Oscillation in Systems of Mathematical Education
W. W. SAWYER, FIMA
In an ideal world, mathematics teaching would be under continual revíew. The arguments for necessary changes would be carefully examined and there would be a slow but steady evolution of content and teaching method. In reality it rarely seems to work like that. Usually, a genuine problem leads to some reasonable, new idea. This idea is then liable to be carried to unreasonable extrernes, witb neglect of equally valid counterbalancing ideas. The present article sketches this process at certain epochs in Russia, the United States and Britain.
In Tsarist Russia only one child in 20 went to school. After the revolution of 1917 the question arose (as ìt has elsewhere in a milder form) - if you intend to educate the entire population, what changes are required? Official articles stressed that education should not consist of children just rnemorising wbat they were told; it should be a creative activity of the children themselves. This would only happen if the children were interested in their work. The existing textbooks were not suitable for creating such interest. Anyone looking at them would be led to believe that "mathematics was the most unecessary subject in the world, dealing only with idle and einpty riddles". 
It was therefore necessary to have books that would relate mathematics to real life. So far the argument is perfectly sound. But things got out of hand. In 1923 it was decided to abolish mathematics as a subject. The whole school programme was reorganised around such themes as Man and Nature, Work and Society. Mathematics was to arise naturally in the study of these themes. I did not work out too welI; Pythagoras Theorem was embedded in a section dea1ing with the Constitution of the Soviet Union, while fractional and negative indices were under Imperialism and the Struggle of the Working Class.
Children brought up under this scheme did not do well at university. Another group was available for comparison. Schools had been set up for adults who had never attended secondary school and wished to enter university. ln these adult classes, mathematics was still taught systematically as a subject, and the aduit students. clearly found this a great advantage when they went to university. 
In 1931 it was decided to return to a strict timetable based on academic subjects. A writer in 1933 maintained thoat the cause of the whole trouble was that they had been too much influenced by the practices of the capitalist countries. It is interesting that, at about the same time, writers in the USA attributed various aberrations in their chools to the infiltration of communist philosophies.
After 1931, the Russians experimented for a time with new arithmetical textbooks, but in 1938 decided to use a revised version of an arithmetic published in 1884.
After this period of osciilation, things settled downi in Russia, and they seem to have established an organisation in which the syllabus evolves in an orderly rnanner, with consultation between schools and universities. In recent times, the Russians have tried to bring in some modern mathematics without sacrificing the traditional skills. The main difficulty appears to be fitting this into the time available for teaching.
The United States
My interest in the topic of this article arose from my experiences in the USA. I had been invited there on the basis of my reputation as an innovator, but found myself in the rather uncomfortable position of tryirig to minimise the damage the reformers were doing.
The. situation certainly called for change. From the 1920 on, professors of education had waged a war on academic sterility. There is such an evil as "academic sterility", and if the educationists had been concerned to find more intelligible and more interesting ways of communicating knowledge, this would have been an excellent thing. However, the movement worked out rather as a determination to do without knowledge. Teacher training consisted largely of very vague and general educational theory. An American teacher, wbo had experienced this, told me that the first course was largely vacuous and the later courses repeated the first course. As a result, teachers did not even understand the elementary mathematics they were teaching. They were largely dependent on the textbooks and the very detailed teaching guides issued by the publishers. Inevitably, teaching by rote without understanding was widespread.
The reformers pointed out, correctly enough, that the teachers' knowledge of mathematics was inadequate (the teachers themselves would have welcomed a more substantial training). However, the críticism took the rather peculiar form of emphasising that teachers were unaware of twentieth century mathematics; they were urged to make themselves familiar with, and to teach, such things as the union and intersection of sets, equivalence. classes, Cartesian products and axiom systems.
The disparity between the disease and the diagnosis puzzled me very much; I was confronted by a movement in which I could discern no coherent philosophy. Eventually I found some facts which, I believe, provide the true explanation of this phenomenon. The American system, it must be remembered, operates at four distinct levels - elementary, secondary, college and university. In 1957, the elementary syllabus involved 8 years of arithmetic; secondary (14-18 years of age) included, at most, algebra, geometry and trigonometry; the colleges provided four years of undergraduate work, while graduate work and research was the business of the uníversities. The "modern mathematics" movement seems to have begun as a complaint by the graduate schools that the colleges were too much concerned with eighteenth and nineteenth century mathematics, and did not adequately prepare those students who were going on to do research work in mathematics for work at university level. As topology, measure theorgy and other subjects developed largely in this century are closely linked to set theory, the emphasis in the modern rnathematics package is readíly understood. In the 1940s, research mathematicians wrote a series of articles in The Mathematical Monthly (the magazine of the college teachers) with titles such as "What is Analysis in the Large" and "What is the Ergodic Theorem?