Mathematics as an Educational Experience
In January 1948, George Baker Jeffery delivered his Presidential Address to the Mathematical Association. It was printed in The Mathematical Gazette 32 (298) (1948), 6-14. We give a version of his address below.
Mathematics as an Educational Experience
by G B Jeffery, M.A., D.Sc., F.R.S.
Presidential Address to the Mathematical Association, January 1948.
Our Association is happy in including within its membership two intermingling streams, those whose interest lies in mathematics as a great branch of learning, and those who see in the subject the possibility of a great instrument of education. Whether his interest comes from the one stream or the other, no English mathematician could fail to regard it as a signal honour be invited to serve the Association as its President.
Among the obligations of the President is to prepare and deliver a Presidential Address. In other places, a presidential address may be a declaration of policy and leadership, given in the first flush of accession to presidential power. The President may be tempted, if he can, to make his address a great display of learning so that his hearers may be convinced of the wisdom of their choice of a President and of the folly of questioning his presidential direction. Your custom has wisely ordained otherwise, and your President may speak only after the steps have already been taken that will ensure that within the hour he will become an ex-President.
That custom has a special advantage for the President, for it gives him a clear indication of what the theme of his discourse should be. From the presidential Pisgah he should declare the Promised Land as he sees it, and as he hopes it will be enjoyed under the leadership of other Presidents. And, as the milk and honey are always relative to the hardships of the wilderness, the President on these occasions may claim some moderate right of reminiscence.
It is now more than half a century since somebody first tried to teach me something of things mathematical, and in the time which has since passed a great change has come over the teaching of mathematics in the schools of this country. Age-old traditions have been brought under assessment and have been radically changed. New life has flowed into the teaching of the subject. Honoured names spring to our minds. In private duty bound, I must remember Thomas Percy Nunn - your former President and my teacher and master. You will bring other names to the list, and happily not a few of them will belong to those still with us as honoured members of our Association. New textbooks have been written, which have left their mark on mathematical teaching. A library of the current school textbooks in mathematics written in this country need fear comparison with such a library from no other country in the world.
All this might have had little effect if it had not been for the devotion of the rank and file of teachers of mathematics. I know of no subject in which the teachers have acted under a stronger sense of professional duty and have devoted more time and energy to the improvement of the teaching of their subject. Our Association, especially through its Teaching Committee, has inspired and sustained this great movement and has provided the chief medium for its corporate expression. The Association has a record of which we may justly be proud.
But pride in the past finds its only justification in duty in the present. My recollection of the discussions at your Council during the past year lends little colour to the view that the work of the Association is abating. Much is under consideration in relation to the teaching of mathematics in grammar schools which has perhaps been our main preoccupation in the past. We have our contribution to make to the burning problem of the modern school curriculum. The persistent questioning of the university curriculum in all its aspects means work for the Association in that field.
Critical times lie ahead for the position of mathematics in the school curriculum. A clearer realisation of the purposes and possibilities of education and a growing knowledge of the course of the development of the young person have broadened our ideas of the kind of experience that a child should gain in school. There is a wealth of new educational material, and the value of the old is called sternly into question. We shall have to face increasing competition from other subjects for the available time in the curriculum. Mathematics is among the "haves" and must expect to attract the envy of the "have nots". Possession is no longer nine points of the law.
Moreover, mathematics is peculiarly open to attack by the anti-intellectual forces that are rampant in our national educational thought. Some would have us believe that effort and struggle are bad in themselves or at least only good for the very good. It is sought to escape from the stern disciplines that teach us that good things demand hard work for their achievement, and we are offered the alternatives of soft options and sagging standards.
Our current controversies on examinations and on the reorganisation of secondary education are all infected by this heresy. No one can have a long experience of examinations without an acute awareness of their imperfections and of the injustice that they not infrequently work. Yet it would be idle to deny the stern discipline they effect by discriminating between sound knowledge and verbiage and between conscientious preparation and slacking. We can unite in our support of the plans for secondary education for all, and for each the form of secondary education best suited to his needs, yet we may deplore the efforts to secure an easy parity by limiting the possibilities of the best to the capabilities of the weakest.
