Mathematics in School and University
On 15 October 1938, George Barker Jeffery delivered his Presidential Address to the London Branch of the Mathematical Association. It was published in The Mathematical Gazette 23 (253) (1939), 26-34. We present a version of Jeffery's address below.
Mathematics in School and University
by G B Jeffery
We have all heard the scheme recommended to Headmasters for their guidance in making the first three reports on a new pupil. The first should be devastating in order to impress the parent with the toughness of the job that the school is up against in remedying the defects in the earlier training of the pupil. The second should be glowing in order to show the progress made when the boy gets to the right school. The third should be very guarded in order to moderate any undue optimism created by the second, and to make it clear that any failure that may lie in the future will be due solely to lack of aptitude or application on the part of the pupil. I fear that some of my colleagues in the universities are prone to adopt a similar line of defence. Certainly one not infrequently hears in college common rooms of the shockingly incompetent teaching given in the schools in this subject or in that. One may sometimes suspect that this is a professional particularisation of the proposition that "all good workmen have bad tools", though every beginner in logic knows that this may not be simply converted to "all who have bad tools are good workmen". Even if the teachers are as bad as some would have us believe, we professors who have had a hand in the training of so many of them had better keep rather quiet about it.
My own opinion, based on observation of the quality of the students who come up to the Universities and on the opportunities I have had of observing the school examinations at close quarters, is that the teaching of mathematics has made very great progress since I was at school, and is now so good that a university professor should criticise it only with considerable hesitation and with some modesty.
It is a great merit of our Association that it brings together those who are concerned with the teaching of mathematics to pupils of different ages and different stages of development in the subject. It helps us to realise, at least within the limits of our own subject, the ideal which many of us have set before us-a unified teaching profession embracing colleges and universities and schools of all grades and kinds-a profession in which the importance of the work of the individual will no longer be measured by the average age of his pupils. Are we not constantly discovering that the special difficulties and problems that arise in the course of our work arise in a modified form in the work of those who are teaching pupils of a different age?
The great merit of a Presidential Address from the point of view of the President is that he is permitted to rove over the whole field of our subject in a way that would not be tolerated in a communication from an ordinary member. I propose to avail myself fully of this privilege. I have no great theme to offer for your consideration but only a few thoughts of no great profundity on the value of our subject to our pupils.
Can we justify the place that mathematics has hitherto occupied in the school curriculum? In the past every boy and girl has spent a quite considerable part of the school time on mathematics of some sort, from the very earliest stages at least up to the School Certificate stage. We have been a sheltered industry teaching a compulsory subject, and we have not been called upon to justify our existence as teachers of other subjects have had to do. It is long since this phase was passed in the universities; now perhaps it is passing in the schools. Next year it will be possible for the first time to take a School Certificate Examination which does not include mathematics, not even a paper on arithmetic. Are we as teachers of mathematics to regard this as loss or gain? Certainly it is a challenge to us to consider anew why we teach mathematics, and in what way we think that the study of our subject is profitable to our pupils.
The old arguments no longer carry conviction. We should like to think with Robert Recorde that Arithmetic is the "Whetstone of Witte". It would be very flattering to those of us who have spent many years in the study of mathematics if we could believe that our minds have been trained and our intellects have acquired a penetrating quality that is denied to those who study lesser subjects. Alas! that this comforting hypothesis should fail so utterly under the test of experience. Are we mathematicians conspicuous among our fellows for the clarity of our thinking on, say, politics? Do the statesmen turn to us to resolve the tangled problems of Europe? They do not, for there is a limit even to the folly of politicians.
Yet I understand that the psychologists permit us to save some- thing from the wreck. There is some gain in learning to think clearly about something. There is at least a faint hope that we may be a little more able and a little more anxious to think clearly about the next thing that comes along.
