences.

1.2.4 Mathematics- As a Tool Subject: Mathematics established

its own goals to pursue. Its mentors of the past engineering,

physical science and commerce-now became no more than its

peers. According to Howard F. Fehr (1996), “If mathematics had

not been useful, it would long ago have disappeared from our

school curriculum as required study”.

1.2.5 Mathematics- An Intuitive Method: Intuition when applied

to mathematics involves the concretization of an idea not yet stated

in the form of some sort of operations or example. A child forms

an internalized set of structures for representing the world around

him.

1.3 Characteristics of Mathematics

Mathematics has certain unique features which one could hardly

find in other disciplines. The following are the important

characteristics of mathematics

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1.3.1 Precision and Accuracy: Mathematics is known as ‘exact’

science because of its precision. It is perhaps the only subject

which can claim certainty of results. In Mathematics the results are

either right or wrong, accepted or rejected. Mathematics can decide

whether or not its conclusions are right.

1.3.2 Logical Sequence: Mathematics also possesses the

characteristics namely logical sequence. The study of mathematics

begins with few well-known uncomplicated definitions and

postulates and proceeds, step by step, to quite elaborate steps. It

would be difficult to find a subject, in which a better gradation is

possible, in which work can be adapted to the needs of the pupil at

each stage, than in mathematics.

1.3.3 Applicability: Knowledge is power only when it is applied.

The study of mathematics requires the learners to apply the skill

acquired to new situations. The knowledge acquired by the

students is greatly used for solving problems. The students can

always verify the validity of the mathematical rules and

relationships by applying them to novel situations.

1.3.4 Generalization and classification: Mathematics gives

exercises in widening and generalizing conceptions, in combining

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various results under one head, in making schematic arrangements

and classifications. It is easy to find instances of successive

generalizations.

1.3.5 Mathematical Language and Symbolism: The language

for communication of mathematical ideas is largely in terms of

symbols and words which everybody cannot understand. There is

no popular terminology for talking about mathematics. In

arithmetic and algebra, the students deal not with facts, but with

symbols. The use of symbols makes the mathematical language

more elegant and precise than any other language. Almost all

mathematical statements, relations and operations are expressed

using mathematical symbols such as +, -, x, ÷, >, <, ∑ , ±, ≠, ∞ and

so on.

1.3.6 Abstractions: Mathematics is abstract in the sense that

mathematics does not deal with actual objects in much the same

way as physics. But, in fact, mathematical questions, as a rule,

cannot be settled by direct appeal to experiment. For example,

Euclid’s Lines are supposed to have no width and his points no

size. No such objects can be found in the physical world.

1.4 Aims of Teaching Mathematics

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The aim of teaching mathematics is the development of

appropriate abilities, appreciations and positive attitudes. The

following abilities, appreciations and attitudes are to be developed

through the teaching of mathematics among students.

Abilities:

• To express the thoughts clearly and accurately

•

Systematic organization and interpretation of data

•

To arrive at conclusions through accurate and

logical

reasoning

• To generalize the concepts accurately

• To have originality in reasoning.

Appreciations:

• To understand the contributions of mathematics to sciences,

social sciences, engineering etc.

• To understand the impact of mathematics on the human

progress and modern civilization.

• To understand the cultural values of mathematics

• To use mathematics for recreational purposes during leisure

time

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Attitudes:

• To gain the ability to express with accuracy and clarity.

• To improve self confidence needed for a sound personality.

• To possess the ability to think independently and originally.

The following are the aims of teaching mathematics:

1.4.1 Utilitarian Aim: We will remain too much handicapped in

our life in case we remain ignorant of mathematics. Utilitarian aim

includes practical utility f mathematical concepts in the life of

every individual. Now-a-days with the advent of automation and

information technology, there is a need to have mathematically

literate workforce that have “belief in the utility and value of

mathematics” (Pollak,1987). Students need to possess knowledge,

skills, flexibility and attitude to change, manage and develop jobs

in the present and in the future. Thus utilitarian aim of mathematics

education must be reflected in instructional material, teaching

process and in assessment.

1.4.2 Bread and Butter Aim: This aim is another side of the

utilitarian aim. Mathematics satisfies bread and butter aim also.

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Every person has to take up some profession/ vocation for his

livelihood. A tailor who stitches clothes, a mason who constructs

houses, a petty businessman who sells articles, etc. do their

procession/ vocation to lead their life in a peaceful way. Every

profession/ vocation is linked with the application of mathematics.

Mathematics is the basis for the knowledge of progress of modern

sciences and technical fields. The modern person enjoys his life

fully with the use of scientific inventions like T.V., telephone, cell

phone, pressure cooker, washing machine etc. In the making/

preparation of every item that is mentioned above, the use of

mathematics is necessary. That is why Bacon states that

Mathematics is the key and door for all the sciences.

1.4.3 Disciplinary Aim: The chief characteristics of the discipline

are simplicity, accuracy, certainty of results, originality, reasoning

and correlation of the teaching of the subject with the problems of

life. All these characteristics are developed to a large extent by the

teaching of mathematics so teaching of mathematics fulfils this

aim of education.

Accordingly to Locke: “Mathematics is a way to settle in the mind

a habit of reasoning. Knowledge follows as a consequence of

reasoning power”.

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1.4.4 Cultural aim: Mathematics has a lot of cultural values. It

helps in the formation of certain habits in the students and helps

them to grow as cultured citizens. For any cultured person the

development of power of reasoning and judgment is the basic

requirement and mathematics develops these qualities in a student.

In addition to development of power of reasoning and judgment

mathematics also helps to develop in the child the qualities of

concentration, thinking, precision, accuracy, self-confidence,

expression etc. Thus mathematics teaching develops all those

qualities in a student so that he/she will become a helpful

individual of the society.

1.4.5 Vocational aim: The chief aim of education is to felicitate

the children to earn their living and to make them self dependent.

To achieve this aim, mathematics is the most important subject

than any other. Any individual who take up any of the vocations

for his livelihood, must have at least workable knowledge of

mathematics, otherwise he cannot lead his vocation and life

peacefully and successfully. That is why mathematics at the

secondary level, includes various topics which are useful to the

future citizens when they enter into life and take any vocation

suited to him/ her.

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1.4.6 Knowledge Aim: Every person must acquire knowledge and

he must become a knowledgeable person. Though knowledge is

unity, every branch of science or arts, becomes full-fledged only

with the application of mathematics. Mathematics gives precision

to them. Thus knowledge of mathematics is a must to any person

who studies sciences or social science. Thus mathematics fulfills

the knowledge aim.

1.4.7 Character Aim: The most important aims of education is the

formation of character. Mathematics education is not an exception

for this. So the study/ teaching of mathematics must fulfill this aim

also. A person who has honesty, accepting his mistakes

unhesitatingly, tolerance to others, impartiality, patient hearing of

others and taking decisions after careful analysis of the situation

and thinking in a critical manner and rationally etc. are the

characteristics of a person who possesses good character. All these

characteristics are imbibed by the study of mathematics. Thus

mathematics fulfills this aim.

There is considerable amount of dependence on mathematics in

every field of technological development. Modern life makes use

of the scientific contributions in all walks of life.

