470 REPORT— 1902. 



used as substitutes for some of the cumbrous formal proofs of propositions 

 such as those in the Second Book of Euclid : for opinions differ as to the value 

 of strictly demonstrative geometry, both for training and for knowledge. 

 Those teachers who do not regard algebraic methods as proper substitutes 

 for geometrical methods might still use them, as well as arithmetical notions, 

 for the purpose of illustrating a proposition or explaining its wider signi- 

 ficance. It is the general opinion of the Committee that some association 

 of arithmetic and algebra with geometry is desirable in all cases where 

 this may be found possible ; the extent to which it may be practised will 

 depend largely upon the individual temperament of the teacher. 



Every method of teaching demonstrative geometry has to face tlie 

 difficulties inevitably associated with any complete and rigorous theory of 

 proportion. In the opinion of the Committee, not merely is Euclid's 

 doctrine of proportion unsuited for inclusion in elementary work, but it 

 belongs to the class of what may be called university subjects. The Com- 

 mittee consider that the notion of proportion to be adopted in a school 

 course should be based upon a combination of algebraical processes with 

 the methods of practical geometry. 



Examinations in Geometry. 



As regards examinations in geometry, the Committee consider that 

 substantial changes in much of the present practice are desirable. In 

 most, if not in all, of the branches of mathematics, and especially in 

 geometry, the examination ought to be arranged so that no candidate 

 should be allowed to pass unless he gives evidence of some power to deal 

 with questions not included in the text-book adopted. Such questions 

 might comprise riders of the customary type, arithmetical and algebraical 

 illustrations and verifications, and practical examples in accurate drawing 

 and measurement. The Committee consider the latter of particular 

 importance when the range is of an elementary character ; some influence 

 will be exercised upon the teaching, and some recognition will be given 

 to the course of practical geometry that should be pursued in the earlier 



stages. 



Arithmetic and Algebra. 



The Committee are of opinion that, in the processes and explanations 

 belonging to the early stages of these subjects, constant appeal should be 

 made to concrete illustrations. 



In regard to arithmetic, the Committee desire to point out what has 

 been pointed out so often befoi'e, that, if the decimal system of weights 

 and measures were adopted in this country, a vast amount of what is 

 now the subject-matter of teaching and of examination could be omitted 

 as being then useless for any purpose. The economy in time, and the 

 advantage in point of simplification, would be of the greatest importance. 

 But such a change does not seem likely to be adopted at present ; and 

 the Committee confine themselves to making certain suggestions affecting 

 the present practice. They desire, however, to urge that teachers and 

 examiners alike should deal with only those tables of weights and measures 

 which are the simplest and of most frequent practical use. 



In formal aritlimetic, the elaborate manipulation of vulgar fractions 

 should be avoided, both in teaching and in examinations ; too many of 

 the questions that appear in examination papers are tests rather of 



