UNIT I
NATURE OF MATHEMATICS
· Meaning and Definition of Mathematics
Mathematics is derived from the Greek letter
‘Mathema’. It means science, learning and knowledge or ‘Mathematikos’ means
fund of learning.
Mathematics means that the science of numbers and
space or the science of measurement and quantity. It is a systemized, organized
and exact branch of science. it deals with quantitative facts, relationships as
well as the problems involving the space and form. It is a logical study of
shape, arrangement and quantity.
•
Mathematics
is a science that involves dealing with numbers, different kind’s of
calculations, measurement of shapes and structures, organization and
interpretation of data and establishing relationship among variables, etc.
•
Pythagoreans
(Pythagoras and his followers) coined the term, Mathematics, which is derived
from the Greek word “Manthanein” means learning ‘Technne” means an art or
technique which means” inclined to learn” or “art of learning”
· Definitions
1. “The Science of quantity”- Aristotle
2. “Mathematics is the Gateway and key to all science” –
Bacon
3. “Mathematics is the indispensable instrument of all
physical researches”- Kant
4. “Mathematics is the queen of sciences and arithmetic
is the queen of all mathematics”- Gauss
5. “Mathematics is the gateway and key to all sciences”-
Bacon
6. “Mathematics is the study of abstract system built of
abstract elements. These elements are not described in concrete fashion”-
Marshal. H. Stone
7. “Mathematics may be defined as the subject in which we
never know what we are talking about nor whether what we are saying true”-
Bertrand Russell
8. “Mathematics is a way to settle in the mind of children
a habit of reasoning”- Locke
9. “Mathematics is the language in which God has written
the universe”- Galileo
10. “Our entire civilization depending on the intellectual
penetration and utilization of nature has its real foundation in the
mathematical sciences”- Prof. Voss
11. “Mathematics is the science that draws necessary
conclusions”- Benjamin Peirce
12. “Mathematics is the science of indirect measurement”-
Comete
13. “Mathematics is the science of number and space”-
Dictionary Meaning
14. “Mathematics is the language of physical science and certainly
no more marvelous language created by the mind of man”- Lindsay
15. “Mathematics is a science of order and measure”-
Descrates
16. “Mathematics is the study of quantity”- Aristotle
17. “Mathematics is an expression of the human mind which reflects the active will, the contemplative reason and desire for aesthetics perfection. Its basic elements are logic and intuition, analysis and construction, generally and individually”- Courant and Robin
18. “Mathematics is engaged in fact, in the profound study
of art expression of beauty”- J.B.Shaw
19. “Mathematics should be visualized as the vehicle to
train a child to think , reason, analyse and articulate logically. Apart from
being a specific subject, it should be treated as a concomitant to any subject
involving analysis and meaning” (National Policy on Education, 1986)
On the basis of above definition-
- o Mathematics is the science of space and number
- o Mathematics is the science of calculations
- o Mathematics is the science of measurement, quantity and magnitude
- o Mathematics deals with quantitative facts and relationships
- o Mathematics is the abstract form of science
- o Mathematics is a science of logical reasoning
- o Mathematics settles in the mind habit of reasoning
- o Mathematics is an inductive and experimental science
- o Mathematics is the science which draws necessary conclusions
- o Mathematics helps in solving the problem of our life
- o Mathematics has its own language- signs, symbols, terms and operations etc.
- o Mathematics involves high cognitive powers of human being
·Nature of Mathematics- Mathematics as a Science, Mathematics as a game,
Mathematics as a language, Mathematics as a tool, Difference
between Mathematical science and basic science
Mathematics as a Science
According to E.E. Biggs, “Mathematics is the discovery
of relationships and expression of those relationships in symbolic form. In
words, in numbers, in letters, by diagrams (or) by graphs. “An Whitehead states
that “Every child should experience the joy of discovery”. Mathematics gives an
easy and early opportunity to make independent discoveries. Today it is
discovery of techniques, which are making spectacular progress. They are being
applied in two fields: in pure number relationships and in everyday problems
like money, weights and measures
Mathematics is a science of space, number, magnitude
and measurement. It has its own language which consists of mathematical terms,
concepts, principles, theories, formulae, symbols etc. it is a systemized,
organized and exact branch of science. Mathematics involves conventions of
abstract concepts into concrete form and it is the science of logical reasoning.
It does not leave any doubts in the mind of learner about theories, principles,
concepts, etc. Mathematics helps to develop the habit of self-confidence and
self- reliance in children and in development of sense of appreciation among
children. It also helps to develop scientific attitude among children.
Mathematics knowledge is exact systematic logical and clear. So that once it is
captured it can ever forgotten. Mathematical rules, laws and formulae are
universal and that can be verified at any place and time.
Mathematics is an intellectual game
Mathematics can be treated as an intellectual game
with its own rules and abstract concepts. Mathematics is mainly a matter of
puzzles, paradoxes and problem solving- a sort of healthy method &
exercise. As a game, it gives better achievement, motivation, mental exercise,
independents, meaningful creativities, interesting, developing and discovering
mentality and 3’H’ Coordination (Head, Heart and Hand).
Mathematics as a language
The most distinguishing characteristic of mathematics
is its unique language and symbolism. Man has the ability to assign symbols for
objects and ideas, and mathematics language is one among the marvelous language
developed by the mind of man. The special feature of the language of
mathematics is defined by the specialty of their numbers and symbols. We are
using language for the purpose of interaction and communication. The language
of mathematics has its own grammar. This might have uses of counting comparing
lengths etc. involved in exchange of commodities required for day-to-day life.
The nature of language of mathematics has been influenced by all these factors
in mathematics, we are using mathematics language. A very important feature of
such symbolic language is that those symbols can be convey conventionary
accepted meanings approved by all societies. The language mathematics has its
own grammar. In fact, symbolism insists on precision and accuracy, straight forwardness
brevity insistence on logic and reasoning, universality etc. are some of the
qualities of the language of mathematics.
According to Lidsay “Mathematical is the language of
physical sciences and certainly no more marvelous language was ever created by
the mind of man”. Mathematical symbols can put the lengthy statements, accurately
and in exact form, in a very brief description. For example, if we wish to say
that the sum of the squares of two sides of a right-angled triangle is equal to
the square of the hypotenuse, then we can write it in mathematical language or
symbolic forms as c2=
a2 + b2.
Mathematics language is symbolic, precise and exact.
