MATHEMATICS FOR CHILDREN. 801 



shrink from initiating him into hard work, if that can be done in 

 a rational way. 



I regard all the sciences as, at least to a certain extent, experi- 

 mental, and, notwithstanding the views of those who would regard 

 the mathematical sciences as a series of operations in pure logic, 

 resting upon strictly ideal conceptions, I believe that we may affirm 

 that there does not exist a mathematical idea that can enter our 

 brain without the previous contemplation of the outer world and 

 the facts it offers to our observation. This affirmation, the discus- 

 sion of which now would carry us too far, may help to a clear idea 

 of the way we should try to convey the first mathematical ideas 

 to the mind of the child. 



The outer world is the first thing the child should be taught to 

 regard and concerning which he should be given as much informa- 

 tion as possible information which he will have no trouble in stor- 

 ing, we may well believe, and from this outer world the first mathe- 

 matical notions should be borrowed; to these should succeed later 

 an abstraction, which is less complicated than it seems. 



Our primary teaching of arithmetic now follows in the tracks 

 of that of grammar, as we might as well say that the teaching of 

 grammar follows in the tracks of that of arithmetic. That is, in 

 either case we teach the child a number of abstract and confusing 

 definitions which he can not comprehend, imposing on him a series 

 of rules to follow under the pretext of giving him a good practical 

 direction, and we force him to learn and memorize these rules 

 whether they are good for anything or not. 



When the child has grown older he is given two or three short 

 lessons a week in science, nine tenths of which, with his fleeting 

 memory, he forgets before the next week's lessons come on. He 

 can not relish anything that is taught him in that way, and it would 

 be vastly better to give him no scientific ideas at all than to scatter 

 them around in such a way, for all teachers agree that a fresh pupil 

 is more easily dealt with and can be taught more satisfactorily and 

 thoroughly than one who has been mistaught. 



When the student has passed through it all and has established 

 himself in life he is apt to look back upon his experiences under 

 such teachings in no very amiable mood, and to regard such matters 

 in the light of barriers that were set up to prevent his getting his 

 diploma with too little work; and even if his profession is one that 

 calls for applications of mathematics he prepares himself with sets 

 of formulas that enable him to dispense with the imperfect instruc- 

 tion he has received. 



When we think of giving a child a mathematical education we 

 are apt to ask whether he has special aptitudes fitting him to receive 



VOL. LV. 56 