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The gist of above aims of mathematics can be given as under:

� The main goal of teaching mathematics is help the students

to enjoy mathematics. It is based on some principles. Based

upon these principles mathematics can be used and enjoyed

in every one’s day-to-day life activities. It is observed that

school is a best place to create interest of students in

mathematics subject. Generating (or not removing) fears of

mathematics can also it is the duty of the teachers to take

away the fears of mathematics in students mind.

� It helps to understand the basic structure of mathematics like

defined terms, undefined terms, formulae, theorems, axioms

and postulates. It also helps to understand the branches of

mathematics like Arithmetic, algebra, geometry and

trigonometry. It offer a methodology for abstraction and

generalization.

� It aims to enable the child to solve mathematical problems

of his everyday life.

� It aims to develop in the child and acquaintance with his

culture.

� It aims at providing a suitable type of discipline to the mind

of the pupil.

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� It aims at preparing the child for technical professions like

accountants, auditors, engineers, cashiers, scientists,

statisticians etc.

� It aims to prepare the child for economic, purposeful,

productive, creative and constructive living.

� It develops in pupil a sense of appreciation of cultural arts.

� It prepares him for elementary as also higher education in

science, economics, engineering etc.

� It develops in the pupils such habits as concentration, self

confidence and discovery.

� It helps the child to follow the maxim, “work is worship”

� It develops in child the powers of thinking and reasoning.

� To develop the learner’s power of expression.

� To enable him to understand and enjoy mathematical

problems.

� To develop in him a scientific and realistic attitude towards

life.

� To bring about all-round, harmonious development of the

personality of the child.

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1.5 History of Mathematics

History is a very useful record of human beings development and

achievement in life. We use it also to prevent ‘reinventing the

wheel’, to examine mistakes committed by our forefathers and also

for self-motivation.

There is no exaggeration in this saying because the development of

Mathematics is the development of civilization. The historical

background of the developmental sequence of mathematics has

been found in the studies and researches of tribal languages and

extinct languages related to them. The studies reveal that the

simplest process of counting might have developed in several

stages to its present systematic level. It has been found that

mathematics is the basis of all systematic knowledge. It has been a

progressive science and also has given guidance to the

development of various subjects, vocations and technology.

1.5.1 Importance of the history of mathematics

Although the history of mathematics has not so far been given its

due place in the curricula, it has its own importance not only in the

study of the subject but also in developing insights into the entire

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human knowledge. Generally, the syllabus of mathematics is

already heavy and lengthy but along with other contents, the

knowledge of history of mathematics also should be passed on the

learners. It can be a source of interest and pleasure to them. Its

importance can be summarized as follows:

1. Mathematics can be presented as a dynamic and progressive

subject, relevant to human development.

2. It will be instructive and interesting; it will remind us of a

glorious past and also teach us how to increase our gained

knowledge.

3. It warns the leaner against making hasty conclusions.

4. Many mathematical topics can be better introduced in the class

by linking them with their development.

5. It can reveal the contribution of mathematics to the history of

human civilization.

6. It reveals, that, at every stage, major or significant development

of mathematics was conditioned by human needs.

7. Most of the terms, concepts and conventions can be properly

understood only with reference to their historical background.

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8. If the teacher reveals his knowledge of the history of

mathematics to the learners they will form a good impression about

the teachers’ scholarship. This helps him to command respect.

9. Graduation of the content of mathematics, correlation of the

subject with other areas of knowledge and the psychological and

logical order of the subject matter-all these can be maintained with

the help of history.

10. The history shows that mathematics is a man made science. It

will thus encourage the learners to contribute something to its

development.

11. It reveals that all the branches of mathematics were developed

in relation to one another. So it guards the learner against

compartmentalization.

12. Some related stories and events, narrated occasionally, can

diminish the monotony of the classroom work.

13. It gives the impression that mathematics has an intimate

connection with other branches of knowledge and hence it should

not be treated as an isolated subject.

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14. It makes students appreciate the progress of man over ages.

They would like to read and hear how the old mathematicians

discovered mathematical facts and tried out their experiments.

15. Interesting anecdotes chosen from the history of mathematical

development can make learning interesting.

1.5.2 A General Review of the History of Mathematics from the

Time of Origin of Human Life:

In the previous chapter mention has been made as to how man,

from the time of origin of human life could sense certain ideas

related to mathematics. It was explained there that it was from the

regular shapes of objects, the rhythm in the arrangement of many

natural phenomena and the systemic rotation of the planets, etc.

that man began feeling a sense of mathematical institution. It was

also pointed out that he derived many ways of action to meet the

issues raised by practical life. For example, it was explained how

man who started to rear cows found out a way to detect whether

certain animals, that went for grazing in forests were lost. Also it

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was pointed out that the technique they adopted resulted in a great

mathematical insight namely one-to-one correspondence.

When man faced the problems of comparing lengths he used his

organ for the purpose as indicated by his using the width of two

figures (inch), the span of the palm, the distance between the left

hand and right hand when they are stretched, the length of the foot,

etc. for the purpose. Standardisation of the units for linear

measurement developed only later because of problems that

demanded accurate measurement. The foot of one person may be

much longer than that of another and to express a distance as five

feet long by one may be only four feet ling for another. There is a

historical anecdote cited to show how a standardised length

representing a ‘foot’ was arrived at by an order of a king of

England. It is said that one day he passed an order that on the next

Sunday the first ten people coming to the church should be asked

to stand in such a way that the front tip of the thumb of each just

touches the hind part of the foot of the person just in front. Then

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using a string the distance of the ten feet was determined. This

string was then divided into ten equal parts and the king declared

that thereafter the length of one such part will be considered as

‘foot’.

Of course the story indicates a modern concept of standardised

units of measurements. Standardised of the units of length

according to the metric system with ten as the basis of demarcation

was a much modern system, now being followed in most part of

the world. Thus a precise system of measurement gradually

developed, reflecting the precise nature of the subject.

During the early stages, people did not know even to count and

there were no number names. That is why one-to-one

correspondence was developed as a technique for comparison of

the number of the number of the members of a group. Gradually in

the place of the stones, etc. used by ancient man to get an idea of

the number of members in groups, they began to use fingers to

count without any number names. A finger was stretched to

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represent one item. If four fingers were to be stretched one for each

item of a group and if all the five fingers were required to represent

another group they could conclude that there are more members in

the latter group than in the first.

To start with, it is told that any number after the first and second

was considered as ‘many’. It may be noted that in the language of

certain uncivilized groups, the number names themselves that were

conceived after centuries literally mean one stone, two stones, three

stones, etc. This is evidence to the argument that numbers were

closely associated with objects. Even now, it is told that certain

aborigins have only two or three number names, meaning one

stone, two stones, and all others in the series are referred to as

‘many’. Gradually man began using all the fingers of both hands

and then ten fingers of two feet together for comparing numbers

using the principle of one-to-one correspondence.

Stories of how man developed the skill for counting and

comparison in terms of numbers, how they are created number

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names, how they arrived at standardised units of measurement of

lengths, areas, volumes, weights, etc. to solve issues relating to day

to day needs associated with various phenomena are evidences for

his intelligent way of thinking and reasoning. The idea regarding

area emerged from the practical need for comparing things and

space with respect to two dimensions. To start with, comparison

might have been made by juxtaposing the object being compared.