Mathematics language is peculiar because
Ø It reduces the length of the statements and brings
clarity and preciseness in expression
Ø Mathematical results are made exact and readily useful
because of the mathematical language
Ø Many complicated problems can be easily solved when
expressed in mathematical language
Ø Later progress of a learner is highly determined by
the learner’s ability to apply the mathematics language and symbols
Ø Most of the scientific inventions are expressed
through mathematical language and symbols
As the list of symbols of mathematics language is so
vast, it is very difficult to prepare a comprehensive list. In the absence of
any clear understanding, they try to cram the statements and try to use these
symbols mechanically, without much understanding. They begin to lose interest
in the subject which appears to them, dull and boring. So, teachers should
train the pupils in the use of mathematical language and symbols. They should
learn to appreciate the beauty, precision and exactness of mathematical
language and symbolism.
Mathematics as a tool
Mathematics is an interdisciplinary language and tool.
Like reading and writing. Mathematics is an important component of learning and
doing in each academic discipline. Mathematics is such a useful tool that is
consider one of the basic more formal education systems. Mathematics is a tool
of science. Mathematics is a tool to
understanding the world one which should work for all students in one
community. Students are using mathematics to justify conclusions and making
relationships between different ideas. Mathematics is more than a tool embedded
deeply in nature of reality.
Science and Mathematics
Mathematics and science have many features in common.
Science provides mathematics with interesting problem to investigate and
mathematics provides science as a powerful tool to use analyzing data. Science
and mathematics both aim to discover the general pattern and relationship. In
this sense they are parts of same endeavor. According to New English
Dictionary,” Mathematics- in a strict sense- is the abstract science which
investigates deductively the conclusions implicit in the elementary conception
of spatial and numerical relations”.30
Difference between Mathematical science and basic
science
|
Mathematics |
Science |
|
Rationalism Reason explanations Hypothetical reasoning Abstractions Logical thinking theories |
Rationalism Reason Explanations Hypothetical reasoning Abstractions Logical thinking Theories |
|
Empiricism Atomism Objectising Materialism Concretising Determinism Symbolizing Analogical thinking |
Empiricism Atomism Objectising Materialisation Symbolising Analogical thinking Precise Measurable Accuracy
Coherence Fruitfulness Parsimony Identifying problem |
|
Control Prediction Mastery over environment knowing Rules Security Power |
Control Prediction Mastery over Problems Knowing Rules Paradigms Circumstance of activity |
|
Progress Growth Questioning Cumulative development of knowledge Generalisation Alternativism |
Progress Growth Cumulative development of knowledge
Generalisation Deepened understanding Plausible alternatives |
|
Openness Facts Universality Articulation individual liberty demonstration
Sharing Verification |
Openness Articulation Sharing Credibility Individual liberty
Human construction. |
|
Mystery Abstractness World Unclear origins Mystique Dehumanised knowledge Intuition |
Mystery Intuition Guesses Day dreams Curiosity Fascination |
·
Nature
of Mathematics
Ø Mathematics is a science of space, number, magnitude
and measurement
Ø Mathematics has its own language which consists
Mathematical terms, concepts, principles, theories, formulae and symbols etc.
Ø Mathematics is a systematized, organized and exact
branch of science
Ø Mathematics involves conversation of abstract concepts
into concrete form
Ø Mathematics is the science of logical reasoning
Ø Mathematics does not leave any doubt in the mind of learner
about the theories, principles, concepts etc.
Ø Mathematics helps to develop the habits of self-confidence
and self-reliance of children
Ø Mathematics helps in the development of sense of
appreciation among the children
Ø Mathematics helps to develop scientific attitude among
the children
Ø The study of mathematics gives the training of
scientific method to the children
Ø Mathematical knowledge is based on sense organs
Ø It gives accurate and reliable knowledge
Ø Mathematical knowledge is exact systematic logical and
clear so that once it is captures it can never be forgotten
Ø Mathematical rules laws and formulae and universal
that can be verified at any place and time
Ø It develops the ability of induction, deduction and
generalization
Ø Mathematical language is well defined useful and clear
Ø It draws numerical inferences on the basis of given
information and data
Ø Mathematical language is applied in the study of
science and in its different branches
Ø It is not only use for different branches of science
but also helps in its progress and organization
Ø Mathematics is a science of precision and accuracy
Ø Mathematics is an intellectual game
Ø Mathematics is science of discovery
Ø Mathematics is a study of structure
Ø It develops critical thinking
Ø It helps in the development of scientific attitude
among children
Ø Mathematical knowledge is exact, systematic, logic and
clear
Ø Mathematical rules, laws and formulae are universal
and well accepted that can be verified at any place and time
Ø
It
develops the ability of induction and deduction
CHARACTERISTICS
§ Precision and Accuracy
Mathematics is
an exact science which is either true or wrong.
Precision means when we approximating the apparent error is named as
precision and relative error means accuracy.
§ Logical Sequence
Inter
dependance of subject assigning mathematics there we can see logical sequence
is subject like arithmetic, algebra, geometry and calculus
§ Applicability
All the results
in mathematics can apply real and actual situations especially the fields of
agriculture, computers, physics, aeronautics, cryptography, economics and also real-life
situations.
§ Generalization and classification
By using
induction method, the results of mathematics can be generalized easily. Based
on the generalization or based on same characteristics we can easily classify
the concepts in mathematics. Eg.-open sets, closed sets
§ It has a language and special symbolism
§ Abstractness
§ Structure of mathematics
A mathematical
structure is a set with various associated mathematical objects such as
subjects, sets of subjects, operations and relations, all which must satisfy
various requirements. The collection of associated mathematical objects is called
structure and the set is called the underlining set.
§ Rigour and logic
§ Objectivity and predictability
§ Free from social influences (universal)
§ Selfcontained
§ Interconnected structures
§ Simplicity and accuracy of reasoning
§ Originality of thinking
§ Similarity to the reasoning of daily life
§ Certainty of results
·
Pure and applied Mathematics
Pure Mathematics
Pure mathematics involves
systematic and deductive reasoning, pure mathematics deals with exact
statements, concepts and theories which are disconnected from perception. Of
course, starting made with perceptual concepts. These are so mathematised that these
ceases to be perceptual. It treats only theories and principles without regard
to their application to concrete things. It consists of all those assertions as
that, if such and such another proposition is rue of that thing. It is
developed on an abstract, selfcontained basis without any consideration of its
possible application if any.