Every boy has his potentialities which he will never fully achieve. The gap between the possible and the actual will be determined partly by the opportunities we give him but mainly by his own moral qualities of application, perseverance and self-sacrifice. It should be the prime purpose of education to support the boy in the development of those qualities and that is not done by soft options and sagging standards.
It seems to me that once we allow our educational policy to be determined by this heresy we have set our feet on the path of national decadence. I am gravely disquieted by the contrast in this respect between the trends of our own national policy and that of at least some of our continental neighbours. It is against this background that I ask you to consider the reasons for the high place that mathematics has hitherto occupied in education at all levels and whether we have good grounds for maintaining as mathematicians that we have an undiminished part to play in the education of the future.
I will spend no time in the elaboration of the arguments touching the usefulness of mathematics at every level of the national life or the necessity of maintaining and developing the great body of mathematical knowledge. There must be a due succession of professional mathematicians and, whatever other duties may rest upon our grammar schools, it is upon them that we must chiefly rely to discover mathematical talent and to give it the opportunity to develop.
I am concerned to discover what part, if any, mathematics can usefully play in the education of ordinary boys and girls who are not destined to become professional mathematicians, who are unlikely to use anything beyond elementary arithmetic in their work and their leisure, and who will probably forget most of their formal mathematical knowledge within a few years of leaving school.
I am disposed to make a bold claim for mathematics. It is a body of knowledge now so vast in its extent that the most learned of us can hope to master only patches of it. But it is more than this. It is a way of thinking so fundamentally a part of our human thought that even those who are most proud of their ignorance of mathematics, when thinking of things far remote from the technicalities of mathematics, are often thinking of those things in a mathematical way. In any piece of constructive and intelligent thought, the mathematician will recognise some of the great concepts of his subject. He will find that the thinker is using these concepts in a mathematical way though without employing the technical language of mathematics. If it is possible to imagine that the thinker adopted rules of thought that rigorously excluded everything mathematical he would find that his intellectual power had gone. Mathematics is not only a branch of technical knowledge, it is one of the modes in which the human mind functions. It is especially characteristic of the more developed forms of thought in which the growing young mind must achieve some measure of proficiency. Elementary mathematics affords great scope for the exercise of these forms of thought in a simple way. That, it seems to me, is the educational value of mathematics.
I am well aware that this claim requires for its proper examination a learning in philosophy and psychology to which I have no title and to which I do not pretend. Any value that there may be in what I have to say will be that of a record of the nature of mathematical thought as it appears to one who has spent much time in mathematical studies, who has often pondered the meaning of mathematics, and who has tried to understand the way in which mathematical understanding grows in his own mind and in the minds of his students. It is therefore liable to all the errors of introspection as well as all the pitfalls that beset the path of the amateur philosopher.
I will begin with our domestic cat who, within his somewhat limited range of concerns, is a highly intelligent representative of his species. I often amuse myself by trying to form a picture of what is happening in his cat's mind. I know that the behaviourists warn me that my picture will not be a true picture. Sometimes I catch the cat looking at me in a way that suggests that he agrees with the behaviourist and is chuckling at the thought of how little I know. For what it is worth this is the picture I see: an activity in constant change, usually at a very leisurely tempo but capable of remarkable acceleration. The pattern of that activity is built up from a relatively small number of ideas. There is the idea of fish, to which belong the taste and smell of fish and the prospect of fierce mastication and comfortable repletion. Another idea belongs to a favourite chair with its memories of warmth and comfort. There is the idea of the ginger cat who lives next door. There is the idea of my wife which is strongly associated with the fish idea. There is an idea of myself with an uncertain and tenuous relation to fish but with a strong negative relation to the chair idea since he and I are strong competitors for its enjoyment. There is an idea of mouse which puzzles me, for I have never been able to decide whether it is singular or plural. I doubt whether that cat has ever faced up to the problem of whether the mouse he chased and missed last night was the same mouse or a different mouse from the one he chased and missed the night before, or whether either of them was necessarily different from the one he chased and ate a week ago. He has a few more ideas. There are not very many of them, and they are all immediately related to his perceptual experience, past or present. The pattern of his mind displays these ideas in changing emphasis and in changing relationship. It is not a contemptible mind, for I observe that by it he directs his conduct in such a way as to get most of what he wants most of the time.
