G H Bryan's Mathematical Association Address
George Hartley Bryan was President of the Mathematical Association 1907-1909. He began the tradition of formal presidential address in 1908 with The uses of mathematics and the training of the mathematical teacher and in 1909 gave the Address of the Retiring President which had no other title. It was published in The Mathematical Gazette 5 (78) (1909), 44-51. We give a version of Bryan's address below.
The Address of the Retiring President
The Report of the Council bears abundant testimony to the ability and energy which that body has displayed in increasing the usefulness of the Association, in particular in bringing it into closer touch with outside interests and other bodies. If this work is continued in the future as it has been of late, there is every prospect that in the course of time our Association will rise to the status and influence of the American Mathematical Society or the Deutsche Mathematiker Vereinigung, and this result will well repay the hours of work and discussion that are undertaken by the body of councillors meeting periodically at King's College.
In this address it is proposed to deal with some of the more serious consequences of England's neglect of Mathematics with especial application to the problem of teaching.
Before doing so it appears desirable to refer to several current topics, and I wish at the outset to say something about our North Wales Local Branch, in the hopes that others will "please copy."
The success of this branch is due to our local secretary, Mr T G Creak, an old Cambridge wrangler and retired schoolmaster. We have over a dozen members and associates, including teachers in secondary boys' and girls' schools at Beaumaris, Bethesda, Colwyn Bay and elsewhere, and an elementary teacher from our immediate neighbourhood who has greatly contributed to the value and interest of our discussions by representing a phase of teaching with which it is often extremely difficult to get into proper touch. We have held meetings up to now three times in the year, and on such occasions these hard-worked schoolmasters give up their Saturday half-holiday and come over by train or boat or on foot to the meeting place, generally Bangor. As a rule, one member is asked to open a discussion on some subject, such as teaching of algebra or the extent to which text-books should be used in teaching. Then we sit round a table and talk. Every meeting impresses me more and more with the originality, ability and general interest shown in their work by those teachers who take part in these discussions. The opening papers have been excellent. In the discussions that have followed the noticeable feature has been that every one has much to say, and everything said has been well worth saying. For example, the conversation turns on the subject of how best to teach a beginner that the product of two negative quantities is positive. Then one of our members puts forward a method he has thought of identical in principle with that used in interpreting complex quantities, and the idea is at once taken up by the other members. One point which these discussions have brought into prominence is the fact that our schools have to prepare for examinations conducted by at least four independent bodies, each under separate control, and each appointing outside examiners who are as a rule not in touch with our system of education. The teachers themselves are willing enough to adopt more practical and intelligent methods of teaching; indeed, they have gone very far in this direction, but the examiners will insist on the old-fangled, useless drudgery, and the efficiency of the classes suffers in consequence. One master was taken to task by his inspector for bothering his boys with simplifying complicated and meaningless algebraic expressions, whereupon he took down a book of recent examination papers and showed the inspector a question of the kind in every one of them.
I give this account mainly as an example of what a local branch can do. It is remarkable what a lot of ideas have been brought up in our informal discussions. At the end of every meeting we have held over abundant material for the next meeting. Surely local branches in other parts of Great Britain would serve an equally useful purpose. There must be people elsewhere who could devote the amount of time necessary for the duties of local secretary, even if they are not quite so free in this respect as our present secretary.
The recent meeting at the Regent Street Polytechnic, at which we had the pleasure of listening to Prof Perry, affords another illustration of the activity of our Association. On reading over the reports of the proceedings the one point about which I am in doubt, despite my attempt to save this particular situation, is how far the members of our Association fully realise the serious danger which is imminent at the present time, namely, the extinction of the English mathematical specialist. We were told that mathematicians must take off their coats and work. I would say that English mathematicians must take off their coats and fight for their lives. At present no one in England has a good word to say for the mathematical specialist. In other countries he is exercising an ever growing influence on the destinies of the nation. We were told at that meeting that to put a mathematical specialist on to teach elementary pupils is like putting a steam hammer on to crack a nut. But the nation which tries to do without the mathematical specialist is like the foundry where they try to drive rivets into boiler plates with a pair of nutcrackers. We still have plenty of steam hammers and riveting machines, which have to content themselves with cracking nuts, as the supply of steam is not sufficient for anything more. Meanwhile, if things go on as they are going, we shall have to buy all our boilers from the German makers. Our present Chancellor of the Exchequer has learnt that if you go over a works in Germany they will probably put a university professor on to show you round.
I have gone very carefully into the question as to how far a mathematician who has had no practical experience can be of use in solving the problems presented by the applications of science to modern industry. The opinion I have formed there is that there is plenty of work to keep mathematicians continually employed in their studies without their having a minute to spare to use their hands except for the purpose of writing. Unfortunately, however, this work doesn't pay. I do not mean that the mathematician cannot make a fortune by doing the work. What I mean is that he will lose the small amount of compensation that he would obtain by giving the same number of hours to elementary teaching, and he will wear himself out far quicker than he would do if he were to forget his mathematics and to level himself down to the standard of a common or garden all round teacher. A man with a sufficiently large unearned income saved by his parents could greatly advance matters, but it is the will of our nation to place heavy obstacles in the way of the man who attempts to live on such an unearned income, even if he gives more than its equivalent back in unpaid work.
