The Euclidean Spook


PRACTICALLY no important English institutions have been designed for the purposes they serve. They were intended for the quite different needs of an earlier age, and have assumed their present form, which may well be rather different from the original one, after a long process of minor changes. Our educational system is not exceptional in this respect; this is particularly noticeable in the secondary schools. Their course of instruction is not guided by any clear conception of what our society requires for the education of its citizens, but has developed gradually in a rather unsystematic manner. The geometry that is taught there is a good illustration of this, and the purpose of this article is to examine briefly how the present state of geometrical teaching came about, to what extent it justifies the claims made on its behalf, and how far it fits in with the needs of a twentieth-century democracy.

We must remember that the inspiration of our secondary schools lay in the classical studies of our old Public Schools and Grammar Schools, and it was therefore natural for the Elements of Euclid to form the foundation of geometrical instruction. Yet although later developments have tended to oust the classics from their dominating position, their effect on mathematical teaching has not been so marked. It is true that the modern geometrical text-book is different from those current at the beginning of the century. Some proofs have been altered to conform more with modern ideas of accuracy. The "proofs" of some theorems whose basis was merely traditional (e.g. that the sum of two adjacent angles is two right angles) are not always given, and the order of presentation has frequently been changed. But for all these superficial alterations the purpose of a school geometry course is unchanged. It is simply to give a systematic development of the proofs of the traditional theorems, and practically nothing is taught except with that end in view.

No one can deny the historical and mathematical interest and importance of the Greek geometry. But that is no guarantee of its suitability, even in its present amended form, for the average secondary pupil. We must remember that Greek mathematics was very different from our own. The modern child can do easily arithmetical problems which would have taxed the powers of the best Greek mathematicians. Algebra was not known in the world of the classics, and this tends to be reflected in their geometry (cf. the set of theorems on areas of rectangles which really have no place at a late stage of the geometrical course, but which should be thrown in as illustrations at a very early stage in the algebra). To the Greeks, geometry was the only subject in which reason and logic were supreme. To-day this is not the case. For instance, the logic of algebra is readily understood by children who find the logic of geometry incomprehensible, whilst the spread of the teaching of science should make it possible (even if it has not done so yet) for reasoning powers to be developed in connection with other subjects. But in spite of this, geometry is still justified as the way of teaching people to think. It is true that people with some mathematical abilities can be taught this way, though that does not mean it is the only method. But the majority of children do not learn to appreciate the notions of theoretical geometry, just because they have never been given any understanding of what reasoning means. How can they be expected to reason logically about dry matters like points, lines and angles when they have never learnt to reason even about things in which they are interested? It is like expecting children to be able to write without teaching them to read. It is really almost incredible that it should be assumed that although children have to be taught reading, writing and practically everything else, they nevertheless learn to think, which is much harder, simply by the light of nature and the examples put before them in a subject whose aims and methods most of them never properly understand.

Of course I do not propose that an attempt should be made to remedy this by teaching children formal logic or philosophy, but a great deal could be done in an informal way. Much of what is required would fit naturally into the subject known as English grammar, and ought to be taught long before any attempt is made to prove geometrical theorems. (Incidentally, one of the first things that the young school-teacher finds out is that a large proportion of his mathematical periods is spent in teaching English.) Naturally this will require new ideas of what to include in grammar lessons, but it should be possible to liven up the subject. The introduction of puzzles of a logical character, and of simple detective stories in which the class should be asked to work out, not to guess, a solution, are two possible innovations, which would not only provide a training of real value, but should commend themselves readily both to teachers and pupils. There is also a rather neglected branch of school mathematics, which can be useful in this respect, and that is co-ordinate geometry, which is usually called "Graphs" and regarded as algebra. You can teach children a lot about using their heads by discussing the deductions which can be legitimately made from a graph. This is an interesting subject, and by a suitable choice of the material presented, it should be possible to impart a great deal of useful knowledge in a palatable way, while the ability to appreciate the meaning of a graph, or any other form of statistical conjuring trick, is an essential part of the modern citizen's education.

The belief in Euclid which most educationists seem to hold is very easy to explain. Those on whom the treatment has been successful naturally endorse it, while those on whom it has failed find themselves in the same position as the courtiers confronted with the Emperor's famous new clothes. Only when the imposture is exposed can we hope to produce a school geometry in accordance with the needs of the day, for the theoretical development of Euclidean geometry takes all the available time in the normal school life. The first point we must emphasise, however, is that many of the results that are now taught are too valuable to omit, and they must be acquired, at least as experimental truths. But we must also provide the further geometrical training that contemporary society demands, and as soon as we reject the traditional justification of Euclid, it becomes reasonable to give this training priority over the theoretical geometry now taught. The war has exposed the limitations of traditional mathematical teaching, as is evidenced by the introduction of the A.T.C. in the secondary schools. I have not the time or the space to go into details about what is required. But there are one or two obvious suggestions. People need spatial perceptions, and one of the main purposes of geometrical teaching should be to develop them, and they obviously require a knowledge of solid geometry. Nor should geometry be divorced from other subjects. For instance, geography includes a great deal of geometrical matter and, in particular, map reading is a subject which could well be taught, while woodwork (or needlework) is a subject with an obvious geometrical content. It is not possible to say here how all this should be built up into a comprehensive teaching programme. It would certainly need detailed thought and careful planning. But the planning must not be, as previously, based on the idea of modifying the existing situation, but should form an independent whole based on the needs of the present and the future, rather than on the traditions of the past.

Eureka, 5.

Reproduced from Eureka 27 pages 18-21.
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