July 7, 1905.] 



SCIENCE. 



complete in those respects which stimulate 

 the powers of accurate (straight) reason- 

 ing. 



It has seemed to me that the present 

 teaching of mathematics is not so effective 

 as that brought to bear on my generation 

 in the secondary schools, and it likewise 

 appears to me that my generation was less 

 effectively • taught to reason, through 

 mathematics, than was my father and his 

 generation. 



I am not a reactionary or one who exalts 

 the past above the present. But I see a 

 reason for the present condition in the 

 extended introduction of analytical mathe- 

 matics and a consequent relegation of con- 

 structive mathematics to a minor place. 

 The introduction of the analytical mathe- 

 matics is not of itself to be regretted, but 

 it seems to have brought with it a change 

 in the method of teaching which is pro- 

 foundly unfortunate. The teacher now 

 feels under requirement to lead a large 

 class over certain ground in a given time, 

 and (to use the concrete example of al- 

 gebra) he finds he may do so by expecting 

 the students to learn the processes by the 

 book, and solve the equations, but he has 

 no time (nor strength, if the class is unduly 

 large) to spend in the work which is really 

 of overshadowing importance — that is, 

 drilling the students to interpret the phys- 

 ical meaning of each pregnant transforma- 

 tion. 



Unhappily this condition has had the 

 support of the science departments (espe- 

 cially of mathematics and physics) in some 

 of our great universities, where it has been 

 held that the equation is the thing and the 

 interpretation of minor moment; and with 

 this support in high quarters, how should 

 we expect the stupefying mechanical 

 method to be banished from the secondary 

 schools. 



But, gentlemen, the equation is not the 



thing. The interpretation of the equation 

 — an understanding of the real meaning of 

 transformations, and a grasp of the rela- 

 tions of things, which lead to sound reason- 

 ing—is the feature of first importance to 

 be derived from the study of mathematics. 



The mental subsoil is stirred in develop- 

 ing physical conceptions of the relations of 

 things, while even the sod may not be well 

 broken in learning the processes of jug- 

 gling equations. Stirring the mental 

 depths often calls for the exertion of the 

 utmost powers of good teaching, but poor 

 teaching is inexcusable, unless, much easier 

 as it is, it may be exacted by the under- 

 manned and overcrowded conditions of 

 some of our schools. 



What constitute first-rate instincts in a 

 teacher of mathematics may be illustrated 

 by an anecdote: 



Some years ago a mature graduate stu- 

 dent who was in one of my college classes 

 asked me if it would not be better to go 

 slower at some places so that the class 

 should thoroughly wiiderstand the relations 

 of things, even if we did not cover the 

 whole subject in the allotted time. This 

 was text-book work and in an engineering 

 subject of analytical character. We were 

 then covering only ten or twelve duodecimo 

 pages per day, but the book was one in 

 which nearly every sentence was charged 

 with important meaning and each mathe- 

 matical expression, however simple or com- 

 plicated, represented some important phys- 

 ical relations. 



The student had been a college instructor 

 with a fine reputation as a teacher of 

 mathematics or mechanics, and since then 

 he has become a professor of engineering. 

 I have understood that he has strongly 

 entrenched his reputation as a man whose 

 students become young men of discreet 

 thought, notable for resourcefulness and 

 character. 