If that hope is justified, then it seems to me that mathematics is a particularly favourable field for its exploitation. Mathematics is usually regarded as a difficult subject, and in some ways it is, but its difficulty arises from the complexity of the patterns it weaves on a simple loom. One has to seek far in mathematics to find an argument that is intrinsically difficult. The arguments are often long sustained. It sometimes requires great ingenuity to construct the argument which bridges the gap between the data and the desired conclusion. But once it is marshalled the argument consists of steps, each of extreme simplicity. The things with which mathematics deals, numbers, quantities, straight lines, curves, are simple to the pure in heart. I know that they are by no means simple to the mathematical philosopher, but his is a different problem, and the difficulties he discovers belong to an order of which, fortunately for him, the schoolboy will remain in ignorance for a good few years at least. To the schoolboy the things of mathematics are simple ideas illustrated on every hand in his everyday life. Moreover, the assertions of mathematics are susceptible of clear and concise statement.
Hence it seems to me that mathematics gives us the opportunity of thinking clearly and in a simple way about some of the simplest notions that the human mind possesses. It is the superlative introduction to the art of clear thinking. This seems to me to be the most important value of geometry at the school stage, and I make one practical inference from this, namely, that work in geometry should be unhurried and that the time element ought not to bulk largely in a geometry examination paper. Different elements con- tribute to the value of arithmetic and algebra. The boy who can make a calculation correctly in five minutes is better than the boy who takes fifteen minutes for the job, and the examination paper may rightly test both accuracy and speed. I do not think that the same is true of geometry to anything like the same extent. A rider is an exercise in clear thought, and most of us find it difficult to think clearly if we are hurried and pressed for time. I am much more interested to know whether the boy can solve the rider given all reasonable leisure than I am in knowing how many he can solve in a given time. It is very easy for the examiner to use the time factor to produce the desired mark distribution, and thus to produce a paper which tests the wrong thing. I think that most School Certificate geometry papers could be shortened with advantage and less required in the time. If the mark distribution went wrong no great harm would be done to anyone.
Clarity of thought is closely related to neatness and clarity of expression on paper. My own standards in this respect have deteriorated sadly from what they once were, yet whenever I get tied up with a mathematical problem I take a clean sheet and start again in my best handwriting and neatest figures. Though the best handwriting is now rather bad and the figures are not as neat as they were, the old method still works surprisingly well. I am glad that this is fully recognised in the schools. From this point of view the work done for the School Certificate Examination is, with rare exceptions, altogether excellent. It shows a marked contrast with the work of the same standard done at the Matriculation Examination. Unfortunately it represents a habit that can be lost. The same candidates appear at other examinations a few years later, and all effort for neatness and decent presentation has disappeared. In fact it seems to be roughly true that the more advanced an examination in mathematics, the more untidy, disorderly and illegible the work will be. No doubt this is due to the fact that most examinations in mathematics test speed as much as they test anything. The man who wants to do well in them must sacrifice everything to speed, and he soon acquires the habits of the overhasty.
We may justify our subject on the ground of its utility. Mathematics is useful knowledge. But he who employs a utilitarian argument must be prepared for the next question - useful for what? Part of the answer is easy to give and is very weighty and convincing. In our industrial and mechanised age some knowledge of mathematics is useful in very many walks of life - in science, economics, engineering, accountancy, and finance. The number of ways in which mathematics enters into modern life is so great and is increasing so rapidly that this in itself is a justification for giving an important place to the subject in the school curriculum. The future career of the boy may not yet be planned, but the chance that it will lie in a direction in which some mathematical equipment will be useful to him is quite high.
We need not adopt a purely vocational view of education in order to justify our subject. If we conceive the business of education to be to make the child at home in the world in which he finds himself, then some knowledge of mathematical things is essential. When we first begin to interpret our sense perceptions in terms of objects situated in space we are recapitulating one of the most practically important discoveries of the human mind - the discovery of space as a means of co-ordinating our perceptions and presenting them to our minds in a comprehensible form. When we learn to count we are repeating for ourselves another of the great discoveries of our ancestors. If we try to clear our minds of every geometrical or arithmetical notion, and then try to think of anything at all, we soon discover how fundamental mathematics is in our common everyday thought. The mathematics we discover in this way is not, of course, the artificial mathematics of the textbook. It is a kind of "natural" mathematics, and the closer we can keep to this natural mathematics in our teaching of the beginners the more valuable our subject will be as an instrument of general education.
