Part 29 (2/2)
ADDITION AND SUBTRACTION FROM ONE TO TWENTY: MULTIPLICATION AND DIVISION
The didactic material which we use for the teaching of the first arithmetical operations is the same already used for numeration; that is, the rods graduated as to length which, arranged on the scale of the metre, contain the first idea of the decimal system.
The rods, as I have said, have come to be called by the numbers which they represent; one, two, three, etc. They are arranged in order of length, which is also in order of numeration.
The first exercise consists in trying to put the shorter pieces together in such a way as to form tens. The most simple way of doing this is to take successively the shortest rods, from one up, and place them at the end of the corresponding long rods from nine down. This may be accompanied by the commands, ”Take one and add it to nine; take two and add it to eight; take three and add it to seven; take four and add it to six.” In this way we make four rods equal to ten. There remains the five, but, turning this upon its head (in the long sense), it pa.s.ses from one end of the ten to the other, and thus makes clear the fact that two times five makes ten.
These exercises are repeated and little by little the child is taught the more technical language; nine plus one equals ten, eight plus two equals ten, seven plus three equals ten, six plus four equals ten, and for the five, which remains, two times five equals ten. At last, if he can write, we teach the signs _plus_ and _equals_ and _times_. Then this is what we see in the neat note-books of our little ones:
9 + 1 = 10 8 + 2 = 10 5 2 = 10 7 + 3 = 10 6 + 4 = 10
When all this is well learned and has been put upon the paper with great pleasure by the children, we call their attention to the work which is done when the pieces grouped together to form tens are taken apart, and put back in their original positions. From the ten last formed we take away four and six remains; from the next we take away three and seven remains; from the next, two and eight remains; from the last, we take away one and nine remains. Speaking of this properly we say, ten less four equals six; ten less three equals seven; ten less two equals eight; ten less one equals nine.
In regard to the remaining five, it is the half of ten, and by cutting the long rod in two, that is dividing ten by two, we would have five; ten divided by two equals five. The written record of all this reads:
10 - 4 = 6 10 - 3 = 7 10 2 = 5 10 - 2 = 8 10 - 1 = 9
Once the children have mastered this exercise they multiply it spontaneously. Can we make three in two ways? We place the one after the two and then write, in order that we may remember what we have done, 2 + 1 = 3. Can we make two rods equal to number four? 3 + 1 = 4, and 4 - 3 = 1; 4 - 1 = 3. Rod number two in its relation to rod number four is treated as was five in relation to ten; that is, we turn it over and show that it is contained in four exactly two times: 4 2 = 2; 2 2 = 4. Another problem: let us see with how many rods we can play this same game. We can do it with three and six; and with four and eight; that is,
2 2 = 4 3 2 = 6 4 2 = 8 5 2 = 10 10 2 = 5 8 2 = 4 6 2 = 3 4 2 = 2
At this point we find that the cubes with which we played the number memory games are of help:
2 4 6 8 10 X X
X XX X
X XX X
X XX X
X XX X
X
X X
X XX X
X XX X
X XX X
X
X X
X XX X
X XX X
X
X X
X XX X
X
X X
X
From this arrangement, one sees at once which are the numbers which can be divided by two--all those which have not an odd cube at the bottom.
These are the even numbers, because they can be arranged in pairs, two by two; and the division by two is easy, all that is necessary being to separate the two lines of twos that stand one under the other. Counting the cubes of each file we have the quotient. To recompose the primitive number we need only rea.s.semble the two files thus 2 3 = 6. All this is not difficult for children of five years.
<script>