MATHEMATICS IN ENGLISH SCHOOLS 167 



kind of "flair." This guessing is a necessary part of mathe- 

 matical thought but the guess has to be verified in some way 

 or other, in geometry generally by deductive reasoning. In 

 solving a geometry rider I may guess or suspect or have an 

 intuition that a certain pair of angles are equal ; I may be 

 unable to solve the rider till I " spot " this and the guess may 

 be half the battle. The other half consists in verifying the 

 hypothesis. Now it is of no avail that a boy should be 

 habituated to verify his hypotheses in geometry unless he be 

 able to M carry over " the habit into real life ; the commonest 

 cause of vulgar errors is the failure to verify hypotheses. But 

 it is very doubtful if the habit is often carried over, and I think 

 that good work would be done if boys were taught explicitly 

 that this habit is formed for them in school in order that they 

 may apply it out of school ; probably very few would realise 

 this unaided. 



Whether the faculty-training view of mathematics be right or 

 wrong, it has held the field during several generations and there 

 has been every chance of ascertaining what it can do. What 

 are the results ? A referendum would certainly lead us to say 

 a failure. Whether or no we have done all that is possible, we 

 have certainly failed in one thing : broadly speaking, we have 

 failed to make mathematical thought enter as a main element 

 into the life of the educated classes. To redeem this failure is 

 work for the present generation of teachers and I suggest that 

 we shall succeed if we think less of the faculty-training and 

 more of the " outlook " aspect of mathematical teaching. 



What is there in the present syllabus to give boys a 

 mathematical outlook? For the average boy the syllabus 

 comprises arithmetic, algebra and geometry and nothing else. 

 My thesis is that the treatment of these subjects should be 

 so remodelled as to leave time for a further range of subjects 

 and a wider field of ideas. Considering the time given to the 

 subject — largely in excess of that given in France or Germany — 

 the amount of mathematics covered by English schoolboys is 

 insignificant. To the faculty-trainer this is a matter of in- 

 difference, as the training can be given as well through a 

 thin syllabus as through any other. From the outlook point 

 of view it is not a matter of indifference. 



More and more the affairs of life are being made amen- 

 able to mathematical treatment and, as it has turned out, 



