34 



SCIENCE 



[N. S. Vol. XXX. No. 758 



experience. The historical parallel, in fact, 

 is strikingly manifest in the form, content 

 and sequence of development of elementary 

 mathematics as proposed in these reforms. 



GERMAN REFORMS^ 



The nineteenth century may be divided 

 into three periods as regards the form and 

 content of mathematical instruction in Ger- 

 many. In the first period, extending from 

 1800 to 1870, mathematical instruction was 

 a mixture of the pure and applied. Ideals 

 were high, efforts were directed toward 

 awakening individual ability, and attempts 

 were made to teach more than is required 

 at present. The candidate for the position 

 of teacher of mathematics must be a spe- 

 cialist capable of original investigation, and 

 as a result we find such names as Grass- 

 mann, Kummer, Plucker, Weierstrass and 

 Sehellbach. 



The opening of the second period, 1870- 

 1890, was signalized by the victory over 

 France, and the assumption by Germany 

 of a more important international position. 

 This period was marked by a separation of 

 pure and applied mathematics. In the 

 schools the feeling prevailed that the de- 

 velopment of the especially gifted pupil 

 was not to be sought so much as that of the 

 average pupil, and consequently greater 

 interest was manifested in the method of 

 instruction. Instead of the early system, 

 a desire was expressed for a systematic 

 graded course in mathematics, keeping in 

 view the ability of the constantly develop- 

 ing pupil. Drawings and models were de- 

 manded, problems were so stated and aids 

 so given that the pupil might see space 

 relations and not depend so largely on the 

 logic of the ancient Greeks. This was a 

 direct result of the teachings of Pestalozzi 



''For a more detailed accoimt, see article by 

 Charles Otterman entitled " A Eeview of Klein's 

 Attitude on the Teaching of Mathematics," Sci- 

 ence, September 13, 1908. 



and Herbart. In this period the standard 

 for teachers was lowered, and there was 

 only required a knowledge sufficient to 

 solve problems of moderate difficulty. 



The third period, beginning with 1890, 

 seems to be characterized by a tendency to 

 again associate pure and applied mathe- 

 matics. That is to say, a teacher is re- 

 quired to be thoroughly familiar with pure 

 mathematics and at the same time to have 

 an extensive knowledge of its applications. 



For many decades the value of mathe- 

 matical training was thought to lie in its 

 formal discipline. Before the revival of 

 learning it was the utilitarian factor which 

 was emphasized, while in the last few 

 decades the majority have reached a more 

 comprehensive notion. Recently Professor 

 Felix Klein, the greatest of living German 

 mathematicians, has shown a deep interest 

 in the problems of the schools, and has 

 taken an active part in their discussion. 

 His views are typical of modern German 

 scholarship, and form the basis of the re- 

 forms instituted and proposed.^ 



According to Professor Klein, mathemat- 

 ical thought should be' cherished in the 

 schools in its fullest independence, its con- 

 tent being regulated in a measure by the 

 other problems of the school; in other 

 words, its content should be such as to es- 

 tablish the liveliest possible connection with 

 the various parts of the general culture 

 which is typical of the school in question. 

 Here, then, it is not a question of the 

 method of teaching, but rather of the selec- 

 tion of material from the great mass fur- 

 nished by elementary mathematics. 



Much of the material of instruction, al- 

 though interesting in itself, lacks connec- 

 tion and is wholly or partially isolated, 

 thus affording only a faulty and indirect 



^ " Neue Beitrage zur Frage des Mathematischen 

 und Physikalischen Unterrichts an den Hoheren 

 Schulen," Klein und Rieeke, Teubner, 1904. 



