Here he must ask himself certain questions. ‘How many apples altogether? How shall I find out? Then I must divide the apples into twenty-seven heaps to find out each boy’s share.’ That is to say, the child perceives what rules he must apply to get the required information. He is interested; the work goes on briskly: the sum is done in no time, and is probably right, because the attention of the child is concentrated on his work. Care must be taken to give the child such problems as he can work, but yet which are difficult enough to cause him some little mental effort.
Demonstrate.—The next point is to demonstrate everything demonstrable. The child may learn the multiplication-table and do a subtraction sum without any insight into the rationale of either. He may even become a good arithmetician, applying rules aptly, without seeing the reason of them; but arithmetic becomes an elementary mathematical training only in so far as the reason why of every process is clear to the child. 2 + 2 = 4, is a self-evident fact, admitting of little demonstration; but 4 × 7 = 28 may be proved.
He has a bag of beans; places four rows with seven beans in a row; adds the rows, thus: 7 and 7 are 14, and 7 are 21, and 7 are 28; how many sevens in 28? 4. Therefore it is right to say 4 × 7 = 28; and the child sees that multiplication is only a short way of doing addition.
A bag of beans, counters, or buttons should be used in all the early arithmetic lessons, and the child should be able to work with these freely, and even to add, subtract, multiply, and divide mentally, without the aid of buttons or beans, before he is set to ‘do sums’ on his slate.
He may arrange an addition table with his beans, thus—
| ⬭⬭ | ⬭ | = 3 beans |
| ⬭⬭ | ⬭⬭ | = 4 ” |
| ⬭⬭ | ⬭⬭⬭ | = 5 ” |
and be exercised upon it until he can tell, first without counting, and then without looking at the beans, that 2 + 7 = 9, etc.
Thus with 3, 4, 5,—each of the digits: as he learns each line of his addition table, he is exercised upon imaginary objects, ‘4 apples and 9 apples,’ ‘4 nuts and 6 nuts,’ etc.; and lastly, with abstract numbers—6 + 5, 6 + 8.
A subtraction table is worked out simultaneously with the addition table. As he works out each line of additions, he goes over the same ground, only taking away one bean, or two beans, instead of adding, until he is able to answer quite readily, 2 from 7? 2 from 5? After working out each line of addition or subtraction, he may put it on his slate with the proper signs, that is, if he have learned to make figures. It will be found that it requires a much greater mental effort on the child’s part to grasp the idea of subtraction than that of addition, and the teacher must be content to go slowly—one finger from four fingers, one nut from three nuts, and so forth, until he knows what he is about.
When the child can add and subtract numbers pretty freely up to twenty, the multiplication and division tables may be worked out with beans, as far as 6 × 12; that is, ‘twice 6 are 12’ will be ascertained by means of two rows of beans, six beans in a row.