Download nature of mathematics

FUNDAMENTALS OF TEACHING MATHEMATICS
UNIT1:PLACE OF MATHEMATICS IN CURRICULUM
A) Meaning and nature of mathematics:
Mathematics is
the
study
of
topics
such
as quantity (numbers,structure, space, and change. There is a range of views
among mathematicians and philosophers as to the exact scope and definition of
mathematics.
Mathematicians seek out patterns and use them to formulate
new conjectures. Mathematicians resolve the truth or falsity of conjectures
by mathematical proof. When mathematical structures are good models of real
phenomena, then mathematical reasoning can provide insight or predictions
about nature. Through the use of abstraction and logic, mathematics developed
from counting, calculation, measurement,
and
the
systematic
study
of
the shapes and motions of physical objects. Practical mathematics has been a
human activity for as far back as written records exist. The research required to
solve mathematical problems can take years or even centuries of sustained
inquiry.
Galileo Galilei (1564–1642) said, "The universe cannot be
read until we have learned the language and become familiar with the characters
in which it is written. It is written in mathematical language, and the letters are
triangles, circles and other geometrical figures, without which means it is
humanly impossible to comprehend a single word. Without these, one is
wandering about in a dark labyrinth."
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Carl Friedrich Gauss (1777–1855) referred to mathematics as "the Queen of
the Sciences.
"Albert Einstein (1879–1955) stated that "as far as the laws of mathematics
refer to reality, they are not certain; and as far as they are certain, they do not
refer to reality."
The
word mathematics comes
the Greek μάθημα
from
(máthēma), which, in the ancient Greek language, means "that which is
learnt",[24] "what one gets to know", hence also "study" and "science", and in
modern Greek just "lesson". The word máthēma is derived from μανθάνω
(manthano), while the modern Greek equivalent is μαθαίνω (mathaino), both of
which mean "to learn".
In
Latin,
term mathematics more
and
in
commonly
English
meant
until
around
"astrology"
(or
1700,
the
sometimes
"astronomy") rather than "mathematics"; the meaning gradually changed to its
present one from about 1500 to 1800.
Aristotle defined mathematics as "the science of quantity
Some just say, "Mathematics is what mathematicians do."
Fields of mathematics
Mathematics can, broadly speaking, be subdivided into the
study
of
quantity,
structure,
space,
and
change
(i.e. arithmetic, algebra,geometry, and analysis). In addition to these main
concerns, there are also subdivisions dedicated to exploring links from the heart
of mathematics to other fields: to logic, to set theory (foundations), to the
empirical mathematics of the various sciences (applied mathematics), and more
recently to the rigorous study of uncertainty.
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Pure mathematics
Quantity
The study of quantity starts with numbers, first the familiar natural
numbers and integers ("whole numbers") and arithmetical operations on them,
which are characterized in arithmetic.
Structure
Many mathematical objects, such as sets of numbers and functions,
exhibit internal structure as a consequence of operations or relations that are
defined on the set. Mathematics then studies properties of those sets that can be
expressed in terms of that structure; for instance number theory studies
properties of the set of integers that can be expressed in
terms
of arithmetic operations.
Space
The study of space originates with geometry – in particular, Euclidean
geometry. Trigonometry is the branch of mathematics that deals with
relationships between the sides and the angles of triangles and with the
trigonometric functions; it combines space and numbers, and encompasses the
well-known Pythagorean theorem. The modern study of space generalizes these
ideas to include higher-dimensional geometry, non-Euclidean
geometries (which play a central role in general relativity) and topology.
Quantity and space both play a role in analytic geometry, differential geometry,
and algebraic geometry. .
Applied mathematics
Applied mathematics concerns itself with mathematical methods that are
typically used in science, engineering, business, and industry. Thus, "applied
mathematics" is amathematical science with specialized knowledge. The
term applied mathematics also describes the professional specialty in which
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mathematicians work on practical problems; as a profession focused on
practical problems, applied mathematics focuses on the "formulation, study, and
use of mathematical models" in science, engineering, and other areas of
mathematical practice.
Applied mathematics has significant overlap with the discipline
of statistics,
whose
theory
is
formulated
mathematically,
especially
with probability theory
Computational
mathematics proposes
and
studies
methods
for
solving mathematical problems that are typically too large for human numerical
capacity.
Meaning and Nature Of Mathematics