•
It is a field of mathematics that
studied entirely abstract concepts.
•
This was a recognizable category of
mathematical activity from the 19th century onwards.
•
Pure mathematics includes systematic
and deductive reasoning.
•
It treats only theories and
principles without regard to their application to concrete things
•
It deals with statements concepts
and theories which are disconnected from perception.
Applied Mathematics
It is the application of pure mathematics in the real world. It considers
those parts of mathematical theories that have certain direct or practical
application to objects and happening in the material world. It plays a great
role in the development of various subjects. Principles of applied mathematics
have seen useful in the investigation of such phenomenon as heat, sound,
astronomy, navigation etc. Making use of pen and paper, we can weigh the earth,
prepare for any space flight, determine the paths of planets, solve complicated
problems of business in the international world. Applied Mathematics thus helps
in solving the intricate problems of the physical world.
In the beginning, man studied only those mathematical structures which
were isomorphic to the structures of relationships in nature and soon turned
into abstraction. According to Dantzigs, “Applied Mathematics is like
wine which becomes pure in course of time”
The relationship is beautifully illustrated by
Synge as follows
Ø A dive from the world of reality into
the world of mathematics.
Ø A swim in the world of Mathematics.
Ø A climb from the world of mathematics
back into the world of reality carrying the prediction in teeth.
Thus, Applied Mathematics acts as a bridge to link pure mathematics with
physical, biological, social science etc. It acts and reacts not only on
science and technology but also pure mathematics Eg: fluid dynamics, space
dynamics, mathematical biology, mathematical economics
Relation
between Pure and Applied mathematics
Pure
mathematics involves systematic and deductive reasoning. Quite often certain
structures are discovered in social and physical sciences and a pure
mathematician makes an effort to develop some parallel structures are
discovered in social and physical science and a pure mathematics makes an
effort to develop some parallel structures in mathematics. Many theories and
structures of pure mathematics later on find many and applications which were
not known at the time of their invention. Taking the example of complex numbers,
we find that these numbers are widely used and they find intensive application
in electricity, radio etc.
Mathematical system
•
Undefined
terms
•
Defined
terms
•
Axioms
or Postulates
•
Theorems
Undefined terms
•
In mathematical
system we come across many terms which cannot be precisely defined.
•
The
choice of the undefined terms is completely arbitrary and generally to
facilitate the development of the structure Eg: point, line, plane, variable,
number…
Defined terms
•
We
defined the other terms of mathematical system in terms of undefined terms.
•
Eg:
angle, line segment, circle …are some terms, which have been defined with the
with the help of undefined terms and defined terms we develop statements
concerning mathematical principles
These statements are of two kinds
- Statements
accepted without proof which are called postulates or axioms
- Statements
which are proved using the undefined terms, definitions, postulates and
accepted rules of logic, such statements are known as theorems.
·
Role of axioms and postulates
Axioms and Postulates
•
Euclid
in his book “ELEMENTS”, presents certain assumptions that are self-evident
truth or obvious and hence no need of proofs is generally.
Axioms
•
That
is an axiom is a mathematical fact to be accepted without proof
•
That
is generally true for any field
•
Impossible
to proove from other axioms.
•
Conformity
by common experience of judgment
•
The
axioms of a mathematical system are like the rules of a game. They are the ‘arbitrary
starting points’ from which a mathematical system can be developed.
•
Oxford dictionary definition: A statement
or preposition which is regarded as being established, accepted, or self-evidently
true.
•
Cambridge dictionary definition: A
statement or principle that is generally accepted to be true, but need not be
so.
•
Eg:
the part is smaller than the whole,
•
Halves
of equals are equal.
•
Things
that coincide with one another are equal to one another
•
The
equals are added to equals the result will be equal
Postulates
A postulate is a self-evident problem assumed without
proof.
Euclid separated 5 items from axioms as those had
special significance in Geometry and called them as postulates.
Specially defined for geometry
Oxford dictionary definition: A
thing suggested or assumed as true as the basis for reasoning, discussion or
believe.
Cambridge
dictionary definition: To suggest a theory, idea etc. As the basic principle
from which a further idea is formed or developed.
Eg:
1. Given two distinct points there is a unique (one and
only one) line passing through them.
2. A finite straight line can be produced indefinitely
3. A circle can be drawn with any center and only one
radius
4. All right angles are equal to one another
5. Parallel postulate: If a
straight line falling on two straight lines makes the interior angles on the
same side of it taken together less than two right angles, the two straight
lines produces indefinitely meet on that side on which the sum of angles have
been found to be less than two right angles.
Theorems
Cambridge dictionary
definition: A formal statement that can shown to be true by logic
Oxford dictionary definition:
A general preposition not self-evident, but proved by a chain of reasoning;
truth established by means of accepted truths.
Eg: Pythagoras theorem,
Fundamental theorem of arithmetic etc.
·
Fundamental branches of Mathematics (Arithmetic, Algebra, Geometry, Trigonometry)-
Origin, nature of content link between the branches
·
Correlation of mathematics with other subjects and real life
Teaching of
mathematics is done keeping in view its correlation with other subjects. It
helps understanding of subject-matter by the students. It is possible to
correlate different experiences and at the same time, allow different aptitudes
and inclinatons to work with coordination and correlation. The mind of the
students works in such manner that he can understand a subject that is being
taught, keeping in view its correlation with other subjects. Such a teaching
brings a healthy development of personality.
TYPES OF CORRELATION
I.
Correlation of different branches of mathematics
II.
Correlation of mathematics with other subjects
III.
Correlation of mathematics with daily life
Correlation of
different branches of mathematics
Mathematics is a sequence subject.
Here the study of one topic depends upon the learning of other topics.
Therefore, it is essential that there must be an orderly treatment of the
subject showing the relationship of one topic with another. Study in any branch
of the subject Mathematics should be so planned that its different topics may
appear to have proper links with each other and learning of one topic may
stimulate and necessitate the need other and studying other topics. The teacher
should try to impart the knowledge of a particular topic only after
establishing sufficient base for the study of that topic. He should also try to
make the students realise the importance of studying other topics in a
particular sequence. For example, L.C.M. is useful in adding and subtracting
factors. It should be taught when the need for adding fractions with different
denominators arises. Similarly in the study of algebra, teaching of formulae
and equations must be made the centre of learning. All the other topics like four
fundamental rules, simplifications, factorization, removal of brackets etc. should
eventually revolve round this centre. Actually, speaking when we teach a
particular topic in any branch of mathematics, we have to base its learning on
the previous related topics and at the same time it is to be made a base for
the study of the topics taught in future. Briefly, the different topics in a
branch must be so correlated as to bring out clearly the objects of teaching
the whole subject. At any juncture, the teaching of mathematics should portray
a clear picture of integration and correlation in such a way that the different
topics of a branch may appear to be the different pearls of one and the same
necklace.