Gradually, when that was not possible in all cases, the idea of two

dimensional measures was developed based upon the linear units.

That is how sq.cm, sq.m, etc. came to be used for the purpose. Of

course, such developments are comparatively ‘modern’.

The concept of volume might have been originally connected with

the need for comparing the capacity of two vessels. As needs

became more complex, scientific reasoning based on linear

measure and square measure might have been extended to three

dimensional units such as cubic centimetre, cubic metre, etc. this of

course might have been a later development.

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The concept of weight and the units for measuring weight, the

concept of time and generation of exact units for its measurement,

etc. are also examples to show, how scientific reasoning was used

to find out solution for issues in day to day life. It is suggested that

all mathematics teachers should gather such information so that

these can be applied at the time of instruction and thus to make the

classes lively.

In short, knowledge of the history of mathematics can be used by

an intelligent teacher to introduce topics in an interesting manner.

1.5.3 Development of Mathematics as a science

We have already told of the saying that mathematics is the queen

of all sciences. In order to deserve this qualification the

mathematicians in due course tried to develop the subject into a

precise, accurate, logically linked discipline. These are qualities

satisfied by all sciences. As a result of the effort to make

mathematical reasoning logical and scientific, its dependence on

concrete objects for even simple concepts began vanishing and

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abstract ideas strongly linked in a logically sound sequence

emerged. To start with, these happened out of intellectual curiosity

to discover newer and newer ideas from the existing ones. Most of

these new discoveries to start with, had more theoretical relevance

than practical utility. This was how theoretical geometry, algebra,

etc. developed. They of course became practically relevant later.

For example, to start with, ancient main in pre-historic times could

appreciate the regularity and precision in the positions and

movement of the heavenly bodies and the beautiful geometrical

forms exhibited crystals. They could also appreciate the symmetry

and rhythm in the natural phenomena like plants, animals, etc. The

arrangement of leaflets in certain compound leaves gave them an

unconscious idea of a ‘series’ which is an important item in

modern mathematics. Generation of axioms and postulates and the

way in which a logically bound system was created has made

geometry a typical science. For example, ‘Euclid’s Elements’ can

be qualified as a logically sequenced system. This is the result of a

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gradual but purposeful development of geometrical concepts

leading to principles and other relations. The theorems deduced

could highlight inductive and deductive reasoning reflected by a

perfect science. Don’t forget that this development has a very very

long history extending over a number of ages in human history.

Euclid himself lived in the third century BC. This history of

development is sure to create in a teacher an insight about the

strength of logical reasoning. Also the application of the

geometrical relations has given rise to many branches of the

subject such as trigonometry, analytical geometry, space science,

etc. The applications of such concepts, principles and processes

were made use of in the study of heavenly bodies (astronomy)

which has helped man to satisfy the curiosity aroused by the

system in the movements, etc. of planets and stars. As all of us

know, this has developed into a very precise and surprising branch

of scientific study about space.

1.5.4 The wide scope of mathematics

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A reference has been made in the previous chapter about the

symbolic nature of the language of mathematics and about how

algebra was used by a person to decipher coded messages. Algebra

played a very imperative part in the progress of mathematics,

science and technology .

The vast number of branches of mathematics and its application in

all other sciences is an indication of the unlimited scope of the

subject. While appreciating the value of the application of

mathematical theories in science and technology and while

enjoying the value of qualities like precision, accuracy, logical

reasoning, etc. we have to think of the very long history behind

development of the subject, especially, during the early stages.

1.5.5 Mathematical Science as the result of

contributions of various nations

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It may also be pointed out that almost all nations of the world have

contributed for the developments mentioned above-India’s

contribution also is very rich; even the single item of the concept of

zero has no other-comparison. It has given the basis for the

development of a number system, especially the denary (base ten)

system followed all over the world. It has enabled us to represent

giantifically large numbers like crore as 1, 00, 00,000 or 107.

In the same way China, Egypt, Rome, Alexandria – in short all the

nations – have contributed to the development of mathematics as a

perfect science. Starting with numeration and notation and the four

fundamental operations it has now developed into scores of

disciplines-abstract as well as practically useful. This fact

highlights the need for understanding the totality called

mathematics. Thus an insight into the long history of the subject

can create in man a feeling of international understanding. The idea

that we are beneficiaries of a large number of nations and our pride

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that we too have contributed much to the science is sure to create

such a value.

Now let us very briefly discuss certain other areas related to the

development of mathematics in the later stages.

1.5.6 A brief discussion on certain topics of importance

1. Metric system of weight and measures: Man used different

stones, seeds, etc. for the purpose of weighing when there was no

scientifically designed measurement system. In India, Ratti was

taken as the basic unit of weight; ‘penny’ was used in England as

the unit of weight and it was considered equal to the weight of 32

wheat seeds. Various limbs and parts of the body were used by

man to measure lengths eg. Cubit, foot, pace, etc. because of lack

of communication there was no uniform system for measurements.

For each nation there was a different basis for measurements. Later

with the evolution of the metric system, uniformity came into

existence, which helped a lot in trade and commerce.

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2. Historical developments of Logarithms: The word logarithm

has been derived from the Greek word ‘logos’ and ‘arithimos’.

Logos means - to reason, to reckon, to calculate. Arithimos means

a number. Thus logarithm stands for calculation of numbers. John

Napier, the famous mathematician of Scotland, invented

logarithms. It took him 20 years to prepare logarithmic tables.

Making use of a logarithmic table one can easily do tedious

multiplications and divisions by performing a simpler process of

addition and subtraction. This saved much time for mathematicians

and scientists. Napier later extracted square roots of various

numbers. Napier also introduced logarithms of sines. Later Briggs

introduced the concept of Characteristics and mantissa.

3. History of geometry: The word geometry has a Greek origin

goes-meaning earth and meton-meaning measure. It has been

proved that Geometry was very much useful to ancient peoples. In

ancient times people used Geometry for surveying, astronomical

studies, navigation and constructing buildings and so on. Geometry

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was actually known as Euclidean geometry. Geometry was

compiled over 2000 years ago in Ancient Greece by Euclid. The

most interesting and accurate geometry text and was called

elements was found and written by Euclid. For more than 2000

years Euclid’s text has been used. Geometry is the study of lines,

line segments, angles triangles, Quadrilaterals, perimeter, area and

volume etc. geometry is entirely different from algebra. Logical

structures were developed and also mathematical relationships are

proved and applied in algebra

4. History of Algebra: The first exposition on algebra was written

in the 3rd century AD. The term Algebra is derived from the Arabic

word al-jabr or exactly means ‘the reunion of broken parts’. He

introduced abacus as solving problems with the help of instrument..

5. Historical background of Computer Mathematics: In the age

of automation man has invented machines for every activity of life

to make his function easy, quick and reliable. Computers are

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invented to help mathematicians. Computers have their origin in

man’s attempts to find ways and means to facilitate calculations.

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

with the help of a stick. This was known as abacus. Erasing on the

abacus consisted in smoothening out dust or sand with hand. Next

attempt was probably a ruled table with small sticks, pebbles or

counters arranged in lines using the principle of position. A single

bead on the first line would represent1; in the next line it would

represent 10 and so on.

