112 



SCIENCE 



[N. S. Vol. XXVIII. No. 708 



least breathing time, and must sometimes 

 at least come to terr, alt for air. ' ' 



The desire to make the university teach- 

 ing of mathematics more practically useful 

 and bring it into live contact, as far as 

 possible, with the whole tendency of mod- 

 ern scientific thought led, on the one hand, 

 to a strengthening of all branches of ap- 

 plied mathematics, not only by courses 

 offered in the universities, but also by such 

 publications as the Encyklopadie, which 

 includes applied mathematics in the widest 

 application of the term ; on the other, it led 

 to reforms in the courses offered to future 

 teachers of mathematics, and ultimately to 

 a thorough investigation of the teaching of 

 elementary mathematics in the secondary 

 schools of Germany. 



The improvement of the teaching of ele- 

 mentary mathematics is the aim of the 

 third and most recent mathematical reform 

 movement in Germany. The reforms 

 proposed in this connection by the com- 

 mittee of the German Association of Natur- 

 forscher und Aerzte, at the Meran meeting, 

 in 190-5, appear to me to deserve very care- 

 ful consideration. They would apply, in 

 this country, to the teaching of mathe- 

 matics not only in the high schools, but 

 just as much in the engineering colleges. 

 For, with the preparation that our students 

 actually have, I am convinced that the best 

 method of imparting a good working 

 knowledge of the elements of analytic 

 geometry and calculus is not through lec- 

 tures, but through actual teaching based 

 mainly on solving problems, that is, by the 

 methods not of the German university, but 

 of the German secondary school. 



The proposals of the committee^ do not 

 change very essentially the number of 

 hours required for mathematics. These 

 are to be: in the Gymnasium as well as in 

 the Realgymnasium, four hours per week 



' See Zeitschrift filr mathematiscTien und natur- 

 wissenschaftlichen Vnterricht, Vol. 36 (1905), 

 pp. 533-580. 



in each of the nine years; in the Oberreal- 

 schule generally four hours per week, in 

 the third and fourth years six hours. The 

 first three years are devoted to common 

 arithmetic and intuitional geometry, the 

 next three years to algebra and geometry 

 carried along together, the last three years 

 to advanced algebra, trigonometry, ad- 

 vanced geometry, conic sections (treated 

 synthetically and analytically) and, in the 

 Oberrealschule, the elements of the cal- 

 culus. Apart from matters of detail this 

 distribution does not vary very much from 

 the practise now followed in the best Prus- 

 sian schools. 



While thus the general program can not 

 be said to constitute a radical departure 

 from existing conditions, the statement of 

 what should be the principal aim of mathe- 

 matical teaching and the indications given 

 for carrying out this aim throughout the 

 whole course'' appear to me as the most 

 important features of the report. In addi- 

 tion to the well-recognized object of mathe- 

 matical teaching to train the mind in rigor- 

 ous logical reasoning the report insists par- 

 ticularly on the training of geometrical 

 intuition and on acquiring the habit of 

 functional thinking. The carefully pre- 

 pared explanations accompanying the de- 

 tailed program for the nine-year course 

 show how these aims should guide the in- 

 struction at every step. The insistence on 

 the idea of the functional relation can not 

 be recommended too strongly to our writers 

 of college text-books, from trigonometry 

 to differential equations. But, as this re- 

 port demands, it should even enter into the 

 very elements of algebra and geometry. 



It should be observed that the committee 

 that prepared this report was not composed 

 of mathematicians only; all branches of 

 science taught in the secondary schools 

 were represented in it; and all these 

 branches received equally careful attention. 

 While the portion of the report devoted 

 'Loo. cit., pp. 543-545, 550-553. 



