THE TEACHING OF MATHEMATICS.

Press, Volume LXI, Issue 11789, 12 January 1904, Page 6

THE TEACHING OF MATHEMATICS.

There was a good attendance in the aatronOiTsy and laaihematics seotioa to hear Mr G-. Kogben'e paper on "The Teaob • iag of 33ementary Mathematics," with special reference to geometry. Mr Hogben heg&a. by referring to the discussion at the Glassy meeting of the -British Association i:» HO*, introduced by a paper read by Pressor Pvrry, and then continued: —'"I'fofesr.Dr I'erry makes usefulness the or iwcii, for teaching niathematic3 ziii to s l . ow thut he docs not take a narrow visw of what he means, he names eight obvious forms of usefulness in the, study mathematics, which it is Dot necessary that I should recapitulate here. Setting aside their utility in passing examinations, which he says is the only form that has not be neglected, and is indeed in itself scarcely a legitimate form of usefulness, we may roughly classify Professor Perry's forma of usefulness under two heads, namely—(l) Educational utility, i.e., in brain development and in the intellectual powers and cukure that this branch of science is able to give; (2) practical utility, i.e., in supplying mental toob to engineers arid others by giving them dexterity in the manipulations of calculations required by their work, and n knowledge <X the principles oa which their applie-i science rests, line watersent mijlit swoai to introduce a conflict in aiiaß, end perhaps therefore in methods of teaching, according to the view of utility that is taken. A narrow view of utility might indeed appear to make- such a conflict inevitable, but after- listening to the eloquent address of Professor 'Davidsoa on Wednesday, we r!:al! lv> all ready to admit that the securing nf intellectual powers is, in the long run. one °f the most practical aims that ■we can keep before us, in the teaching of mathematics. Still in the higher stages a conflict may_ exist, or rather a parting of j the ways, when the pure mathematician and the engineer begin to pursue different- lines of study. JCet Uβ ask . ourselves the : question—. "Does- this conflict between what ia educationally useful and what is immediately practically useful, exist' in the teaching of elementary mathematics?" Or would you treat alike, in th« early stages the future mathematician, the future engineer, and the man who takes mathematics merely nH a part of an ordinary liberal education? ily reply to thafc is a distinct affirmative. This answer is already given in the teaching of elementary arithmetic, although that teaching would be vastly improved by considerably extending the use of the concrete in forming the basis for the fundamental Ideas and by making more extensive use of those ideas in their immediate application to easy practical examples. Euclid-and geometry ia the science (or the philosophy) of the concepts of certain- standards tp which actual physical facto may be." referred, and from which, as definitions (or quasi definitions, for you cannot define a primary concept), and as axioms assumed as universally true, the ■whole- science by a process of deductive mooning is founded. It is not a physical science, but a scieace of abstractions from certain .physical facto. Its value, both for cdacatioDcl purposes and for. practical purpo?»!i of life, among other things for the. (whom I take as a type of the practical man •■who , must use mathematics us a tool for his work), will depend upon tb; soundness of the coucepta upon which it is founded, .ard upon the degree in which ita. conclusione can be applied to practical u*c. I Jthesefore assume (a) thr.t the teaching .of deductive geometry ;'iicjui«l hi preceded by a careful examinatioa of the physical facts ppon which, its waiepts rest, that; is by a course of what i» called experiipental geometry; (b) that nil teaching of geometry uhould include soma deductive geometry as the generalisation which is scarcely possible without it widens the intellectual outlook, and gives an immensely; increased breadth of power in. dealing with, merely practical questions; (c) that the' propositions evolved by the'deductive method should be- tested in order to ascertain how far they are applicable to the facts of nature; and (d) that these propositions should then te actually used for the solution of all kinds of practical questions. It does not follow;, that the whole of tach should be completed; before the next, stage is begun; for instance, it Is cot necessary that the pupils should finish" & complete course of experimental geometry before beginning deductive reasoning, but in each "subditision of the subject experimental work should precede deductive demonstration and so on. The double difficulty that has to be faced by the boy (or girl) beginning Euclid, viz., dealing with unfamiliar subject matter, and being introduced to & new method o< reasoning, and the lack of interest consequent upon this double difficulty, are largely answerable for failure and delay u> reaching the stage at which he is allowed to use his knowledge. It is simply absurd that boys or girls that have been studying geometry for two or three years should leave school without knowing anything about similar triangles or the ares, of & circle and its sections, or the volumes of solids other than rectangular solid*, or without having even, partial knowledge of the matter, pome of the ordinary fallacies that have marked the orthodox system should disappear as soon as possible. Among these is'the sharp artificial line that has been drawn -between geometry on the one hand* and arithmetic and algebra on the other. Arithmetic most enter largely into experimental 'geometry, for it is a, science of magnitudes, end therefore, of measurements, and algebra being merely generalised arithmetic should be used in the generalked proposition of geometry whereever its use ia appropriate or of practical advantaged

The address was Illustrated by t number cf practical examples. 'Professor Carslaw, of Sidney University, said that wlten lie •rmed in New South Wales ia the beginning of last year, he found considerable uncertainty amongst teachers as t-> the attitude they should adopt with regard to the changes that- were being made at home ia the teaching of elementary mathematics, and no doubt the same--un-certainty existed in New Zealand. He traced the meaning of those changes, and ■aid that vrhilo to a certain extent they vera due to the demands of leading engineers for a more practical the prevalent feeling amongst leading pure mathematicians had also exercised inflathe men folding that some cbat>ge_ would be beneficient. The rcgulatio&s - now being drawn up by - the University of Cambridge owed much to the influence of Professor Fbrsyth. The changes recommended 1 by the committee of that University affected, to ft certain extent, the teaching of arithmetic, algebra, and trigonometry, bat the most far-reaching of the changes were those recommended in the teaching ot geometry. The Cambridge University, -whilst, leaving teachers froe to adopt the old Enclidian- ayetem, also left them free to adopt toemote modern method, a feature of which was that beginners wuuld in ftll cases spend some time in practical and experimental courses, and in their later study of geometry a free use -would be made of such illustrations. It would be a great advantage throughout Australasia if the recommendations of Uie

Cambridge University -were considered, and to a great extent adopted; the changes would require to be made gradually, ana the" alteration meant that teachers in the primary and secondary schools would receive a suitable training ia their work. The following resolutions were carried: — Moved by Professor Cook, and seconded by Mr Sairky, Inspector-General, Brisbane—"Thr.t It be a recommendation to ir.c examining bodies of Australia and New Zealand that they base the regulations for their public esaminatiocs on the syUabu3 adopted by the University of Cambridge." Moved byProfefsor Braggis, and recondod by Mr Hogben—"That in order that ths new syllabus be properly appreciated and correcily tcught, it is necessary that provision for the training of mathematical teacher.l should be made, and that such prevision should be made by the Universities and training Colleges of Australasia in the division of chemistry, and mineralosrr."