802 POPULAR SCIENCE MONTHLY. 



it. Do we ask any such questions when we talk of teaching him to 

 read and write? Oh, no! we all acknowledge that reading and 

 writing are useful, practical, and indispensable arts, which every 

 human being not infirm or defective should learn. Now, elemen- 

 tary mathematics, which represents a tolerably extended equip- 

 ment, is no less useful and indispensable than the knowledge of 

 reading and writing, and I assert further, what may seem para- 

 doxical to many, that it can be assimilated with much less fatigue 

 than the earliest knowledge of reading and writing, provided al- 

 ways that instead of proceeding in the usual way and giving lessons 

 bristling with formulas and rules, appealing to the memory, im- 

 posing fatigue, and producing nothing but disgust, we adopt the 

 philosophical method of conveying ideas to the child by means of 

 objects within reach of his senses. The teaching should be wholly 

 concrete and applied only to the contemplation of external objects 

 and their interpretation, and the instruction should be given con- 

 tinually, especially during the primary period, under the form of 

 play. Nothing is easier than this, then, in arithmetic; for instance, 

 to use dice, beans, balls, sticks, etc., and by their aid give the child 

 ideas of numbers. 



Do we do anything of this kind? "When I was taught to read 

 and write I knew how to write the figure 2 before I had any idea 

 of the number two. Nothing is more radically contrary to the 

 normal working of the brain than this. The notion of numbers 

 up to 10, for example should be given to the child before ac- 

 customing him to trace a single character. That is the only way 

 of impressing the idea of number independently of the symbol or 

 the formula which is only too ready to take the place in the mind 

 of the object represented by it. 



When a child has learned to count through the use of such 

 objects as I have mentioned he may be taught what is called the 

 addition table. This table can be learned by heart easily enough, 

 but when we reach the multiplication table we come upon one of 

 the tortures of childhood. Would it not be simpler and easier to 

 make the children construct these tables, instead of making them 

 learn them? 



Let us first take the addition table, and suppose that we trace 

 ten columns on suitably ruled paper, at the top of which we write 

 the first ten numbers, for example, and then write them again at the 

 beginning of a certain number of horizontal lines (Fig. 1). Let us 

 suppose, too, that we have a box divided into compartments arranged 

 like the squares in our table, into which we put heaps of balls, beans, 

 or dice corresponding to the numbers indicated in the table. The 

 child will take, for example, two balls from one compartment and 