If without disrespect to my cat I may take my picture of what goes on in his mind as indicative of the nature of thought at its most elementary level I may ask what happens as thought is developed to higher levels. In the first place the range of perceptual experience is vastly increased and is reinforced by second-hand experience. My idea of the Battle of Hastings has probably little correspondence with anything that was seen or heard by anyone at Senlac in 1066, but that is my error. At this level I am still thinking of the external world in terms of the knowledge of it that would have come to me through my senses had I been present at all places and at all through my senses had I been present at all places and at all times in its history. So far my thought does not differ essentially from that of the cat.
In contrast to the cat mind, I consider my own mind as I am thinking of, shall we say, the theory of functions of a complex variable. I will not suppose that I am attempting to evaluate a troublesome integral because, by a process of which I shall have something to say later, I would in that event probably have come back to the cat level by creating an artificial perceptual experience for myself by making certain marks on paper and then looking at them. Let us suppose rather that I am contemplating the theory and the way in which the assumptions we make as to the small-scale structure of the function condition its large-scale structure and lead, for example, to the majestic generality of Cauchy's theorem. My mind is moving in a world of ideas which is far removed from perceptual experience, and it moves with freedom and with a measure of mastery.
This contrast exhibits the nature of intellectual development. At the lower level the mind is limited to ideas which are directly related to perception and through perception to objects and relations in the external world. At the higher level the mind is liberated into wider fields in which there many ideas that are not directly derived from perception and which have no necessary correspondence with anything in the external world. The power that comes from such a liberation is easily understood by a mathematician who knows, for example, how much more perspicuous some plane geometry becomes when it is set in a three-dimensional space.
It is interesting to try to trace some of the processes by which this liberation is achieved. An early but important stage is the recognition of similarities between different things - similarities sufficient to support similar judgments. Two dogs are recognised as different and yet for some purposes the same, and a third is immediately recognised on first acquaintance as a bow-wow. The aggregate of the dogs of the child's acquaintance grows by imperceptible stages until it becomes the variety canis or perhaps more - the class of all dogs that are or have been or might be.
We all know that the logical analysis of the mathematical concept of class presents problems of considerable difficulty and depth, yet we see this concept emerging in the mind of the young pupil without crisis and developing easily and naturally. It may never attain the subtlety of Bertrand Russell, but it may nevertheless at any particular stage be held with a degree of precision sufficient to support clear thinking.
It is impossible to exaggerate the importance of this development, for it is the necessary antecedent of any form of logical thought. It is also a fruitful source of the proliferation of ideas. Our first notions of class arise from the classification of ideas directly corresponding to perceived objects on the basis of their observed similarities. This implies the analysis of the ideas in terms of abstract qualities. This in turn opens the way for the definition of classes by intention, and the development of logical thinking both in its deductive and inductive phases.
Notice that in the course of this development we have acquired a great stock of ideas which do not correspond directly to perceived objects - ideas of abstract qualities, classes, and the relations of classes. It is the ability to think in a world of ideas of this order that distinguishes my mind from that of the cat.
One of the great educational values of mathematics is that it affords opportunities for experience in simple thinking of this sort. Our first lessons in geometry may be very concrete and deal with a few triangles drawn on paper or cut from cardboard. We can proceed by easy stages to consider the triangle and argue to the conclusion that the sum of the angles of any triangle is two right-angles. The value of the experience is largely lost if the study never passes beyond the concrete, though for a reason to which I shall refer later one must be careful not to get too far from the concrete too soon.