Correlation of
mathematics with other subjects
§ Correlation of mathematics with Physical science and Biological
sciences:
Some quotes are given
below which give an idea of correlation of mathematics with Physical and
Biological sciences.
“All scientific
education which does not commence with mathematics is, of necessity, defective
at its foundation” Comte
“A natural science is
a science only in so far as it is mathematical”. Kant
“Mathematics is the
indispensable instrument of all physical research” Berthelot
“Mathematics is the
gate and key of sciences” Roger Bacon
§ Correlation between mathematics and Physics
For
higher education in physics, a good knowledge of mathematics is quite helpful.
All the laws of physics are expressed in mathematical language. To solve
numerical problems in physics we need a good knowledge of mathematics. In
physics we have to be mathematically accurate e.g., launching of satelites on
the moon required right amount of thrust in rockets, accuracy in time and
speed, angle of launching, shape to provide minimum friction and so many other
calculations. Working of a machine is possible when there is proper adjustment
of its components. Principles of Physics are presented in a workable form. Some
more examples are as under:
S = ut +
V=u + ft
Where the symbols
have their usual meanings
G = K
The
law of lever is based on the simple mathematical principle of balance of the
sides of an equation. The units of measurement are frequently used in physics. Tables
of specific heat, latent heat and melting points are prepared with the help of
mathematics.
§ Correlation between mathematics and Chemistry
According
to J. W. Mellor, “It is almost impossible to follow the later developments of
physical or general chemistry without a working knowledge of higher
mathematics”.
All
chemical combinations are governed by certain mathematical laws. All chemical
compounds have their constituent elements in a definite ratio, eg. In synthesis
of water, we take two atoms of hydrogen and one atom of oxygen under suitable
chemical conditions. Chemical equations are balanced by counting the number of
atoms on either side.
In the structure of atom, there are
some set relations concerning electrons, protons etc. valencies of elements
have a mathematical base. Molecular weights of organic compounds are calculated
mathematically.
§ Correlation between mathematics and Biology
"
In mathematics we find the primitive source of rationality and to mathematics
must the biologist resort for means to carry on their visit us " A. Comet
Mathematical principles find many applications
in observing and interpreting certain phenomena. Defer nominal are described,
classified and compared for generalizing and deriving a biological law. Life
processes are the most intricate of all-natural phenomena e.g., composition of
animal and plant cells is studied and expressed clearly and concisely in a
simple intelligible language. Mathematics provides a brief and precise
expression of ideas.
Moreover,
study of Biology depends upon its branches - Biophysics and Biochemistry. Both
these branches use mathematics on a wide scale. Recently biomathematics has
also started to grow and this is an important field of study for Biologists.
Various
experiments in Biology also need analysis. For many a complex problem the
solutions can be found only be used of methods of statistics. Just 8 used this
method for measurement of Inheritance of stature of children and stature of
their parents. He found this Koi option to be 2/3.
The Schultz-
Borissoff Law after the Action of Enzymes such as pepsin and rennin is
expressed by the formula
x-K√F.gt
where x= amount of
substance transferred.
t= time of transformation
f= concentration of enzyme
g=initial concentration of substrate
(e.g. albumin of milk) K is a constant.
§ Correlation between Mathematics and Engineering
Mathematics is considered to be the foundation
of Engineering. Engineering has been defined as, " the art of directing
the great sources of power in nature for the use and convenience of man ".
For
admission to any engineering courses only those students are eligible who have
of mathematics course at the qualifying class. In various engineering courses
we deal with surveying, levelling, designing, estimating, construction etc. and
for all these knowledges of mathematics is essential. In all the branches of
Engineering such as civil, mechanical, electrical etc. mathematical principles
are directly involved. It is possible to increase the durability of a product
by the application of geometrical principles. Mathematical knowledge can also
be used in verification of results in engineering. All this is enough to bring
to fore the close relation between mathematics and engineering.
Mathematics forms the foundation for
architecture. Geometrical concepts like measurements, area, volume, angle,
symmetry, proportion, Golden Ratio, co-ordinate geometry is used in modern and
ancient architecture. One can observe mathematical principles in the great
Constructions which always attracted human minds. The Great Wall of China, the
pyramids, Taj Mahal, the parthenon and the Colosseum are a few among them.
§ Correlation between Mathematics and Agriculture
Agriculture as a science is a highly depend on
mathematics. We can attain desirable targets only if we have a calculating
attitude. It is the need of the time that a common agriculturist should be so
calculated as to be able to compare the amount of return he is likely to get by
putting in his labour and money.
In agriculture we have to make various land
measurements and for this purpose a knowledge of mathematics is essential. Thus
we find a lot of correlation between Mathematics and Agriculture.
§ Correlation between Mathematics and other Sciences
We have discussed the correlation between
certain branches of science and mathematics in the preceding paragraph but
science is a very wide subject and it includes many subjects. Mathematic has
played a very important role in building our Civilization by perfecting all Sciences.
Mathematics is the science of all Sciences and art of all arts. Are most are
branches of science have a great deal to do with mathematics. Graph and chart,
curves etc. all are based on principles of mathematics. In Astronomy and Hydromechanics,
it is the principle of mathematics that works. Thus, we can say that there is a
lot of correlation between science and mathematics.
§ Correlation between Mathematics and Social science
Though
the relationship between mathematics and social science is not very strong yet
it is quite strong. In Social Sciences we come across graphs, charts, curves
etc. thus we can say that Social Sciences draw a lot from mathematics.
