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POPULAR SCIENCE MONTHLY. 



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FIG. 9. 



eight. To make the demonstration, we have only to group the 

 nine match-sticks as in the figure (Fig. 9) below. We might demon- 

 strate in a like way that half of twelve is seven by cutting the Roman 

 numeral XII in two, leaving the upper part visible. Such pleas- 

 antries have a pedagogical value, because the paradox is precisely 



of a kind to attract 

 the attention of the 

 child, and he will al- 

 ways afterward be 

 sure not to fall into 

 the trap. 



The side of this 

 kind of instruction 

 on which I insist 

 most is that, given 

 under the form of play, it is free from every sort of dogmatic char- 

 acter. No truth should be imposed on the child; on the contrary, 

 he should be allowed to discover it as a fruit of his own activity. 

 He will be thoroughly impressed with the truths which he has thus 

 found out himself. They had better be few at first; the impor- 

 tant thing is for him to know them completely. 



The instruction should also be essentially objective and free from 

 all abstraction. The absence of abstraction should, however, be 

 rather apparent than real. Abstraction is indeed one of the ele- 

 ments that contribute most to give mathematical science a fearful 

 air to outsiders, and yet it is most usually a simplification of mat- 

 ters quite the contrary of what is generally supposed. It is, in 

 fact, such a simplification and so necessary that we all make it as 

 if by instinct, and the child makes it, not in mathematics only, but in 

 all the considerations of life. 



Thus, when I want to give the child his first idea of the number 

 two I put two beans in his hand and let him contemplate them. He 

 gets a perfect notion of the collection two. Yet, if you look at 

 them a little closer and he himself looks at them closer he will find 

 that the two beans, whatever else they may be, are not identical, 

 for there exist no two objects in Nature that are not different. So 

 when the child introduces this idea of collection into his mind in 

 a wholly instinctive way, by identifying the things he sees, he begins 

 to perform abstraction. This abstraction delivers him from all the 

 complications and all the annoyances that come to him from the 

 contemplation of real objects. By the philosophic process of ab- 

 straction it has been possible to construct all the sciences, and espe- 

 cially the science of magnitudes. 



The ideas I have been setting forth in outline are not mine, and 



