848 ^ REPORT— 1902. 



classical control, and tlie masters assume as an axiom that ' when we are learning 

 Latin grammar we are learning English.' 



A change in the head-masterships of the schools is necessary, both in the 

 interests of modern science and in those of modern languages. These schools have 

 heen too long under classical and clerical control. 



4. Joint Discussion with Section A on the Teaching oj" Mathematics, 



5. On the Teaching of Elementary Mathematics. By A. W. Siddons, M.A. 



A little more than twenty-five years ago the British Association and the 

 Association for the Improvement of Geometrical Teaching (now called the Mathe- 

 matical Association) worked at this question ; the ends aimed at were not gained 

 in their entirety, but much improvement resulted in the teaching of the subject. 



That movement has taught us that attempts to secure any but moderate re- 

 forms are doomed to failure, and we see to-day from private and published corre- 

 spondence that mathematical masters generally are anxious for moderate reform, 

 but dread any hasty change. 



The Mathematical Association Committee recommends that a first introduction 

 to o-eometry, and to each new branch of geometry, should be experimental with 

 the use of instruments and numerical measurements and calculations. This recom- 

 mendation, which is by no means new, seems now to meet with almost universal 

 approval; but, in order to encourage this work, the Report suggests that elemen- 

 tary geometry papers should contain questions requiring the practical use of 

 instruments. 



So far as deductive geometry la concerned, there seem to be four alterna- 

 tives:— 



1. To have no one syllabus placed in the position of authority. 



2. To replace Euclid by one standard syllabus. 



;{. To modify Euclid by omission and readjustment. 

 4. To retain Euclid in its present form. 



The Orst of these .alternatives would make teaching difficult in classes of boys 

 who have previously used diflerent books following different orders, and also in 

 classes of boys preparing for dill'erent examinations which adopted diH'erent 

 schedules: any proposal to abandon the use of one definite authority would meet 

 with the strongest possible opposition, and would probably cause Euclid to be 

 granted a new lease of life. 



The Mathematical Association Committee has recommended the adoption of a 

 modified Euclid ; they considered the time not yet ripe for the proposal of a 

 standard to be finally adopted in place of Euclid. 



Though the Committee considered it wise to retain a modified Euclid for the 

 present, I regard the proposal and adoption of a new standard as the next step 

 to be undertaken, and I hope that the Mathematical Association Committee, 

 consisting as it does of practical teachers and examiners, will draw up a tentative 

 syllabus, to be criticised by schoolmasters and others, and finally submitted to this 

 body for confirmation. Such a syllabus should leave great freedom to the teacher 

 and text-book writer ; it should be so elastic as to allow of changes from time to 

 time, but suiliciently precise to prevent chaos. 



The modifications proposed in the Mathematical Association report include : — 



1. The omission of some propositions which do not help on the course, or 

 •which should be regarded as axiomatic. 



2. Improved methods of proving other propositions, including the use of 

 hypothetical constructions. 



3. The addition of a few propositions. 



4. The adoption of Playfair's axiom and the * limit ' definition of a tangent. 



