July 9, 1909] 



SCIENCE 



47 



which the observation of the child should 

 be directed, and with snch practise, numer- 

 ation and figuring will follow of necessity. 

 The aim of arithmetic should be chiefly to 

 emphasize those arithmetical magnitudes 

 which relate to practical life, and these 

 should be presented to the child as objects 

 of observation. 



In order to determine the subject matter, 

 it is necessary to consider that positive 

 phases of the subject considered relate to 

 practical life and in how far through the 

 consideration of the subject matter can the 

 moral and the cultural side of the child be 

 nurtured. 



The present lesson plans expect too much 

 of the observational and comprehensive 

 powers of the child, and it is certainly not 

 a loss to the child if in the first-year arith- 

 metic is not taught. In emphasizing the 

 fundamental idea that numeration and 

 arithmetical operations are chiefly to be 

 acquired through the observations of arith- 

 metical magnitudes, Fitzga expressly states 

 that he does not approve of the rigorous 

 handling of the subject matter. This, he 

 says, naturally assists the child in acquiring 

 thoughtless habits. 



The conception of a thing only becomes 

 clear when it is the abstraction of many 

 similar and clear presentations. For this 

 reason a teacher should not develop prin- 

 ciples during a lesson in arithmetic, but 

 should discourage the use of principles 

 even by the brighter pupils, who have 

 grasped these principles during the process 

 of development, for the weaker pupils will 

 immediately use them, and will not trouble 

 themselves to acquire them through a proc- 

 ess of logical reasoning. The teacher who 

 wishes to bring his pupils to a clear com- 

 prehension of arithmetical problems can 

 not develop any principles, but must wait 

 until the child shall see the principles him- 

 self in the progress of his work. 



Instead of the rigorous treatment of the 

 subject, and the development of principles, 

 Fitzga has, therefore, divided the separate 

 parts of the subject matter into elements 

 from which follows a logical arrangement 

 based on the idea of repeated observations 

 and presentations necessary to form clear 

 conceptions of the different arithmetical 

 operations.^^ 



He also points out that it is necessary 

 to choose examples in such a way that the 

 context of the problem is comprehensible to 

 the child. For this purpose it is necessary 

 to bring to the mind of the child the vari- 

 ous relations of practical life, out of which 

 the examples are taken, and this can be 

 done by giving a system of logical observa- 

 tion lessons. 



The method of giving observational les- 

 sons as the basis of mathematical instruc- 

 tion has been explained in considerable 

 detail by Jackman." The article referred 

 to is primarily a plea for the correlation of 

 mathematics and physics, on the ground 

 that mathematics has become so isolated 

 that it is universally considered as the bug- 

 bear of the curriculum. This, it is stated, 

 is due to the fact that the problems deal 

 with subject matter with which the pupil 

 has no concern, and mathematics has thus 

 become simply a science of empty processes. 



The question of the hour in the teaching 

 of mathematics is not how do pupils in 

 their thinking develop ideas of exact quan- 

 titative results, but ideas of what quantita- 

 tive results are worth developing. 



The unfortunate position in which math- 

 ematics finds itself is also partly due to the 

 mistaken idea that a constant manipulation 

 of mathematical formulas has a peculiar 

 disciplinary value, whether they give any 

 decidedly useful results or not. Great 



" For courses in detail see his book, mentioned 

 above. 



-" Jackman, Educational Review, Vol. 25. 



