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SCIENCE 



[N. S. Vol. XXVIII. No. 713 



the imagination, which analysis does not 

 do. But in the use of analysis, the first 

 step, namely, the formulation of a problem, 

 is really concrete. This, too, is neglected 

 in our usual courses. Our examination 

 papers are full of questions which involve 

 simply the analytic processes — the differ- 

 entiation, the integTation, the twisting and 

 turning of equations, while much less at- 

 tention is paid to the formulation in mathe- 

 matical language of practical problems. 

 Our students, therefore, when they meet a 

 practical problem, are unable to select or 

 judge of the correctness of the data, and 

 even if they can do this, are unable to 

 formulate the data as a preliminary to the 

 solving of the problem by the use of the 

 mathematical machine. 



One of the great defects which I find in 

 students of mathematics is one already re- 

 ferred to, namely, that they do not in- 

 terpret their equations. The average stu- 

 dent who has completed his mathematical 

 course, for instance, has not the slightest 

 conception of what a parabola is. I make 

 this statement advisedly, because I have 

 tested it again and again for years. If he 

 could tell you what a parabola really is in 

 his mind, he would probably tell you that 

 it was a curve of more or less beauty 

 represented by letters. Perhaps he could 

 tell you what the letters are, but give him a 

 concrete problem and he would convince 

 you immediately that he did not know 

 what the letters mean. 



4. Another defect, as it seems to me, in 

 our present methods, is the lack of training 

 in mental operations. In the good old 

 days mental arithmetic was taught, but 

 that seems to have gone out of fashion, 

 with so many of the other good old 

 methods. Ask the ordinary graduate of 

 our mathematical courses to tell you the 

 square of 20.75 without using pencil or 

 paper and he will look at you open- 

 mouthed with astonishment, but if he had 



really grasped the meaning of the binomial 

 theorem and had learned to do a few 

 "sums" in his head, any grammar-school 

 boy would, of course, be able to give the 

 result immediately. 



5. Another reason for poor results is, I 

 believe, inadequate class-room methods, and 

 especially the use of the lecture system. 

 In Germany, where the students in the uni- 

 versities have had the advantage of a 

 thorough preliminary training, they may 

 be able to appreciate lectures on mathe- 

 matical subjects, although I doubt even 

 this in the case of the average student. 

 For students in our American universities, 

 however, I believe that lectures in mathe- 

 matics are almost useless, except for a very 

 small number of students; and yet, I am 

 told that even in some of our high schools 

 mathematics is taught to a considerable 

 extent by lectures. The lecture system is 

 easy for the teacher. It involves no cross- 

 questioning, no endeavor to discern what 

 is going on in the student's mind, no adap- 

 tation of question with the object of put- 

 ting him on the right track. 



Again, some mathematical exercises are 

 conducted by sending the students to the 

 board, each with a problem to solve, and 

 then marking that on the correctness of 

 their work. Occasionally a formal expla- 

 nation of his problem is required of the 

 student. This, again, seems to me to be a 

 mistaken method. Many a student can go 

 through a demonstration of a principle, or 

 solve a problem by substitution in a 

 formula, while knowing nothing of the real 

 meaning of the subject. In my opinion 

 class-room instruction should be conducted 

 by the Socratic method— by question and 

 answer— the teacher endeavoring to put 

 and keep the student upon the right track 

 by showing him what he can do for him- 

 self if he wiU only learn how. 



6. Reference has been made to the kind 

 of teachers of mathematics. Personally I 