Next came Napier bones. In 1642, Pascal devised the first

mechanical computer. He attached cylinders with notched wheels

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

system was so arranged that one complete rotation of the unit

wheel would turn one-division of the tens wheel and so on. It was

an adding machine. Leibnitz introduced in this, a device to perform

multiplications and divisions.

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In 1944, the world’s first electronic brain originated; thereafter it

was improved and was introduced in all the fields of study, work

and business we can think of.

1.5.7 Contributions of Renowned Indian

Mathematicians

The height which mathematics is occupying today and the progress

which it has made through the ages are all due to the dedicated and

sustained work of many great Indian mathematicians also. The life

history and contributions of some great Indian mathematicians

such as Aryabhata, Bhaskaracharya, Brahmagupta and Ramanujan

are presented below.

Aryabhata

Aryabhata was the first among the great Indian mathematicians. He

lived from 475 to 550 AD near Patna. He was the first person to

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present arithmetic, algebra and geometry in his astrological

calculations.

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 of

mathematical knowledge. They are considered to the landmarks in

the history of the subject.

Notation System

The origin of notation system is as old as the man himself. Number

sense is something innate in man. It is believed that animals and

birds also have number sense. The primitive man was able to

differentiate one object from two but could not tell one and one is

two.

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

notches, cuts in the trees, lines on the ground, pebbles etc for the

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purpose. The notation system originated and developed differently

in different countries.

The numeral ‘1’ perhaps meant one lifted finger.

‘Two’ was represented by two fingers or lines. If we write two

lines without lifting the pen, it becomes ‘Z’ which ultimately

changed to 2. Or it becomes µ the numeral used by the Arabs or

Persians.

Similarly if we draw three lines without lifting the pen, it becomes

3 or Ɯ

Babylonians

The Babylonians used wedge-shaped symbols. One was

represented by V , ten was represented by < and hundred by V<.

Roman System

The Roman system is based on the idea of counting fingers or

lines. Thus I, II, and III represented one, two and three

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respectively. V probably represented the whole hand. To avoid

clumsy I I I I they wrote I before V i.e IV, the symbol gave rise to

the idea of positional value. Then the symbols VI, VII, VIII etc.

The symbol X was perhaps the combination of two fives.

Hindu-Arabic System

The notation system 1,2,3... can be called Hindu-Arabic System.

This system was originated by Hindus, perfected and transmitted to

the west through the Arabs. Some ancient symbols carved on stone

are found.

I II + 6

One Two Four Six

At some other places, some such symbols are found:

- = ± 7 ?

One TwoFour Seven Nine

The symbol 0 was used to denote vacuums. The word zero comes

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from Arabic ‘Sifar’ which was translation of the Hindu word

‘Shunya’. Arabs also made certain modifications in hindu

numerals. During 13th century and after the Hindu-Arabic system

spread all over the world.

1.6 The Place of Mathematics in Everyday Life

A little reflection will show what predominant role mathematics

plays in our everyday life and how it has become an indispensable

factor for the progress of our present day world. It is the pivot of

all civilization. Everybody has to calculate his income and balance

his family, budget, although only a few of them undergo any of the

university courses. This is the subject which indisputably forms the

very basis of entire world’s commercial system. It is a contributory

factor in the prosperity of the human race. There is no science, no

art and no profession where mathematics does not hold a key

position. The accuracy and exactness of a science is determined to

a major extent by the amount of mathematics utilized in it. Even

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social sciences like economics, psychology, geography etc make

abundant use of mathematics. The gigantic works of construction

of dams, bridges, other works of architect, building of ships, aero

planes etc are possible only because of the quantitative science.

Even medical men have to measure the doses, the blood pressure,

the beat of pulse, the bodily temperature etc. most of the natural

sciences and philosophy are to be studied on mathematical lines

and without the study of mathematics there would be no

improvement in them.

In the universe it is commonly seen that even uneducated people

use mental mathematics in their day-to-day life activities. Most of

the people appreciate the richness of mathematics. And many

students are trying to relate the knowledge of mathematics

knowledge in their life It helps the students to get inspired and be

motivated.

In Southern part of India, women draw kolams (complex figures

drawn on the floor in front of their houses every day with the help

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of white powder or rice powder. These kolams are same like the

rangolis in the northern part of India, but usually they use without

colour. Each and every day they draw a new kolam. A great variety

of kolams have been created and drawn daily in front of their

houses. Even often they conducted kolam competitions also. The

structure of these kolams, like symmetries, closed curves etc are

based on mathematics. Also, art, architecture like Taj mahal,

temples etc and music also some of the examples of cultural

development of mathematics.

Even nature also embraces mathematics completely. The sun rises

and sets at the specified moment. The stars appear at fixed time.

Mathematics runs in the veins of natural sciences like physics and

astronomy. This subject is inextricably incorporated with world

and the natural phenomena.

Arithmetic, the language of commercial activity: algebra which

gives the idea of functional dependence and generalization:

geometry which teaches logical thinking and natural design, all

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these combine to produce a very valuable literature of

interpretation, control and progress. We understand the world

better. Graphical representation of numbers is becoming very

common. Mathematics is home decoration designs: measurement

and contraction: in banking and business” in protection of life and

property; in painting and art, is playing a vital role.

He devised Algebra so simplify arithmetical problems. For

measurement he invented geometry. To find the position of high

mountains and stars, Trigonometry was invented. The most salient

feature of natural phenomena is change; the most important branch

of mathematics – Calculus was invented to measure change. To

measure social phenomena, he created Statistics. Mathematical

knowledge is thus indispensable and no one can deny the truth that

no development would been there in any one’s life, if he has no

mathematical knowledge.

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1.7.1 Defects and possible Remedies in the present day

teaching of Mathematics

It should be frankly admitted that the mathematics teaching in

today’s life is far from the conventional teaching. A literate person

fails to calculate while making payments to a shopkeeper for the

articles purchased by him.

Everybody has a complaint against the teaching of mathematics. It

is dull, boring, difficult and useless from the point of view of the

learner. “It is too remote from life to interest the students.” The

teachers complain of excessive workload and lack of facilities in

the form of aids and equipment.

Teachers’ Qualifications:

Now-a-days most of the private school teachers are not sufficiently

capable in the subjects apprehensive. Without proper qualifications

and proper training, they fail to do justice to the subject. This is not

a sufficient criterion to allow him to continue with the teaching of

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his subject. An adequate, high qualification with proper training

,the teacher develops self-confidence in him and serves as a source

of inspiration to his students. The teacher must be mature in his

subject. Professional training should equip him to attain desirable

standards in teaching. He must possess real knowledge and insight

into, the processes of mathematics and their effective teaching.

Teacher’s burden:

Now-a-days many teachers are overburdened on all sides like

teaching, assigning the students’ work, paper checking, etc. He

cannot adopt new techniques of teaching, and prepare for effective

methods, as he has no spare time. His burden does not allow him

time to remove individual difficulties. It should be reduced to

enable him to show his originality and innovative.

Teacher’s Salary:

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Many teachers who are working in private schools get very low

salary and their economic position is not good. He remains worried

and unsatisfactory person. So, he cannot give his best to the

learners. He often runs after other activities to supplement his

income. In these hard days, he must be suitable paid.

