ON TEACHING ELEMENTARY MATHEMATICS. 477 



mechanical facility than of clear thinking or of knowledge. The ideas of 

 ratio and proportion should be developed concurrently with the use of 

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

 after the notion of fractions has been grasped. Methods of calculation, 

 accurate only to specified significant figures, and, in particular, the prac- 

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

 simple functions should be begun as soon as the student is capable of 

 understanding the general nature of the functions tabulated ; for example, 

 the use of logarithms in numerical calculation may be begun as soon aa 

 the fundamental law of indices is known. 



In regard to the early stages of algebra, the modifications (both in 

 teaching and in the examinations) which are deemed desirable by the 

 Committee are of a general character. 



At first, the formulffi should be built on a purely arithmetical founda- 

 tion, and their significance would often be exhibited by showing how they 

 include whole classes of arithmetical results. Throughout the early 

 stages, formula and results should frequently be tested by arithmetical 

 applications. The arithmetical basis of algebra could be illustrated for' 

 beginners by the frequent use of graphs ; and the pi-actice of graphical 

 processes in such cases can give a significance to algebraical formulas' 

 that would not otherwise be obtained easily in early stages of the subject. 



In passing to new ideas, only the simplest instances should be used 

 at first, frequent refei'ence still being made to arithmetical illustrations. 

 Advance should be made by means of essential development, avoiding the 

 useless complications of merely formal difficulties which serve no other 

 purpose than that of puzzling candidates in examinations. Many of the 

 artificial combinations of difficulties could be omitted entirely ; the discus- 

 sion of such as may be necessary should be postponed from the earlier 

 stages. Teachers and examiners alike should avoid matters such as curious 

 combinations of brackets ; extravagantly complicated algebraic expres- 

 sions, particularly fractions ; resolutions of elaborate expressions into 

 factors ; artificially difficult combinations of indices ; ingeniously manipu- 

 lated equations : and the like. They have no intrinsic value or import- 

 ance ; it is only the mutual rivalry between some writers of text-books 

 and some examiners that is responsible for the consideration which has 

 been conceded to such topics. 



General Remarks. 



If general simplification either on these or on similar lines be adopted, 

 particularly if graphical methods are freely used, it will be found possible 

 to introduce, quite naturally and much earlier than is now the case, some 

 of the leading ideas in a few subjects that usually are regarded as more 

 advanced. Thus the foundations of trigonometry can be laid in connec- 

 tion with the practical geometry of the subject-matter of the Sixth Book 

 of Euclid. The general idea of co-ordinate geometry can be made familiar 

 by the use of graphs ; and many of the notions underlying the methods 

 of the infinitesimal calculus can similarly be given to comparatively 

 youthful students long before the formal study of the calculus is begun. 