There are two conclusions that I draw from this view. The first is as to the extreme importance of the right teaching of very young children on these matters. We who are concerned with older pupils are sometimes impatient of the great care that is now given to the first steps in number work, and to the geometry of shapes and objects taught at the kindergarten stage. Yet it seems to me that careful work at this stage may well be a determining factor in the future development of the child, and that if this work is rushed by the expedient of supplying the child with a set of ready-made rules we are permitting a structural defect in the very foundations of our work.
My second conclusion is that simple three-dimensional geometry should be given a larger place in our school work, for it is three-dimensional space rather than two-dimensional that we discover through our perceptions. One of the best ways of learning the simple geometry I have in mind is through handicraft. The boy who can plane up a piece of wood and test it for winding and squareness has grasped in a particularly vivid way the substance of half-a-dozen significant propositions in solid geometry. The mathematical teacher will find the teacher of handicraft a useful ally in other ways. A good cabinetmaker has a fund of geometrical knowledge. Practice in the making and in the interpretation of working drawings brings a practical appreciation of much good geometry. Quite apart from its other values a craft which, like woodwork, admits of considerable precision in shape and measurement, is a valuable means of acquiring a grasp of geometrical truth.
Another ground for our faith in the value of our subject is to be found in what I believe to be a fact, namely, that a considerable proportion of our pupils enjoy mathematics. The view of education as a preparation for life sometimes leads us to assume that life begins at the school-leaving age. It is by no means clear that something which contributes to the enjoyment of life at the age of sixteen is on that account less valuable than something which contributes to the enjoyment of life at fifty. And so the mere fact that so many of our pupils enjoy mathematics seems to me to be part of the justification for teaching it. Even so, in a way we perhaps do not full understand, we are helping the pupil to prepare for his later life. Just as free and enjoyable activity of the body provides a particularly beneficent form of physical exercise, so free and enjoyable activity of the mind seems to contribute to our intellectual growth.
It is perhaps worth asking why boys and girls of school age should find enjoyment in mathematics. The joy of mathematics two-fold in its character. It arises partly from the artistic satisfaction we have in contemplating mathematical results. I have in mind the great theorems of pure geometry and some of the broad and fundamental theorems of analysis. Of the same kind is the satisfaction we find in the beauty of form in argument - the delicate touch and the ingenious turn that show the master mind. Now I do not think that there is very much in school mathematics that has this quality of mathematical beauty. Perhaps we come nearest to it in geometry in such simple results as this, that no matter what triangle I take its medians are concurrent. In my own mind this quality is usually associated with great generality, and a high degree of generality is usually inappropriate to the school stage.
But there is another source of satisfaction in mathematics that is more nearly akin to the crafts - the satisfaction of attempting something, and doing it, and knowing that one has done it. Mathematics presents opportunities of this sort more freely than most other subjects of the curriculum. A mathematical problem presents a short range objective; you either solve it or you do not; and when you have solved it, you are usually quite certain that you have solved it, though you may occasionally be mistaken. On the other hand, the answer to a question on history may have any value between perfection and complete inaccuracy or error. The value of the answer is relative to the stage of development of the pupil. A question could be set in identical terms at a school certificate examination and at a degree examination. An answer which would be regarded as approaching perfection in the one case would be quite inadequate in the other.
This characteristic of mathematics creates a special problem in the conduct of examinations. It is easy to frame a paper for a competitive examination in mathematics. All one has to do is to include a certain number of questions which are beyond the powers of the weaker candidates and a certain number that only the really brilliant ones can tackle. The circumstances of a School Certificate Examination are quite different. Here every question is within the powers of the average well-prepared candidate. It is therefore to be expected that a considerable proportion of the candidates will obtain substantially full marks. This would not conform with the normal distribution nicely balanced about 50 per cent., which is so dear to the heart of some examining bodies. As far as mathematics is concerned this seems to be a quite arbitrary fashion imported from other subjects in which it may have some justification, and often too lightly accepted by mathematical examiners.
