It is a commonplace that pictures leave a deeper impress on the human mind than abstractions and, in spite of its abstract nature, that is also true of many part? of mathematics. A proof of a theorem or the construction of a counter-example is often suggested by pictorial considerations, and that is true of algebra and analysis as well as of obviously geometrical subjects like topology. (For entertaining examples see Professor J. E. Littlewood’s book [1].) There is obviously a connection between a developed geometrical intuition of that kind and the kind of spatial experience which ought to be furnished by a school mathematics course. No one would deny its importance and the writers of textbooks on ‘modern’ mathematics are well aware of the need to stimulate geometrical intuition. What has been called in question is the relation between that kind of geometrical intuition and the formal treatment of the euclidean plane based, albeit insecurely, on Euclid’s Elements. Indeed, the traditional treatment has been abandoned with indecent haste and without serious question. In Part I of this paper we make a plea for a re-consideration of the case for some kind of formal geometry as an indispensable part of the education of the more mathematically able children.

At the end of the nineteenth century David Hilbert made the decisive step in the foundations of geometry (for the historical development see [1]): a road was opened to a theory of geometric structures. In spite of the fact that one spoke (and speaks still today!) strangely enough of ‘implicitly’ defining the geometrical concepts through the axioms, it became clear in the sequel that the axiom system of Hilbert constitutes an ‘explicit’ definition: not of course for ‘point’, ‘line’, etc. but of the concept ‘euclidean geometry’ (or ‘euclidean space’).

The number π signifies the ratio of circumference to diameter of a circle, and the definition depends upon the fact that the value of this ratio is independent of the size of the circle. We can determine a similar ratio for any shape by replacing circumference by perimeter and diameter by width.

The definition of a group normally used in the classroom is that of a set of a finite or an infinite number of elements, for which an operation is defined, subject to the four conditions of closure, associativity, identity and inverse. The group is usually specified by giving its group table. However, this is not very practical for a finite group of large order and is never possible for an infinite group. Also, the group table often contains redundant information and so is not very efficient.

Whatever may be the future of Colleges of Education after the re-organisation indicated in the White Paper Education: a framework for expansion, the great majority of teachers will continue to be trained in these colleges. In nursery, infant and junior schools most teachers are women, and the rate of turnover due to marriage is very high. In recent years, in order to staff the schools, allowing for this ‘wastage’, nearly one-third of the teaching profession has had to be in training at any given time, and rather more than two-thirds of these have been women. Clearly there are not enough young people leaving school with good academic qualifications to meet the needs of the profession, even if they wished to teach. In general, the more academically able choose not to teach, but rather think in terms of a career which is more remunerative and a Higher Education which is allegedly more ‘respectable’.