The story has been repeated ad nauseam of the mathematician who made a discovery, and who stated that its great charm was that it could never be of any use to anyone. There are two sides to that story, and the commonly accepted version is the wrong one. The mathematician probably knew that he would get the kicks in any case. What pleased him was the hope that the man who gave him them would not pocket the halfpence. It is quite possible, moreover, that if the mathematician lived in the present day he would find his discovery put to practical use.
Any mathematician who will try and get into contact with the English practical man will discover widespread ignorance of the most elementary ideas. Not very long ago I found in the columns of the English Mechanic the old exploded fallacy about the oar being a lever of the second class, and numerical results given which were based on this fallacy. It never occurred to these good people that had their calculations been correct, the simplest way of propelling a boat would be to do away with oars altogether and to sit in the seat and tug away at the rowlocks. A man had a gas engine and a well, and he wanted to pump water out of the well with his gas engine. He consulted a "practical" engineer, and was told this was impossible. So he himself had to get the belting and pulleys necessary to gear the engine down. He invited the engineer to see the machine actually at work. I believe the engineer still retained the same scepticism that the old Pisa professors had after they had seen Galileo's stones dropped from the Leaning Tower.
English firms who have engines ordered abroad for special gradients send ordinary engines instead, simply for want of a little elementary mathematical knowledge.
I believe that the hard things which are said about mathematicians in England are due to the fact that people in general cannot distinguish - it is impossible that they should distinguish - between a wrangler and a mathematician. By a wrangler I mean not only a Cambridge wrangler, but any man who undergoes a rigid course of mental training to enable him to pass a difficult examination in mathematics. One peculiarity of this examination is that the syllabus goes a long way past the dividing point at which the mathematician and the physicist branch off in different directions. Another peculiarity is that it covers a number of subjects each of which, to be properly mastered, would take the whole number of hours that the candidate usually can devote to the entire syllabus. A third peculiarity is that greater importance is attached to the candidates' powers of devising artificial dodges for the solution of tricky problems than to his knowledge of first principles and their direct applications. The examinations set for the Indian Civil Service illustrate these peculiarities in a considerable degree. I believe that if the candidates, instead of having to face these advanced papers, were required to show a thorough knowledge of one such subject as Theory of Groups, Higher Analysis or Theory of Statistics, and Probability, they would be much better fitted for the Indian Civil Service. In the last examination only six out of the thirty-five top candidates took Advanced Mathematics, while twenty-four took Greek History.
Personally, I think that properties of circles, triangles, and trilinear co-ordinates are carried too far, both in our Gazette and elsewhere. They do not produce the kind of student who can solve at a glance a very simple problem on the catenary or cycloid, such as every practical man should be able to solve, without using pages of algebraic formulae.
The dynamics of the typical wranglership examination can best be described as "Dogmatics." I have myself to teach students this subject for examinations of the wranglership type. The typical wrangler believes or tries to believe that all bodies are perfectly smooth, and he has to prove, or attempt to prove, results quite different from anything which he will ever see in a physical laboratory or elsewhere.
Thus is produced a man who is called a specialist and condemned as incompetent to teach. He is a wrangler, but not a specialist.
On the other hand, a man who has taken a general practical course in mathematics and science becomes none the less a good all-round man, none the less competent teacher if his ideas have been broadened and extended by reading such a book as Hilton's Groups, or Carslaw's or Bromwich's Series, or one of the other modern treatises published at Cambridge or elsewhere. I maintain that the successful teacher requires his ideas to be enlarged in some direction beyond the syllabus of the typical B.Sc. examination, if he is not to run in blinkers all his life. If this be done, we shall train up a body of teachers each of whom is a specialist in his own little way, and some of them will in course of time do their quotum of research, but these teachers will in no case have ruined their careers by making themselves unpractical or unsympathetic with the work they have to do to earn their daily bread. They will in fact be much more like the teachers who are turned out in every civilised country except Great Britain, and who certainly succeed in producing intelligent citizens.
The specialist teaching elementary classes has been compared to the racehorse drawing a cart. If this analogy is correct, it is nothing less than sinful to bribe the horse, by offers of fodder in the form of scholarships, to train as a racehorse, if the supply of fodder is to be cut off as soon as the race has been run. The alternative I propose is to give the horse a bit of a run over some rough or newly explored piece of ground. When he has been put into the shafts, I maintain that he will be much better able than before to pull his cart over any unexpected obstacles.