The views of some thinkers on this
relationship are quoted below:
"
The Social Sciences mathematically developed are to be the controlling factors
in civilization " W. F. White
"
All great scientific discoveries are but the reward of patient, pains taking
shifting of numerical data "- Lord Kelvin
"
Analytical and graphical treatment of statistics is employed by The Economist,
the philanthropist, the business expert, the actuary and even the physician,
with the most surprisingly valuable results, while symbolic language involving
mathematical methods has become a part of Well-nigh in every large business
" L. Karpinsky
Today collection of statistics is very
important for Social Sciences. Without these statistics it is not possible to
have a character study of any subject. It is mathematics that provides these
statistics. In fact, that statistics is nothing but an integral part of
mathematics. In the following lines we shall try to assess the correlation
between different Social Sciences and mathematics.
§ Correlation between mathematics and economics
In
Economics, mathematical principles and language is freely used to describe and
interpret to social phenomena.
According
to Marshall " the direct application of mathematical reasoning to the
discovery of economic truths has recently rendered great services in the hands
of Master mathematician"
In
economic we study the earning and spending factors of a society, population, production
etc. In Economics statistics plays an important role. The application of
statistical methods is most helpful in economic forecasts. Trade cycles, Trends
of exports and imports, volume of trade etc. all are presented statistically
the theory of probability applied to Assurance is simply a mathematician of
economic problem. The word constant, average, ratio, variable etc. are used by
every magazine, periodical that deals with Economic problems. These are
actually mathematical terms which are used in mathematics.
The
Businessman who is a leader in the field of Economics has to face problems
involving simple and compound interest, installment payments and accounting. He
makes use of statistical methods for making economic forecasts.
The
correlation between mathematics and economics is so vast that now a days in
many a university, a course in Advanced Mathematics is included as a part of
degree courses in economics. This is because of the fact that mathematics plays
an important part in economics.
§ Correlation between mathematics and psychology
Without using mathematics psychology is only a
flight of imagination.
In
the works of Herbert " it is not only possible but necessary that
mathematics be applied to psychology ".
Utility
of psychology has increased tremendously in the modern world because of
application of mathematics in the study of psychology. Various statistical
methods are being used to interpret the psychological data and experimental
psychology has become highly mathematical as it deals with intelligence
quotient, standard deviation, mean, mode, coefficient of correlation etc.
For example,
I. Q.= M.A X 100
C.A
Where abbreviations have their usual meanings.
§ Correlation between mathematics and logic
Logic
is the scientific study of the conditions of accurate thinking and valid
influence. All laws of logic are based on experiments and verification. D'
Alombert says, " geometry is a practical logic, because in it rules of
reasoning are applied in the most simple and sensible manner ".
Mathematics is the only field of knowledge where the logical laws can be
applied and results verified without any personal bias.
W. C. D. Whetham
" mathematics is but the higher development of symbolic logic ".
C. J. Keyser "
symbolic logic is Mathematics; Mathematics is symbolic logic ".
Pascal says. "
Logic has borrowed the rules of geometry. The method of avoiding error is
sought by everyone. The logicians profess to lead the way, the geometers alone
reach it, and aside from their science there is no true demonstration ".
Logic
makes us exact and systematic in use of our language while mathematics makes us
exact and systematic when we translate these thoughts into action. A person
having a sound knowledge of logic cannot be led astray by any jugglery of facts
in mathematics and vice versa.
§ Correlation between mathematics and philosophy
According
to Herbert " the real finisher of our education is philosophy but it is
the office of mathematics to ward off the dangers of philosophy ". Philosophy deals with abstract ideas.
Mathematics tries to draw attention on the ideas that are practicable and
worthwhile. According to A.N. Whitehead " philosophers when they have
possessed a thorough knowledge of mathematics, have been among those who have
enriched the science with some of its best ideas ". Mathematics helps the
philosophers in shifting truth from falsehood, attainable from the
unattainable, facts from fiction and so on.
According to J.S. Mill " Mathematics will
ever remain the most perfect type of Deductive method in general; and
applications of Mathematics to the simpler the branches of Physics, furnish the
only school in which philosophers can effectively learn the most difficult and
important portion of their act, the employment of the laws of the simpler
phenomena for explaining and predicting those of the more complex ".
Though to a common man it appears that there
can't be any relationship between such widely different branches of knowledge,
but the fact is that mathematics occupies a central place between neutral
philosophy and mental philosophy. Philosophers find orderly and systematic
achievements of unambiguous truths. The philosopher can really on mathematics.
Truly speaking, it is the mathematical development of sciences that saved
philosophy from degenerating into pure sensationalism and imagination.
§ Correlation between mathematics and fine arts
Mathematics
is the pivot of all arts. An object is beautiful when depicted in correct
geometrical proportions. That why the Greeks, the greatest geometers of the
age, where are so successful in Arts sculpture. Mathematics itself is a great
art of study of harmony and symmetry. The following views Express the
relationship between mathematics and fine arts.
Leibnitz " Music
is a hidden exercise in Arithmetic of a mind and conscious of dealing with
numbers ".
Pythagoras " Where harmony is, there
numbers ".
John Arbuthnot "
truth is the same thing to the understanding as music to the ear and beauty to
the eye "
The
concept of mathematics light geometric shapes, symmetry, translation,
tessellation, proportion etc. can beach seen in various art forms. The famous
painting of Leonardo da Vinci" Mona Lisa " was drawn using Golden
rectangle. Music and dance are two art forms which essentially include
mathematics. The basic concept of music, ' Swarasthana' is determined by
proportions. The whole number proportion of Sa re gama pa dha ni sa are repeated
to be 24, 27, 30, 32, 36, 40, 45 and 48. The musical instruments are
constructed based on mathematical ideas. In dance, every step (rhythm) is a
number which follows a specific pattern and symmetry. One can say different
mathematical shapes and concepts like a straight line, right angle, parabola,
parallel line, etc. In hand movements in different dance form.
§ Correlation between mathematics and geography
Geography deals with the study of physical
conditions of a particular country or society. It deals with the study of
rivers, canals, mountains, population etc. In this we also try to heights and
draw charts and graphs. In all these, mathematics plays a prominent role. Thus,
there is a great application of mathematics in geography. The areas of the
earth's longitude and latitude, calculation of time at various places, rotation
of the earth leading to the formation of days and nights and Seasons, moments
of winds, falling of rains, factors influencing climate of a region etc. all
depend upon mathematical calculations. Mathematics helps in drawing and
understanding of maps. Graphs of various kinds are frequently used in
geography.
Relief
maps are based purely on mathematics. Geology 2 considered as a separate branch
of science influences Geography to a very large extent. The geological studies are not possible
without the proper use of mathematical knowledge.
