March 13, IflOo.J 



SCIENCE. 



409 



there is a movement for the enlargement 

 of the strong secondary schools, by the 

 addition of the two years of junior college 

 work and by the absorption of the last two 

 or three grades of the primary schools, 

 into institutions more of the type of the 

 German gymnasia and the French lycee;* 

 in favor of this movement there are strong 

 arguments, and among them this, that in 

 siieh institutions, especially if closely re- 

 lated to strong colleges or universities, the 

 mathematical reforms may the more easily 

 be carried out. 



The fimdamental problem is that of the 

 unification of pure and applied mathe- 

 matics. If we recogTiize the branching 

 implied by the very terms 'pm-e,' 'applied,' 

 we have to do with a special case of the 

 correlation of different s^lbjects of the cur- 

 riculum, a central problem in the domain 

 of pedagogy from the time of Herbart on. 

 In this ease, however, the fundamental so- 

 lution is to be found rather by way of 

 indirection— by arranging the curriculum 

 so that throughout the domain of elemen- 

 tary mathematics the branching be not 

 recognized. 



The Primary Schools.— Would it not be 

 possible for the children in the grades to 

 be trained in power of observation and 

 experiment and reflection and deduction so 

 that always their mathematics should be 

 directly connected with matters of thor- 

 oughly concrete character? The response 

 is immediate that this is being done to-day 

 in the kindergartens and in the better ele- 

 mentary schools. I understand that seri- 

 ous difficulties arise with children of from 

 nine to twelve years of age, who are no 



* As to the mathematics of these institutions, 

 one may consult the book on ' The Teaching of 

 Mathematics in the Higher Schools of Prussia ' 

 (New York, Longmans, Green & Co., 1900) by 

 Professor Young, and the article (Bulletin Amer. 

 Math. Soc. it), vol. 6, p. 225) by Professor 

 Pierpont. 



longer contented with the simple, concrete 

 methods of earlier years and who, never- 

 theless, are unable to appreciate the more 

 abstract methods of the later years. These 

 difficulties, some say, are to be met by al- 

 lowing the mathematics to enter only im- 

 plicitly in connection with the other sub- 

 jects of the curriculum. But rather the 

 material and methods of the mathematics 

 should be enriched and vitalized. In par- 

 ticular, the grade teachers must make wiser 

 use of the foundations furnished by the 

 kindergarten. The drawing and the paper 

 folding must lead on directly to systematic 

 study of intuitional geometry, including 

 the construction of models and the ele- 

 ments of mechanical drawing, with simple 

 exercises in geometrical reasoning.* The 

 geometry must be closely connected with 

 the numerical and literal arithmetic. The 

 cross-grooved tables of the kindergarten 

 furnish an especially' important type of 

 connection, viz., a conventional graphical 

 depiction of any phenomenon in which one 

 magnitude depends upon another. These 

 tables and the similar cross-section black- 

 boards and paper must enter largely into 

 all the mathematics of the grades. The 

 children are to be taught to represent, 

 according to the usual conventions, va- 

 rious familiar and interesting phenomena 

 and to study the properties of the phe- 

 nomena in the pictures : to know, for ex- 

 ample, what concrete meaning attaches to 

 the fact that a graph curve at a certain 

 point is going down or going up or is hori- 

 zontal. Thus the problems of percentage 

 —interest, etc. — have their depiction in 

 straight line or broken line graphs. 



* Here I refer to the very suggestive paper of 

 Benchara Branford, entitled ' Measurement and 

 Simple Surveying. An Experiment in the Teach- 

 ing of Elementary Geometry' to, a small class of 

 beginners of about ten years of age {Journal of 

 Education, London, the first part appearing in the 

 number for August, 1899). 



