The Mathematical Association’s News-letter No. 11, February 1967, contained this paragraph:

In all the ferment of ideas for syllabus reform that have come to the notice of the main body of mathematics teachers in this country over the last ten years or so, there is the least agreement on those concerned with the teaching of geometry. Largely through the efforts of the Mathematical Association in its earlier years, the cry “Euclid must go” is an empty phrase in Britain today; for forty years or more we have managed with a pseudo-Euclidean compromise, but now many teachers, dissatisfied with this, are experimenting with fresh approaches. The purpose of this article is to state some of the objectives of geometry teaching, and to examine some of the possible treatments in the light of these objectives. It is confined to a consideration of the CSE/GCE O Level course, both for those whose formal mathematical education ends there and for those who will go on to further study.

There is a natural argument about teaching topology in schools which goes something like this. Here is a modern subject that is flourishing at the research level, it is truly geometrical in flavour (more so than geometry itself we gather) and its elementary notions are more primitive in the learning of the young child (according to Piaget)f than the ideas of triangle, rectangle or circle. Moreover, we understand that, as well as having an intrinsic interest and beauty in itself, this subject has elegant applications in other branches of mathematics, such as the calculus, which we think of as part of the traditional syllabus in schools. Surely, then, there is the strongest possible case for bringing this topic to the attention of the children while their geometrical intuition is still uninhibited by perfectly straight lines and perfectly round circles and to give them the advantage of the more intuitive proofs of the theorems in calculus which topology can furnish.

The exponential law of increase has been described by Professor D. J. de Solla Price as “the fundamental law of any analysis of science”, and considered to hold over long periods of time [1]. The same writer suggested that low-grade work doubled in about ten years, but that the period taken was about twice as long for very high quality activity. It is useful to remember that a doubling every fifteen years corresponds to a multiplication by about a million in three centuries, or an annual increase of nearly five per cent.

In Spring 1963 the first National Mathematical Contest here was organized by Mr. F. R. Watson (now of the Dept. of Education, Keele University). The papers were those of the Mathematical Society of America and Society of Actuaries (past papers are published in the two “Contest Problem Books” (Random House)). About 1000 people, aged 14-18, took part. There were no prizes, only the interest, for teacher and pupil, of submitting to an American test in which even those of modest talent can put up a satisfactory performance.

In the first war, not only did computers not exist, but the labour of calculating trajectories by a step by step process was regarded as quite prohibitive, and approximate formulae were the only resource. In the end these proved surprisingly accurate. The methods inevitably worked on approximations valid in the first instance for small values of a parameter, ø, say. But it was found possible to find formulae correct to the fourth or fifth order of ø, and as a result there emerged the principle that “In Ballistics 25° can be a ‘small’ angle”; results for 25° could be correct to 0·1%.

F. Bowman has found the following pretty result.

The minimum value of the definite integral

for real a is given by I = 1/(n + 1).

The study of convex polyhedra from a mathematical point of view began in ancient Greece well over two thousand years ago. It is believed that Euclid’s “Elements” was written to inspire a wider appreciation of the properties of the five regular polyhedra or “Platonic solids”, as well as to place their study on a secure mathematical foundation. The discovery of convex polytopes (the analogous figures in four or more dimensions) and their investigation by the Swiss mathematician Ludwig Schläfli just over a century ago, led to a renewed interest in polyhedra, and since that time hundreds of papers on this subject have been published. However, there are still many unsolved problems, and the purpose of this article is to state a few of these. These problems have widespread appeal because they can be readily understood with very little previous mathematical knowledge, and although they may be quite difficult, there is always the possibility that someone with a very minimum of mathematical background will devise a method of approach that leads to a solution.