§ Correlation between mathematics and history
The
relationship between mathematics and history is reciprocal. History helps
mathematics to know about various mathematician who were pioneers in their
field and enriched Mathematics by their contributions. History also provides
the information about the origin and development of mathematics.
Mathematics
helps history in regards to calculation of dates and days etc. Of various
Historical events. Does there is a close contribution between mathematics and
history.
§ Correlation between mathematics and drawing
In
drawing we come across many a branch such as a geometrical drawing, memory
drawing, figure drawing etc. In geometrical drawing various principles of
geometry and frequently used does we find a close correlation between drawing
and mathematics. If we consider it the other way round we find that a good
drawing is needed to draw a good geometrical figures in the study of
mathematics. The exactness of a figure can be measured only with the help of
mathematics.
§ Correlation between mathematics and languages
No subject can be taught without the help of
language. This is also true of mathematics. The principles of mathematics are
expressed through the medium of language. Thus, it is the language that helps
mathematics. Is the language that is spoken in the mathematics class or written
in a book is not correct, the exact idea of the subject-matter shall not be
conveyed. This shall have a wrong impression on the mind of students.
Justice
language help mathematics also there is the story of mathematics helping the
language. Teaching of mathematics develops an attitude of exactness in the
students. This exactitude help them to write the language correctly. Thus, it
may be said that the relationship between the two is quite significant.
§ Mathematics in nature
Mathematical
concepts like symmetry, similarity, Fibanacci series, Golden Ratio, proportion,
different geometrical shapes and sizes are found in nature. The concepts of
symmetry and similarity can be observed in the butterflies, flowers, leaves
etc. The nautilus shell follows the concept of Golden ratio, it can be observed
in the human body also. The beehives are always hexagonal in shape. The
branches of trees, arrangement of petals in a sunflower, pineapple etc. include
the Fibonacci series. The mathematics in nature cannot be exclusively listed.
Correlation of
mathematics with daily life
§ Correlation teaching of mathematics with other aspects of life
Mathematics
cannot be taught in isolation. Utility of this subject is no confined to the classroom
of the school only. It has an important bearing on various aspects of life. As
far as possible, while teaching mathematics, a reference should be made to its
use in actual life. Whenever we go out for shopping and purchase certain
things, we get the measured and weighted. In this activity, it is the
mathematics that plays its role. The student should, therefore, be explained
the utility of mathematics in practical life. The teacher of mathematics issued
also try to explain to the students the practical application of the principles
of mathematics. He may also show that big electrical installation and bridges
etc., and explain to the students that all these has been made possible only
with the help of mathematics. This would impress upon them the utility of the
subject matter of the life and the correlation between actual life and
mathematics. This would be scientific as well as an interesting method.
Mathematics
is an indispensable part of our daily life. One starts a day by using
mathematics when he adjusts his time for various activities, so that he can
join his duty at the right time. While calculating wages, planning the
expenses, estimating the balance, buying or selling things, doing banking etc.
mathematics is used in one or another form.
Even
a housewife applies Mathematics by preparing the family budget. The taste of
food depends on the appropriate quantity or ratio of the ingredients it needs.
The total quality of food prepared in a home is proportional to the number of
family members and the amount each member requires. In order situations we can
see the application of mathematical concepts.
Our
leisure time activities are also related with mathematical knowledge. An
individual to whatever field he belongs- agriculture, carpentry, tailoring,
business, banking, share market - use the knowledge of mathematics at each and
every second of his life. In this Complex society, it is sugar that a person
who is weak in mathematics will easily be cheated. In our day-to-day
communication, we use mathematical language to make our ideas clear and
precise. Thus, mathematics is in every sphere of life.
·
Evolution of Mathematics as a discipline: Development as a science,
History of Mathematics from ancient period to 20th century
History of mathematics gives a detailed description of the main trends
in the development of mathematics throughout the ages. Knowledge of history of
mathematics is essential for teachers and students of mathematics
Teaching of history of
mathematics helps us in the following ways:
·
For
proper understanding of the subject.
·
For
getting knowledge about all happenings in the area of mathematics.
·
For
getting a knowledge of time sequence.
·
For
developing interest in mathematics study.
·
For
having an idea about history of Civilization
·
For
knowing the independence of the subject.
·
For
knowing the practical value of mathematics in our life.
·
For
developing appreciation of the contribution f great mathematician
A general review of the history of mathematics from the
time of origin of human life
·
It
was explained there that it was from the regular shapes of objects, rhythm in
the arrangement of many natural phenomena and the systematic rotation of
planets etc. that man began a feeling a sense of mathematical intuition.
·
When
man faced the problem of comparing lengths, he used his organs for the purpose
as indicated by his using the width of two fingers (inch), the span of the
palm, the distance between the left end of the left hand and right end of the
right hand when they are stretched., the length of a foot etc.
·
The
study of mathematics as a “demonstrative discipline” begins in the 6th
century BC with the, Pythagoreans, who coined the term “mathematics” from the
ancient Greek meaning” subject of instruction”.
·
The
Greek mathematician Eratosthenes was able to calculate the Earth’s diameter
using a rod stuck in the ground and the rule of three. And he did it several
centuries before it was demonstrated that the planet is round.
·
Johannes
Kepler (1571-1630), a mathematics teacher in Austria, was the first to state
that orbits of planets were not circular round the sun, but elliptical.
·
The mathematics
that we know in the modern world has its roots in ancient Mesopotamia, Egypt
and Babylonia.
·
Then
it was developed in Greece, China and in India..
Ancient Mathematics
SUMER/BABYLONIA(4000-3000 B.C.E)
·
Sumerian
and Babylonian mathematics was based on a sexagesimal or base 60 numeric system.
3000 B.C.E
·
The Egyptians
were the first people to develop a numerical system that was based on the
number 10.
·
Hieroglyphic
numerals developed in Egypt.
300 B.C.E
·
The major
Greek progress in Mathematics
·
During
this era, Euclid wrote the Elements, a compilation of theorems, axioms in
Algebra and postulates and theorems in Geometry
·
With
this he gained the title, FATHER OF GEOMETRY.