It is high time that we should insist on our mathematical teachers possessing at least the knowledge of their subject necessary to read one of the advanced books referred to. A student of mine who had been teaching mathematics for some time in a school told me he was going over to Germany, and asked my advice. I advised him to follow a course of university lectures there, but he said he would not be able to understand them. Such a man is not fit to teach. Again, I constantly hear of students who have neglected their mathematics in order to take honours in Chemistry, Latin, or Botany, and who then try to teach mathematics in the schools. Then the schools wonder at their large number of failures.
The number of members of the Council and Committees of our Association who have taken Part II of the Mathematical Tripos, bears abundant testimony to the valuable work which specialists are able and willing to do when they turn their thoughts to questions connected with elementary teaching, It would be a strong argument in defence of the specialist, if statistics were drawn up showing the proportions of Part II. men serving on these committees. [Since writing this I have noticed that three out of the six delegates appointed by our Association to confer with the delegates of the Public Schools Science Masters are men who to my personal knowledge have distinguished themselves.]
I have dwelt on this subject, as I believe that the extermination of the specialist is one of the greatest calamities that can befall our country. Even an invasion by Germany might improve matters in this one particular respect, if in nothing else.
Coming next to the training of the ordinary citizen who goes to school - whether to the elementary or to a secondary school - and who does not intend to go to the university, nor to take up physics or engineering, what mathematics ought to be taught to him? I think we are mostly now agreed that he ought to learn to calculate and measure properly. It must not be forgotten, however, that constructive geometry can be made as dull and uninteresting as anything else. If you put a boy down to verify a lot of geometrical propositions by drawing figures with a hard pencil, where he can see by his own reasoning that the results must be true, you will be doing that boy more harm than good.
Another important feature of a boy's education is that he should be able to properly estimate the chances of an uncertain event. Everything human is uncertain. The boy who decides to embark on a certain career simply because some successful man in that profession happens to be earning £1000 a year is a gambler, pure and simple. He only ceases to be a gambler when he takes the element of chance into account, and estimates his probability of overcoming the odds that are against, him by his individual exertions. I am glad to see that a chapter on probability has been introduced into a recent arithmetic book. The subject can be taught experimentally as well as theoretically. It exemplifies cases in which fractions are more useful than decimals, and it should therefore be welcomed by some of our members who have strong views on this point.
I am afraid that many of the students who are now training for elementary teachers will not only never join our local branches, but they will fail to teach our working classes how to measure and calculate properly. Till this year I have had to teach such students. They sometimes had to leave their class to pass an examination in music, and we were told that a certain Mus. Doc. had described their performance as the best he had ever heard. But as for bringing the least bit of common sense to bear on a practical question, that was in many cases quite out of the question. There were notable exceptions, but their general idea was that the whole object of the mathematics classes was to enable them to answer stereotyped questions in a stereotyped examination paper.
A cook once asked us the meaning of "dactyl" and "spondee," and said that her little sister at the Board School had been asked to write a sentence containing one of those words.
The want of practical elementary experience in calculating and measuring is everywhere apparent. The tailor who cuts one's coat according to his cloth fails to make that coat fit simply because he has never performed laboratory experiments on the stretching of cloth in different directions when weights are suspended from it. He ought also to study geometrical models of the various surfaces that can be formed by the deformation of textile fabrics. I believe that in Germany such things are taught. But England is not Germany.
In conclusion, the mathematician of the future, if he is to save himself from destruction, must be a fighting man. He must fight against the mass of superstition and prejudice that has grown up against him. He will have no time to tinker with syllabuses or to squabble about petty matters of detail with his brother mathematicians. He must convince the British public that he is a power in the country, that he is anxious to use his power for the benefit of the country, and that the country cannot get on without him.
["The Parting of the Ways." - The following remarks were thought of after the address was communicated; they seem however to deal with an important point which has not been previously touched. The task which the joint Committee of our Association and the Science Masters have undertaken will necessarily involve, at some stage or other, the classification of mathematical and physical subjects into three more or less defined groups; the first containing such subjects as ought to be taught to all would-be students whether of mathematics or of science, the second including subjects required for the study of mathematics only, and the third those required exclusively for the study of experimental sciences. This classification however involves a system of watertight compartments, and is bound to be a failure if rigidly insisted on. There are very few branches of mathematics which have not some application, and that usually a very important one, to science; very few branches of physics, a knowledge of which is not of great use in illustration of some mathematical theory. One teacher will want one thing put in the first group, another will want another, and in the end the curriculum instead of being simplified will be made heavier and more indigestible than before. The only remedy is to give greater scope for individuality on the part of teachers, so that one teacher may exploit a part of the subject which he considers important, while another is laying stress on another part. Instead of a hundred schools each taking its pupils over the same ground, we want them, as soon as the necessary minimum of common knowledge has been covered, to branch off on as different lines as possible, so that if a group of their pupils be selected at random after they have left school, they will be found in the aggregate to possess a wide knowledge of mathematics or physics, and not all to possess a common knowledge of an infinitesimal fraction of these subjects. No education be efficient which does not turn out all sorts and conditions of men.]
Last Updated January 2021