CATERING FOR THE EXTREMES.
In Mathematics in School for May 1994, Canon Eperson, who taught Alan Turing at Sherborne School, refuted an accusation that Turing's time at Sherborne had been harmful to the budding genius. The article makes it quite clear that Turing had in fact received a mathematical education ideal for a future leading mathematician.
At one end of the academic ability scale, it is widely recognized that some special treatment is required for pupils whose interests and abilities are completely non-academic. (1.) At the other end of the scale, there is much less recognition of the disastrous frustration that can result if thought is not given to the needs of those with academic talents well above the average.
The article about Turing reminded me of the mathematical education that some of us were fortunate enough to receive in the 1920s. It was based on a policy that had been thoroughly tested and had proved highly satisfactory. It seems to have been recognized and used only in a very restricted sector - the mathematically strongest public schools.
There is no reason why this policy should not be used much more widely. It is extremely tantalizing that this possibility seems to be almost completely ignored so that to-day hundreds of potential mathematicians are denied the opportunity to develop fully. I am convinced that comprehensive schools could be quite as successful in producing first-rate mathematicians as the best public schools were in the past.
It may be objected that the resources available to schools are far below what they should be and that teachers already have too much to do. This is true. However, as will appear later in this article, the point of the approach in question is that teacher involvement is minimal.
A comparison of mathematics with music may be helpful. It is well known that Mozart showed remarkable musical ability at the age of four and in fact embarked on a career of performing in public at the age of six. It would clearly have been both cruel and stupid if some bureaucratic government had insisted that he should go to school at the age of five and do five-finger exercises along with the rest of the class. No syllabus can possibly meet the needs of such a pupil; what was needed -- and what, fortunately, happened in fact - was to liberate him altogether from every kind of syllabus.
Few people recognize that mathematicians can be equally precocious. The infant Gauss had somehow taught himself arithmetic, and before his third birthday pointed out an error in his father's bookkeeping. Emmy Noether, the first woman to be recognized as a mathematician of the highest rank, when she was four years old, pointed to a chair and said, "There are 4 rows of 12 brass pins on the back of this chair, so there must be 48 altogether." These of course are extreme examples, but we should realize that we may encounter young mathematicians anywhere in the range from genius, to gifted, to talented, to simply above average, and that we should try to cater for their special needs.
There is of course one great difference between the 1920s and to-day. Then a graduate, wanting a career involving mathematics, had to decide whether to become an actuary or a teacher. A school might well have a staff containing several eminent mathematicians. We shall have to analyze what contribution these teachers made. It lay mainly in choosing the policy to be followed, hardly at all in the actual teaching, as will appear in the cases now to be examined.
We now turn to the approach that worked so well with Turing and other mathematicians, and to the justification of my earlier remark that it did not impose extra duties on teachers. Canon Eperson, the author of the article on Turing, quotes his report on Turing's work in 1928;-
Further on, Eperson writes
J.E. Littlewood was one of the outstanding mathematicians of the present century. In his book, A Mathematician's Miscellany, there is a chapter, A Mathematical Education, describing his years at St. Paul's School, when he was in S.Macaulay's class for Mathematics. (Macaulay became an F.R.S. in 1928.) Littlewood writes
He had read nearly all of this by the time he was seventeen-and-a-half. He gives some statistics showing the overwhelming success of pupils raised under this regime.
I have first-hand experience of what growing up in such a system meant. I won a scholarship to Highgate and started there in January 1925. In many ways it was a traditional school for the upper classes; however it had one very striking feature, extreme academic flexibility. It was possible to go to the headmaster, Dr. J.A.H. Johnston, and explain that you wanted your studies to develop in a particular direction, and he would organize some arrangement to make this possible. There were 10 mathematics sets. The set you went into depended solely on your mathematical knowledge. (This suited me down to the ground. I was keen on mathematics, science, English and Latin, and had an appalling record in French, history and geography.) A chunk of mathematics that could be covered in one term was assigned to each set. If you mastered it, you would be promoted to the set above. Thus in one year you could win three promotions. Anyone with a strong inclination towards mathematics could finish the work of the top set by the age of 15, and then go into the Headmaster's Specials. This usually consisted of about 6 boys of varying ages (in a school of 600). One of these boys would just have won a scholarship to Oxford or Cambridge; another would be preparing for a scholarship examination. There would be one or two 15-year-olds who had just come in to the Specials, and others who had been there for a year or two. We were left to get on with reading mathematical textbooks appropriate to the stage we were at. The headmaster was usually busy reorganizing something. He would drop in occasionally to see how things were going. I still have a copy of the big, old-fashioned Differential Calculus by J.Edwards, in which he had put ticks against the examples he thought it most important that I should do. Most of the books we read had answers in the back. I never remember handing in any work for correction. The only examinations we had were external. At the age of about 16 we took London Matriculation. We used to put on bowler hats and proceed to the People's Palace in Mile End Road where the examination was held. Later we took Higher School Certificate. The books I read had much in common with those listed by Littlewood, though I am sure I did not read such an incredibly large amount. However my reading did include some substantial items, such as more than half of Jeans' The Mathematical Theory of Electricity and Magnetism. What can be said, in all three cases, is that a significant fraction of a university honours mathematics course had been mastered by the time we left secondary school. There is so much mathematics to be learnt, and in a life there is so little time to do it, that it is a tragic waste if the highly important early years are not used to the full.