Teacher’s attitude:

Maybe, he does not have genuine love for his subject and

profession. He may have been forced by circumstances to take to

this profession. He remains on the lookout for a better job and

leaves the profession as soon as he get an opportunity to do so. He

lacks faith in the utility of the subject, and therefore, cannot create

interest among the students. Only really anxious and willing

individuals should be allowed to join this profession by

introducing a check at the time of selection, a teacher’s love for his

job and the subject should also be ascertained before giving him

his duty.

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Lack of purpose:

The students do not recognise the purposes behind the study of the

topics of mathematics. The particular and general aims of every

topic should be emphasised effectively. The teacher has to be

careful so that no student ever comes to think that these aims can

be attained through easy, soft and amusing work. If the work lacks

purpose, it is the teacher’s duty to make it purposeful. The purpose

should be attractive to stimulate the students to work hard. This

misconception should be uprooted from the minds of the parents

and pupils that most of the mathematics taught in the schools is not

purposeful.

Method of teaching:

The teacher clings to traditional methods, because these offer the

path of least resistance. The powers of thinking, acquiring

knowledge, understanding, creating interest about the topic and

retention are not thus developed in the students. If the students are

not showing any interest in the subject, it can be created not by

blind memorising, but by shifting the methods. There is spoon-

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feeding and the daily dose of mental work is much more than the

student can comfortably swallow and digest. The authorities run

after showy results which are obtainable only through cramming.

They have no appreciation for good mathematical teaching. There

is no emphasis on though, understanding, initiative, judicious study

and power. The remedy necessitates a fundamental change in

values and methods. Intelligent understanding should be the

guiding principle.

Rigour in study:

Any student who is discouraged, he does not make much progress.

Classroom atmosphere should be charged with freedom and

encouragement. The child should be given the opportunity of self-

education as far as possible. The teacher should not become a hard

task master, but should be a sympathetic helper and guide. The

emphasis should be on understanding, liking and interest.

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Large classes:

It is a general defect. No individual attention can be paid. It

becomes difficult for the teacher to establish close contacts with

the students. He cannot easily judge the capacities of the

individuals. This defect can be removed only by limiting he

number of students in each class upto a maximum of forty-five.

Practical Aspect:

The practical and application aspect of knowledge is not generally

emphasised knowledge given in the class-room is divorced from

practical life. The subject loses its appeal, as it is taught in an

abstract, dry and uninteresting manner. The affinity between

mathematics and life should be discovered and put to use. The

students should feel that they are getting something of direct

practical value. Mathematics should be taught as a part and parcel

of their daily life.

Mathematical language:

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Mathematical symbols have their own meanings and have their

own significance which the teachers generally fail to bring home to

the students. The meanings behind these symbols and their

historical background should be clarified to the students. Some

assignments may be given for the clarification of their meanings

and use.

Syllabus:

Some people say that the syllabus is defective, because it is heavy

and lengthy. The greater defect of the syllabus is that it does not

provide hints and instructions for teacher’s guidance. The teacher

cannot deal with the syllabus effectively, because most of the

details are left to him. It may be a bit lengthy, but it must lead to

understanding. In that case, the students will not mind a little over-

work.

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Text-books:

The traditional style of the syllabus also affects text-books

adversely. The authors have not been able to get rid of dogmatism

and traditionalism. The illustrations and problems given in the text-

books are divorced from actual life. These have been mainly

written on synthetic and deductive lines, whereas the psychology

of the child and the nature of the subject require them in analytic

and inductive forms. The material Is made available in a

readymade form which goes against thinking, discovery and

originality. The present day books promote cramming and to not

lay stress on understanding. Their style is seldom interesting and

impressive. They do not provide suggestions that may facilitate

learning. Text-books should give a brief history of the

development, possibilities of correlation, applications in practical

life, use of aids, plays, activities, projects, etc., concerning every

topic. They should abundantly present diagrams, sketches,

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illustrations, etc. The arrangement of the subject matter should

both be logical and psychological.

The Students:

There may be some defects in the students of the subject. The

subject demands regularity. It is a sequence subject, and if a

student is absent even for a few days, the sequence is broken and

he fails to comprehend the subsequent steps. An irregular student

cannot pull on well in this subject. Similarly, irregularity in home

work also makes the students lag behind. With the present –day

methods of teaching and the criteria of judgement of progress, the

students form misleading notions about their intelligence. The

crammers excel whereas the intelligent ones may suffer. The

present brings frustration for some able students, and there is in

store frustration for the crammers in the future. So, in fact, the

majority suffers. Nervous and rash students are also not likely to

do well in this subject. The very sight of the examination paper

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upsets and puzzles them. Moreover, this subject demands whole-

hearted concentration which some students may not be easily able

to give.

It is the joint responsibility of the home and the school to keep the

students regular in attendance and home work. If there is some

unavoidable absence, special and separate coaching arrangements

for some time are desirable. If a proper approach in its teaching is

adopted only really intelligent pupils will come to the forefront. To

remove their nervousness and confusion, the teacher should try to

develop self-confidence in the students.

Child-Centric Approach:

Teaching has been subject-centric and not child-centric. The child

has been treated as a miniature adult. Knowledge is thrust on him.

It has been presumed that all the students of a class have the same

capacity; same tastes, aptitudes, power of grasp and speed of work.

There has been no consideration of individual differences of the

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children. The child has been adjusted to the subject, whereas it

should have been the reverse. The child should not be subordinated

to the subject. He should be given as much as he can assimilate.

His interests, likes and dislikes, capacities, and difficulties and

aptitudes should be uppermost in the teacher’s mind. Let him be

given as much s he can assimilate, neither more nor less.

Libraries and laboratories:

The organisation of mathematical laboratories is yet awaiting a

start. The authorities have not paid any attention to this mast urgent

aspect to make the teaching effective. The library should offer

books of general interest and also books on the methods of

teaching. Similarly the laboratory should provide for B.B.

instruments, charts, models, instruments and various other

materials. The mathematics room should look as such.

Ban on short-cut methods:

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The use of short-cut methods has been banned totally, but these are

of great value and must be employed wherever possible. The

quickest, shortest and easiest methods of solving-problems should

be popularised. The desire to save the time and effort is natural.

Quick methods of calculation are much wanted in actual life also.

To meet this demand certain special instruments and processes

have been invented by mathematicians. The students should be

acquainted with these instruments and processes.

Examinations:

The teaching has been highly influenced by the examinations

which are full of defects. The main aim is to get through the

examination rather than to understand and grasp the subject. When

a defective system of examination dominates teaching, the latter is

not going to show any improvement. Some students by heart a few

selected topics meticulously to score a passing mark and a major

portion of the syllabus is left untouched. There is emphasis on

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guess papers rather than a real teaching. Then there are heavy

failures in this subject. Consequently the students’ enthusiasm and

interest are curbed. Examination is a matter of chance of more in

mathematics than in any other subject. It does not depict the true

picture of a candidate’s ability and work. A crammer may get the

upper hand in the examination, and an intelligent student may not

get his due.

This dominance of examination should go. The system needs a

thorough overhauling. The annual examination and essay type

examination should not be allowed to remain all important. We

should introduce true test of intelligence and understanding. Its

results should be reliable and valid. At the same time, it should not

remain a horror for the young learners. Though a high pass

percentage may retain a criterion yet it should have a sound

examination system as its basis. Success should no longer be a

matter of chance and cramming.

