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?" These articles were intended to make the college teachers aware of what graduate work in mathematics entailed. In this setting, the programme was entirely rational. The trouble came when Sputnik made mathematical education a national issue, and the reformers seemed not to realise that any amendment was required if their recommendations were to meet the needs of students of different types and widely different ages. The almost universal acceptance of these recommendations was particularly disastrous for young children, the abstractions of graduate school being the exact opposite of the concrete experiences needed by the young;
The extreme instability of the situation was due to the extreme mathematical ignorance of most teachers. Many felt that something wrong was happening, but they did not have the knowledge or the confidence to stand up to a university lecturer who assured them (correctly) that everything they taught was known befor 1640 AD and (incorrectly) that it was totally out of date.
The "modern mathematics" movement in the USA has run the usual course of a movement that has swung too far to one side - frorn the fashionable craze to a dirty word. It has given way to the cry - "back to basics" - the wish to return to a golden age of mathematics teaching, which never existed.
Certain conclusions are suggested by the American experience.
(a) One maxim current in the USA in 1957 was "if it is good mathematics, it wiII be good pedagogy." The idea was that clear, logical ideas should be easier to understand than vague, illogical ones. Accordingly, as recent abstract matflematics is supremely logical, children should meet its concepts as soon as possible. Whatever may be the value of the maxim itself, this deduction is certainly false. It is very much easier to see some dots on graph paper than to appreciate a subset of a Cartesian product, and so far as I can see, the latter verbiage adds absolutely nothing to our understanding at the level of elementary mathematics. (It is appropriate for graphs involving Hilbert spaces.) The ultra-scholarly presentation is particularly harmful in primary education, because the unfamiliar terminology conflrms the belief, held by many teachers, that mathematics is an impenetrable mystery. The object of early rnathematics teaching is to emphasise simplicity, not erudition. Children should come to associate the symbols of arithmetic with familiar, real situations, about which they can think confidently.
b)The relation of American colleges and universities resembles that of our sixth forms and universities, and at this end of thc school curriculum, modern mathematics is indeed appropriate. Many students experience a shock (sometimes fatal to their mathematical deelopment) when they first experience mathematics as presented by a university lecturer. Sixth form teaching should prepare them for this experience. Students should become aware of the long historical deveiopment of mathematics to its present abstract, axiomatic form. Most classical mathernatics can be understood in relation to the physical world - arithmetic and algebra to counting and measuring, geometry and trigonometry to sizes and shapes, calculus to velocities, areas and volumes. Modern mathematics has been distilled front this by a process akin to the isolation of vitamins from food. By explaining this process, schoolteachers can help students to understand those mathematicians who are totally immersed in the most recent phase of the subject.
(c) The thumbnail history of mathematics given in (b) above suggests a moral in regard to applications. If, historically, the physical universe taught us classical mathematics and so provided a foundation for modern mathematics, in applications the sequence usually is reversed. Some modern theory throws light on a problem; formulated in classical terms, which is applicable to reality. A valid exposition of any recent mathematical topic should try to indicate both the older mathematics from which it arose and the applications to which it can lead. A careful study of this kind should always be made. The fact that a theory is highly prized at the research level is not, in itself, sufficient justification for including it in a general education. If the theory should happen to be very teachable and stimulating at a school level, that would be a valid reason for giving some account of it.
The American movement originated with research mathematicians; its commonest phrase was, "This is not modern." The earlier reform movement in Britain originated in the classroom; its commonest phrase was: The boys do not understand this: (Girls schools played a creditable part in the movement, but the reports tended to visualise the situation in boys boarding schools.)
The situation in Britain in the late nineteenth and early twenieth centuries was very different from that today. In many of the secondary (private) schools, the mathematics teachers had more distinguished university records than many of the university lecturers and professors.*
*H. M. Cundy, in the present century, is the last example I know of this being so.
They wanted to teach what they knew, and so made especial provision for the mathematically gifted to forge ahead to the Iimit of their powers. Their classes contained the same intellectual spread as today's comprehensive school, and they were sufficiently sensitive to the difliculties of the average and slower pupils to seek for new, more informal ways of communicating mathernatics. Informality was certainly needed. In 1860, Euclid, in its original form, was the standard geometry textbook. In 1867, J. M. Wilson, at the suggestion of his headmaster, wrote an alternative treatment of geometry and the book caused a great sensation. However, owing to the iron conservatism of the universities and the civil service examiners, it was not until 1902 - 35 years later! -that such treatment was officially acceptable. In the meanwhile teachers had organised themselves in the AIGT (Association for the Improvement of Geometrical Teaching), renamed the Mathematical Association in 1897. Similar movements were taking place in other countries and influenced all of Europe and overseas countries subject to European influences.
