SCIENCE. 



[N. S. Vol. XXII, No. 549. 



ence, it is not only an advantageous school 

 course for preparing students for college, 

 but it is a preferable course for those 

 numerous young people who can not go 

 through college. 



Many of the errors of teaching in the 

 universities are the result of an undiscrimi- 

 nating chasing aft«r the popular cry, and 

 many spurious pedagogical ideas are still 

 propagated which were long since laid by 

 the competent leaders. I presume that the 

 same condition is found in the secondary 

 schools. What engineering teacher can do 

 his duty who does not understand the truly 

 simple relation between theory and prac- 

 tice, and where can he find it better ex- 

 pressed than when reading the inaugural 

 dissertation of Professor Eankine ? What 

 secondary school teacher can teach his best, 

 whatever may be his instinctive capacity, 

 who has not read Montaigne's essays on 

 teaching or Spencer's little book on educa- 

 tion, or who has not absorbed the point of 

 view of some of the great teachers through 

 adequate biographies? 



Of the elementary mathematics taught in 

 the schools, I have just said that the factor 

 of accuracy in application is often omitted, 

 or, if it is not actually omitted, it is largely 

 neglected. 



Contact with men entering college shows 

 that: 



The arithmetic class is taught the rules, 

 but not the reasoning upon which the rules 

 are founded or the overshadowing impor- 

 tance of impeccable accuracy in numerical 

 results. 



The algebra class is taught to transform 

 (or, as I may call it, juggle) equations, 

 but little thought is bestowed where the 

 greatest thought belongs— that is, to the 

 physical meaning of each form that is 

 produced. This fault, I must admit, is 

 not missing from the universities, and is 

 propagated in the schools by association. 



The geometry class is apparently taught 

 by rote, and even where a show is made of 

 encouraging the originality of the pupils 

 it is likely to be more an illusion than a 

 fact. 



Large classes encourage teaching for the 

 average mass, rather than the stirring of 

 each individual as must be done to create 

 the fullest results. Apparently in few 

 mathematics classes are the pupils taught 

 to scrutinize and check the results of their 

 labors by means appealing more to the 

 common sense than to cut and dried meth- 

 ods. Many years of observation with col- 

 lege students have shown me that the atten- 

 tion of secondary school pupils is seldom 

 drawn to such useful processes for check- 

 ing numerical results (which were taught 

 with fidelity to our fathers) as 'casting out 

 the nines ' ; and the worst of it is that the 

 pupils have not been taught even the 

 simple philosophy of our decimal system 

 which might enable them to work out the 

 processes for themselves. 



If the causes that contribute to allow the 

 pupils to reach the end of the secondary 

 school training with their originality sleep- 

 ing, their normal sense of accuracy lost 

 and their best accomplishment in mathe- 

 matics a parrot-like following of hackneyed 

 method in familiar problems— if the causes 

 from which these conditions spring are 

 anchored in overcrowded classes, then it is 

 your duty and privilege to cry aloud for 

 more air, more breath of life, more chance 

 to teach each living individual instead of 

 the average of an inert class. 



Mathematics is a tool— a powerful sys- 

 tem of logic, an aid to reasoning— which 

 confers power and advantage on the indi- 

 vidual in proportion to the fullness of his 

 possession. The value of mental discipline 

 obtained while accomplishing that posses- 

 sion is inestimable. And the teacher 's aim 

 ought to be to make that possession most 



