NATURE 



457 



THURSDAY, MARCH 20, 1902. 



REFORM OF THE TEACHING OF 

 MA THEM A TICS. 

 British Association Meeting at Glasgow, 1901. Discus- 

 sion on the Teaching of Mathematics, which took place 

 on September 1 4 at a Joint Meeting of two Sections, 

 Section A, Mathematics and Physics, Section L, 

 Education. Edited by John Perry. Pp. vi + loi. 

 (London : Macmillan and Co., Ltd.) Price 2s. net. 



PROF. PERRY'S views concerning the teaching of 

 mathematics have been expounded often in the 

 columns of Nature and elsewhere, and have aroused a 

 great deal of interest. It was a happy idea to bring 

 about a discussion of these views at a joint meeting of 

 the mathematical and educational Sections of the British 

 Association. The report of that discussion has been 

 issued as a separate small volume. It contains the in- 

 troductory address delivered by Prof. Perry, a specimen 

 syllabus prepared by him for use in training colleges, the 

 discussion that followed the address, written remarks on 

 the subject communicated by several teachers who did 

 not take part in the discussion, and Prof Perry's reply. 

 The discussion was restricted to what may be called a 

 possible school range of mathematics ; the teaching of 

 advanced mathematics at the Universities was scarcely 

 tou ched upon. As an outcome of the discussion, a com- 

 mittee of the British Association was appointed 



" to report upon improvements that might be effected in 

 the teaching of mathematics, in the first instance of 

 elementary mathematics, and upon such means as they 

 think likely to effect such improvements." 



The movement thus initiated is important. Many of 

 the more enlightened teachers of mathematics in Britain 

 have long been dissatisfied with the conditions under 

 which they have been compelled to work ; and efforts in 

 the direction of reform have been made, in particular by 

 the Association for the Improvement of Geometrical 

 Teaching. The smallness of the results achieved by 

 those efforts is to be traced mainly to the action of the 

 chief examining bodies; but there are not wanting signs 

 that these bodies are now less complacently conservative 

 than they have been in the past. It is earnestly to be 

 hoped that the indicated change of attitude may turn out 

 to be real. The handy publication under notice should 

 be of great assistance to those who are agitating for 

 reform. 



The movement is in great part a reaction against the 

 pedantic and unpractical character of the habitual teach- 

 ing of mathematics in schools. This character belongs 

 both to the methods of presenting particular subjects and 

 to the order in which the subjects, and the parts of the 

 subjects, are studied. The methods and the order now 

 in vogue are not, of course, a system devised purposely ; 

 they have been arrived at gradually, and are sanctioned 

 by tradition. The principles of the proposed reform may 

 be stated broadly as follows : — (i) The way for every 

 abstract generalization should be prepared by intelligent 

 practice of comparatively concrete processes, chosen so 

 as to stimulate inquiry ; '2) whenever a choice of alterna- 

 tive orders is offered, that order should be preferred 

 NO. 1690, VOL. 65] 



which conduces best to the application of mathematics 

 to practical needs. A subsidiary principle of reform 

 would be that all purely artificial and unimportant de- 

 velopments should be omitted. The principles thus 

 stated will not perhaps excite much opposition ; yet far- 

 reaching and radical changes would be required to give 

 effect to them. One result that maybe expected to follow 

 from such changes would be that the study of the elements 

 of mathematics would become abetter preparation than it 

 is at present for the study of the more advanced theories. 

 The mathematics of most of our elementary text-books is 

 felt to be almost as much out of touch with modern 

 mathematics as with everyday life. This conviction has 

 had some share in promoting the movement in favour of 

 change. 



As regards geometry, the thoroughgoing adoption of 

 the above-stated principles of reform would lead in the 

 first place to the introduction of courses of practical work 

 in drawing and measuring, which would precede and 

 accompany the study of demonstrative geometry ; and it 

 would lead in the second place to a great simplification 

 of that study. That simplification would consist partly 

 in the abandonment of Euclid's " Elements " as a text- 

 book. The difficulty of finding a generally acceptable 

 substitute for Euclid was fatal to the proposals of the 

 Association for the Improvement of Geometrical Teach- 

 ing, and the controversy between the supporters and the 

 opponents of Euclid has been a main feature of all dis- 

 cussions concerning the improvement of the teaching of 

 mathematics in this country. Now it may be contended 

 that we ought not to attempt to present geometry to 

 children as a formal system deduced from a minimum 

 number of assumptions ; but that our objects should be 

 to impart knowledge of the properties of geometrical 

 figures and to cultivate power of geometrical reasoning. 

 Upon the general adoption of this view of the true 

 objects of geometrical teaching, the above-mentioned 

 difficulty would disappear ; for it is safe to assert that 

 these objects can be secured better without Euclid's book 

 than with it. In this matter the French are far ahead of 

 us. Their text-books of geometry have been evolved by 

 a natural process of development from the "Geometric" 

 of Legendre ; and text-books similar to those of France 

 are in use in Germany, Italy and the United States. It 

 is far from my purpose to suggest that British teachers 

 should now adopt as a substitute for Euclid a translation 

 of a French book or any other foreign book. In all 

 these books too much attention, as I think, is paid to the 

 development of a formal logical system, with the result 

 that much space is occupied in proving propositions that 

 are sufficiently obvious without proof. We have yet to 

 work out for ourselves the method that is best suited to 

 our educational needs and national temperament. This 

 cannot be done without freedom from the tyranny of an 

 orthodox standard ; nor can it be done in a hurry. 

 Changes will have to be made gradually. In this con- 

 nexion we may note that the memorial sent by some 

 schoolmasters to the committee of the British Association 

 (N.ATUKE, January 16, 1902, p. 233) points out a promising 

 method of beginning to make changes. If the chief ex- 

 amining bodies would now sanction such a departure 

 from existing practice as is there proposed, progressive 

 reform would become possible. 



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