It appears to me that the development of mathematical thought deserves careful and systematic study for the light that it would throw on the development of thought generally. We need a psychological rather than a logical analysis of mathematics, and that is a task I am not competent to undertake. I can only offer the observation that whenever I consider one of the great principles of mathematical thought I can trace its effects on the structure of our general thinking. For example, an aggregate may have properties apart from those of the elements which compose it. It may in particular have unattained bounds or limits. The aggregate of the ideas of all the dots and lines that I could draw on paper is an aggregate in which each element corresponds to a perceived object. But the limits of appropriate sub-aggregates are Euclidean points and lines which have never been perceived. Here again the logically difficult seems to be the psychologically easy, for the intelligent fifth-form boy finds no great difficulty in conceiving the point that is without dimensions and the line that is infinitely thin. I have sometimes tried my younger friends with the argument that if a line gets thinner and thinner, by the time it becomes a Euclidean line there is nothing there at all. Therefore, Euclid is all about nothing. I find that they instinctively reject my logic, though they have not the technical skill to define its fallacy.
This is no fanciful illustration, for it corresponds to something of the deepest significance in our intellectual life. We take our knowledge of the real world as it comes to us through perception and from it we derive conceptions of things that are approached but never attained in the real world. They are ideals. It is when we turn these ideals back again upon the real world so that we set side by side the picture of the world as it is and a picture of the world as it might be that we have the beginning of progress and endeavour. This is the field of the greatest achievements of the human spirit and some of our earliest opportunities of learning to think in this way come through simple mathematics. If the mind in maturity is going to function in this kind of way, the first experiences of this sort are important.
Perhaps the most obvious contribution of mathematics to our general thought is in the field of number, and its associated fields of quantity, measurement, equality and inequality. Later this afternoon this meeting is to consider this topic from the point of view of the teacher, under the able guidance of Professor Neville, and I shall say less about it than I otherwise would have done. The extent to which these concepts enter into every department of our thought needs no emphasis. It is hardly less obvious that they are necessary for the refinement of logical thought. The quantities of logic recognise only none, at least one, all but at least one, and all. The proposition "Some men are mathematicians" inevitably leads to the question " How many men are mathematicians?", and the reply which leaves us guessing between one and the total male population of the world is somewhat vague. In view of your later discussion I will make only the passing remark that here again the logically difficult appears to be the psychologically easy. Boys and girls acquire without great difficulty an adequate mastery of these mathematical concepts. It often seems to me that the intelligent schoolboy's conception of the cardinal number five is not very far from that implied in the Frege-Russell definition, though he would not understand the terms of the definition. It is, I believe, a good deal nearer to the mark than, for example, the abstraction of the quality of fiveness from all the groups of five concrete objects that he has known.
I want next to refer to what appears to me to be the invariable characteristic of mathematical thought at all levels. It is concerned not with the nature or origin of our ideas but with the way in which our minds function in relation to ideas. If I tackle a new mathematical problem my mind is filled with a complex jumble of loosely related ideas, and I am bemused and uncomfortable. Eventually I discover that I can sort out these ideas and relate them in a pattern in such a way that the whole becomes cogent and significant, and then the problem is solved. Where does the pattern come from? Is it inherent in the ideas and awaiting discovery, or do I supply it from a stock of patterns with which I am familiar and compress the ideas within it with whatever violence may be necessary? Why do I need a pattern at all? Is it because my mind cannot function in relation to more than a certain limited measure of complexity and that I am accordingly in a perpetual struggle after a simplicity that is within my competence? These are questions that I must to you unanswered, but I have no doubt that we are effective in our thinking very largely according to the measure in which we are able to attain this tidiness of mind. We can, of course, have superficially tidy minds just as we can have superficially tidy bedrooms, but these constitute no argument against the value of good order as a practical aid to effective work in any sphere.
A great educational value of mathematics is that it affords ample opportunities in a relatively simple field of ideas for the attainment of this tidiness of mind. The mathematician acquires great skill in discerning order in complexity. He can marshal the steps in an algebraic calculation. the product of two projective pencils will stand out to his eye from a complicated diagram. He can order the sequence of propositions to build up a great mathematical system. The mathematician is liable to the failings of his craft. He may exalt tidiness even above godliness. He may, and often does, come to live in an oversimplified world, but he can think effectively in mathematics, and the vast structure of modern mathematics is the measure of what he can achieve.
