446 REPORTS ON THE STATE OF SCIENCE. 



facility than of clear thinking or of knowledge. The ideas of ratio and 

 proportion should be developed concurrently with the use of vulvar 

 fractions. Decimals should be introduced at an early stage, soon after 

 the notion of fractions has been grasped. Methods of calculation, accu- 

 rate only to specified significant figures, and, in particular, the practice of 

 contracted methods, should be encouraged. The use of tables of simple 

 functions should be begun as soon as the student is capable of understand- 

 ing the general nature of the functions tabulated ; for example, the use 

 of logarithms in numerical calculation may be begun as soon as the funda- 

 mental law of indices is known.' 



Recommendations of Mathematical Association Committee.^ 



' 1. The Committee consider that there is considerable danger of the 

 true educational value of arithmetic and algebra being seriously impaired 

 by reason of a tendency to sacrifice clear understanding to mere mecha- 

 nical skill. 



' 2. In view of this the Committee recommend : — 



■- ' (a) That easy viva-voce examples should be frequently used in both 

 arithmetic and algebra ; 



' (b) That great stress should be laid on fundamental principles ; 



' (c) That, as far as possible, the rules which a pupil uses should be 

 generalisations from his own experience ; 



' (d) That, whenever practicable, geometry should be employed to illus- 

 trate arithmetic and algebra, and, in particular, that graphs should be 

 used extensively ; 



' (e) That many of the harder rules and heavier types of examples 

 which examinations alone compel us to i-etain in a school curriculum 

 should be postponed.' 



' With these as guiding principles the Committee are led to make the 

 following suggestions : — 



' Arithmetic. — 3. That, as a preparation for contracted multiplication, 

 pupils should, in multiplying, from the very first be taught to begin with 

 the highest digit of the multiplier. 



'4. That first principles should be carefully taught before vulgar 

 fractions are begun, special stress being laid on factors with index nota- 

 tion and the use of the signs -f , — , X , -=-, and of brackets. 



' 5. That, as far as possible, prime factors should be used in finding 

 H.C.F., L.C.M., and square and cube roots. 



' 6. That much time should not be spent in teaching complicated 

 vulgar fractions. 



' 7. That, in order to facilitate the teaching of decimals, a scale 

 divided decimally (both in inches and centimetres) should be used for 

 the actual measurement of lines ; the scale should be subsequently used 

 for the determination of areas and volumes. 



'8. That in teaching decimals concrete examples should be taken 

 from the metric system. 



' 9. That approximate work with decimals should be introduced early, 

 and that recurring decimals should be treated in this way, the theory of 

 recurring decimals being postponed to a later stage. 



' Extract from the Report of a Committee of the Mathematical Association on 

 the Teaching of Arithmetic and Algebra. Mathematical Gazette, July 1902. 