Comparing this revolution with the modern mathematics movement in the USA, we find its main negative
feature in the reactionary obstinacy of the universities here. The two great strengths of the movement were that
it came from the centre, from good mathematicians engaged in classroom teaching, and that it embraced all aspects
of mathematics and its applications. In the USA most mathematicians were totally uninterested in physics and engineering.
On the other hand, at the
The official acceptance of the new approaches led to an outburst of creative activity and the appearance of many excellent textbooks. Sylvanus Thompsons "Calculus Made Easy" (1910) paved the way for the introduction of intuitive calculus early in the secondaxy school: Perhaps because of the 35-year-long preliminary discussion, little or nothing was introduced that later needed to be undone. A delay of 35 years is a high price to pay for such immunity from mistakes. In a time when reforms are mooted, a milder conservatism, to act as a gentler brake on change, is necessary if wild oscillations are to be avoided.
It would be naïve to suppose that, after these changes, mathematics was taught well in alL schools. All one can say is that the lead given by the most experienced teachers in their writing of textbooks was a good one.
The brake of an intelligent scepticism was insufficiently strong in the next wave of change, when the rise of the "new maths", after a brief Darwinian struggle, led to the predominance of the SMP. The director of the SMP expressed alarm in his reports for 1962-63 and 1963-64 (see reference , pages 15-16 and page 55). The SMP had been designed as an experiment, but it. was being widely accepted before its results had been carefully assessed. He condemned the freedom of schools in England to make such changes, and with this view I must disagree. I have worked in countries where there is monolithic central direction, and the evils of such a system are incomparably greater. Teachers lose. their sense of responsibility and initiative for adapting to local conditions, and when a mistake is made it is catastrophic. The disasters of the "modern maths" phase in the USA were in fact produced by directives of the central government.
The SMP books have many virtues. They have inherited from the earlier revolution the tradition of lively teaching, which an American author praised as "the art of teaching mathematics with a light touch." They contain much recreational mathematics, which helps pupils to enjoy the subject. They contain a number of simple yet interesting puzzles, which encourage pupils to attack problems and gain confidence from success. They emphasise understanding. All of these are essential parts of good teaching; their purpose is to provide the psychological foundation for the acquisition of skills, and it here that the SMP's greatest weakness becomes apparent. The deliberate rejection of skill acquisition runs like a thread through their reports: "we have constantly tried to shift the emphasis towards mathematical ideas and away from manipulative techniques" (reference , page 19); "the acquisition of techniques is best left until the post-O-level stage" (page 20); "some teachers have been most agreeably surprised at the changed atmosphere in the classroom as a consequence of the concentration on discovery and ideas rather than technical skill" (page 8); "we saw in the teaching of mathematics a. social function, which should in some measure displace the technical, skill-acquiring emphasis of traditional subjects" (page 217).
The postponing of techniques to the sixth form is a major, psychological blunder. Elementary algebra abounds in small details, of no great intellectual interest, which have to be got right - that 3x2 does not mean (3x)2, that l/(x+y) is not (1/x)+(l/y), and so forth. Children of 9 or 10 years of age are far more willing to cope with such details than adults of 16 or 17, and indeed the teaching of arithmetic can be enlivened by bringing in an element of algebra - the search for unexpected coincidences in arithmetic and the explanation of them.
The absence of even the most elementary skills distorts the whole structure of the SMP works. The explanation of row-by-column multiplication for matrices is essentially a confidence trick. The supermarket explanation could equally well give row-by-row or column-by-column multiplication as in the oldest treatises on determinants. The algebra needed to understand (and indeed to discover) this process is simply the ability to substitute in the equations
x2 = ax1 + by1, y2 = cx1 + dy1
the values of x1 and y1 given by
x1 = 2x0 + 3y0 , y1 = 4x0 + 5y0
and observe the resulting coefficients. (In school work it would be wise to avoid the subscript notation.)
In order to see that areas are changed in the ratio ad-bc when the matrix acts, the SMP expects pupils to guess this result by considering the six special cases
This is both harder and less instructive than finding the area. of the parallelogram to which a unit square is transformed, for which pupils need only to know the area of a rectangle and a right-angled triangie. In the same way, in SMP Book 4, pupils are expected to guess the formula for the matriox inverse to
It is much to solve than the equation
ax + by = u, cx + dy = v
In any case, where is the logic in teaching matricies to boys and girls who are unable to perform the simplest algebraic operations that inevitably arise in any application of matrices?
