May 22, 1913] 



NATURE 



305 



TEACHING OF MATHEMATICS IN 

 GERMANY. 1 

 T N previous issues we have referred to papers on 

 -*■ English education in mathematics which were 

 laid belore the 1912 International Conference on 

 Mathematical Teaching. We have now before us, in 

 five volumes, the German contribution to that con- 

 ference. They give an account of mathematics at the 

 primary and secondary schools, at the universities, in 

 technical education, and in training colleges for 

 teachers. They deal mainly with Prussia, but include 

 also the non-Prussian parts of the German Empire, 

 with an occasional reference to Austria. 



Germany also has its reform movement in mathe- 

 matics, and most of the changes that have been made 

 lie to the credit of a body which bears the euphonious 

 name of "der Damnu," into which its full title 

 " Deutscher Ausschuss fur den Mathematischen und 

 Naturwissenschaftlichen Unterricht " has for the sake 

 of brevity been telescoped. This body was formed in 

 1907, by the united action of a number of voluntary 

 scientific associations, and is playing much the same 

 part that the British Association committee has played 

 in this country. 



The aims and the present position of the movement 

 are well illustrated by a scheme of teaching proposed 

 by Dr. Schimmack for the Oberrealschule. The 

 scheme covers the nine school years between the ages 

 of nine and eighteen. 



In geometry the scheme begins in the manner to 

 which we are now accustomed in England, with 

 measuring, drawing, practice with instruments, and 

 work which familiarises geometrical concepts. 



It is noticeable that there is two years' work in 

 geometry before algebra is begun. This procedure, 

 so excellent because of the more abstract and difficult 

 nature of algebra, is not the result of the reform 

 movement, but has long been the practice in Ger- 

 manv. It is a promising sign that the report of the 

 curriculum committee of the Headmasters' Confer- 

 ence advocates this procedure, and gives us leave to 

 hope that in this matter England will follow Ger- 

 many's lead. 



Algebra, then, is begun in the fourth school year, 

 geometry having been begun in the second. It leads 

 off well, with signless quantities, and it is rather a 

 pity that the subject is not carried on for a year or 

 so with such quantities before the distinction between 

 positive and negative quantities is introduced. 



The trigonometry of right-angled triangles is to be 

 introduced in the sixth year, as is also " projective 

 geometry " (or cross-ratio geometry). The former 

 proposal would find much support in this country ; 

 the value of the second item is not so clear. 



In the seventh year the calculus is begun, differ- 

 ential and integral at the same time, a proposal which 

 manv in England will approve. Not so manv will, 

 however, approve of Dr. Schimmack's relegation of 

 arithmetical and geometrical series to their proper 

 place beside the calculus. 



The scheme closes with "discussion of the founda- 

 tions of geometry," too metaphysical perhaps for most 

 boys ; but we must remember that the scheme is 

 intended for the Oberrealschule, and that less ambi- 

 tious schemes would be appropriate to the Gvmnasium 

 and Realgymnasium. 



An important note is appended to the scheme to 

 say that throughout the course geometrical figures 

 are to be thought of as variable and not rigid, and 



chen Un 



:erricht in Heut'chland. 

 :he Unterrichtstkommis 

 F. Klein. (Leipzig and 



1 " Abhandlungen iiber den 

 ver.anlasst durch die Internationale ftlathematis 

 sion." In twenty-five parts. Herausgegeben von 

 Berlin : B. G. Teuhner, tooo-12.) 



" Berichte und Mitt*ilungen veranlasst durch die Internationale Math 

 matische Unterrichtskommission." In four parts. (Leipzig and Berli 

 Teubner, 1910-1?.) 



. G. 



NO. 2273, VOL. qi] 



that attention is to be directed to the interdependence 

 of the parts as the form of the figure changes. 



The consideration of this scheme goes to justify our 

 English reformers in their view that they are not 

 sacrificing thoroughness to the desire to cover ground. 

 It is much that the Germans, with their love of be- 

 ginning from the very foundations, should declare for 

 '• functional thinking " from the start, for the intro- 

 duction of trigonometry at fourteen or fifteen, and 

 of the calculus at fifteen or sixteen, and should feel 

 it possible to prune the course sufficientlv to allow- 

 that. 



The language of these volumes is at times curiously 

 heavy. One happens on long, long sentences the 

 meaning of which cannot be extracted by ordinary 

 reading ; they have to be logically dissected. ' And such 

 sentences are frequent. Is the accusation true that 

 command of language is spoilt by a mathematical 

 training? Or have the Germans spoilt their language 

 by the replacement of foreign words by sesquipedalian 

 words of home manufacture? Whatever the cause, 

 these volumes contain also, the germs of better things. 

 "Der Damnu" has been referred to. "Die Imuk" 

 is a portmanteau word for die Internationale Mathe- 

 matische UnterrichtsKommission, and "Der Datsch " 

 for der Deutsche Ausschuss fur Technische SCHulen. 

 Some extension of this idea will quickly reduce the 

 most unwieldy sentence to manageable size. 



THE ASSOCIATION OF TEACHERS IN 

 TECHNICAL INSTITUTIONS. 



THE annual conference of the above association 

 was held at Bradford during Whitsuntide. The 

 address of the president of the association (Mr. P. 

 Coleman, Northern Polytechnic, London) dealt mainlv 

 with the organisation of technical education, the value 

 of "internal" examinations for technical students as 

 compared with "external" examinations, and the 

 London University Commission report in its bearing 

 upon polytechnic work. He remarked in reference to 

 this : — 



"The report and the recommendations based thereon 

 unfortunately show a bias that can only be due to a 

 complete misconception of the work and standing of 

 the London polytechnics. . . . The reasons given in 

 the report appear to be based on insufficient evidence, 

 and at variance with the facts as known to those 

 who have a close acquaintance with the polytechnics." 



Mr. Coleman urged the development of "non- 

 vocational courses in the technical schools, partly 

 because in many towns these schools are the only 

 suitable institutions in which to hold such courses for 

 adult students (whether technical students or not), 

 and also in order to brinsr home to students whose 

 main interest is necessarily the study of science or 

 technology that "the work associated with their 

 future occupation should not lead them to forget everv 

 other means of culture." As a practical measure in 

 this direction, he suggested that the technical institu- 

 tions " should definitely associate with themselves the 

 University Extension Lectures of the locality, or such 

 work as that of the Workers' Educational Associa- 

 tion." 



Papers were read to the conference upon vocational 

 education, by Mr. Arthur C. Coffin, director of educa- 

 tion, Bradford ; co-ordination within a county area, by 

 Mr. F. W. Cook, chief officer for technical education 

 for the West Riding of Yorkshire ; and the corporate 

 life of technical institutions, by Mr. W. Hibbert, 

 Regent Street Polytechnic, London. A number of 

 sectional meetings were held, attended by teachers of 

 special subjects, at which questions such as the quali- 

 fications for the registration of teachers, the syllabuses 



