of full-time study to the age of fourteen and the subsequent four years of part-time instruction are considered as a single whole.

Up to the age of eleven or twelve there need be no distinction between the curriculum for boys and for girls a simple course of "nature study" based, wherever possible, upon open-air observations and the cultivation of the school garden, is best for both sexes. It should reach some such standard as is here assumed as the starting-point for the four-years' course in a Secondary School.1 A friendly understanding between the Elementary Schools and the Secondary Schools to which their brighter pupils are transferred as "junior scholars" would secure all necessary continuity in the instruction without necessarily fettering the individuality of the teachers in either type of institution. A similar understanding between the teachers in the Elementary and Continuation Schools is still more desirable. It should, indeed, be regarded as essential to a well-organized system; for, without it, the present obstacles to efficient science-teaching for the bulk of the population will still remain. Good working arrangements between the two types of institution being postulated, it should be possible to devise properly graduated schemes of work which, while taking account of the widely differing circumstances and always limited opportunities of the pupils and the varying interests of the teachers, would yet supply to each age the kind of teaching in science that best meets its special needs.2

1 See Hodson, Broad Lines in Science Teaching (Christophers, 1909), Ch. III., for a syllabus of this type drawn up by Miss C. von Wyss. 2 The very important Report of the Prime Minister's Committee on the position of Natural Science in the Educational System of Great Britain (1918 [Cd. 9011]) has appeared while this book was in the press. The Committee's findings seem to be in substantial agreement with the argument of this chapter.

CHAPTER VI

MATHEMATICS

BY JAMES STRACHAN, M.A., B.Sc.

FROM the time of Plato to the present day the place of Mathematics in a liberal education has never seriously been disputed. The teacher of Mathematics has not been obliged, like his classical colleague, to defend his subject from external attack; nor has he, like the teacher of Geography, for instance, had a struggle to rescue it from undeserved neglect. The well-known existence of numberless applications of Mathematics to commercial and industrial affairs, increasing with and largely contributing to the progress of civilization, carries sufficient conviction of the importance of the subject to the man of the world; while the dictum of Bacon that Mathematics " make men subtile " sums up a widely accepted view that satisfies those who are more concerned for the mental and moral than for the material results of education.

It is, then, little matter for surprise that the movement for the reform of mathematical teaching, which took shape towards the end of the nineteenth century, should have been mainly directed from within. The teacher gifted with some power of self-criticism could not rest satisfied with the easy acceptance of his practice by an indulgent, because unenlightened, public opinion. He was bound to ask himself whether there was any

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evidence amongst the pupils that had passed through his hands of that subtlety which the study of Mathematics was alleged to produce, and on a fair review of his experiences as a teacher he was compelled to admit that, whereas stupidity and fatheadedness as the outcome of his ministrations were of constant occurrence, the production of subtlety was phenomenally rare. Mathematics as a cure for a "wandering wit" might be all very well for a man of the intellectual capacity of Bacon, but somehow it seemed to fail in its effect when applied to the ordinary schoolboy or girl. Nor when he looked at his subject from the utilitarian standpoint was he any happier in an attempt to justify his existence. He could not honestly say that to 90 per cent. of his pupils, apart from arithmetic, the stuff he was teaching was ever likely to be of material service-an impression which conference with mathematicians engaged in the practical applications of the subject only served to intensify. Whether or not this view was unduly pessimistic, it was abundantly clear that such results as were produced were in no way commensurate with the labour involved in producing them.

There were thus two respects in which the results of mathematical teaching fell short of expectation. In spite of that, there could be no doubt that the subject lent itself admirably to treatment by the most approved methods of class-management. Every step taken, every demonstration given by the teacher, was followed by copious examples worked by the pupil. There ought to be progress made; for it was impossible of belief that the mass of mankind and womankind should

be such hopeless blockheads as to be unable to progress in a study of such vital importance to the human race. "The study," said the schoolmaster, "is dull."

"True," said the stern educationist, "but that is no disadvantage. All the better discipline." But here the schoolmaster joined issue with the educationist. He felt instinctively it ought not to be dull. It might be hard, but give it interest and he would undertake that, however hard the road, he could carry his pupils along with him. What is taught in a school has no right to be dull. The question, he began to see more clearly, was not whether the subject was of such a nature that it might be expected to produce some specified mental effect, or whether if studied now it might be of material service in after life, but whether, as now presented, it possessed sufficient inherent interest to make its study possible at all; and if not-for as regards the mass of pupils the answer was obviously in the negative-if not, whence could the necessary interest be obtained, and how could it be introduced?

The experiments in mathematical teaching of the last twenty years may as a whole be regarded as an effort to find a solution to this problem. As happens in all exploration, the pioneers have often missed the way, misled by false impressions or held back by preconceived ideas. The danger of confounding interest with amusement is always present, if not with the pioneers themselves, at all events with their less clear-thinking followers. At various times the reform movement has been brought into temporary disrepute by an excessive devotion to so-called practical work or by alliance with the extreme heuristic school. In no subject have the fetters of tradition been more firmly riveted on the teaching than in Mathematics, and, even with full allowance for the salutary effect of a wise conservatism upon any movement of reform, there can be little doubt that the reform of mathematical teaching has

been unduly delayed by hesitation in breaking with the past; so that much good new wine has been spilt by the attempt to preserve it in old bottles.

The reform movement began with Geometry. The Association for the Improvement of Geometrical Teaching, which later developed into the Mathematical Association, was founded as far back as 1871, and has carried on propaganda work continuously from that time to this. In those days Geometrical Teaching in schools was dominated by Euclid; in fact, so far as the vast majority of pupils was concerned, it was nothing else. The virtues of Euclid are so well known and acknowledged as hardly to need recapitulation. As an exercise in pure deductive logic, no text-book in the world has ever approached it. The intellectual value of its requirements of absolute precision of language and authority for every statement made can hardly be over-estimated. The value, however, is contingent firstly on the pupil's making adequate progress in the subject, and secondly on his transferring the habits acquired in geometrical reasoning to his practice in everyday life. That the former contingency was not in general realized, was shown by the large proportion of pupils who failed to display any independence in the exercise of Euclidian reasoning, and took refuge in memorizing the steps, and at times even the very words and letters, of Euclid's propositions. The other contingency, that of transference of acquired habits to other fields of activity, is a phenomenon as to the existence of which psychologists are not yet agreed.

On the other hand, regarded as a text-book for the modern schoolboy or girl, Euclid's Elements left much to be desired. Assuming a pupil to have successfully struggled through the whole text of Euclid as now