But have we been talking about things that are essentially mathematical and have relevance only to algebra and geometry? Or have we been talking of things that have their due relevance to straight thinking to whatever end it is pursued? If so, we have one more ground for holding that mathematics, quite apart from its technical content, is an essential part of the structure of our thought. Mathematics, or some activity that provides equivalent opportunities, is a necessary part of the experience in which the young mind develops.
I am not thinking specially of advanced mathematical studies, for the experience may well begin in the nursery school with the orderly arrangement of material objects and the first steps in number work. Young children find a peculiar satisfaction in order and orderliness, and they generally enjoy mathematics until we force their pace beyond their capabilities.
The value of mathematics is enhanced if the search for order is consciously pursued, and if the generalised tidiness of mathematics is exhibited as including ordinary tidiness as an important and particular case. Careless paper work does not assist the pupil in the discovery of mathematical order or in the solution of mathematical problems. A muddled blackboard is a singularly inappropriate medium for the illustration of mathematical ideas.
I have suggested that the development of thought consists very largely in the acquisition of the power to deal with ideas that are not directly related to perception, and that the more developed our thought becomes the more tenuous the relation between our ideas and anything that can be heard or seen. Mathematical genius might be conceived as an escape of the human spirit from the trammels of the flesh into a rarefied space of its own making. Certainly there are mathematical occasions on which any attempt to link thought with perception leads only to confusion. You will not get very far with the study of hyperspace until you give up trying to see what four lines meeting at a point and each perpendicular to the other three look like. But these are the ways of mathematical genius, and for most of us flights into the rarefied atmosphere of abstract thought must be short and we must frequently return to solid ground.
Continued abstract thought for most of us leads to a certain tension that is not entirely due to unfamiliarity. Abstract ideas are apt to be fleeting and difficult to hold before the mind. We seek to anchor them to the more permanent ideas which are in direct relation to perception. Sometimes it is possible to do this in a simple and direct way as, for example, when I draw a graph of a function or make a model of a tetrahedron. Sometimes my thoughts have developed so far away from the perceptual that these simple expedients are no longer possible. Then I relieve the tension by creating a situation which will give rise to a perception which can serve to support my abstract idea. I invent a symbol for my idea and the idea acquires a pseudo-concreteness in the symbol. I feel that I have re-established my relations with the real world, and I am ready to begin the whole process over again. My new beginning is, however, different from my old because my thought is based on symbols and not on perceived objects, and I may forget that they are symbols.
This perpetual effort to externalise our abstract ideas has important consequences. The musician and the painter do more than explore the variety of sense perception. They are thinkers who evolve highly abstract ideas and who have the skill to create sights and sounds which serve them giving perceptual support to their ideas. To the creator the painting or the symphony is an apt symbol of his thought. Every time he sees it or hears it afresh it brings back his thought, not in the static manner of mechanical reproduction but in the dynamic way of a living and changing thing. The mystery of art is whether and to what extent the work of art induces in our minds something of the thought that was in the mind of its creator - a thought which by its very nature is incapable of representation in material form.
There have been moments when I have been disposed to claim that among the many facets of our subject there is one which shows mathematics as a great medium of art comparable to music. This is not because of any resemblance of structure between a symphony and a mathematical theory; so far as I can discover there is none. Behind the sounds of the symphony there is the mind of the composer which has conceived things that transcend all sound and sense. He has heard the music that echoes through the temple of the human spirit, and he has sought to catch some of the tones and to fix them in the notes of his score. As the sounds of his symphony fall upon our ears something stirs within us and we begin to hear faint echoes of that greater music which no orchestra can play.
There is something behind the symbols of a great mathematician, something as distinct from those symbols as great music is distinct from the vibrations of air. The mathematician has been with the musician in the temple and each is striving to express in his own medium what he has found there-striving and imperfectly achieving.