"Mathematics is the Science of number and Space"

"Mathematics
is
the Science
of Measurement,
quantity
and
Magnitude."
These definitions clearly indicate that Mathematics is an accepted
Science which deals with the quantitative aspect of our life and
knowledge.

Mathematics has been recognized as one of the Central Strings of
human intellectual activity throughout the centuries.

The Word "Mathematics" has been used in two distinct and different
senses. i.e. One as a method used to solve the problems of quality,
space, order etc. and the second as a set of laws or generalizations
of truths that are discovered.

Mathematics is a tool specially suited for dealing with abstract
concepts of any kind and there is no limit to its power in this field.
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
In other words, as in mathematics we find results at the abstract
level with the help of process of reasoning, therefore mathematics
may be regarded as a science of abstract forms.

As expressed by Hindi Mathematicians, "Mathematics(Ganita)
means the Science of Calculation."
From the above opinions we include that Mathematics is a Science of
Quality and Space. It deals with the questions and problems involving
size, portion, area, time interval, distance etc. It is also a science of
calculation involving the use of number and symbols.

It deals with the relationship between magnitudes. It deals with
Numerical part of Man's life.
In this way it is a Systematic, organized and exact branch of science,
which deals with abstract concepts.
NATURE OF MATHEMATICS
1. Mathematics, like everything else that man has created, exists to
fulfil certain human needs and desires.
2. The dominant intellectual interest of mankind shows that,
mathematics appeals very powerfully to mankind.
3. It involves man's high cognitive powers.
4. It has its own language-signs, symbols, terms and operations etc.
5. It has its own tools like intuition, logic, reasoning, analysis,
construction, generality and individuality etc.
6. It helps in drawing conclusions and interpreting various ideas and
themes.
7. Mathematics is the science of inductive, deductive reasoning.
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Inductive-reasoning means when a particular property is true in a
sufficient number of cases, then we conclude that it will true for all
similar cases.
Deductive-reasoning is based on axioms, postulates, self-evident
truth, undefined terms and definitions.
1. "Mathematics is a way to settle in the mind as habit of reasoning."
2. Mathematics can be used in two distinct and different senses i.e.
(i) The truth that are discovered and (ii) The methods used to learn
truth.
3. Mathematics is of two types- Pure and Applied Mathematics
Pure Mathematics deals with theories and principles without regard to
their application to concrete things.
Applied Mathematics is the practical side of pure mathematics.
That is why each and every invention and discovery in the field of
physical, biological or social sciences owe much to applied
mathematics.

"It is the science of number and space."

"It is the science of measurements, quantity and magnitude."

Rationality, accuracy, originality of thinking, certainty of results,
transfer of learning in terms of similarity to the reasoning of day to
day life and verification are the vital components of mathematics.
Mathematics is an expression of the human mind reflects the
active will, the contemplative reason and the desire for aesthetic
perfection. Its basic elements are logic and intuition, analysis and
construction, generality and individuality.
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 BRANCHES OF MATHEMATICS
1)ARITHMATIC
:ART
OF
COMPUTING—NUMBERS
/CALCULATION
2)ALGEBRA:AL-JABAR-AL MUQABALAH---THE OPERATION OF
TRANSFORMING A QUANTITY FROM ONE SIDE TO OTHER
SIDE OF EQUATION---E.G---indices-graphs-directed numbers
3)GEOMETRY:measurement of earth-euclidian geometry
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