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The Study of Mathematics

The Study of Mathematics

page 203

The Study of Mathematics

The present century is a period of considerable expansion in the activities undertaken in our schools, and with each additional development it has been necessary to curtail the time allotted to those subjects already established. There appears to have been a change in emphasis too; the instruction has become more specific and practical with a strong leaning to vocational training. The older aim of a liberal education as a process of mental discipline is in danger of being lost; indeed, to use such a term as this to describe the aim of education is to be quite out of favour. With the increasing competition of interests in the schools there is constant need for their evaluation as elements of a liberal education, and the particular interest in this essay will be mathematics.

One approach is through the experimental work which has been done to find how far the training gained in any one subject is transferred from that activity to related activities. At first the conclusion was that there was no transfer, though this opinion appears later to have been modified somewhat. As at present constituted, the mental tests applied by teachers and others, are not sufficiently reliable to discriminate between subjects in this matter. page 204 The technique of the intelligence test has too much of the character of a snap judgment. It neglects the power of sustained effort to which serious study habituates the individual, the subconscious mental action which follows such study and frequently leads to the solution of a problem in a flash of inspiration.

What is to me the significant outcome of this experimental work is the need it shows for the overhaul of the method of teaching. Mathematics is regarded as an easy subject to teach; and it is easy if the result is judged by the ability to go through the motions of solving a quadratic equation or writing out the proof of a geometrical theorem. The comparative ease with which such technical facility can be acquired makes it a 'good' examination subject, but if the present fire of criticism destroys this vicious tendency it will have served a worthwhile purpose. What is so often missing is the realisation that the practice of mathematics calls for the use of the whole mind and not just the manipulative faculty and the logical faculty. That such a presentation of a mathematical topic is possible is very well shown in a recent study by H. P. Fawcett.1 This is an account of a course in geometry where the aim was not a list of theorems learned nor the ability to solve riders but an understanding of the nature of proof. The material for discussion was not limited to the geometrical situation, but covered the whole range of the pupils' knowledge and experience, and they undoubtedly did achieve an appreciation of the value of precise and critical thinking. The account makes most stimulating reading and shows the educational process at its highest level.

In a book2 by the American historian of mathematics, page 205 Professor Florian Cajori, a different and novel approach is adopted. It is a review of the 'judgments of the prominent men of the ages' relating to the place of mathematics in liberal education. Every endeavour was made to collect judgments on both sides of the question. His summary shows a five to one verdict in favour of the high value of mathematics and, omitting the mathematicians, a result of over three to one in favour. In his discussion of the judgments there is this significant passage:3 'Educators denying the great value of mathematics in liberal education are in a minority; their arguments have been considered by some of our best thinkers, but found to be unconvincing. The bulk of their arguments refer to defective teaching of mathematics and, therefore, call for better teachers, but not for the virtual elimination of algebra and geometry from the required courses in secondary schools.'

The new syllabus for mathematics in New Zealand secondary schools raises in acute form the problem of the place of mathematics in liberal education. In the notes prefacing the syllabus the stress is on the utilitarian value of mathematics—mathematics as required for the physical sciences, for industry, for commerce. Such applications are valuable and have contributed largely to the material advantages this age has over the past, but this evaluation does scant justice to mathematics' claim as a worthwhile element in education. The danger lies in the over-emphasis of this aspect, since what we are aiming to do in our schools and universities is surely not to produce merely technicians but citizens, able and willing to take their full part in the cultural and material development of community life.

Mathematics has contributed to human knowledge two things: firstly, the method of deductive reasoning as it is developed in technical mathematics; and secondly, the page 206 mathematical description of nature. The prevailing tendency is to play down the first aspect and thus neglect one of the great achievements of the human mind.

The method of thought developed in mathematics is that of rigorous thinking, which demands a highly critical attitude towards the details of any demonstration; an argument is either perfect in every detail, in form as well as substance, or it is wrong. Undoubtedly in many, if not most, of the situations of everyday life such a standard is unattainable, but as a standard its value is indisputable. The simplicity of the mathematical situation is ideally suited to show the pattern of such thinking. The satisfaction obtained on the completion of such a demonstration, the feeling of certainty in the conclusion are experiences of the greatest value for the individual.

A further important feature is the unique status of the individual mind. No authority is recognised but that of one's own intelligence. It calls for scrupulous honesty of thinking and constant attention to detail.

Let us now consider two characteristics of the mathematical method—its abstractness and its constant striving after generality. That the mathematical method is abstract is the source of its simplicity. We thereby free ourselves from the complexities and ambiguities inherent in the impressions we receive through our senses from the world round us. It teaches us to search for the essential elements in any situation—essential, that is, for the purpose in hand. It is this stage that is probably the most difficult and of which the tentative nature has constantly to be stressed. The intuitive element in mathematical discovery shows itself most clearly here. It is often denied that intuition has any place in mathematics, and the part it does play is certainly obscured by the deductive form of mathematical proofs. The deductive character of the mathematical page 207 method is the source of the truth of mathematical conclusions, but this should not be allowed to obscure the fact that intuition and construction are the driving forces behind mathematical discovery.