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Even after removing the above mentioned defects completely there

will be a scope for improvement. Concerted efforts will have to be

made for a pretty long time to set right the present deplorable

affairs. The best remedy is the devoted teacher. His personality can

over-shadow all the handicaps. The subject possesses tremendous

practical, disciplinary and cultural values. Its importance cannot be

over-looked. It should be taught as it should be taught.

1.8 Developing Speed and Accuracy in Mathematics

Speed and accuracy is indispensable for effective mathematics

learning more than in any other subjects.

We cannot satisfy one for the other. Students often employ

wasteful and inefficient procedures for learning mathematics. They

fail to be systematic and orderly in their mathematical work. They

do not take time for deliberate reflection before starting their work.

Quite often they do not think independently. Most of the time, they

do not have confidence in their own intellectual powers to produce

original and independent work. They easily get distracted and

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allow their work to be interrupted. They are careless in reading,

listening and written work. All these factors do not help the

students to develop speed and accuracy.

The teachers of Mathematics will have to make conscious efforts

to develop accuracy in mathematics among his students. The

following ways will be helpful in this regard.

Memorisation and habit formation

The students quite often make mistakes in numeral computation.

Therefore, all the fundamental computations should be thoroughly

memorized and habituated so that the required response to any

number of situations becomes automatic. Drill is an effective

means of memorization and habit formation. Also, knowledge of

principles helps to make pupils remember facts and to form habits

quickly. However, care should be taken so that the students do not

resort to rote memorization; instead they should be trained in

meaningful memorization of principles and formulae.

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Use of Oral questions

Oral questions can be asked to make the method of solving a

problem clear before they are asked to solve a problem in the

writing form. Once the method is clear, the student will be able to

solve it accurately.

1.8.1 Developing the habit of understanding and

analyzing the problems

Students make mistakes because they do not understand the

statement of the problem or analyse it properly as to what is given,

what is to be found out, what relationship exists among the given

data and so on. Many students look for clues in the problem before

deciding about the operation to be adopted (Example: altogether

for addition, how much left for subtraction, etc).or method to be

followed. Frequent practice in understating and analyzing the

problem through stimulating and thought-provoking questions can

help the students in a very significant way to increase accuracy in

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problem solving.

By encouraging the students to make correct

statements:

If the students are not able to write correct statements, he is likely

to develop the habit of inaccuracy. The correct statements and their

right sequence lead to the correct solution to the problem. This

should be made known to the students and adequate training in this

aspect could considerably enhance the accuracy of their

mathematical work.

Neat work, legible handwriting and proper posting of

figures:

The students make mistakes mainly due to inaccuracy in numerical

computation; computational errors occur because of shabby work,

illegible handwriting, improper posting of figures, overwriting etc.

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the teacher must personally take care to see that the students do

the work neatly, have legible handwriting, place the numbers

according to their place value specially while doing numerical

operations and avoid overwriting. Students may be asked to assign

separate space or column for rough work and rough should be done

neatly and systematically. Otherwise pupils will find it difficult to

retrace the work.

Copying all the figures correctly:

It is very common that students make mistakes while copying

problems and figures from the textbook, or from blackboard or

from question papers resulting in accuracy. There is also the

possibility of making errors while carrying over results from one

page to another. The students should be trained in early stage of

primary education itself at the habit of copying the numbers

correctly and to check each time whether the copied numbers are

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right or wrong. This habit will increase the accuracy in doing

mathematics.

Habit of verification of results:

The habit of verification of results is one of the important means of

ensuring accuracy. Not only should numerical calculations be

checked by the students, but also, all the forms of thinking leading

to the solution should be checked and verified for accuracy.

Absurd results can be avoided if the students are accustomed to ask

themselves the question “Is the answer probable, or reasonable?”

The habit of assuming the answer to be right leads to inaccuracy.

The students should be trained to develop habit of verifying not

only the last result, but also all the steps leading to it.

Encouraging correct answer and discouraging

inaccurate answers:

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This is based on the psychological principle of reward and

punishment. Positive reinforcement provided to the students on

getting the right answers help the students in regarding accuracy as

a thing to be acquired because it is rewarded.

Employing diagnostic test and remedial measures:

Diagnostic tests will find the causes of inaccuracy and the teacher

can provide suitable remedial programmes for the removal of

errors, leading to greater accuracy.

1.8.2 WAYS AND MEANS OF DEVELOPING

SPEED:

The following measures are helpful in developing speed.

Developing accuracy:

Inaccuracy is the greatest obstacle in the development of speed.

Once accuracy has been developed, speed can be developed.

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Inaccuracy cannot be accepted in mathematics. The students

should do it correctly and do it with speed. In mathematics, both

speed and accuracy go together.

Providing time limit:

Students may be asked to complete the assignments or solve

mathematics problems within a limited time. The teacher can

reward the first few students who finish the task correctly within

the specified time limit. This will motivate other students to finish

the task within a limited time and will help in increasing speed. If

given a lot time, students will not develop speed, but they will

develop accuracy.

Use of short-cut methods and formulae:

Another important means of increasing speed is the use of short-

cut methods and formulae in solving mathematics problems.

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Wherever possible, students should be encouraged to use short cut

methods and formulae to solve mathematical problems.

Drill and practice:

Drill and practice help in fixation and memorization of facts. This,

in turn, helps in increasing speed.

Use of symbols and mathematical language:

Speed can be increased by making the statements in a concise and

precise form using mathematical language and symbols. The

students should be discouraged from long and elaborate verbal

description of unnecessary details. Their relevant steps and figures

should be avoided.

Use of Calculators:

Speed in numerical calculations can be interested through the use

of calculators. However, adequate care should be taken in the use

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of calculators. Calculators should be introduced only after the

principles underlying the fundamental operations are made clear to

the students.

The following suggestions can be useful if adopted by the students.

- Form the habit of studying mathematics at a regular time, in

a quiet place, with concentration, without interruptions and mental

distractions, keeping the mind alert and active.

- Do independent work by planning beforehand and giving

plenty of time for scientific thinking and by analyzing difficulties.

- Work out the questions that bother you. Make them clear

and specific by analysis. Often the answer will suggest itself.

- Learn fundamental concepts, formulae and principles; but be

sure you understand their meanings and can use them correctly.

- Work carefully, systematically, regularly and neatly.

- Develop the habit of expressing verbal statements in

symbolic form.

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- Sketch graphs or diagrams, wherever possible. This often

makes it easier to understand the problems.

- Do one step at a time. Check each step and the final answer.

Although speed has to be developed, there must be no hurry as

haste makes waste.

1.8.3 Arousing and Maintaining Interest in

Mathematics:

It is well known fact that students will work most diligently and

effectively at tasks in which they are genuinely interested.

Therefore, one of the important tasks of the teacher of mathematics

is to create and maintain interest among his students. Perhaps, it is

one of the most difficult problems encountered by the teachers of

mathematics.

The students will show greater enthusiasm for the work for which

they are highly motivated. Thus the motivation has two aspects: (1)

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creating or arousing interest (2) to keep continuously maintain the

interest of the work.

Interests are motives which serve as important influences in

producing both activities and attitudes that are most favourable for

learning. A strong interest in mathematics tends to produce a

positive attitude towards mathematics and such an attitude would

in turn enhance the desire to learn mathematics in a more

productive way. Thus the development and maintenance of interest

in mathematics becomes an important concern of the mathematics

teacher.

