59^. 



NA TURE 



[October lo, iqoi 



As instances of differences of opinion as to what should be 

 inchuled in a subject and how the subject should be taught, the 

 discussions on the teaching of elementary mathematics and of 

 botany may be cited. In each case a whole morning was 

 devoted to the expression of expert opinion and the statement of 

 experience in relation to the subject under discussion. For the 

 discussion of the former subject, a joint meeting was arranged 

 with the mathematical depirtment of Seciion A, and for the 

 latter a joint meeting was held with the Section of Botany. 



The Teaching of Mathematics. 

 In urging a reform of mathematical teaching, Prof. Perry 

 remarked that he would teach mvthematics — at all events ad- 

 vanced mathematics — in different ways to different students. 

 In any case he thought the system of teaching boys elementary 

 mathematics as if they were all going to be pure mathematicians 

 must be altered. We taught all boys what is called mathemati- 

 cal philosophy that we might catch in our net the one demigod, 

 the pure mathematician, and we did our best to ruin all the 

 others. In his experience there was scarcely any man who might 

 not become an advancer of knowledge, and the earlier the age 

 at which you gave him the chances of exercising his individuality 

 the better. Educate through the experience already possessed 

 by a boy : look at things from his point of view — that is, lead 

 him to educate himself Throtigh his whole mathematical course 

 let him be taught through his own experiments, and do not call 

 it waste of time to plot the stream lines, for example, after the 

 algeb'aic academic answer of a problem has been arrived at. 

 The unpractical nature of mathematical teaching, he held, caused 

 men to leave common sense out of their teaching, and he in- 

 stanced the great continental Polytechnics, where an elaborate 

 course of many months, or a year, was often devoted to a 

 subject, of which the general principles could be grasped in a 

 practical course of a few weeks. 



■■\.ll advocates of orthodox methods seemed willing to sacrifice 

 every form of usefulness of mathematics to the mind-training 

 inherent in a perfect logical system — a huge complex deduced 

 logically from simple fundamental truths. Where would be 

 the harm in letting a boy accept the trutlf of many propositions 

 of the first four books of Euclid, partly by faith, partly by trial ; 

 of giving him the whole fifth book by simple algebra ; and in 

 letting him assume the sixth book as axiomatic ? He would 

 allow him, in fact, to begin his severer studies v/here he was 

 now in the habit of leaving off; and would let him put aside 

 nuich more than is usually done, so that he would get quickly 

 to the solution of partial differential equations and other useful 

 parts of mathematics. He had been speaking of the training 

 of the mathematician, and he might be wrong ; but as to the 

 educational training of the man who was to use his mathe- 

 matics in the study of pure and applied physical science, he 

 had no doubt whatever of the importance of skipping judiciously 

 in all early mathematic work. In these days all men ought to 

 study natural science, and in such study they required to have 

 the knowledge of algebraic formul.v and the power to use them ; 

 to be familiar with the use of logarithms in computation ; with 

 the use of squared paper, and with the methods of the calculus. 

 He held that dexterity in these is learned by quite young boys, 

 and he felt sure that such dexterity could not hinder, and could 

 only further, the mathematical study of the e.xceptionally clever 

 student. 



Mathematics was a powerful weapon to unlock the mysteries 

 of Nature. If a man knew how to use the method, that 

 would be enough ; he could leave to others, who delight in 

 that, the forging and complete study of the weapon. The 

 average young engineer might be made to possess a power 

 of using the methods of mathematics, which would be as 

 easy to him as reading or writing or using any hand tool — a 

 power which would never grow rusty, because it would be exer- 

 cised every day of his life ; and his present hatred of mathe- 

 matics and theory of engineering was leading to disaster. 

 Higher mathematics had become a very useful thing. As in 

 the case of all other generally useful things, the complete study 

 of its philosophy in the orthodox manner was not a necessary 

 part of the school or college curriculum. In concluding his 

 remarks. Prof. Perry defended himself against the charge which 

 his engineering friends had brought against him, that he had an 

 exaggerated notion of the importance to all men of possessing a 

 love for mathematics. 



The discussion upon the paper was commenced by Prof. 

