Part 30 (1/2)
The repet.i.tion soon becomes monotonous, but the exercises may be most easily changed, taking again the set of long rods, and instead of placing rod number one after nine, place it after ten. In the same way, place two after nine, and three after eight. In this way we make rods of a greater length than ten; lengths which we must learn to name eleven, twelve, thirteen, etc., as far as twenty. The little cubes, too, may be used to fix these higher numbers.
Having learned the operations through ten, we proceed with no difficulty to twenty. The one difficulty lies in the _decimal numbers_ which require certain lessons.
LESSONS ON DECIMALS: ARITHMETICAL CALCULATIONS BEYOND TEN
The necessary didactic material consists of a number of square cards upon which the figure ten is printed in large type, and of other rectangular cards, half the size of the square, and containing the single numbers from one to nine. We place the numbers in a line; 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. Then, having no more numbers, we must begin over again and take the 1 again. This 1 is like that section in the set of rods which, in rod number 10, extends beyond nine. Counting along _the stair_ as far as nine, there remains this one section which, as there are no more numbers, we again designate as 1; but this is a higher 1 than the first, and to distinguish it from the first we put near it a zero, a sign which means nothing. Here then is 10. Covering the zero with the separate rectangular number cards in the order of their succession we see formed: 11, 12, 13, 14, 15, 16, 17, 18, 19. These numbers are composed by adding to rod number 10, first rod number 1, then 2, then 3, etc., until we finally add rod number 9 to rod number 10, thus obtaining a very long rod, which, when its alternating red and blue sections are counted, gives us nineteen.
The directress may then show to the child the cards, giving the number 16, and he may place rod 6 after rod 10. She then takes away the card bearing 6, and places over the zero the card bearing the figure 8, whereupon the child takes away rod 6 and replaces it with rod 8, thus making 18. Each of these acts may be recorded thus: 10 + 6 = 16; 10 + 8 = 18, etc. We proceed in the same way to subtraction.
When the number itself begins to have a clear meaning to the child, the combinations are made upon one long card, arranging the rectangular cards bearing the nine figures upon the two columns of numbers shown in the figures A and B.
A B +----+ +----+
10
10
10
20
10
30
10
40
10
50
10
60
10
70
10
80
10
90
+----+ +----+
Upon the card A we superimpose upon the zero of the second 10, the rectangular card bearing the 1: and under this the one bearing two, etc.
Thus while the one of the ten remains the same the numbers to the right proceed from zero to nine, thus:
In card B the applications are more complex. The cards are superimposed in numerical progression by tens.
+----+
10
11
12
13
14
15
16
17
18
19