Some of my distinguished predecessors have ably dealt in their Presidential addresses with questions that are most closely related to the aims of our Association, namely, the teaching of Mathematics in Schools. Others have sought to speak on subjects in which they were actively interested. I have followed this precept and accordingly I propose to address you upon a subject closely related to my own scientific interests, namely, the exploration of our universe. Naturally, I cannot claim to speak with authority over its whole range.

Let us begin with a lesson I gave to IC (aged 12) three weeks ago. We are trying to construct a safe method of encoding words. We begin with a simple substitution alphabet in which the 26 letters are paired off with the numbers from 1 to 26.

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 Several major experimental courses have been pioneered and in addition unpublished courses on similar lines are being used in some of the 20,000 high schools and junior schools throughout the nation.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.

The first thing to be said about what is commonly called “modern” mathematics is that it is not really so modern as one might suppose. A great deal of it was in fact created more than fifty years ago. What gives it an air of novelty is that only in the last generation has it had a serious effect on the undergraduate teaching of mathematics in our universities, and only in the last few years has its impact on school mathematics begun to be felt. Some such time-lag is natural and inevitable ; new mathematics is usually created as a research tool, and so long as it is nothing more there is no compelling reason to seek the simplest way of expounding it or to re-examine more elementary mathematics in its light. It is when one starts to teach a new subject that simplification begins, and when this has been accomplished one begins to appreciate the possible advantages of the new point of view at a more elementary level.

Some seven years ago an attempt was made to tell something of the story of Francis Walkingame, the author of The Tutor’s Assistant, which was first issued in 1751 and became the most popular arithmetic text book for over a century afterwards. Recent contributions to the Gazette shows that this author and his famous book are still unknown to many readers. Interest was aroused by Dr. T. M. Flett’s description of Edward Winder’s Exercise Book, which was clearly based on an edition of The Tutor’s Assistant of about 1830. In his description of John Stewart’s Exercise Book, written at Dundee some thirty years earlier, Mr. W More remarked that the rules were the same as Winder used and quoted additionally the Rule of False in very similar words to those used by Walkingame. This example is interesting as showing that Walkingame’s book was used in Scotland at the turn of the century. Richard Hartley’s 1846 letter gives a list of the later sections in Part I (Integers) of Walkingame and shows that he had just started Part II on Vulgar Fractions.

The Editor is under the impression that I am conversant with all the experimental work in this country that is directed towards modernising the syllabus in schools. This is a fallacy. A great deal is going on, but only a small part of the iceberg shows above the surface (an unsuitable analogy, however, for a topic so often associated with hot air!). But the genuine innovator is usually modest and self-critical; when discovered, and the question put: “Why don’t you send an account of this to The Gazette, it is just what is wanted” the invariable reply is : “Oh, this is a very amateurish effort, surely there are lots of others making a much better job of it”. There have always been experimenters, because of the nature of the English temperament and the absence of a central syllabus (though that would probably make little difference) ; and until the last War there was no lack of able mathematicians in the schools, as is shown by the large number of original (and consequently unsuccessful) textbooks published during the past hundred years. Forty years ago Mr. Hope-Jones was teaching probability and pleading for its adoption by all ; it has been available in several examination syllabuses since the War. Others advocated the earlier introduction of trigonometry or vectors, but use of these in a geometry paper could be dangerous ; nowadays an examiner would hesitate to mark-down a candidate who used vectors instead of congruent triangles in solving a problem. It is not uncommon, of course, for the unenterprising teacher to say that if such-and-such a modern topic were in the examination syllabus he would be glad to deal with it, but at present he dare not risk his pupils’ future careers, etc. But a new examination syllabus, though desirable, is neither a necessary nor a sufficient condition for the success of an experiment. There appears to be some acceptance of what might be called the First Law of Motivation, namely, that every boy (and master?) continues in his state of rest, or of uniform motion along the tramline, except insofaras he may be compelled by externally applied force or an external examination to change that state.

Several books on Group Theory show that the isomorphism between the group of proper rotations of a cube and P4, the group of 24 permutations on four arbitrary symbols, can be established by a consideration of the four diagonals of the cube. To each arrangement of these diagonals corresponds a unique position of the cube and therefore a unique rotation from one position as reference to that position. The extension of this result to other polyhedra does not appear in the books that I know.

Teaching a class of 33 girls any subject whatsoever would always be interesting, but when the subject is mathematics and when the exact nature of the course is left to the teacher, things become exciting! This was the case during the last term at a Girls’ Grammar School in Leicester where I was given a free hand to tackle any aspect of mathematics with the form 5Sc. Their average age was 15 (exactly: we had laboriously evaluated it the previous term when considering mean values!) and their ability was very high. Although they were still one year from O-level, I thought this to be an excellent opportunity for trying some “Modern Mathematics.”

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 discussion of the principles of teaching any subject must be a two-sided affair. Inasmuch as such principles involve matters of syllabus content, the professional practitioner of the subject is concerned; but on the other hand the educationalist with his knowledge of such things as the motivation of students, the stimulation of their interest, the ways in which concepts are learnt and understood by them, and so on, must also be concerned.

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.

As Geometry was developed in recent centuries, it became realised that some of the more interesting results were not metrical in nature, and that even some of the metrical theorems were immediately derivable from more general non-metrical ones. For example, mid-point theorems were often derivable from harmonic theory, and confocal conics from general tangential pencils. It seemed inelegant as well as unnecessary to use the ladder of a metric to achieve non-metrical results and then to kick it away as if it had never existed; a far more satisfactory procedure would be to develop pure geometry from pure non-metrical axioms.

A recent conference, held at Southampton early in September, marked the end of the first stage, and of the first twelve months’ work, of this Project. Its objectives are now well-known, namely, the experimental introduction of entirely new mathematical syllabuses in certain schools, together with the setting of appropriate G.C.E. papers, and the writing of text-books and teachers’ guides. But it may be of some interest to fill in some further details of the work.