THE VALUE OF MATHEMATICS IN EDUCATION
Mathematics enters into every walk of life. It has application even in the simple processes of counting, measuring and weighing the tangible things of the material world. Hence, unlike any other subjects, mathematics is a medium and an instrument uniquely adapted to the understanding and the control of natural and social phenomena. A consideration of the ‘postulational’ nature of mathematics, as a field dealing with relationships between idealized concepts’ or ‘absolute truths’, would indicate that it is uniquely adapted to stimulating pupils to think. Mathematics affords excellent opportunities for developing the higher mental processes in the form of generalizing relations and applying these generalizations to other situations and thus creating certain attitudes and patterns of behavior congenial to life in a democratic social order.
In order to appreciate the importance of mathematics in the secondary school curriculum one must distinguish between its ‘aims’ and ‘values’. An ‘aim’ is the conscious purpose that one keeps in mind while one acts. One acts because one wants to realize a purpose. ‘Values’ are the actual results that one obtains while one acts or after the action is over. One may have one aim but may acquire many values. This does not mean that the aim and the values are always different. In fact, in many cases one acquires what one aim at while one acts. We realize many values by teaching mathematics but we do not necessarily aim at realizing them when we decide to teach it in our schools. Hence one should not mix up aims with values.
Generally, many people mix-up the aims and the values of teaching mathematics. It is obvious from the answers to the question, ‘Why do we teach mathematics? One teacher says that the ‘aim’ of mathematics instruction is to promote the power of concentration. Another teacher says that a ‘value’ of mathematical instruction is he progress in the ability to concentrate. Truly speaking, promoting the power of concentration is not a fundamental aim of teaching mathematics.
The values of teaching mathematics.
What then are the values of teaching mathematics in secondary schools? “Mathematics is said to have, for example, disciplinary value in habituating the pupil to accuracy of statement and closeness of reasoning; it has utilitarian value in giving command of the arts of calculation involved in trade and arts; cultural value in its enlargement of the imagination in dealing with the most general relations of things; even religious value in its concept of the infinite and the allied ideas”. John Dewey, democracy and education. P.287. perhaps these statements are general in nature; and for this reason we shall discuss each more fully with illustration in the following pages.
When we attribute practical, disciplinary, cultural and religious values to mathematics, we are confronted with two questions: firstly, are these values related to the field mathematics only? Necessarily it need not be so. However the fact that mathematics shares with other subjects certain desirable values is not the fault of the subject. Besides, it is a fact that among all the subjects of the curriculum, mathematics has the maximum utilitarian value cultural value and religious value. Secondly is it possible to realize these value by familiarizing pupils with certain facts, concepts, propositions and principles in mathematics? It is answered by John Dewey. He remarks that “mathematics does not accomplish such results, because it is endowed with miraculous potencies called values”. John Dewey, Democracy and Education.p.287& p.158. These values are merely possibilities to be realized through effective instruction in mathematical topics.
The Practical Values of Mathematics.
The practical values of mathematics fall into two categories:
i. Utilitarian value (useful to an ordinary individual) and
ii. ‘knife and fork’ value (useful as a ‘tool’ subject).
Utilitarian Value.
Referring to the utilitarian value of mathematics, J.W.A.Young wrote, “There is no subject, except the use of the mother tongue, which is so intimately connected with everyday life, and so necessary to the successful conduct of affairs”. (J.W.A Young, The Teaching of Mathematics in the Elementary and the Secondary School. P.13). To a coolie who calculates his wages as well as to a finance minister who frames the national budget a knowledge of mathematical facts and principles is necessary. “The need of a good command of arithmetic by the skilled mechanic, by the up-to-date farmer, by the progressive professional man, by the successful merchant, and by the efficient housewife, is so obvious as to need no discussion. ( Lindquist, Theodre, Modern Arithmetic Methods And Problems. P.30) observes Theodore Lindquist. No wonder, mathematics formed the vital part of the curriculum even in the ‘Pial’ schools and ‘Patasalas’ of ancient India and in the ‘Academies’ of ancient Greece.
The principles of fundamental operations, the calculations with units of measurements, the meaning of graphic representation, the derivation and solving of simple equations, the concept of negative number, the algebra of the formula, various geometric forms etc., are indispensable ‘tools’ in the intellectual equipment of every intelligent citizens of a democratic country, if he is not to “feel the stigma of ignorance of the common things that the educated world talks about or reads about”. (David Eugene Smith, as quoted in The Teaching of Mathematics in the Elementary and the Secondary School J.W.A.Young, p.391). If one desires, as one should, to take an active part in social life and in the business of government, one should acquire a certain amount of mathematical facts, concepts and principles.
At this juncture one may question the use of identities, factorization, and a hundred odd geometric propositions like ‘ the sum of the three angles of a triangle is two right angles’, for an ordinary man? This question has to be considered from a broad point of view.