 Hudson. He said that a too common fault in teaching mathe- 



NO. 1667, VOL. 64] 



malics consisted in allowing the pupil to learn by heart pro- 

 positions, formulae and rules, instead of using them as a means 

 of training the reisoning powers. He trusted that Prof. Perry 

 did not really wish to recommend that method, but he was 

 afraid that its advocates might quote Prof. Perry in their 

 support. Elementary teaching should be so conducted as to 

 prepare for more advanced teaching ; nothing should have to 

 be unlearnt. Geometry should be based on the observation 

 and handling of models of solid figures, and thus could be 

 begun at a much earlier age than was generally supposed. 

 Prof. Forsyth criticised the vehemence of the attack which 

 Prof. Perry had made upon the mathematician while sympa- 

 thising lo a considerable extent with his aims. He pointed out 

 that subjects do not necessarily progress on the lines of direct 

 usefulness, and that very many of the applications of the theories 

 of pure mathematics had come many years — sometimes centuries 

 — after the discoveries themselves ; the weapon had lain ready 

 to hand, but the man had not been there to use it. He also 

 indicated briefly his views on the teaching of elementary mathe- 

 matics, and advocated the inclusion of a course on practical 

 geometry early in the pupil's career. With this suggestion, that 

 the pupil should be led to pure geometry only after he had been 

 accustomed to handle and to work with the figures with which 

 geometry is concerned, all the suljsequent speakers cordially 

 agreed. Prof. Forsyth further desired to point out the need (^f a 

 proper system of training teachers so that when they began their 

 profession they would not have to devote their time to practising 

 upon their earliest pupils the method that happened to suit their 

 own particular temperament. Major MacMahon — the president 

 of Section A — joined in the discussion, but confined his remarks 

 to the subject of elementary teaching without entering upon the 

 more important questions raised by the address, which he had 

 already dealt with in his opening address to the Section. 

 Prof. Riicker said that there seemed to be a general agreement 

 among all the speakers that, in the case at all events of the 

 younger children, the teacher ought to approach the subject as 

 far as possible from the concrete side. He also held, with Prof. 

 Perry, that a somewhat rapid advance was advisable in the 

 first case, the various qualifications with which the general 

 statements had to be guarded being entered upon later. While 

 not attacking the system of examinations — which had come in 

 for severe criticism by other speakers — he considered that it had 

 its weak points, but that it was a necessary part of our educa- 

 tional apparatus. Prof. Silvanus Thompson and Prof. Henrici 

 were in entire accord with Prof. Perry ; and the latter expressed 

 the hope that qualified mathematicians would prepare text-books 

 upon the lines laid down in the address. Prof. Everett pointed 

 out the need of distinguishing between technical and liberal 

 education, and Prof. Miall criticised the system in which the 

 needs of the pupil and teacher were sacrificed to the demands of 

 the examiner and inspector. Mrs. W. N. Shaw spoke upon 

 the bearing of the discussion on the education of girls ; and 

 there also joined in the debate, Mr. J. Parker Smith, M.P., 

 Prof. Greenhill, Prof. AKred Lodge, Prof. Minchin, Mr. E. M. 

 Langley, and others. 



An immediate result of Prof. Perry's address has been the 

 appointment of an influential committee of the Association, 

 with Prof. Forsyth as president and Prof. Perry as secretary, to 

 report upon improvements that might be effected in the teaching 

 of mathematics. 



The Teaching of Botany. 

 The joint discussion on the teaching of botany was held in 

 the rooms of Section K, Prof. Bayley Balfour being in the chair. 

 Mr. Harold Wager introduced the discussion by reading a 

 paper on the teaching of botany in schools. He said that more 

 attention should be paid to methods of teaching if the subject 

 was to take its proper place in the school curriculum as a part 

 of the general scientific training. Too much time should not be 

 spent in mere descriptive work ; and the use of the compound 

 microscope should not be encouraged. The right selection of 

 topics was important. Such subjects as experimental plant physi- 

 ology, the structure and germination of seeds, and the structure 

 and function of the flower were specially to be commended. A 

 good grip of fundamental principles and not an imperfect 

 acquaintance with a vast number of facts was wanted in school 

 teaching. 



Prof. Bower read a paper on the teaching of botany in uni- 

 versities. He also urged that the use of the microscope in 

 schools should not be allowed. It should be left to the uni- 

 versity course. Thoroughness in special branches should be 



