INTRODUCTION:
Mathematics plays a vital role in the day to
day life. It is a very important subject. Therefore before imparting and
transmitting its knowledge it is necessary to understand that ‘what is
mathematics?’ and its nature etc. There are various definitions of mathematics
has been interpreted and explained in various ways. Mathematics deals with the
quantitative facts and relationships as well as with problems involving space
and form.
Though mathematics has been with us for
more than 5000 years, the subject has never been made as lively as it is today.
The pace of mathematical discovery and invention has accelerated amazingly
during the last few decades. It has been said that mathematics is the only branch
of learning in which theories of two thousand years old are still valid.
MEANING AND DEFINITIONS
OF MATHEMATICS:
The dictionary meaning of mathematics is
that ‘it is either the science of number and space or the science of
measurement, quantity and magnitude’.
According to Webster’s dictionary
“Mathematics is the science of number and there operations inter relations,
combinations, generalizations and abstraction and of space configurations and
generalizations.”
“Mathematics may be defined as the subject
in which we never know what we are talking about not whether what we are saying
true.”------Bertrand Russel.
“Mathematics is the gate and key of the
sciences”---Roger Bacon. Neglect of mathematics work injury to all knowledge
since who is ignorant of it cannot know the other science world. And what is
worse, men who are thus Ignorant are unable to perceive their own ignorance and
so do not seek a remedy.”
“Mathematics is the language in which god
has written the universe. ----Galileo.
“Our entire civilization
depending on the intellectual penetration and utilization of nature has it s
real foundation in the mathematical science.”-----Prof.Voss.
According to
Locke-“Mathematics is a way to settle in the mind of children a habit of
reasoning.
On the basis of above definitions we can
say or conclude that,
·
Mathematics is the science of Space and
Number.
·
Mathematics is the science of calculation.
·
Mathematics is the science of measurement,
Quantity and magnitude.
·
Mathematics is a systematized, Organized
and exact branch of science.
·
It deals with quantitative facts and
relationship.
·
It is the abstract form of science.
·
It is the science of logical reasoning.
·
It is an inductive and experimental
science.
·
Mathematics is the science which draws
necessary calculations.
NATURE OF MATHEMATICS:
Mathematics is the gate
way of all science. In school those subject which are included in the
curriculum must have certain aims and objectives on the basis of which its
nature is decided. Now we are in position to conclude the nature of
mathematics. The nature of Mathematics are enlisted in the following points,
·
Mathematics is an exact science.
Mathematical knowledge is always clear, logical and systematic and that may be
understood easily.
·
It is the science of space, numbers,
magnitude and measurement.
·
Mathematics involves conversion of
abstract concepts into concrete form.
·
It is the science of logical reasoning.
·
It helps the man to give exact
interpretation to his ideas and conclusion.
·
Mathematics is that science which is by
product of out empirical knowledge.
·
Mathematical propositions are based on
postulates and axioms from our observations.
·
It may exhibit abstract phenomenon into
concrete. Thus abstract concepts may be explained and understood with the help
of mathematics.
·
It is related with each aspect of human
life.
·
Mathematical knowledge is developed by our
sense organs therefore it is exact and reliable.
·
The knowledge of Mathematics remains same
in the whole universe, everywhere and every time. It is not changeable.
·
The knowledge of mathematics has no doubt.
It provides clear and exact response like yes or no, right or wrong.
·
It involves inductive and deductive
reasoning and can generalize any proposition universally.
·
It helps the self evaluation.
CHARACTERISTICS OF
MATHEMATICS:
Mathematics has certain
unique features which one could hardly find in other disciplines. The following
are the important characteristics of mathematics.
1) LOGICAL SEQUENCE: The study of mathematics begins with a 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. Mathematics learning always
proceeds from simple to complex and from concrete to abstract.
2) Structure in
mathematics: In English language
structure denotes ‘the formation, arrangement, and articulation of parts in
anything built up by nature or art’
It seems reasonable to assume
then that a mathematical structure should be some sort or arrangement,
formation, or result of putting together of parts.
For example, we take as the fundamental
building units of a structure the members a,b,c,…. Of a non empty set ‘S’. We
hold together these building units by using one or more operations.
The familiar operations
of addition denoted by +, and multiplication denoted by X, of natural numbers
are operations on set N of natural numbers. Subtraction is not an operation on
the set of natural numbers since the difference of two natural numbers may not
be a natural number(Example:3€N, 3-6=-3₵N) But subtraction is an operation on
the set ‘I’ of all integers.)
3) PRECISION: Mathematics is known as an ‘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.
Mathematics can decide whether or not its conclusions are right. Mathematicians
can verify the validity of the results and convince others or its validity with
consistency and objectivity. This holds for all not only the experts in
mathematics.
Even when there is a new emphasis on
approximation, mathematical results can have any degree of accuracy required.
Although precision and accuracy are distinctly different as criteria for the
measures of approximation, they can be most effectively discussed when
contrasted with each other. The most effective measures of both precision and
accuracy are in terms of the errors (positive or negative) involved. The
precision of a measure or a computation is evaluated in terms of the apparent
error. The accuracy of a measure or a computation is evaluated in terms of the
relative error or percent of error made.
4) ABSTRACTNESS: 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.
Euclid’s geometry describes an imaginary world which resembles the actual world
sufficiently for it is a useful study for surveyors, carpenters and engineers.
Infinity is something that we can never
experience and yet it is a central concept of mathematics. Our whole thinking
is based on the assumption that there are infinitely many numbers, so that
counting need never stop; that there are infinitely many fractions between
0 and 1, that there are
infinitely many points on the circumference of a circle etc.
Again someone whose thinking was
essentially physical might refuse to believe in negative numbers on the ground
that you cannot have a quantity less than nothing. Still more, such a person
would refuse to believe in the square root of minus one.
5) 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. For
example, the distinction between a number and a numeral could head the list. A
number is a property of a set; that property tells how many elements are there
in a set. A numeral is a name or a symbol used to represent a number.
A teacher ought to be very careful to use
correct terms, since this helps children to learn and think better. It is
important that a student understands the distinction between a number and a
numeral so that he may realize the differences between actually operating with
numbers and merely manipulating symbols representing those numbers. This manipulating
symbols representing those numbers.
Without language, we cannot talk about
anything. Mathematical talk consists of making use of mathematics symbolism.
Understanding mathematics is realizing what symbolism corresponds to the
structure that has been abstracted. The
process of speaking of the mathematical language runs as follows: an
abstraction process, followed by a symbolization process, followed again by the
learning of the use of the symbols.
The use of symbols makes the mathematical
language more elegant and precise than any other language. For example, the
commutative law of addition and multiplication inb real number system can be
stated in the verbal form as ‘ the addition and multiplication of two real
numbers in independent of the order in which they are combined’.
This can be stated in a concise form as:
a +b =b +a and a X b = b X a, where a and b are elements of R. Almost all
mathematical statements, relations and operations are expressed using
mathematical symbols such as +, -, X, %,<, >, ≤ , √, ∑, € and so on.
