January i8, 19 12] 



NATURE 



397 



purely abstract treatment failed to disclose the close rela- 

 tions of mathematical ideas with the physical experience in 

 which the abstractions took their origin. That Euclid has 

 any relation to the problems of actual space was seen by 

 the majority of those who suffered under this s\stem only 

 at a later time, if at all. The relations of symbols with 

 the concrete, and the economy of thought involved in their 

 use, remained for the most part unappreciated ; such 

 appreciation came, if at all, as die product of later reflec- 

 tion on the part of a very few of those who had attained 

 to some facility in the manipulation of the symbols. 



Mais nous avons change tout cela. The modern 

 methods of teaching appeal in the first stages to those 

 interests which are strongest in the majority, instead of 

 running atilt against the most undeveloped sides of the 

 minds of the pupils. Geometry, the science of spatial rela- 

 tions, is introduced by the observational and experimental 

 >tudy of the simplest spatial relations, verification by actual 

 measurement playing an important part ; the abstract treat- 

 ment in accordance with the deductive method being 

 relegated to a later stage. The interests of the average 

 boy are rather practical than theoretical, therefore, it is 

 thought, he must be interested with space relations on their 

 practical side. He is not interested in formal logic, there- 

 fore he must not be bored with learning a chain of 

 theorems of which the object is not apparent to him. He 

 is not usually ingenious, therefore, it is thought, no 

 demands must be made upon him which require ingenuity. 

 He does not readily move in the region of abstract 

 symbolism, therefore he must be introduced to the use of 

 symbols only in an arithmetic manner, in which the con- 

 crete implications are prominent. Laborious exercises in 

 algebra, in which expertness in the manipulation of symbols 

 is the object to be attained, should, it is thought, be for 

 the most part omitted. 



Owing in large measure to the activities of the Mathe- 

 matical Association, a considerable transformation in the 

 methods and in the spirit of mathematical teaching has 

 already taken place in many of our schools, and the 

 changes in the direction indicated by the newer ideals are 

 no doubt destined to have even more far-reaching effects 

 than at present. However, the old mechanical methods of 

 tf^aching still linger on in many of our schools, in which 

 conservative traditions are notoriously difficult to eradicate. 

 The detailed discussions, both in print and viva voce, which 

 arise in connection with the work of our association may 

 hr' of inestimable value in directing aright the detailed 

 'l.-velopment of the reformed methods of teaching. 1 hope, 

 also, they may prove useful in the direction of checking 

 those one-sided exaggerations which are always apt to 

 arise in connection with activities in which the objects to 

 be attained are various, as they must be in the case of so 

 many-sided a branch of education as the one with which 

 v\c are concerned. Some degree of compromise, without 

 undue sacrifice of principle, may often reasonably be made 

 in adapting the teaching so as to take account of the 

 widely diverging future careers in prospect for different 

 classes of pupils. 



It may, I think, be safely maintained that, the better 

 the theory underlying the method of instruction may be, 

 the more exacting will be the demands made upon the 

 skill, the knowledge, and the energy of the teacher. My 

 own early recollections of learning mathematics call up 

 memories of the classical master, without any real know- 

 I'dge of, or real interest in, the subjects, hearing repetition 

 of propositions of Euclid, or setting a long row of sums 

 ill algebra, monotonous in their sameness. Somehow a few 

 of us managed to learn .something, but I trenibU' to think 

 what would have been the results, had the said classical 

 master attempted to teach in accordance with the newer 

 m"thods. For the success of the teaching in accordance 

 with th'- reformed methods, a high degree of efTiciency on 

 tile part of the teacher is essential if the results hoped for 

 .u-e to be attained, and even if those results are not in 

 some respects to fall short of what was reached under the 

 older system. The teacher must possess a high degree of 

 >kill in presenting his material; he must have a broad 

 knowlcdgf of the subject, reaching much beyond the range 

 which he has directly to teach ; he must have skill and 

 alertness in handling a class, that skill having been 

 developed by definite training, but, of course, presupposing 



NO. 2203, VOL. 88] 



a natural capacity for the kind of work. Some of the 

 failures of which one hears, of the newer methods to pro- 

 duce satisfactory results, may probably be traced to a 

 tailing short on the part of the teaching in one or more 

 of the points I have indicated. 



