COUNTING BY TENS
(For more advanced classes)
In the store ten objects are sold for one cent, e.g.:
(10 beans), one cent for each ten.
One ten = ten, 10.
Two tens = twenty, 20.
Three tens = thirty, 30, etc.
From forty on (in English from sixty on) the numbers are more easily learned because their names are like simple numbers with the ending -ty (Italian -anta).
Charts should be prepared (rectangular in shape) on which nine tens appear arranged one under the other; then nine cards where each ten is repeated nine times in a column; finally, numerous cards with the unit figures 1, 2, 3, 4, 5, 6, 7, 8, 9, to be fitted on the zeros on the cards where the tens are repeated nine times.
10—10—20
20—10—20
30—10—20
40—10—20
50—10—20
60—10—20
70—10—20
80—10—20
90—10—20Some difficulty will be experienced with the tens where the names do not correspond to the simple numbers: 11, 12, 13, etc. The other tens, however, will be very easy. When a little child is able to count to 20, he can go on to 100 without difficulty. The next step is to superimpose the little cards on the first chart of the tens series, having the resultant numbers read aloud.
Problems: Problems are, at first, simple memory exercises for the children. In fact the problems are solved practically in the store in the form of a game; buying, lending, sharing with their schoolmates, taking a part of what is bought and giving it to some other child, etc. The store exercises should be repeated in the form of a problem on the following morning. The children have simply to remember what happened and reproduce it in writing. Problems are next developed contemporaneously with the various arithmetical operations and computations (addition, multiplication, etc.). The teacher explains the operations starting with the problem, which becomes for the children a very amusing game. The problem, finally, becomes an imaginative exercise: "Suppose you are going to the store to buy," etc., etc. We can ultimately arrive at real problems that require reasoning. In the store the teacher illustrates the various operations on the blackboard, using simple marks at first: "You have bought 2c worth of beans, at three for a cent. Let us write that down: III—III. Then let us count. 1, 2, 3, 4, 5, 6. There are six.
III III Well, then, 3 + 3 = 6. We can also say: 2 groups of III equals 6; twice, three, six; two times three, six; 2 × 3 = 6. How much is 3 + 3? How much is 2 × 3? How much is 3 × 2?"
The following morning, when the written problem is given, the child should have before him for reference the computation charts with all the combinations possible.
The transition to mental computation will come after this and not before.
| III III | ||
| Well, then, | 3 + 3 | = 6. |
Sample Cards
(Addition)
| 1 + 1 = 2 | 2 + 1 = 3 | 3 + 1 = 4 |
| 1 + 2 = 3 | 2 + 2 = 4 | 3 + 2 = 5 |
| 1 + 3 = 4 | 2 + 3 = 5 | 3 + 3 = 6 |
| 1 + 4 = 5 | 2 + 4 = 6 | 3 + 4 = 7 |
(Multiplication)
| 1 × 1 = 1 | 2 × 1 = 2 | 3 × 1 = 3 |
| 1 × 2 = 2 | 2 × 2 = 4 | 3 × 2 = 6 |
| 1 × 3 = 3 | 2 × 3 = 6 | 3 × 3 = 9 |
Subtraction in the same way. The development of these various operations followed logically on the practical exercise in the store, where multiplication proved to be a product of sums, division, a process of successive subtractions.
In our classes we have arithmetic lessons every day. The afternoon practice in the store prepares for the theoretical lesson of the following morning. Accordingly, on the day when the practical exercise occurs, there is no theoretical lesson and vice versa.
The decimal metric system applied to weights, measures and coinage is taught in the same way. The store should be equipped with scales, weights, dry and liquid measures, etc. All kinds of coins should be available, including bills up to $20 (100 francs). Work in the store should continue to be not only a help toward arithmetical computation but also toward the preparation for practical life. For instance, when cloth is sold, some attention should be given to its actual market value; its qualities should be emphasized by feeling, etc.; and the child should be taught to observe whether the storekeeper has given him the right amount and the right quality. Money changing should be made ready and easy. The money which the children spend at the store should be earned by them as a reward for their application to study and their good behavior.