200 B.C.E
·
Archimedes
derived a range of formulas in Geometry including the area of circle, surface
area and volume of sphere and the area under a parabola
140 B.C.E
·
Trigonometry
of Hipparchus developed
·
He
is considered as the founder of Trigonometry
Middle Ages
830 C.E
·
Arabic
Algebra and Indian numerals came to western Europe through the writings of
Muhammed ibn Musa Al- Khwarizmi
·
One
of his principal achievements in algebra was his demonstration of how to solve
quadratic equations by completing the square, for which he provided geometric
justifications.
·
He was
the first to treat algebra as an independent discipline and introduced the
methods of “reduction” and “balancing” (the transposition of subtracted terms
to the other side of an equation, that is, the cancellation of like terms on
opposite sides of the equation)
·
He
has been described as the Father of Founder of Algebra
·
The
term Algebra itself comes from the title of his book (the word Al- Jabr meaning
“completion” or “rejoining”)
1202 C.E
·
Leonardo
of Pisa, also called Fibonacci, wrote Liber Abaci, a book filled with
arithmetical and algebraic information which he had collected during his
travels
·
The best-known
contribution of Hindu Mathematics to Modern mathematics was the decimal
position system.
1489 C.E
·
Johannes
Widmann was a German Mathematician. The + and – symbols were first appeared in his
book Mercantile Arithmetic.
1514 C.E
·
Vander
Hoecke was the first to use the + and – signs in writing algebraic expressions
Modern Era
17th Century
·
John
Napier and others greatly extended the power of mathematics as a calculatory
science with his discovery of logarithms.(1614)
1619 C.E
·
Rene
Descartes invented cartesian coordinate system and developed analytic geometry.
1629 C.E
·
Fermat
together with Pascal began mathematical study of probability.
·
Pascal
invented Pascaline and an early mechanical calculator
Pascal is also known for Pascal’s triangle.
1736 C.E
·
The most
important mathematician of 18th century was Leonhard Euler who
started the graph theory and differential geometry.
1799
·
Carl
Friedrich Gauss German mathematician, generally regarded as one of the greatest
mathematicians of all time of his contributions to number theory, probability
theory, planetary and the theory of functions.
·
In 1799,
he proved the fundamental theorem of algebra
1874
·
John
Venn introduced Venn diagram this became a useful tool in set theory
1975 C.E
·
Benoit
Mandelbrot introduced the theory of Fractals. He published the Fractal Geometry
of Nature in 1982.
·
According
to him Fractal is “ a ruff or fragmented geometric shape that can be split into
parts each of which at least approximately a reduced size copy of the all”
·
The branch
Fractal Geometry in mathematics is coming under measure theory.
Development of Number system
|
35000 BCE |
|
African |
First notched tally bones |
|
2700 BCE |
|
Egyptian |
Earliest fully-developed base 10 number
system in use |
|
2600 BCE |
|
Sumerian |
Multiplication tables geometrical exercises
and division problems |
|
1200BCE |
|
Chinese |
First decimal numeration system with place
value concept |
- ·
Pythagoras
is known as Father of Mathematics
- ·
Egyptians
used lotus flowers to represent 1000
· Our measurement of time is based on : sexagesimal number system
? Development of mathematics is the Development of Civilization- critically
evaluate
? Development of mathematics as a science
? History of mathematics from ancient period to 20th century
? Explain the contributions of mathematicians in different century
Aryabhatta
· He gave the formula
for the volume of a pyramid as one third of the product of the base area and
height.
· He suggested the use
of letters to represent unknown.
· Aryabhatta worked on the approximation for pi(π), and may have come to the conclusion that π is irrational.
· He gave the identity (a+b)^2= a^2+2ab+b^2
· He declared that number of days for a year is 365.3586. Modern scientist confirmed it as 365 days 5 hours 48 minutes 46 second.
· He explained how to
find square root and cube root.
· He gave the formula
for the area of triangle.
· He gave the formula
for the area of circle.
· In his memory and honour the first satellite launched on the space by india was given the name Aryabhatta.
Bhaskaracharya II
· Known as Bhaskara II
to avoid confusion with Bhaskara I.
· Bhaskaracharya wrote 'Sidhanta Siromani'. It is divided into four namely Lilavati, Vijaganit,
Goladhyaya and Grahganit.
· For the first time he
introduced the idea that dividing a number by zero will result in infinity.
· A proof of the Pythagorean Theorem by calculating the same area in two different ways and then cancelling out terms to get a^2+b^2=c^2
· In Lilavati, solutions
of quadratic, cubic and quartic indeterminate equation are explained.
· Solutions of indeterminate
quadratic equations (of the type ax^2+b =y^2
· Solved quadratic
equations with more than one unknown, and found negative and irrational
solutions.
· Calculated the
derivatives of trigonometric functions and formula.
Brahmagupta
· Bhaskaracharya designated
Brahmagupta as “ Gem of the circle of mathematicians" (Ganitha Chakra
Chudamani”).
· He was the first
mathematician to provide the formula for the area of a cyclic quadrilateral.
· His work the “Brahmasphudsidhanta”
contained many mathematical findings written in verse form. It had many rules
of arithmetic which is part of the mathematical solutions now. These are “
positive number multiplied by positive number is positive” , “ A
negative number multiplied by a positive number is negative” , “
A negative number multiplied by a negative number is positive”.
Srinivasa Ramanujan
· 1729 is known as the Ramanujan number. It is the sum of the cubes of two numbers 10 and 9. For instance, 1729 results from adding 1000 (the cube of 10) and 729 (the cube of 9). This is the smallest number that can be expressed in two different ways as it is the sum of these two cubes (1^3 +12^3, 9^3+10^3). Interestingly, 1729 is a natural number following 1728 and preceding 1730.
· He also discovered the
properties of the partition function.
· Goldbach’s conjecture
is on of the important illustration of Ramanujan’s contribution. The statement is
that “Every even integer greater than 2 is the sum of two primes”. Ramanujan and
his associates founded that “Every large integer could be written as the sum of
4 primes” (43=2+5+17+19)
Major contributions
of Renowned International Mathematicians
Euclid (Father of
geometry)
· Euclid was an ancient
Greek mathematician from Alexandria who is best known for his major work,
“Elements”
· Euclid proved that √2
(the square root of 2) is an irrational number
· Euclid’s axioms
o It is possible to
draw a straight line from any point to any point
o It is possible to
extend a finite straight line continuously in a straight line.
o It is possible to
create a circle with any center and distance(radius).
o The whole is greater
than a part.
Pythagoras
·
He invented the terms odd and even.