A mathematics teacher should be well versed with the means and

techniques of arousing and stimulating interest in mathematics.

Some of the devices are discussed below. However, a teacher can

always devise techniques which are most appropriate for his own

students.

Element of novelty: Students become interested in things which

are new and exciting. Though the possession of background

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information related to the new work tends to intensify the interest,

novelty is more compelling than familiarity. The mathematics

learning should arouse the curiosity of the students; should satisfy

their thirst for knowledge and should help in appreciating the

beauty of mathematics. The teacher has to arrange the mathematics

activities in a manner most suitable for the students level of

understanding.

Ensuring students’ understanding: Inability to understand

makes the students restless and listless leading to general loss of

interest. Students tend to remain interested in those things which

they understand well and which they can do most successfully. A

reasonable degree of competence should be ensured keeping in

mind the nature of the content presented and capacity of the

students to understand.

Intellectual challenge: The mathematical work presented should

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not be too simple. It should present a continual intellectual

challenge to the students, devoid of drudgery and boredom. This is

possible by organizing the various mathematical activities depends

upon the maturity of students.

Improving problem-solving ability : Genuine interest in

mathematics probably depends upon the problem solving aspect of

the subject. Mathematics teaching presents the students with an

abundance of problems every day. These problems equip the

students with modes of thought and techniques which enable them

to solve the problems successfully. Each successful solution

provides the student with a sense of achievement, a feeling of

satisfaction and joy. This could act as a driving force to seek

similar experiences, pursuing tasks of the same kind. Therefore,

the mathematics teacher should plan problems with care and

should make sure that the students are able to solve them

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successfully so that the students gets the positive reinforcement to

proceed further.

Use of incentives:

Incentives such as marks, rewards of various kinds may serve to

build motives or “inner drives” and thus promote genuine interest

in mathematics. However, use of incentives does not guarantee

this. They should not be used indiscriminately or thoughtlessly.

Unless used judiciously, they can do more harm than good. The

success of the use of such devices depends largely upon the kinds

of incentives used and the ways in which they are used. A teacher

can make use of devices such as mathematical games and contests,

tricks, puzzles, and other recreations, multisensory aids, projects,

information about the application, values and utility of

mathematics, historical notes on mathematics etc. as incentives.

Emphasizing the practical application of mathematics:

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The practical application of mathematics in daily life situations and

real life situations, provides an important means of stimulating

interest. Many topics that the students learn in high school

mathematics have immediate relevance for solving problems that

arise in daily life. For example, while teaching topics such as

simple interest, compound interest, recurring deposit, discount,

percentage, stocks and shares, direct and indirect variations,

mensuration etc., the teachers should not fail to stress their

importance from this point of view. The mathematical principles

help the students in understanding and interpreting laws of nature

and environment. This knowledge also could sustain the interest of

the students.

Use of audiovisual aids and practical work:

Mathematics teaching can be made more interesting though a

variety of sensory experiences than through mere talk and chalk

method. The use of audiovisual aids provides a variety of sensory

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experiences which wil l help in making abstract mathematical

concepts concrete and meaningful. Moreover, it facilitates better

understanding of the subject and thereby makes it more interesting.

It is desirable to use experiments and laboratory work to verify

mathematical truths and discover mathematical laws and

principles. It arouses the intellectual curiosity of the students and

helps to maintain interest in the subject.

Using mathematics for fund and recreation:

Mathematics provides enough opportunities for fund and

recreation. This can be used as an effective means of stimulating

interest in mathematics. Mathematical games and puzzles serve as

interesting setting for mathematical principles. But care should be

taken to see that they do not present distorted ideas of the nature of

Mathematics. Mathematics clubs are also forums where interesting

learning can take place through fun filled activities.

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References to histor ical development of the mathematical

concepts: The teacher can use historical anecdotes relating to the

development of mathematical concepts and ideas as a device for

stimulating interest among the students. The life history of

mathematicians, the painstaking efforts taken by the

mathematicians to discover mathematical facts, and the patience

and perseverance exhibited by the mathematicians in their pursuit

for search for mathematical truths can be appealing for the students

to pursue their mathematics learning with interest and enthusiasm.

A resourceful teacher can think of many more such devices which

are especially suitable for the students whom he teaches. However,

selection of devices should be guided by the level of achievement,

the intellectual maturity of the students as well as attitude and

aptitude of the students. Above all, the teacher himself should be

highly motivated to teach mathematics with a high level of interest

in the subject. Only such teachers could inculcate interest in the

students to learn.

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Arranging for field work and field trip:

Field work and field trips are the two important techniques for

stimulating interest in students. Both field work and field trip

initiate the students to the real world and provide first hand

experiences in the practical application of mathematics learning. A

teacher, wherever possible, could make use of these techniques for

arousing interest in mathematics.

1.9 Vedic Mathematics

Vedic Mathematics is a unique method of solving problems in the

use of fast calculations. It is a unique system. Vedic mathematics

helps all kinds of mathematical problem of all kinds to be solved

easily and efficiently. This wonderful method has been discovered

by Sri Bharati Krishna Tirtha Swamiji (1884-19600 of Govardhan

peetha, Puri.

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Vedic Mathematics is Generally based on sixteen sutras (formulas)

and thirteen Sub-sutras. It deals with numbers and also with

advanced theories such as calculus, simultaneous equations,

solving differentiation and integration problems. These sutras will

enrich the skills of solving mathematical problems. By memorizing

these simple sutras one gain confidence in rapid mathematical

computation and solving mathematical problems intellectually.

Vedic Mathematics mostly deals mainly with various Vedic

mathematical formulae / algorithms. Vedic mathematics is useful

for solving even very difficult problems mentally. Vedic

Mathematics is very easy and much simpler to understand than

traditional Mathematics. By using, with the help of Vedic

Mathematics we can solve problems in a single step and also

problems can be solved faster than a calculator. Vedic mathematics

helps to solve mathematical problems very much faster than the

traditional methods of solving problems. In Vedic Mathematics,

most of the calculations can be solved from left to right. This is

opposite to traditional method of solving mathematical problems.

In traditional method one can start to calculate problems from right

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to left. Vedic mathematics makes mathematics an easy one. This

also creates an interest among students. Vedic Mathematics is

considered as a magical method of fast calculation. It is a very

unique system based on simple rules and principles which facilitate

all kinds of mathematical problems to be solved easily and

efficiently.

Now – a – days many are using Vedic Mathematics with pleasure.

Vedic mathematics can be used to solve mathematical problems

without pressure but with much pleasure. Vedic Mathematics is an

ancient technique, developed in India. Even some of the

prestigious institutions in Europe, England Britain, the US,

Australia etc are being started to use vedic mathematics. Even

today, the NASA scientists have been applied vedic mathematics

in the area of artificial intelligence.

1.9.1 Origin of Vedic Mathematics

His holiness Jagadguru Sankaracarya Sri Bharati Krsna Tirthaji

Maharaj of Govardhana Matha, Puri (1884-1960) had written

VEDIC MATHEMATICS or ‘Sixteen Simple Mathematical

Formulae from the Vedas’. It is the result of the of the author, who

durring the course of eight years of highly concentrated mental

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endeavor . He intuitionally visualised some of the fundamental

mathematical truths and principles.

