In the preface to the first edition (1897) of his book [1] on the theory of finite groups, William Burnside wrote: ‘The subject is one which has hitherto attracted but little attention in this country; it will afford me much satisfaction if, by means of this book, I shall succeed in arousing interest among English mathematicians in a branch of pure mathematics which becomes the more fascinating the more it is studied.’ He returned to this point in his presidential address delivered to the London Mathematical Society on 12 November 1908.

In 1952 I was invited, with teachers of mathematics from many other schools, to a reception hosted by the Cambridge University Press. The guest of honour was A. W. Siddons, and the purpose was to celebrate the fiftieth anniversary of the initiation of Godfrey and Siddons’ first book: Elementary Geometry. There can be few authors who survive to enjoy such an occasion; fewer still whose book remains in print and continues to sell well after fifty years. But this was a book which marked a turning-point in the teaching of mathematics, and set a standard for mathematical education which remains relevant as we approach the end of the century.

My first contribution to the Mathematical Gazette appeared over half a century ago in October 1943 (why does ‘half a century’ sound more impressive than ‘fifty years’?) so the editor thinks I ought to contribute to the centenary issue. Looking back over the time I have spent studying and teaching mathematics, of all the changes that have taken place the one I most regret is the eclipse of geometry. As readers should know, the Association was originally founded as the ‘Association for the Improvement of Geometrical Teaching’ (in 1871), as in those days the only geometrical textbook in use was Euclid’s Elements. Splendid a job as Euclid did some 2300 years ago in bringing together and systematising the mathematical knowledge of his time (not all geometry, as some of his Books are really disguised algebra), it was never intended for the instruction of the young (although I suppose it was the world’s longest running best-seller!), and was doubtless little understood except by the ablest pupils.

The day I was asked to write this article, I was visited by an old friend. ‘Look at this Geoff’, he cried, showing me a periodical from a sister association in which a university mathematics lecturer told how he had solved a trigonometrical problem which had baffled all his engineering colleagues. ‘It’s the kind of thing we used to do when we were fourteen!’ Actually, this was not strictly true: I did it at fifteen and I was running a year in advance of my age group. It is indeed very easy to throw up one’s hands in despair about the deficiencies of today’s students and teachers. Yet, while there is much to cause concern, it is important that, if we are to take a historical view, as your editor requested, then we should bear many facts in mind other than what was achieved fifty years ago by a restricted number of pupils in a small part of our educational system.

Many questions were raised by the imposition of the National Curriculum; many of them specifically related to mathematics. In particular, the future of curriculum development is called into question and, under that heading, the future of projects. The question may be put bluntly as follows: ‘If there had been a national curriculum or syllabus in the 1950s and 1960s, would it have been possible to set up curriculum development projects, either “officially”, under the Schools Council, or independently, like the Nuffield Project, or the Mathematics in Education and Industry Project (MEI), or the School Mathematics Project (SMP)?’

David Hilbert (1862–1943) was one of the most outstanding mathematicians of the modern era. At the International Congress of Mathematicians in Paris in 1900 he presented twenty-three major research problems which he felt would be important for the development of mathematics in the twentieth century. These problems all seemed very hard, but in bringing them to the attention of other mathematicians Hilbert felt the need to stress that this should not be used as an excuse to put off trying to solve them.