Meaning, Nature, Characteristics & Scope of Mathematics

Meaning, Nature, Characteristics & Scope of Mathematics

  Meaning of Mathematics:

Mathematics is the science of measurement, quantity and magnitude. Mathematics is known as “Ganita” in Hindi which means the science of calculation. Developing children's abilities for mathematics is the main goal of mathematics education. The narrow aim of school mathematics is to develop 'useful' capabilities, particularly those relating to numeracy–numbers, number operations, measurements, decimals and percentages. The higher aim is to develop the child's resources to think and reason mathematically, to pursue assumptions to their logical conclusion and to handle abstraction. It includes a way of doing things, and the ability and the attitude to formulate and solve problems.

Mathematics is the most closely related subject in our daily life. Its knowledge is exact, systematic logical and clear. Mathematics involves the process for intellectual development of mental faculties. It is not that mathematical knowledge is needed only by engineers, doctors or business personals. Even the smallest citizen of society such as laborers and workers need the basic knowledge of mathematics. Besides the mental ability, mathematics develops some quality like concentration, truthfulness, seriousness and reasoning. Thus, in the words of Locke it is rightly said that, “Mathematics is a way to settle in the mind the habit of reasoning”.

Definition of Mathematics:

      “The abstract science which investigates deductively the conclusions implicit in the elementary conceptions of spatial and numerical relations, and which includes as its main divisions geometry, arithmetic, and algebra” - Oxford English Dictionary, 1933

          “The study of the measurement, properties, and relationships of quantities and sets, using numbers and symbols” - American Heritage Dictionary, 2000

     “The science of structure, order, and relation that has evolved from elemental practices of counting, measuring, and describing the shapes of objects” - Encyclopedia Britannica

   Nature of Mathematics

Based on the various definitions, the nature of mathematics can be listed as follows:

     

Ø  Mathematics – A science of discovery:

Mathematics is the discovery of relationships and the expression of those relationships in symbolic form – in words, in numbers, in letters, by diagrams or by graphs. According to A.N.Whitehead (1912) “Every child should experience the joy of discovery”. 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 early opportunity to make independent discoveries.

            The children must not only have opportunities for making their own discoveries of mathematical ideas, but they must also have the practice necessary to achieve accuracy in their calculations. Today it is discovery techniques, which are making spectacular progress. They are being applied in two fields: in pure number relationships and in everyday problems, involving such things as money, weights and measures.

Ø  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.

Mathematics – The art of drawing conclusions:

            One of the important functions of the school is to familiarize children with a mode of thought which helps them in drawing right conclusions and inferences. According to J.W.A. Young a subject suitable for this purpose should have three characteristics:

1.        That its  conclusion are certain. At first, at least it is essential that the learner should know whether or not he has drawn the correct conclusion.

2.      That it permits the learner to begin with simple and very easy conclusions to pass in well graded sequence to very difficult ones, as the earlier ones are mastered

3.      That the type of conclusions exemplified in the introductory subject be found in the other subjects also, and in human interactions, in general.

These characteristics are present in mathematics to a larger extend than in any other available subject.

Ø  Mathematics – A tool subject

It could be more elegantly expressed as “mathematics, handmaiden to the sciences”. From the beginning, down to the nineteenth century, mathematics has been assigned the status of a servant. Then in the nineteen century, mathematics attained independence. It achieved a completeness and internal consistency that it has not known before. Mathematics continued to be useful to other disciplines, but now it is dependent upon none of them. With its new found freedom, mathematics established its own goals to pursue. Its mentors of the past- engineering, physical science and commerce – now became no more than its peers.

Mathematics has its integrity, its beauty, its structure and many other features relate to mathematics as an end in itself. However, many conceive mathematics as a very useful means to other ends, a powerful and incisive tool of wide applicability.

Ø  Mathematics – A system of logical processes

Polya suggested that mathematics actually has two faces. One face is a ‘systematic deductive science’. This has resulted in presenting mathematics as an axiomatic body of definitions, undefined terms, axioms, and theorems. Mario Pieri stated “Mathematics is a hypthetico-deductive system”. This statement means that mathematics is a system of logical processes whereby conclusions are deduced from certain fundamental assumptions and definitions that have been hypothesized. This has been reinforced by Benjamin Pierce when he defined mathematics as ‘The science which draws necessary conclusions’. The student draws the inferences from the premises, provided the premises are true. In mathematics, granted the premises, conclusion follows inevitably. For example:

“When two lines intersect, vertically opposite angles are equal”

(the premise)  are vertically opposite angles.