In many schools, teachers supplement the SMP work by a totally disconnected course in traditional algebra. The greatest harm is done in schools with inexperienced teachers, who do not realise that anything needs to be added to the SMP material. The SMP themselves have of course recognised the need for supplementing their original books.
Sorne schools follow the MEI (Mathematical Education for Industry) syllabus, in which the need for a balance between modern concepts and traditional skills was recognised from the start. .
A missing revolution
A regrettable feature of the past 30 years has been the cornparative neglect of an irnportant experience, described in the Scottish Education Departments Report on Secondary Education in 1947 - surely one of the most outspoken reports ever published. "We are in no doubt that Mathematics in Scottish Schools needs a drastic overhaul. . . . It is the great central mass of boys and girls, ranging from the Cs well up into the B group, who have fared badly, and the dullness and futility of nuch school teaching of the subject has been thrown into relief by the remarkable interest shown and progress made by many of these same pupils in the mathematical work of the Air Training Corps". W. Flemming, who quoted this report in 1955,  went on to remark, "Many teachers were in fact amazed at the keen interest taken in rnathematics by Service personnel during the war. The approach there was usually from the practical problem to the mathematics involved, and the problem secured the interest". I would add that the practical problem, by giving a concrete realisation of the mathematics, also helped students to understand what the mathematics meant.
The Scottish report is particularly significant in that it does not report on some small local experiment but on the actual experience of thousands in the Second World War. There is little doubt that a major breakthrough in the teaching of mathematics to the more physically active and practically minded pupils would result if the schools could achieve a similar embedding of mathematics in reality, by actual tasks that called for correct calculations and design. Undoubtedly there are schools in Britain where pupils design and carry out projects involving mathematics, but (so far as I know) there is no organisation for exchanging information and suggestions for such work.
A movement for relating mathematics to the real interests of children does of course exist in
primary schools and has been described in books such as Mathematics in Primary Schools and films such as I do and I understand. In
competent hands this approach marks an outstanding advance. It can be mishandled when teachers without mathematical
knowledge or insight pick up the phrase that "children should be actively exploring the environment"
and get the impression that alI that is needed for a good mathematics lesson is for children to be running round
in the playground. Actually this approach calls for sequences of activities, systematically designed to form mathematical
concepts and give practice in necessary skills. Ideally, there should o also be unanticipated activities, suggested
by the children
Adaptation to Ability
Half a century ago, the educational system was heavily weighted in favour of the academically gifted and very little imagination was shown in teaching the less academic. Time has taken its revenge, and the present tendency is to condemn as elitist any concern for the academically abler student. This is a mistake in the opposite direction; there should be concern for the full developrnent of indìviduals of every kind. This does not mean that all pupils should be doing the same thing. Men may be created equal but they are not created congruent. The problem would probably solve itself if schools offered a free choice between mathematics in a practical setting, as recommended by the Scots in 1947, and a theoretical course ìn which pupils were encouraged to read ahead at their own rate. Mathematicians, like musicians, are notoriously precocious and self-propelled. From the age of 9 or 10 years on, children showing mathematical talent should be encouraged to read an work for themselves, beginning with algebra and geometry, and probably covering trigonometry and some calculus by the age of 14, as many children, who were privileged to have this opportunity, did in the past. A list of recommended reading should be prepared; many of the better older books would be suitable for this purpose. Coordinate geometry should be begun as early as possible. It gives excellent practice in the algebraic statement of a problem, it provides a good foundation for matrices and it has important applications, for instance in multivariate analysis and in stability of oscillations.
Those who desire educational changes often confine their activity to some form of political pressure. But it is posible to influence education without waiting for the cumbrous process of national readjustment. For many years now I have been running voluntary rnathematics clubs, with memberships anywhere in the rangefrom half-a-dozen to 300, and I am at present doing such work in accordance with the philosophy described in the previous. paragraph. The effects of such clubs, in preserving and stimulating interest ìn mathematics and in demonstrating the levels at which boys and girls are able to work, are out of all proportion to the time required for the actual meetings.
This article first appeared in The Bulletin of the Institute of Mathematics and its Applications October 1978, 14, pages 259-262 published by The Institute of Mathematics and its Applications Telephone 01702 354020
We are grateful for the permission of the Institute for permission to reproduce the above article.
Copyright © W. W. Sawyer & Mark Alder 2000
Version: 22nd March 2001