July 9, 1009] 



SCIENCE 



45 



inherent in the subject, Professor Newcomb 

 goes on to say that in the teaching of ele- 

 mentary mathematics, especially arithmetic, 

 care should be taken to embody mathemat- 

 ical ideas in a concrete form. The difficulty 

 with the beginner is that he has no clear 

 conception of the real significance of the 

 subject which he is working upon. Figures 

 and algebraic symbols do not represent to 

 his mind anything which he can see or feel. 

 So long as this continues his work consists 

 of formal processes which have no corre- 

 spondence in the world of sense. 



Although a single experience may suffice 

 to establish certain conceptions, it does not 

 follow that the mind can apply these con- 

 cepts in reasoning. It is one thing to know 

 what a thing is but quite a different matter 

 to handle it. This suggests that the diffi- 

 culty in the teaching of elementary mathe- 

 matics may be somewhat obviated by show- 

 ing the mathematical relations among sen- 

 sible objects. 



To illustrate, not much progress was 

 made in the study of imaginaries in algebra 

 until Gauss and Cauehy conceived the idea 

 of representing the two elements which 

 enter into an algebraic imaginary by the 

 position of a point in a plane. The motion 

 of the point embodied the idea of the varia- 

 tion of the quantity, and the study of the 

 subject was reduced to the study of the 

 motion of points; in other words, an ab- 

 straction was replaced by a concrete repre- 

 sentation. The result of this simple repre- 

 sentation was that an extensive branch of 

 mathematics was created, which would have 

 been impossible if the abstract variable of 

 algebra had not been replaced by the mov- 

 ing point of geometry. If such concrete 

 representation is essential for expert math- 

 ematicians, it is obvious that immature 

 pupils should be offered the same ad- 

 vantage. 



In arithmetic, it is suggested that graphic 



methods be used throughout by way of 

 illustration and explanation, lines being 

 drawn to represent the numerical magni- 

 tude of the quantities involved. Actual 

 measurements, hoivever, should not he 

 made, hut magnitudes should be estimated 

 hy the eye. 



This statement is especially noteworthy^ 

 as the idea implied seems to reconcile the 

 differences between the ultra-practical and 

 the ultra-logical extremes. The great dan- 

 ger in the laboratory method is that it will 

 develop manual dexterity at the expense of 

 intellectual power, or, from the ethical 

 standpoint, that it will sacrifice the ideal 

 to the material. The one pedagogical prin- 

 ciple universally recognized, and the one 

 on which it is claimed that the laboratory 

 method is based, is that instruction should 

 proceed from the concrete to the abstract. 

 However, with the extensive laboratory 

 equipment suggested by the advocates of 

 the laboratory method, there must be a/ 

 tendency, through lack of time if for no 

 other reason, to remain with the concrete 

 without making any sensible advance to- 

 ward the abstract. Professor Newcomb 's 

 suggestion regarding the graphical depic- 

 tion of relations without measurement, vis- 

 ualizes the idea to be presented without 

 waste of time or involving the question of 

 accuracy of measurement. The latter ob- 

 viates the essential objection to graphical 

 methods raised by Professor Halsted and 

 others, and thus goes far toward meeting 

 all demands, both critical and practical. 

 Its simplicity, and the fact that this method 

 has been, and is, in constant use, is also in 

 its favor, and must appeal to all teachers 

 who are interested in the progress of their 

 pupils rather than in the exploitation of 

 novel ideas. 



It is also suggested that for at least one 

 half the sums given in arithmetic, there be 

 substituted a course of calculation of sizes, 