At a higher level, in the realm of theories, the abstract approach is one of the significant features of present day mathematics, as for instance in abstract algebra. As with other branches of knowledge, mathematics has undergone an amazing expansion in recent times. New ideas and theories are developed and soon each becomes a growing-point for further developments, to such an extent that the student may well be overwhelmed by the wealth of material. But again the abstract approach is producing some order out of the mass of detail. Abstract algebra illustrates the result as well as any. We have at the base the concept of integer, and from this we proceed to rational number, real number, complex number, algebraic number, hyper-complex number, each being the source of ideas of great significance for other fields of mathematics. What have these concepts in common? By a study of structure the mathematician arrives at an abstract development in which the previous separate developments are but incidents whose particular features can be expanded as required. Incidentally it is the particular type of structure discovered here that has proved of assistance to the mathematical physicist in developing his theories of sub-atomic phenomena.

The other quality of the mathematical method mentioned above, its constant search for generality, leads also to ultimate simplicity. Take, for example, the problem early encountered in algebra of solving two linear equations in two unknowns having numerical co-efficients. The elementary method of successive elimination leads at once to the solution and this procedure can be applied to three page 208 equations in three unknowns, and so on. When, however, we consider, instead of the particular cases with numerical co-efficients, the general equation of each order with literal co-efficients, we discover that the solutions are built up in a regular way from the co-efficients and we are able to set up expressions which cover all the cases treated individually before. Nor is this the only gain. The particular combination of symbols used in this, called a determinant, is found to occur in other connections and its systematic use leads to a considerable gain in economy and simplicity in these other cases also.

Again from the field of geometry: the ordinary Euclidean geometry is a study of those properties of geometrical figures which are unaltered under rigid translations and rotations. If we now discard the requirements implied by the word rigid and consider what properties are invariant when a certain degree of stretching is permissible (for example, we can think of an ellipse being derived from a circle) we have projective Euclidean geometry which includes ordinary Euclidean geometry as a special case. A more vivid way perhaps of regarding the changes to which a figure can be subjected in this case is to consider lines drawn to all points of the figure from some fixed point not in the plane of the figure, the corresponding figure being the locus traced out by these lines on another plane not necessarily parallel to the plane of the given figure. We can go still further. Imagine a figure drawn on a sheet of rubber and the sheet deformed in any way without tearing it. The properties invariant under the mathematical counterpart of such drastic changes are the subject matter of topology, the rigid motions and the projections mentioned above being very special cases of topological transformations. This branch of mathematics is one of the significant growing points at present.

page 209

The abstract character of mathematics seems to be regarded by many as making it unsuitable as a part of the education of the young. The belief seems to be that because it is abstract it is therefore detached from life. Nothing could be further from the truth. It is simply the ideal handling of the problems of life, just as the artist may idealise the human form or a landscape. The central ideas of mathematics, the concepts upon which its theories are built, are merely the chief ideas which give life its order and rationality. Thus at the basis of much mathematics there are the concepts of variable and constant, which are the essence of the everyday notions of changes and fixedness. Then the all-pervading concept of function, indicating a correspondence between two or more variables, aptly expresses our sense of the interdependence of the elements that constitute our experience. The concept of group, fundamental in the modern abstract algebra previously mentioned, typifies much that we understand by form and structure.

Although clearly these ideas in their pure form are unsuitable for elementary instruction, yet they must definitely dominate it, if it is to have coherence and value for the student. This raises the question of what one writer calls the need for 'the humanisation of the teaching of mathematics.' The matter and spirit of the subject must be so interpreted and presented that it appeals to the whole mind. The aura of austerity that surrounds much of the instruction must and can be expelled. The thrill of exploration and discovery, the sense of satisfaction at a piece of work worthily completed are too often lost in the interests of a gain in technical facility and premature rigour.

The charge is often made against mathematics that the pattern of thought employed is too rigid and makes the student incapable of the type of analysis required, for page 210 example, in the social sciences, where the relationships expressed by 'tendency' play such a large part. The concepts and methods of statistical mathematics provide the necessary basis for the quantitative treatment of such fields of human activity and could well have a larger place in mathematical instruction in New Zealand. It is pleasing to note that a beginning has been made in the new secondary school syllabus. But care and thought are going to be required to ensure that the instruction does not lead only to arithmetical and manipulative facility. There is an even greater danger here since the ideas involved and their significance are less well known than those of the usual mathematical topics.

It cannot be too strongly emphasised that the notions upon which statistical method is based—probability, frequency curve, frequency surface, correlation and regression, sampling distribution, confidence interval—are mathematical entities and so are in a form suitable for use in precise thinking. Because of the deductive form of the mathematical method, of which statistical method is but a part, it is incumbent on the user to ensure that a particular procedure is an adequate characterisation of a given physical situation. Statistical method provides concepts which greatly facilitate our thinking in the fields to which they are applicable but they will not do this thinking for us. An appreciation of the value of this branch of mathematics is definitely growing, but disillusionment will result unless this is accompanied by the precise and critical thinking called for in the mathematical method.

A valuable field that is too much neglected is that of the history of mathematics. The lives and work of the men who contributed to the development of the ideas and methods are a fruitful source of interest and stimulus. These too will impress most vividly the dynamic quality page 211 of mathematical thought. Mathematical instruction can too easily become concerned mainly with the content of the subject and fail to convey the spirit and method. An extensive knowledge is not enough—to that must be added an appreciation of mathematics as one of the great enterprises of the human mind.

1 Fawcett, H.P., The Nature of Proof. New York, 1938.

2 Cajori, F., Mathematics in Liberal Education. Boston, 1928.

3 loc. cit., p. 164.