August 28, 1908] 



SCIENCE 



259 



They very generally lack the power to 

 do anything with the mathematics which 

 they have been taught. 



With reference to the reasons for this 

 state of things, I venture to state what 

 seem to me to be some of them, and the 

 suggestions which have occurred to me by 

 which possibly the results might be im- 

 proved. 



1. In one of the previous papers a state- 

 ment was made that many students who 

 studied advanced algebra in the technical 

 schools had not studied algebra in the pre- 

 paratory schools for the two years previous. 

 This illustrates what I believe to be one 

 failing in our so-called system of educa- 

 tion, namely, the lack of continuity. The 

 remedy is to reform and simplify the cur- 

 riculum, and to unify and simplify the 

 entrance examinations to our colleges and 

 technical schools. So long as these en- 

 trance examinations are so extended and 

 cover so large a range of subjects, our pre- 

 paratory schools will be unable to carry 

 out their true purpose, which is, as it seems 

 to me, no less and no more than that of 

 all education, namely, to train a man 

 thoroughly in a few things and to give him 

 the power to do some little thinking for 

 himself and to take up new subjects with- 

 out assistance. 



2. The great inherent difficulty which 

 teachers of mathematics as well as teachers 

 of every other subject meet with is the 

 attitude of the student, and his inability 

 to realize the seriousness and the im- 

 portance of his work. I am fond of ex- 

 pressing my view in regard to this by the 

 statement that the school is not a restau- 

 rant, but a gymnasium; not a place where 

 a student comes to be filled up, but a place 

 where he finds apparatus and the instruc- 

 tion, by making use of which he may 

 strengthen his mental muscles. 



The manufacturer can take his raw ma- 

 terial and shape it into the form which 



he desires. The raw material of the 

 teacher is the student, but the teacher can 

 not take this material and shape it ; he can 

 only show it how it can shape itself. I 

 believe, however, that much may be done 

 in impressing upon students the proper 

 attitude which they should take toward 

 their work, and by a proper cooperation 

 between teachers and parents, which is un- 

 fortunately lacking as a rule in this 

 country, and the responsibility for which 

 must largely fall upon the parents. 



3. I believe that one cause of the poor 

 results in mathematical teaching is that too 

 great a stress is laid upon analysis. Mathe- 

 matics is, of course, divided into geometry 

 and calculus, using the words in their 

 widest sense. Geometry is concrete; and 

 the mind perceives the steps in a geo- 

 metrical demonstration. This branch, the 

 oldest branch of mathematics, however, has 

 been largely supplanted by the modern 

 analytic methods which have been de- 

 veloped during the past three centuries, 

 largely to the detriment, it seems to me, of 

 the educational results obtained. Analysis 

 is abstract— it is a powerful machine, an 

 invention for doing certain things. Into 

 one end of the machine we put the data; 

 we turn the crank, and the result comes 

 out with absolute correctness so far as is 

 warranted by the data. Now I believe that 

 too much stress is laid on these analytical 

 processes; that the student is not urged to 

 visualize his results, to express them 

 geometrically and to interpret his equa- 

 tions. I warmly second the remarks of 

 Professor Ziwet with reference to descrip- 

 tive geometry, which I believe should be 

 treated as a branch of mathematics and 

 taught more thoroughly, as it is taught in 

 Germany. For my part, I derived as much 

 benefit from my study of descriptive 

 geometry, and afterward from the study 

 of projective geometiy, as from any other 

 mathematical studies. These studies train 