The contribution of the teachers.
We now return to a question touched on earlier - what service did the teachers of the 1920s do for their pupils, and how can their contribution be replaced in the different circumstances of to-day?
(1.) Their greatest service was being responsible for letting the young athematicians take control of their own lives and forge ahead at their own rate. It is very hard for someone who has not had the experience of independent mathematical development to realize both the desirability and the tremendous possibilities of such
(2.) They indicated the books that might profitably be read.
(3.) Like Eperson, they "stood by" to help if needed. Littlewood writes
What is the situation in these regards now?
(1.)Readiness to let the gifted forge ahead depends on the vision of principals and heads of departments.
(2.) Any school requiring advice as to suitable books for private reading could easily obtain this from the Mathematical Association or from some other interested organization. Naturally, there arises the question of how to obtain the books. Many schools already have inadequate resources for getting even the regular textbooks. The number of books required for independent reading would be relatively small, as perhaps only one percent of the pupils would be involved in such work. Public libraries could give some help. I believe too that if it became known that a serious effort was being made to educate the next generation's mathematicians, there would be many mathematicians and some organizations eager to lend or donate books.
3.) This activity is less vital. It is clear from the records that pupils did not require assistance in understanding and learning the principles and processes of mathematics, but only in solving some exceptionally difficult problem. It would not be fatal if such help were entirely lacking. We all meet from time to time some particular problem we cannot solve, and we deal with it much as a mediaeval army dealt with an impregnable castle. We go round it and on, no doubt with the hope that it may yield at some time in the future. However it may be possible to locate some source of occasional assistance, perhaps a student at university who plans to become a teacher, or a retired mathematician in the locality.
In any scheme for allowing individual progress the following conditions need to be satisfied; -
1.) It must not involve any extra work for teachers.
2.) It must not in any way interfere with or disrupt the work of other pupils.
3.) The work on advanced topics must not lead to forgetting of the mathematics in the regular syllabus, with the danger even of failure in the usual examinations.
We have already seen that condition (1) is automatically satisfied. As to (2), we should offer the scheme only to a pupil with a temperament that allows him or her to work quietly in a library or at the back of the classroom without attracting attention. In regard to (3), it should be stipulated that the pupil, from time to time, takes a test on the material of the usual curriculum, and is allowed to continue reading independently only if a mark of 95 % or more is obtained. If these conditions are observed, there should be no fear of the new departure being wrecked by some disaster.
The Scale of the Problem.
I have been among those trying to lessen the gap in mathematical education. Since 1977 interested secondary school pupils have been coming to our home on Saturday mornings. They usually start at the age of 14 and finish when they leave the Sixth Form. The work has frequently been well up to university level, and some of the students have already begun distinguished careers. I am however under no illusion that such individual efforts are in any way of adequate magnitude. This is apparent in two ways. On the one hand, the number one can help by such efforts is a minute fraction of the number who could benefit from it. On the other hand, with the morning sessions at intervals of a week, it is much harder for the students to recall the details of work done some time previously. What is required for mastery is that work should be done day in, day out, and learning reinforced by working many examples. Only in this way can the material become really ingrained in the mind. No one can offer this continuity of work except the schools.
(1.)For ways of meeting the needs of the least academic pupils see my article Mathematics, Emotions and Things. Mathematics Teaching. March 1993. pp. 6-19.
We are grateful for the permission of the Association to reproduce the above article.
© W. W. Sawyer & Mark Alder 2000
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