We are living in an experimental age in the teaching of Mathematics, and new topics and methods by which to introduce them to our pupils exercise the minds of the majority of teachers. Opinions and preferences are as numerous as the alternatives. The following article has been submitted to the Modern Trends Subcommittee and is offered as one way of introducing linear programming and matrices.

Ladies and gentlemen—It has become customary for the President of the Association, when a university teacher, to speak of some aspect of mathematics in which he or she is an expert. When the Council of the Association did me the very great honour of inviting me to preside over this conference, the first, I believe, ever to be held in Oxford or Cambridge, I decided to break away from tradition. This decision was strengthened when I knew that I was to have the privilege and responsibility of addressing you in the Shel-donian, which is furnished with no apparent visual aids.

For some years courses of instruction on the programming of digital computers have been offered at Letchworth College of Technology. The courses of interest to schools and school teachers fall into two categories:

• (a) courses of instruction specifically designed for sixth-formers and

• (b) courses designed for school teachers.

Tonight I propose to say something about a man who is now recognised as a pioneer mathematician of his time, but I do not intend to talk about his work as a creative mathematician. I shall leave that to others who are far better qualified to do so. As a young man I studied pure mathematics under Young and G. H. Hardy, but most of my work as a professional mathematician has been in mathematical physics, and I cannot say anything significant about subjects that I ceased to work on some thirty-five years ago.

It is proper and desirable, at most of our meetings, for speakers to talk on mathematical topics to which they may themselves have made significant contributions. In general at such meetings we shall all understand the broad outline, but the details are likely to be obscure to most of us. The presidential address, I have always thought, is in rather a different category. The speaker should ensure that his audience, though originally lacking deep knowledge of the chosen topic, should go away with the impression that they not only understand every word but could also reproduce the arguments, say to a class of sixth-form mathematicians. Since our association is mainly concerned with the refreshment of school mathematics and mathematicians, I hope to talk about a part of numerical mathematics which could be taught in school, which uses “standard” mathematics, and which springs from standard mathematics. I start with a rather lengthy introduction on the general principles of error analysis.

When, as a newcomer to the profession, I started to teach elementary Mechanics, I taught it more or less as I had been taught myself. It was a classroom subject, one which drew on the experience of the pupil certainly, but one in which actual experiment was never attempted. The ideas seemed to me to be fundamentally simple, and to be so interlocked and so few in number that it ought to be possible to communicate them to a class in a very short time. I need hardly say that I was rapidly disillusioned. I began to be faced with “Why?” questions that I could not answer convincingly, and with questions like “What is the difference between momentum and kinetic energy?” and “Is it sine or cosine?” which revealed a complete failure to understand. I found boys dividing their answers by thirty-two, because they were confused by my somewhat haphazard use of gravitational and absolute units. One boy made it plain to me that he believed that my Mechanics had no practical bearing on the world in which he lived. Much later another boy said that Mechanics to the mathematician was substituting in the right formula. I realised that some fundamental re-thinking was called for.

There has been a great interest in the possibility of teaching ‘modern mathematics’ in schools ever since the Southampton Conference in 1961 when Professor Bryan Thwaites and his team put forward their plan for modernising school mathematics, especially for the period spanning the four years before ‘0’ level. Their reason for the change was that, since during that period the child’s mind is most intellectually alive, a lot of dull, heavy manipulative work would do far more harm than good, whereas they felt that ‘modern mathematics’ would be more intellectually stimulating and enjoyable. Since then, several Examination Boards have produced alternative syllabuses for ‘0’ level, containing set theory, group theory, etc. More and more schools have been doing the two-year course leading up to this examination, and the first papers were set last year.

Today, school mathematics syllabuses all over the world are being revised: the teaching world is in a state of ferment. From Illinois to Cracow, from Wellington (N.Z.) to Wellington, Shropshire, there are more than whispers of change. The suggestions for the actual changes vary tremendously: in Cracow, it is for the introduction of lattice structures and axiomatics; in the USA it is logic and truth tables (or was that last year’s model?). In Britain, some introduce groups, while others believe fervently in motion geometry; some ph their faith on vectors, while others tackle the problems of the negative integers. All are less than satisfied with the present syllabuses; but how many are concerned about how it is taught? In SME we are trying to see both sides of the coin and as the work develops it is becoming clear that we need to think carefully about both.

From any given polyhedron another can be derived by reciprocating with respect to some sphere. The most rewarding cases arise when the sphere is either the inscribed or the circumscribed sphere of the first polyhedron. In this way every Archimedean polyhedron gives rise to a new polyhedron, all of whose faces are congruent. Taking the simplest Archimedean solid, the truncated regular tetrahedron, we may derive a solid which has been called* both the “triakis tetrahedron” and the “tristetrahedron”. This solid has twelve isosceles triangles as faces, which meet by threes and sixes in a total of twelve vertices. It enjoys a certain importance in crystallography; for example, the mineral eulytine occurs in crystals which are basically triakis tetrahedra. We shall here consider some of the solids formed by stellating the triakis tetrahedron.

If equilibrium is about to be broken by rotation about the center of the disc, i.e. slipping around both pegs, the forces are as shown.

Statistics is not a strange subject on its own, but should come in naturally in planning and discussing experiments in Physics. Boys must be trained not to expect to get the answer but to get an answer with a certain degree of accuracy. I was lucky to find a group of 15 year olds who didn’t ‘know’ the formula for the period of a simple pendulum. Experiments on this lead to the revision of many of the topics in my statistics programme.

Text-book miscalculations are regrettable, but they do no more harm than causing the student or the teacher to spend some extra time checking his answer. Errors in principle or in argument on the other hand, whether they be on the part of an author or a teacher, are more serious, especially when they are as plausible as the one which it is the purpose of this article to expose.