There is another aspect of this process of the externalisation of abstract ideas which is of special significance to us as teachers of mathematics. It is important that we should learn to think abstractly, but most of us have only a limited capacity for sustained abstract thinking. No doubt we learn by practice and experience, but the pace of our learning cannot be forced. Under any attempt to force the pace the mind protects itself by substituting the symbols for the ideas and proceeding to juggle with the symbols. This is a perpetual danger and is present at all stages. It makes the teaching of mathematics a task of great delicacy. If we make our approach crudely practical we rob our pupils of the opportunity of developing their powers of abstract thinking. If we make our approach too theoretical our pupils take refuge in symbol shoving, and the result is much the same. The broad lines of the technique that should be followed are clear though much care is needed in its detailed application. Mathematical thought, like any other kind of thought, must begin close to perception. Accordingly our first approach should be concrete. We should, however, begin to build up abstract ideas as soon as our pupils are ready for them and proceed at a rate that is within as soon as our pupils are ready for them and proceed at a rate that is within their powers. Abstract ideas should be buttressed by a wealth of concrete illustration and example. Symbols should be recognised as symbols and not allowed to obscure the ideas for which they stand.
Perhaps a word may be said in favour of symbol shoving. It is a peculiarly satisfying human experience to attempt a task of some difficulty and to know beyond a peradventure that one has succeeded. The child has many opportunities of this sort on the physical side. He tries to walk and is proud of his early successes. He attempts to vault a bar in the gymnasium and he lands safely and not too clumsily on the other side. He sets out to climb Snowdon and has the satisfaction of sitting on the cairn. In handicraft he has similar opportunities in manual dexterity. But once he has learnt to read he has relatively few opportunities of this sort on the intellectual side. Whatever task you set him in history, geography or French, he can make some kind of a shot at it, but however well he does there is always room for improvement. Failure is seldom absolute and success is never complete. But set him a sum to work or a mathematical problem to solve. He either discovers how to tackle it or he does not. He either gets the right answer or the wrong. He either succeeds or he fails. At a stage at which interests are not developed to the point at which they can sustain long-continued effort, there is a value in these short-term objectives that can be attained or missed within the hour.
But symbol shoving should not be mistaken for mathematics any more than crossword solving (which is of course an intriguing and delightful form of symbol shoving) should be mistaken for literature. That this mistaken identification has often been made is the misfortune of most of our university teaching of mathematics and of that part of our school mathematics which falls under the shadow of the university scholarship. The emphasis is placed almost entirely upon problem solving and there is little attempt to discuss the philosophic nature of the elements with which mathematics deals or to exhibit mathematics as the achievement of great minds. One technique follows upon another and the student frequently gets no more from his study than an amazing facility in the manipulation of symbols.
The time has come when this rambling discourse should draw towards its close. I have been concerned to convince you that as teachers of mathematics you are teachers of a great subject; that you stand in a long and honourable succession; and that it is within your power to give something to your pupils that they can receive from no other source. I have striven to find an expression for my own idea of mathematics that will call an answering response from you. The knowledge and the skills that the study of mathematics imparts find many and varied applications and that is sufficient to ensure that mathematics will continue to hold an important place in the curricula of our schools and colleges. But our ultimate concern is not for the teaching of mathematics but for the training of young minds and the education people. We wish for them that they may have full and useful lives and that they may be well-equipped to face all the problems that life may bring them. From whatever point of view we examine the intellectual power that makes man great and capable of great things we find elements within it which are essentially mathematical so that we are impelled to the conclusion that mathematics is part of the very substance and structure of human thought. You will not suppose that I have suggested that mathematics is the whole of that substance and structure and that there are not other elements equally essential and of equal value. But mathematics has the peculiar value that, rightly conceived and rightly taught, it affords the opportunity at its earliest and most elementary stages for the child to think in the ways in which a man must be able to think.
And so, as your President, I would encourage you all in your work in our schools and colleges; and for our Association I would wish that it may long continue to be a power for good in education.
G B J.
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