The aim of this paper is to illustrate developments in the teaching of mathematics of its day; emphasising, when reasonable, lines of succession. The concentration is on English teaching over the past 350 years although this is developed from its European context. The paper concerns itself with teachers rather than teaching and is intended as a modest reminder of our lineage. Much is owed to Professor Taylor’s two books on the Mathematical Practitioners, and not a little to my own Gazettes.

In any consideration of the Open University it is as well to recall the words of Lord Crowther its Chancellor. He said “We are open first as to people…. We took it as axiomatic… that no formal academic qualifications would be required for registration as a student. Anyone could try his or her hand and only failure to progress adequately would be a bar to continuation of studies…. We are open as to methods…. Every new form of human communication will be examined to see how it can be used to raise and broaden the level of human understanding…. We are open finally to ideas.”

Despite its title, this article is not intended as a clarion call against miserliness, but rather as a plea for a greater use of non-parametric methods in the teaching of statistics to sixth-forms.

In a well-known theory of infinite series, if a series converges it is taken to represent the number to which it converges, while if a series does not converge it is not thought to represent any number. Hidden behind this well-known theory, there are some intriguing examples of the representation of numbers by series which puzzled mathematicians of the 18 c. and 19 c. When taken individually the examples might be dismissed as being distinctly odd, yet because of the substantial degree of consistency which exists between them they have been studied extensively. Early disputes about the examples centred on the question of whether there is any sense in which the series could be regarded as representing numbers. Yet those who accepted such a series and used it in their work tended to ask the simpler question, what number does this series represent?

A practical method is described which uses the result

a series which we shall denote by S8 . The approach follows the same general lines as that in a previous note (The Mathematical Gazette, Vol. LIV, October 1970, p. 261) though at a more sophisticated level. We carry out the process first and discuss error analysis, as well as the series itself, afterwards.

Two years ago I was in Paris for the New Year. The shops were gay with decorations, and polyhedra were obviously in fashion. I was particularly impressed with a lampshade which was a feature of many of the decorations; it was in the form of an icosidodecahedron, the pentagonal faces being dimpled, and with “petals” added, so that the whole resembled the flower of a Golden Chalice (Solandra grandifloral). Eventually I found a little shop by Notre Dame which would sell me one; I found myself the purchaser of a small box containing twenty strips of card, about 35 cm by 7 cm, which I was assured would make up into the lampshade I required.

Undergraduate classes of physics and applied mathematics, in their studies of introductory relativity theory, are often presented with a special-relativistic calculation of the perihelion advance of planetary orbits. The result comes to be one-sixth of the gen-eral-relativistic value and serves as a demonstration of the inability of special relativity to account for the residual effect observed in the case of Mercury. As well, the calculation provides a rather pleasant example (perhaps more from the teacher’s viewpoint than the student’s) of two-dimensional relativistic particle dynamics.