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.

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.

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.

To design a sixth form mathematics course suitable for grammar schools one must keep in mind the nature of secondary school education. In earlier days, grammar schools gave a specialized academic training to a small number of boys, whereas nowadays, with a much larger population, such schools are expected to train boys as useful citizens in a more complex social and economic world, giving them a breadth of knowledge and a depth of understanding.

Schools differ very widely, even if we limit ourselves to that section of them dealing with pupils from 12 to 16 years of age which we commonly call “main” The pupils in the schools differ even more, and it is difficult to make generalizations about them, but nevertheless some guiding principles may be laid down in considering the kind of mathematics to which they are suited.

Recent changes in the content of some Secondary School courses in mathematics have been brought about chiefly to bridge a discontinuity which is often found between the mathematics presented to undergraduates and the examination requirements for sixth forms. It is mathematicians who have been the leading advocates of change at this stage. There has been no such pressure from Secondary Schools to Primary, though the discussions on new Secondary programmes have roused a considerable interest among Primary teachers and have already begun to influence the changes which are taking place in so many Primary Schools and at such great speed.

Our major Colleges of Technology, and in particular, the Colleges of Advanced Technology, have had long and successful experience in preparing students for the General Honours and Special Honours degrees of London University. In some Colleges these are still the principal Mathematics courses which they offer, but in others significant developments have occurred. To put these in their proper setting, it is necessary to review briefly some of the events which have occurred during the past 20 years.

The role of Training Colleges and University Departments of Education in the training of mathematics teachers makes their attitude towards modern mathematics of more than parochial interest. The working conditions of these two types of institution, however, are so very different that it is impossible to discuss them as a single unit. This article has therefore been prepared by collaboration, with two main sections. In spite of these differences of structure, however, the general attitudes of mathematics staff in the two types of institution have so much in common that it has been possible to write a concluding section expressing these views. It may be that this final section will be the most characteristic part of this contribution to the symposium.