· He discovered that any odd number (say 2n+1) can be expressed as the difference of two squares: 2n+1 = (n+1)^2- n^2
·
Pythagoras Theorem
·
Pythagoras discovered harmonic progression in
the music scale.
George Cantor
· He is best known as
the inventor of set theory.
· For Cantor, sets are
collection of objects that can have finite or infinite elements
· Major works by George
Cantor includes uncountable sets, the Cantor set, infinite set, convergent
series, number theory and function theories,
He defined the cardinal and ordinal numbers and their arithmetic.
·
Role of Mathematics in school curriculum in India- Recommendations of
various Committees and commissions (NPE, NCF, KCF)
NPE 1968
·
The major reform
in curriculum for all stages of school education came after NPE 1968 as per the
report of Kothari commission
·
A common
curriculum for class 1 to X was prepared at national level with adjustments
according to local needs
·
General maths was compulsory
subject upto class X and at secondary level an advance mathematics was there as
optional subject
·
General maths-
Arithmetic, geometry, simple algebra
·
Advance maths-
integers, quadratic equation, logarithm, coordinates geometry
·
“Education will have
to be streamlined to facilitate modernization of production, services and
infrastructure. Besides, to enable the young people to develop enterprisal
ability, they must be exposed to challenges of new ideas, old concepts have to
be replaced by new ones in an effort to overcome the resource constraint and
input dynamism”
·
At UP level
Numbers,
fractions, decimal fractions, money, measurement, idea of simple geometry,
unitary method, simple interest-ratio proportion
·
At secondary level
Number system,
sets, irrational number, complex number- Indices and logarithm, Algebra-
expression, equalities, factors, quadratic equations, geometry-theorems, properties,
proofs and application- mensuration, Discount- shares-graphs-compound interest-
Banking- introduction to Trigonometry, Statistics
NCF 2005
·
For class I to V
Geometry
(shapes and spatial understanding), Number and operation, Mental Arithmetic,
Money, Measurement, Data handling, Pattern
·
For class VI to
VIII
Number system
and playing with numbers, Algebra (introduction and expression), Ratio and
proportions, Geometry (basic ideas 2D and 3D) understanding shapes, symmetry,
construction, mensuration, Data handling- introduction to graphs
·
For class IX to X
Number system
Algebra
Co-ordinate
Geometry
Geometry
Mensuration
Statistics and
probability
Trigonometry
·
According to the
NCF 2005 , the main goal of mathematics education in school is the
“Mathematisation” of a child’s thinking
·
The NCF envisions
school mathematics as taking place in a situation when
1.
Children learn to
enjoy mathematics rather than fear it
2.
Children learn
“important” mathematics which is more than formulas and mechanical procedures
3.
Children see
mathematics as something to talk about to communicate through to discuss among
themselves to work together on
4.
Children pose and
solve meaningful problems
5.
Teachers are
expected to engage every child in class with the conviction that very one can
learn mathematics
The NCF also list the
challenges facing mathematics education in our school are
1.
A sense of fear
and failure regarding mathematics among a majority of children
2.
A curriculum that
disappoints both a talented minority as well as the non-participating majority
of the same time.
3.
Lack of teacher
preparation and support in teaching of mathematics
4.
Crude methods of
assessment that encourage the perception of mathematics as mechanical
computation problems, exercise, methods of evaluation are mechanical and
repetitive with too much emphasis on computation
The NCF recommends
·
Shifting the focus
of mathematics educations from achieving ‘narrow’ goals of mathematical content
to ‘higher’ goals of crediting mathematical learning environments where process
like formal problem solving, use of heuristics, estimation and approximation
optimization, use of patterns, visualization representation, reasoning and
proof making connections and mathematical communication …
·
Engaging every
student with a sense of success while t the same time offering conceptual
challenges to the emerging mathematician
·
Changing models of
assessment to examine students mathematisation abilities rather than procedural
knowledge
·
Enriching teachers
with a variety of mathematical resources
Organisation of the curriculum
·
Pre- Primary
All learning occurs
through play rather than didactic communication
Rather than
the rote learning of number sequence, children need to learn and understand, is
in the context of small sets, the connection between counting and quantity
·
Primary
Having
children develop a positive attitude towards and linking for mathematics at the
primary stage is as important as developing cognitive skills and concepts.
Mathematical
games, puzzles and stories developing a +ve attitude and in making connections
between mathematics and everyday thinking.
Besides
numbers and number operation the importance must be given to shapes, special
understanding, patterns measurement and data handling
·
Upper Primary
Here students
get the first taste of the application of forceful abstract concepts that
compress previous teaching and experience
This enables
them to revisit and considerable basic concepts and skills learnt at the
primary stage, when is essential from the point of view of achieving universal
mathematical literacy
·
Secondary
Students now
begin to perceive the structure of mathematics as discipline
They become
familiar with the characteristics of mathematical communication carefully
defined terms and concepts
the use of
symbols to represent them precisely stated propositions and proofs justifying
particularly in the area of geometry
·
Higher Secondary
The aim of
mathematics curriculum at this stage is to provide students with an appreciation
of the wide variety of the application of mathematics and equip them with basic
tools that enable such application
·
Assessment
Board
examination be restricted.
So that the
minimum eligibility for a state certificate is numerically, reducing the
instance of failure in mathematics
KCF
(2007)
·
Social Justice
·
Awareness on
environment
·
Citizenship
·
Awareness of
science and technology
·
Nationalism
·
Awareness of one’s
right
·
Scientific temper
·
Cultural identity
·
Vocational skills
·
Resistance
·
Construction of
knowledge
·
Critical Education
Mathematics
·
To analyze and
interpret the world on the basis of numbers
·
We present the
numerical relations inherent in natural phenomena through algebraic
equation
·
It becomes
possible to arrive at minute forms of knowledge and accurate predictions from
this angles
·
Maths is a
numeric- centered language
·
Maths has an
independent logical trajectory other than the practical numerical calculations
Why do we learn mathematics?
·
Mathematics that
is required in daily life Eg: Basic calculations, percentage, measurements.
·
Useful for higher studies.
E.g.: Trigonometry, statistical data interpretations, algebra and geometry
·
Deeper ideas into
the complex details of mathematics E.g.: Proof of geometrical principles
Direct, Concrete experience👈👉Explanation through language
👆👇 👆👇
Formulating ideas using pictures👈👉Making use of mathematical symbols
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