In fact the mark distribution in mathematics does not differ very much from that in other subjects. This result is achieved by framing the examination so as to test two elements distinct from mathematical skill and knowledge. Firstly, the examination is used to test powers of presentation, though in a restricted and somewhat artificial way. A substantially successful answer to a geometry question may receive less than full marks because the candidate has omitted a step. In such a case we are not deducting marks because of something the candidate does not know. On the contrary, there is a presumption that he does know it, otherwise he could not have solved the problem. We are deducting marks because of something which the candidate ought to have known and stated, but which he knew and did not state. Now after what I have said in another connection you will not expect me to rebel very violently against this method of adjusting the mark distribution. From the educational point of view the art of the clear presentation of mathematical results is at least as important as the acquisition of mathematical knowledge. If, however, we are going to reflect this view in the examination we should take into account the orderliness and clarity of the whole presentation, and not concentrate merely on the inclusion of all necessary details.
The second method of adjusting the mark distribution is much more powerful, and consists in operating the time factor. If the average mark runs higher than we expect, we ask the candidates to do more in the time. The method is fatally powerful and easy to apply, and in my opinion examiners have come to rely upon it to far too great an extent. A very high premium is placed on speed, and though speed has its value in mathematics, its undue cultivation is inimical to the deeper values of the subject.
Special problems are presented by the boys and girls who are either very good or very bad at our subject. Most boys and girls of normal intelligence who are prepared to work with that moderate degree of application appropriate to healthy young persons of their age find that our subject comes quite naturally to them. There are others who find it difficult, and who progress only at the cost of really hard work. There are some who, whatever may be the reason, find the subject of extreme difficulty and make very little progress in it. In the school examinations we have so far endeavoured to meet this situation without completely abandoning the idea that mathematics should be a compulsory subject. It has been permissible to offer a qualifying paper in arithmetic instead of the full subject. This device has no doubt effected a partial solution, but it is disconcerting to find that there are still a certain number of candidates who fail hopelessly on even this modified test, but who nevertheless show distinct merit in other directions.
We shall probably all agree that a great deal of time is wasted by boys and girls on subjects at which they are no good and never will be any good, and that it would be to the advantage of everybody concerned if they were allowed to turn their attention to other subjects in which their labours might be more profitable. The problem is, however, not quite so simple as that, for there is a value in the discipline of relatively uncongenial subjects. We have to learn sooner or later that we cannot go through life devoting our whole time to enjoyable tasks. Most of us have to spend some part of our time on jobs that we do not find particularly enjoyable, and if we are wise we get them done without making too much fuss. A boy who finds a subject difficult, but who sticks to it and wins through to some measure of success, thereby gains a great deal. It is a difficult and individual problem to decide whether the time has come for a boy to drop a difficult subject and to turn his energies in other directions, or whether he will gain more by sticking to it. If the examining bodies now abolish mathematics as a compulsory subject for the School Certificate, they are not solving this problem; they are passing it on to the schools, and I for one believe that it is only within the school that the problem can be rightly solved.
Our very good pupils present us with a problem of a different kind, the problem of specialisation. I think that a great deal of nonsense is talked on this subject. With some "specialisation" is a term of abuse, and is contrasted with "broad education", which on examination is usually found to be heavily over-weighted on the linguistic side. If specialisation means that each of us should discover his special gifts and then focus his activity in the direction of those gifts, then one might well claim that specialisation was one of the most important functions of education. All the same I think that specialisation in mathematics at the school stage can be excessive, and in any case should be undertaken with caution. As a professional mathematician I feel that I must always be on my guard against the essential limitations of my subject-a mathematician tends to live in a black and white world of propositions, each of which is either true or false. If from time to time he ventures to peep out into the real world he finds it extraordinarily difficult to frame any proposition which is either completely true or entirely false. His subject is essentially an abstraction. He must never forget that it is an abstraction, and how much it leaves out of account. Hence it seems to me essential that a mathematician should cultivate and keep alive some other interest, secondary perhaps to his main interest, but effective for the purpose of keeping him in touch with the real world of people and things.
