Revolutionary ideas about the teaching of high school “math” have been fermenting in the United States since the 1950s. Several major experimental courses have been pioneered and in addition unpublished courses on similar lines are being used in some of the 20,000 high schools and junior schools throughout the nation. In examining these ideas the Englishman needs to consider them in the context of American mathematical education, and this in turn needs to be viewed against the background of American education, which is different in detail, method and philosophy from the English system.

It has been suggested in The Listener lately that industrial mathematicians may have something to contribute to mathematical education by describing some of their work in articles to be published in The Mathematical Gazette. As the personal approach is often more graphic and interesting, I have decided to set down an outline of some of my experiences as a member of an industrial research group.

A great deal of attention has recently been given to the problem of revising both school and university syllabuses of pure mathematics. Though applied mathematics has not escaped attention, it has been discussed less intensively. It is therefore of interest that in his inaugural lecture, Professor Bonnor subjected the London University syllabuses in this subject to searching criticism on the ground that it gives a possibly unnecessarily thorough training in classical theoretical physics. He noted that, excepting Imperial College, most of the subject matter contained in the London B.Sc. Special Examination contains ideas which are eighty years old, or older.

A short course on the simple mathematics of satellites and space travel was tried as an experiment with boys who had just completed their “A” level examinations. They were “B” stream boys who had taken the Cambridge G.C.E. in Mathematics as one subject. The contents of the course may be of interest to others with time to occupy after examinations, or as useful illustrations of normal work in both pure and applied mathematics.

When a point source of light is placed low over one side of a gramophone record and the eye is diametrically opposite it at about the same height, then the bright reflections on the record take the form of a rectangular cross, symmetrically placed. The direct image of the source is coincident with the centre of the record.

Figure 1 shows a vertical cross-section of a bell floating freely in water in a circular vessel, both the gas pressure inside the bell and the atmospheric pressure above being p. The problem, which appeared in the Mechanical Sciences Tripos, Part I, 1957, is to find an expression for the distance δy the bell rises when the pressure in the bell is increased δp by the admission of gas through a small pipe.

The object of this article is to draw attention to a point in connection with the above topic which, as far as the memories of the books I read and of those which I have more recently read vicariously or had summarised to me, seems to have been entirely overlooked. This is the existence of what I term the “index interval”, associated with, and clearly discernible in, the differential equation itself, the recognition and use of which both simplifies the presentation and removes the obscurity surrounding points in connection with the intrusion of logarithmic terms into the solution of some equations of this type.

The formation of a Joint Mathematical Council for the United Kingdom has recently been announced, and readers of the Gazette may be interested to know the reasons for setting up such a body, and what it hopes to achieve.