NATURE OF MATHEMATICS

Nature of Mathematics

     

1.4.1        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 (E.E.Biggs, 1963). 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.

1.4.2   Mathematics –An intellectual game

Mathematics can be treated as an intellectual game with its own rules and without any relation to external criteria. From this viewpoint, mathematics is mainly a matter of puzzles, paradoxes, and problem solving – a sort of healthy mental exercise.

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:

  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.

  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

  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.

1.4.3   Mathematics – A tool subject

It could be more eleganty 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. John.J.Bowen (1966) in an article titled ”Mathematics and the Teaching of Sciences” stated  “Not all students are captivated by the internal consistency of mathematics, and , for everyone who makes it a career, there will be dozens to whom it is only an elegant tool”.

According to Howard F.Fahr (1966), “If mathematics had not been useful, it would long ago have disappeared from our school curriculum as required study”.

1.4.4 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)

            This conclusion is deducted from the premise ‘vertically opposite angles are equal’. Thus the process of deductions involves two steps: first, replacing the real premises by hypothetical ones; second, making a mathematical inference from the hypothetical working premises. Therefore to think mathematically is to free oneself by abstraction from any particularity of subject matter to make inferences and deductions justified by fundamental premises. It involves logical reasoning. By reasoning we prove that if something is true then something else must be true. However, the validity of the conclusions rests upon the validity and consistency of the assumptions and definitions upon which the conclusions are based. The teacher should make the students realize this point of view.

Polya described the second face of mathematics by saying ‘Mathematics in the making appears as an experimental, inductive science”. There has been a growing emphasis on the experimental side of mathematics. The generalizations follow as outcomes of the observations of mathematical phenomena and relationships. It s based on the principle that if a relationship holds good for some particular cases, it holds good for any similar case and hence the relationship can be generalized. Such a process is called inductive reasoning. For example the student generalizes that the ‘sum of the angle in a triangle is 180^O’ after having observed this property in a number of triangles. Thus a generalization, a rule or a formula is arrived at, through the careful observation of particular facts, instances and examples. Many of the mathematical definitions and rules can be generalized through induction. The teacher of mathematics has to provide an adequate number of examples requiring the student to observe so that the relationships are explicit leading to generalization.

1.4.5 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.

            The first step in the learning of any mathematical subject is the development of intuition. This must come before rules and stated or formal operations are introduced. The teacher has to foster intuition in our young children, by following the right strategies of teaching.