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.

As recently as ten years ago, professional mathematicians did most of their work at a desk, using paper and pencil. Today’s mathematician still sits at a desk, but facing a cathode-ray screen. The paper and pencil are still there, but a lot of the mathematician’s activities now involve use of the computer. In particular, powerful computer packages like Mathematica and Maple advertise themselves as systems for ‘doing mathematics on a computer’. This rapid transformation has changed the nature of doing mathematics in a fundamental way. The computer does not simply ‘assist’ the mathematician in doing business as usual; rather, it changes the nature of what is done. In particular, the logical structure of mathematical reasoning carried out with the aid of an interactive computer system is different from the structure of the more traditional form of mathematical reasoning. This article is a logician’s view of the ‘new maths’ of the computer age.

To most of the outside world, mathematics appears to be a closed book: everything known, no loose ends, no mysteries. From within the subject, it is hard to understand how such an impression can have arisen, for in almost every direction today’s mathematician sees unsolved problems and wonderful new ideas. There seems to be little danger that mathematics will either grind to a halt in the face of insuperable obstacles, or become defunct when the Final Theorem is proved and nothing remains to be discovered.

In school we learn classical mathematics, the mathematics known before the year 1900. It has a direct relation to the actual world. Arithmetic is related to counting and measurement, algebra to general truths about calculations, geometry and trigonometry to the sizes and shapes of objects, calculus to the velocities of moving objects and a host of other applications. Students have no difficulty in knowing what it is about and what it is for. Students who go on to modern mathematics, the mathematics of the present century, either in university or by private study, find it considerably different. There is no immediate contact with reality, and they may find it hard to visualise what is being discussed.

Nowadays many children meet the idea of a locus at an early age. This is because of the great (and deserved) success of LOGO as an educational tool to introduce geometrical ideas to young children. The challenge is to devise lists of instructions to make a ‘turtle’ move in a certain way. Originally the turtle would have been an actual mechanical object moving on the floor. It is now more usually an image on a computer screen. The mechanical turtle held a pen which marked out its path; the computer version* whether it looks like a cartoon turtle or an arrowhead (or even remains ‘hidden’) leaves as its trail a series of dots showing where it has been. The end product drawn on the floor or the screen is thus seen as the path of a point moving according to a prescribed set of rules. In this way children are encouraged to think of any geometrical diagram—whether it is a square or a hexagon, a rectangle or a house, a star or a circle—as a locus. Geometry is built up from collections of points.

The development of algorithms has been one of the principal concerns of mathematics throughout its history, but now that machines can perform algorithms more efficiently than people our attitude to algorithms has to adjust. It is hardly possible for a school to implement the National Curriculum in mathematics without having the computer language LOGO among its resources, but the curriculum's prescriptions are minimal, and it is quite possible to introduce the opening steps of the language and then do no more. What further use can be made of the language? In this article I show how LOGO can provide a systematic approach to some traditional problems of combinatorics.

Doublet tasks consist of changing one word to another by altering single letters at each step to make a different word. The two words at the beginning and end of such a chain must, of course, be the same length, and they should be related to each other in some obvious way. They must not have identical letters in the same positions. All words in the chain should be common English words, proper names excluded. A perfect solution has a number of steps equal to the number of places the given words differ. For example: COLD, CURD, CARD, WARD, WARM. If a perfect change is not possible, the best solution is the shortest chain. For playing doublets as a game with two or more players, Lewis Carroll invented a set of scoring rules to determine who wins.

Over 60 years ago an article with the above heading appeared in the Mathematical Gazette, shortly after the celebration in 1932 of the centenary of the birth of Charles Lutwidge Dodgson, better known and admired universally as the author of the Alice books, Lewis Carroll.

At that time a literary critic expressed his opinion that ‘only a mathematician could have written the Alice books’, perhaps because they contain several references to arithmetic. In Wonderland, Alice met the irascible Ugly Duchess. ‘If everybody minded their own business’, the Duchess said in a hoarse growl, ‘the world would go round a great deal faster than it does’.

Mathematicians of every age have been seen to criticize the proofs of their predecessors or of their contemporaries as ‘not being rigorous’; and often those which they proposed as replacements for the defective proofs were in their turn considered inadequate by the succeeding generation. This apparent repeated putting into question of mathematicians' arguments at length led some people to think these arguments contained a hidden flaw which meant that a proof could never be found unassailable by any criticism.

Since there is far more mathematics even quite elementary mathematics than anyone can carry in their working memory, it must follow that results which are unused will be forgotten, and are then often rediscovered after many years. A tour of some old haunts which are now being opened up again would therefore seem not out of place in a centenary number of the Gazette. Tourists, however, will need some preliminary briefing if they are to enjoy the tour fully.