Hence  are equal (the inference)

           

Ø  Mathematics – An intuitive method

            Intitution implies the act of grasping the meaning or significance or structure of a problem without explicit reliance on the analytic apparatus of one’s craft. It is the intitutive mode that yields hypothesis quickly. It precedes proof; it is what the techniques of analysis and proof is designed to test and check. It is a form of mathematical activity which depends on the confidence in the applicability of the process rather than upon the importance of right answers all the time.

            Intuition when applied to mathematics involves the concretisation of an idea not yet sated in the form of some sort of operations or example. A child forms an internalized set of structures for representing the world around him. These structures are governed by definite rules of their own. In the course of development, these structures change and the rules governing them also change in certain systematic ways. To anticipate what will happen next and what to do about it is to spin our internal models just a bit faster than the world goes. It is important to allow the student to express his intuition and check and verify its validity. When mathematics is taught in a very formal way by stating the logical rules, and algorithm, we remove his confidences in his ability to perform mathematical processes. Teachers quite often provide formal proof (which is necessary for checking) in place of direct intuition. For example, to check the conjecture, 8x is equivalent to 3x+5x, a formal rigorous statement as the following,

“By the commutative principle for multiplication, for every x, 3x+5x=x3+x5. By the distributive principle, for every x, x3+x5=x(3+5). Again by the commutative law, for every x, x(3+5)=(3+5)x or 8x. So for every x, 3x+5x = 8x”, could dampen the students’ spirit of intuition and interest. It is up to the teacher to allow the child to use his natural and intuitive ways of thinking, by encouraging him to do so and honouring him when he does.

           

Characteristics of Mathematics:

The characteristics of mathematics that makes it unique among other subjects are:

Ø  Logical sequence : Thus logical sequence becomes a main characteristic of mathematics. To put in simple words, the study of mathematics begins with few well – known uncomplicated definitions and postulates, and proceeds, step by step, to quite elaborate steps.

Ø  Structure : Based on this definition mathematical structure should be some sort of arrangement, formation, or result of putting together of parts. The familiar operations of addition denoted by +, and multiplication denoted by x, of natural numbers are operations on set N of natural numbers.

Ø  Precision and accuracy : 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, accepted or rejected. There is no midway possible between the right and wrong. Mathematics can decide whether or not its conclusions are right. Mathematicians can verify the validity of the results and convince others of its validity with consistency and objectivity. This holds not only for the expert, but also for anyone who uses mathematics at any level.

Ø  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 settles 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.

Ø  Mathematical Language and Symbolism : Mathematical language and symbols cut short the lengthy statements and help the expression of ideas or things in the exact form.  Mathematical language is free from verbosity and helps into the point, clear and exact expression of facts.

Ø  Applicability : Knowledge is power only when it is applied. The study of mathematics requires the learner to apply the skills acquired to new situations. The knowledge acquired by students is greatly used for solving problems. The students can always verify the validity of the mathematical rules and relationships by applying and verifying the mathematical ideas. The knowledge and its application, wherever possible, should be related to daily life situations. Concepts and principles become more functional and meaningful only when they are related to actual practical applications. Such a practice will make the learning of mathematics more meaningful and significant.

Ø  Generalization and classification : Mathematics gives exercises in widening and generalizing conceptions, in combining various results under one head, in making schematic arrangements and classifications. It is easy to find instances of successive generalizations. 

Ø  Mathematical Systems : Mathematics includes many components which are inn themselves mathematical structures or mathematical systems. A typical mathematical system has the following four parts: undefined terms, defined terms, axioms and theorems.

Ø  Rigor and logic : It goes without saying that logic is an important factor in mathematics; it governs the pattern of deductive proof through which mathematics is developed. Of course, logic was used in mathematics centuries ago. During the last few decades there has been great emphasis on the analysis of the logical structure of mathematics as a whole. The presentation of mathematics in rigorous form is ill – advised. Mathematics must be understood intuitively in physical or geometrical terms. This is the primary pedagogical objective.

Ø  Simplicity and complexity : The mathematician desires the simplest possible single exposition. Through greater abstraction, a single exposition is possible at the price of additional terminology and machinery to allow all of the various particularities to be subsumed into the exposition at the higher level.

SCOPE OF MATHEMATICS

Ø  In the field of teaching

Ø  In the field of research

Ø  In the field of industry

Ø  In the field of banking and finance

Ø  Application in daily life

Ø   Applicable in science and technology  etc.