The connection between thought and its expression in symbolism is perhaps more important in mathematics than in any other subject, for the complexity of mathematical thinking soon makes reference to symbolism an indispensable method of learning and discovery This is true both in the historical development of the science and in the experience of the individual student. Without visible symbolism to act as a scaffolding, the sheer complexity of the subject matter results very soon in an incoherence and breakdown of the thinking process. While this is true to some extent of all subjects of any high degree of complexity, in mathematics a new and extremely interesting element enters into the situation.

My function in this discussion is the humble one of attempting to supply a rough scaffolding into which the durable parts of the edifice will be fitted by the other three speakers.

The teaching of algebra has never been dominated by a single work in the way in which Euclid for so long a time dominated the teaching of geometry For that reason, perhaps, the revolts in geometry have corresponded to quiet reforms in algebra, and the problems involved, though equally important, have never received the same amount of attention. On the teaching of the subject in schools, excellent books exist; there is Sir Percy Nunn’s classic, and more recently, a book by Durell, and much relevant matter in Godfrey and Siddons, Teaching of Elementary Mathematics.

The method of introducing logarithms given in the Algebra Report consists in taking the powers of 1·1, which are easily calculated and are so closely packed that common logarithms can be deduced by proportional parts correct to three decimal places. It forms an interesting set of lessons for post-certificate pupils to discuss why this is so, and to extend the method with a view to obtaining greater accuracy Furthermore, one of the most important principles in pure mathematics is amply illustrated—that of sandwiching a function between two bounds. It is assumed that the expansion of loge(l + x) is known by the pupils and also its differential coefficient.

1. Professor C. E. Weatherburn’s paper in the Gazette, 1936, p. 320, gives an interesting proof by vector-analysis of the condition for a twisted curve C to have as binormals the principal normals of another curve C1.

It may be of some interest to give a few bibliographical notes on the subject and to mention new or not generally known properties of certain of these pairs of curves.

The County Cricket Championship table that appeared on 16th June disclosed a paradoxical state of affairs, which, when examined closely, is shown to depend upon the neglect of an important statistical principle used in the calculation of death-rates, comparative wage-rates, index numbers, and other data not usually associated with our national game. Sussex and Yorkshire had each completed seven matches, and each had won six and lost one. As for the unfinished matches, Sussex had four times led on the first innings and once been behind, while Yorkshire had been behind on the first innings in the only two unfinished matches they played. Thus on the complete matches the two counties were exactly equal, while on the unfinished ones Sussex had done better.

My part in this symposium is to deal with the introduction of the fundamental ideas of algebra, as far as the stage usually known as the “Certificate stage”. As I shall attempt to show later, this is an extremely vague phrase, but it will be sufficient for the present to note that I am speaking of that part of algebra which is a compulsory subject for the great majority of secondary school pupils and that it is no part of my plan to deal with the needs of universities or any specialised developments of the subject. These needs, in my opinion, should not be allowed to influence unduly teaching at the “compulsory” stage.

[At the evening meeting of the Annual Meeting of the Mathematical Association on 5th January, 1937, Mr. A. W. Siddons took the chair, and in calling upon Professor Wolff for his paper said: Professor Georg Wolff came to England in 1913, and I believe I am right in saying that Harrow was the first school to which he came and that the last educational institute he was at in England was this Institute of Education in Southampton Row, where he talked with Sir Percy Nunn. When he was in England on that occasion Professor Wolff spent much of his time learning about the teaching of mathematics in this country. He has more recently been lecturing at Columbia University, New York.]