At the present time it is not possible to form any precise 

 estimate of the actual effects of the recent reforms in 

 mathematical teaching. It will only become possible to do 

 so when the confusion incident to a state of transition has 

 passed away. That in many quarters the gain has already 

 been considerable I have no doubt. I have no doubt that 

 the principles underlying the newer methods are sounder 

 than those which formerly held sway. I have no doubt 

 that it is right to proceed from the practical and concrete 

 side of the subject, rising only gradually to the more 

 abstract and theoretical side. But the adoption of more 

 correct principles is only one step ; their actual transla- 

 tion into practice gives r'se to many difficulties and to 

 many dangers, some of which have most certainly not 

 been altogether avoided. The process of change has as 

 3'et not been one involving pure gain. 



A perusal of some of the current treatises on " practical 

 mathematics " has led me to think that in some quarters 

 the purely practical side of mathematics is unduly 

 emphasised. The teaching should, without doubt, com- 

 mence with this side, and should never lose touch with it ; 

 but the study of mathematics must be pronounced to be a 

 relative failure as an educational instrument if it fails to 

 rise beyond the purely practical aspect of the subject to 

 the domain of principle. Purely numerical work, calcula- 

 tion with graphs, problems in which the data are taken 

 from practical life — all these are excellent up to a certain 

 point, and they form the right avenue of introduction to 

 scientific conceptions. But if this kind of work is unduly 

 prolonged, and too exclusively practised, it tends to develop 

 a one-sided mechanical view of the capabilities of mathe- 

 matical methods, and the study ceases to be in any real 

 sense educational. Such practical work is only educational 

 when it precedes, and leads up to, a grasp of general prin- 

 ciples, and when it is employed to illustrate such principles. 

 I do not wish in the least to depreciate the importance of 

 mathematics as providing the tools for a vast variety of 

 applications useful in various professions. This side should 

 never be lost sight of in school work. But the most 

 important educational aspect of the subject is as an instru- 

 ment for training boys and girls to think accurately and 

 independently ; and with this in view the more general and 

 theoretical parts of the sulijin t should not be entirely 

 sacrificed either to the exiij;. lu v of providing useful tools 

 for application in after-life or to the supposed need of 

 sustaining interest in the subject by a too anxious 

 adherence to its concrete and practical side. 



I gather that, in some of the current teaching of practical 

 mathematics, a kind of perverse ingenuity is exhibited in 

 evading all discussion of fundamental ideas, and in the 

 elimination of reference to general principles. Instead of a 

 skilful use being made of practical methods to lead up to 

 general methods and illuminating ideas, practical rules 

 seem sometimes to be m.ul.' the end of all things. I have 

 been told, for example, tli.ii the use of logarithms is some- 

 times taught to students wlu) at no time attain to a com- 

 prehension of what a logarithm really is, or of the grounds 

 upon which the rules for the use of logarithmic tables rest. 

 Students who are in the habit of employing, for purposes 

 of calculation, formula? the origin of which they do not 

 understand have entered upon a path which will inevitably 

 lead to disaster, not only as regards their mental culture. 

 but also in the practical domain. If mathematics is de- 

 graded to the level of a set of practical rules, of which the 

 grounds are not understood, for dealing with practical 

 problems of special types, the unscientific character of such 

 a study will .ivenge itself even on the practical side of life. 

 .\ student who proceeds on these lines will fail to arrive 

 at those points of vi( w ih.it are not only the most stimu- 

 lating mentally, but of wliich the attainment is really 

 essential for success in applying mathematics to practical 

 matters. The practical applicalions of mathematics are 

 much too varied to be capable of being confined within 

 the range of anv number of prescribed rules and formulte. 

 Practical problems will be found constantly to arise in 

 connection with professional work which are not quiK» on 



