MATHEMATICS FOR CHILDREN. 809 



that objects material bars, for example can be selected so as to 

 represent numbers by their length. He can be readily made to un- 

 derstand that if he has one bar three and another five inches long 

 he can obtain the sum of these lengths, in what we might call a ma- 

 terial way, by placing them lengthwise, one at the end of the other 

 an essentially practical notion and easily carried into effect. If we 

 take a line and mark a starting point on it, calling it zero, then meas- 

 ure off segments on it representing the bars we have been talking 

 about one after another, we can get the sum represented by the 

 length of the two segments. If, instead of measuring three plus 

 five inches I measure three plus two I reach another point. If, 

 instead of adding two and three, I wish to take one of the bars or 

 numbers away (3 2), or subtract, the operation will be easily per- 

 formed by measuring the two in the opposite direction. The differ- 

 ence will be represented by the length that is left. If we try to 

 form the quantity 3 5 in arithmetic we can not do it; but in pro- 

 ceeding in this method and measuring back on the bar we get to a 

 point back of the original starting point which represents this differ- 

 ence say two inches behind where we began. Here we have in 

 the germ the whole theory of negative quantities, concerning which 

 thousands and thousands of pages have been written. Yet we find 

 that by carefully graduating our lines we can make it intuitive and 

 accessible to a child who has learned that the common operations of 

 addition and subtraction can be represented with material objects. 

 The generation of negative and positive quantities follows quite 

 naturally. 



These examples, I think, are sufficient to show that we might 

 considerably enlarge the field of the investigations within reach of 

 the child. For this purpose a small amount of very simple mate- 

 rial, which we can vary as we please, is needful. The first element 

 of this material is paper ruled in squares, a wonderful instrument, 

 which everybody dealing with mathematics or with science gener- 

 ally should have. It is of special pedagogic use in giving children 

 their first ideas of form, size, and position, without which their early 

 instruction is only a delusion. Add to this paper dice, buttons, 

 beans, and match-sticks things always easy to get and we have 

 all the material we need. 



There is no amusement, however puerile it may appear, not even 

 a play of words, that can not be utilized in teaching of this sort. 

 For instance, when your child has learned his addition table, if you 

 put him to a demonstration, assuming to prove to his comrades that 

 six and three make eight, his curiosity will be excited, and you may 

 be very sure that, once his attention has been given to this amuse- 

 ment, he will never forget that six and three make nine and not 



