170 SCIENCE PROGRESS 



handling algebraic expressions. Now this statement does not 

 imply that teachers try to make boys manipulate algebraic 

 expressions without understanding what they are doing. I am 

 not speaking of ordinary bad teaching ; but I am saying that 

 a great number of very competent teachers are inspired by 

 a wrong ideal. They want boys to understand in order that 

 they may manipulate correctly, whereas their ideal should be 

 just the reverse. The ultimate aim should be not manipulation 

 but understanding and outlook. 



English education is dominated by examinations. Examiners 

 cannot test outlook and they can test understanding only by 

 testing manipulation. Teachers have to supply what examiners 

 demand. It is not surprising then that many teachers have 

 mistaken the means for the end. A vast amount of time is 

 spent on purely mechanical work : highest common factor, 

 fractions and factors beyond the types needed for practical 

 purposes, needlessly heavy equations, together with all sorts 

 of artifices and elegancies which are to the average boy as 

 pearls to swine. 



Now if a boy be certainly destined for a career in which 

 he will be bound actually to make use of mathematical manipu- 

 lation, a case might possibly be made out for drilling him at 

 school to a high degree of dexterity in the technique of algebra, 

 just as a student who aspires to be a professional pianist must 

 devote an astonishing number of hours to the technique of 

 playing the piano. But for the moment we are considering 

 the case of the general student of mathematics, the non-specialist. 

 We may assume that the average man, not connected with any 

 mathematical or scientific profession, finds practically no occasion 

 in the affairs of life to enter into the details of an algebraic 

 calculation ; it is even more certain that if such an opportunity 

 present itself to him exceptionally the opening is declined, in 

 spite of (perhaps because of) the heavy drill that has darkened 

 his schooldays. On the other hand, those of us who believe in 

 a mathematical training are profoundly convinced that such 

 general mathematical ideas and modes of thought as may be 

 wrought into the mind by a suitable course of instruction 

 are an asset of permanent cultural value. It is these ideas and 

 modes of thought that we regard as a necessary element of a 

 liberal education ; manipulative dexterity on the other hand 

 we look upon as a purely specialised technical accomplishment. 