Such abstract facts as mentioned above may not have a narrow utilitarian value. But they are required for the study of other subjects as well as in certain vocations; hence, students who may specialize in mathematics or go for technical education need to learn them. However, one cannot predict which individual will become a mechanic, a farmer or an engineer when pupils are in secondary schools. Therefore, these facts have to be learnt by all students in secondary schools. Therefore, these facts have to be learnt by all students in secondary school. The attempt to compartmentalize mathematics into that which is useful for immediate or most common purposes, and for other subject and vocations, and thus to give instructions separately, is neither meaningful nor possible when the whole treatment of the subject is expected to be elementary, practical and vocational-biased. In fact, the real point connected with the question concerned is not the use of those abstract facts to the ordinary man but the selection of problems relating to these as well as the adaptation of suitable methods of instruction so as to make the maximum practical use of them.
Besides, if such question arises in mathematics, questions of the same nature may be asked about the other subjects in the curriculum, say the sciences and social studies. How many of us have at any time directly used the bundles of facts we learnt in History, Geography and even in the Sciences? J.W.A.Young has rightly observed, “while there are few occasions on which the ability to solve a quadratic equation is directly useful in business, it would be hard to imagine anywhere the ability to scan hexameters, to name the parts of a flower, or to give the date of the battle of Waterloo would be of practical benefit”. (J.W.A.Young, The Teaching of Mathematics in the Elementary and Secondary School P.14). To think of mathematics in terms of immediate or hand-to-mouth use is to take a superficial view of the function of mathematical study. “To condemn branches of mathematics because their results cannot be obviously applied to some practical purpose is short sighted. ( Jourdain, E., The Nature of Mathematics, p.17) . Indeed, if man had rested on his laurels, perhaps he would not have emerged from barbarism. Browning pertinently asks: ‘Man’s reach should exceed his grasp, else what’s heaven for?’.
Mathematics as a ‘tool’ subject.
Mathematics is an efficient and necessary ‘tool’ for the study of other subjects. It is literally indispensable for the study of subjects such as physics, chemistry and astronomy, and no informed person would question its instrumental value is this connection. Thus, mathematics has utilitarian value as well as it acts as a tool subject.
11. Disciplinary Value of Mathematics.
Definition – From time immemorial many people believe that mathematics disciplines certain power of the mind; and powers acquired through mathematics help the individual in other situations or in the study of other subjects. The history of education is replete with names of educators who held that mathematical training would give pupils “greater power to manipulate numbers and quantitative relations, capacity for thought and abstraction in mathematical problems, greater power of concentration and ability to analyze a mathematical situation by studying the known conditions and discerning a relations aiming them which will lead to the value of the unknown element”. To illustrate,
Plato wrote over the portals of his Academy: ‘Let no one ignorant of geometry enter here’. “He esteemed geometry and arithmetic highly as a means of compelling the mind to reason about abstract things”. Another Greek philosopher Socrates, maintained that the study of mathematics up to a proper point trains pupils in attention and concentration. These views and others of the same kind indicate that many people consider the mind to be a storage battery that can be charged with mental energy by mathematics, and the power thus stored can be utilized on other occasions when required.
Cultural value of mathematics.
The term ‘Culture’ is commonly understood as social heredity, – ‘the fundamental capital of civilization’. A subject may be said to have cultural value when it has general usefulness. It must enable a “person to understand his social and physical environment, to converse more fluently and intelligently about various topics and to meet various situations better”. (J.H.Minnick, Teaching of Mathematics in the Secondary Schools, pp.38-41).
Mathematics has general usefulness. For instance, it helps a banker to solve his financial problems, to a carpenter to cut two pieces of timber to fit them at right angles, and to a statistician to compute a correlation etc. Even to understand the meaning of a simple formula in a book on industries, or a graph in a daily paper, a mathematical method and the language and symbolism of mathematics are indispensable. Thus, mathematics has various importances in the development of various sciences and even in the evolution of modern civilization. Hence it has a cultural value.
Mathematics with its order, rhythm, symmetry and harmony resembles the fine arts, such as music, poetry and sculpture and it has a beauty unique in nature. As a thing of beauty, mathematics is a joy for many people, at least for those who do not learn it for the purposes of examination.
Mathematics provides opportunities for intellectual development and intellectual pleasure. For intellectual pleasure, mathematical concepts like the ‘irrational numbers’ of the Hindus, ‘Theory of Numbers’ of Ramanujam, and the whole field of Greek mathematics would not have come into existence
Thus, mathematics has general usefulness. It facilitates thinking, and provides intellectual pleasure and enables one to experience the joy of achievement. Hence, it is said to have a cultural value.
Mathematics as a subject has enormous amount of value in education. This can experienced only if it is taught through appropriate teaching methods.