Side by side with this conviction I hold another as to the extreme value of the time spent by a boy or girl in the sixth form. It is a time of great development and opportunity. So many subjects can be treated in a way that is impossible lower down the school. That these invaluable years should sometimes be spent on mathematics to the exclusion of all else is to me the unforgivable sin of education. I know the pressure of the university scholarship system, but this is not a matter on which the schools should leave the last word to the universities. I know that some schools make a genuine effort to meet the wider needs of their mathematical specialists. I know that many more cover their scheme by a feeble pretence of a weekly essay, or an odd class in German, that fades out rapidly as the scholarship examination approaches. I would far sooner have a boy with a three-subject Higher School Certificate from a school which encourages hobbies and crafts than the typical product of the scholarship class. He has a better foundation on which to take a university course in mathematics. He may know less mathematics, but he has a better furnished mind.
After the manner of those who prefer the minding of other people's business to their own, I have spent most of my time and all your patience in speaking of mathematics in the schools. May I excuse myself by recalling what I said at the beginning of this address as to the close resemblance of our problems. Looking back over what I have written, I see nothing that I have said about school mathematics and school examinations that could not be said of university mathematics and university examinations.
There is one difficulty which I believe is more acutely felt in the universities than in the schools. It is in connection with the selection of the mathematical topics to be included in any particular course. There is a tendency to think of mathematical knowledge as arranged in a sequence. In any undergraduate course you begin at the beginning, and get as far as you can in the time available. In a growing subject any such procedure is bound to lead to an ever-widening gap between undergraduate teaching and current mathematical thought. That gap is much wider than it ought to be. We should be more ready than we are to revise our schedules frequently, to scrap the things whose importance has diminished and to include things which are more significant in relation to the present state of the subject. One has only to glance at the chapter headings in the best school textbooks of to-day and compare them with those of the corresponding books of a generation ago to see that the teachers of mathematics in the schools are alive to the necessity of this kind of adjustment.
Perhaps this problem is felt most acutely in the universities in connection with mathematics as part of a three-subject degree in Arts or Science. Very often the schedules have been conceived as the first part of a full honours course, and consist of the more elementary parts of Algebra, Calculus, Geometry, Statics and Dynamics. The development of sixth form mathematics has robbed this work of a great deal of its interest. For the undergraduate of a generation ago this course had a large content of new and important ideas. Now it would hardly be an exaggeration to say that the student meets no mathematical idea of major importance for the first time in this course. His time is largely spent in working harder problems on his sixth form work. Neither is the content of the mathematical course particularly well adapted to the modern study of the mathematical aspects of science. It ought to be possible to devise a course on mathematics in a three-subject degree which puts less emphasis on manipulation and more on ideas. It is not easy to see how to do this in practice, and as far as I know the problem is unsolved in the English universities.
May I conclude by saying something about the transition from school to university mathematics. I have already indicated that in my view the best preparation for an honours degree course in mathematics is a Higher School Certificate including other subjects. As a general rule students who have this kind of preparation do best at the university. There are, however, some exceptions, and one meets cases in which a student with a good school record in the subject fails conspicuously at the university, while sometimes the reverse holds good. No doubt most of these cases are to be explained in terms of the development of the individual, but I think that there is a difference of emphasis between school mathematics and university mathematics which almost has the effect of making them different subjects. I have already referred to the artistic and the craft elements in the appreciation of mathematics, and said that for good reasons it is to be expected that the craft element should predominate at the school stage. At the university the emphasis shifts and we are more concerned with the logical and philosophical aspects of our subject. We are entitled to hope that the craftsman is an artist and that the artist is a craftsman, and usually it turns out so, but there is a lesser kind of craftsmanship which never develops into artistry. That, I believe, is the secret of most of our disappointments among undergraduates.
I do not think that this difficulty can be resolved by any adjustment of syllabuses. I think it can best be met by advice and guidance in the schools quite outside the framework of examinations. In advising a boy to embark on a mathematics course at a university, the master should try to see beyond the mere cleverness that can carry a boy far in school mathematics. He should try to discover whether he has a real feeling for the subject and the beginnings of that appreciation of that underlying spirit which makes mathematics something greater than all its tricks.
We can face all these problems with a cheerful confidence. There never has been a closer co-operation between school and university or a more lively consideration of their common problems than we have to-day. We may find a reasonable pride in the part that our own Association is playing in this good work.
G B Jeffery.
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