The Vedas are well-known as four in number Rg, Yajur, Sama and

Atharya, but they have also the four Upavedas and the six

Vedangas all of which form of divine knowledge .

The following are the four Upavedas: Veda

Upaveda Rgveda

Ayurveda Samaveda

Gandharvaveda Yajurrveda

Dhanurveda Atharvaveda

Sthapatyaveda

In this list the Upaveda of Sthapatya comprises and all visual arts.

Swamiji naturally regarded mathematics calculations and

computations to fall under this category.

Swamiji could attract large audiences. He could speak for several

hours at a stretch in Sanskrit and English.

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We are told in his preface by Swami Sankaracharya that he

contemplated to wrapup all the diverse branches of mathematics

such as trigonometry, astronomy, Statistics etc., with these basic

Sutras. That comprehensive application of the sutras could not be

left by him.

The ancient system of mathematics is Vedic Mathematics.

According to Sri Bharati Krsna Tirthaji, vedic Mathematics is

based on sixteen Sutras. With the help of Vedic mathematics any

one can solve 'difficult' problems or huge sums without much

difficulty. The problems can be calculated by mentally is one of

the simplicities of vedic mathematics.

By using Vedic mathematics we have many advantages. One can

use or discover his or her own method to solve their problems.

There are many methods to solve problems. One can select any

method according to his/ her convenience. This helps the students

to be more creative.

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1.9.2 The History of Vedic Mathematics

Vedic Mathematics was born in the Vedic Age, but it was buried

under centuries of wreckage.

The former Shankaracharya (a major religious leader) of Puri,

India, Bharati Krishna Tirthaji delved into the ancient Vedic texts

and established the techniques of this system in his pioneering

work - Vedic Mathematics in 1965. This is considered the starting

point for all work on Vedic Mathematics. It is said that after

Bharati Krishna's original 16 volumes of work illustrating the

Vedic system were lost, in his final years he wrote this single

volume, which was published five years after his death. Jagat Guru

Bharti Krishanji had worked very hard for eight years to get all this

knowledge.

1.9.3 SIXTEEN SUTRAS AND THEIR COROLLARIES

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Sutras

1.Ekaadhikena Purvenna (also a corollary)

2.Nikilam Navathascaramam Dasathah

3.Urdhva-thiryagbhyam

4.Paravarthya Yojayet

5.Suniyam Samyaasamuccaya

6.(Anurupiye) Suniyamanyat

7.Sankalaana – viyavakaland-bhiyam (also a corollary)

8.Puraanapuranbhyam

9.Calana-Kalanabhiyam

10.Yavathuunam

11.Viyastisamasthih

12.Sesanyankena Caramenna

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13.Sopantyadvayamanthyam

14.Ekanyunena Purvena

15.Gunithasamuccayah

16.Gunakasamucscayah

Sub-sutras or Corollaries

1.Anurupiyena

2.Sisyate Sesasamjinah

3.Adyamadyenantya-manthyena

4.kevalaih Sapthakam Gunyath

5.Vesthanam

6.Yavathuunam tavathuunikrtya vargancha yojayet

8.Anthyayorthasakepi

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9.Anthyayoreva

10.samuccayagunitah

11.Lopanastapanabhyam

12.vilokanam

13.Gunithasamuccayah Samuccayagunitah

1.9.4 Advantages of Vedic Mathematics

There are obviously many advantages in the system of Vedic

Mathematics.

• Vedic mathematics can be used to stimulate creativity in all

types (gifted, average, below- average) of students.

• Vedic mathematics helps the slow-learners to understand the

basic concepts and solve mathematical problems easily.

• Many students don’t like mathematics. But vedic

mathematics helps to create students in mathematics so that

they can solve the problems easily.

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� Vedic mathematics helps to reduce the burden of

remembering more mathematical tables.

� When compared to the conventional method it enables

faster calculation. Thus, the time that one gets saved with

the help of using Vedic mathematics problems can be

used solved very easily.

� Vedic mathematics helps to increase concentration and

speed to solve more problems very fast.

� It encourages solving problems mentally without using

paper and pen.

� Vedic mathematics saves time.

� A dreadful subject of many students is converted into a

playful and blissful subject.

� Vedic mathematics helps students too participate and win

competitive exams.

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� Mathematical problems can be calculated very faster than

the traditional method of solving problems with the help

oh vedic mathematics.

� It creates interest towards mathematics and will be

beneficial throughout lifetime.

� It helps the people to guess the answer intelligently.

(Getting the answer without actually solving the

problem).

� It is a magical tool to solve mathematical problems. It

reduces the finger counting and improves mental

calculation.

� Saves time during examination.

1.10 NEED & IMPORTANCE OF THE STUDY:

Mathematics is the most important & compulsory subject in our

present school curriculum. From Multi millionaires to daily

labourers have been using mathematics in one or another way.

Therefore everyone should have the knowledge of mathematics.

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But today most of our younger generations are completely

depending upon technology. Technology has been diminishing

their creativity. Because of many barriers, they cannot depend

upon technology. They should know Vedic mathematics to solve

mathematics problem very easily & quickly. It is the duty of

teachers to preserve our traditions as well as lead our students to be

self-dependent & solve the mathematics problems with confident.

Mathematics being a compulsory subject of present curriculum,

and also getting basic mathematics education is each and every

child’s right. It is the duty of the teacher to give quality education

to all students.

In the present day mathematics, many students do not like

mathematics subject. They require more effort in understanding

and solving mathematical problems. But ith the help of vedic

mathematics we can change students mind. They can understand

the basic concepts and solve the problems without much effort but

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with much interest.

Today interest in the Vedic mathematics system is increasing in

many people. Some of the mathematics teachers are looking for

something better. Now-a -days using vedic mathematics as well

as the effects of learning Vedic Mathematics on students.

Today, many schools and even universities use Vedic mathematics

as an alternative system of mathematics in modern mathematics.

Modern mathematics has established methods and allows the use

of calculators. In the case of Vedic math, it is flexible and

encourages the use of arithmetic, geometry & trigonometry. This

may contribute to brain development in children.

With the help of Vedic mathematics students can score high marks

and also excel in competitive examinations. In the present

scenario, all the competitive examinations contain Mathematical

aptitude sessions, in which students should score good marks . If

the candidate or student is going to solve or calculate problems in a

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traditional manner he has to spend a lot of time for completing that

particular examination.

If one uses Vedic mathematics in a proper way, then he can solve

mathematical problems in very fast. And also he can save a lot of

time in completing examination. Now Vedic Mathematics plays a

significant role in Arithmetical, Algebra, geometry statistics and

alsoin the theory of equations etc.

However, much research is still ongoing, especially in India to find

ways to facilitate the application of Vedic mathematics in calculus,

geometry and calculus.

Keeping the above points in mind, this topic has been selected by

the investigator to serve the students’ community & also give

awareness about Vedic mathematics.

CONCLUSION

In this chapter the researcher justified the research title with

Introduction, Nature and characteristics of Mathematics, History

of Mathematics, Importance of teaching and learning mathematics,

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Vedic Mathematics, Advantages of Vedic Mathematics etc and

need and importance of the study.

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