A Study on the Process of the Multiplication

Abstract

The multiplication is said to be the progressive calculation of the second stage and to develop from the addition which is the progressive calculation of the first stage. What is the process of this development? In what points are they different from each other? As to these questions the following formula α+α +α +...+α=a.m seems to be very suggestive, but it tells us psychologically nothing. This, article intends to pursue the reasons why many aα in the left side can be represented by one α in the right, why the parenthesized m becomes to be able to rank with the α, and why the plus signs change into a multiplication sign, etc.
The subjects were 15 boys of 7-8 years old.A certain number of dishes,each of which contained several beans, were presented to the subjects and they were asked to tell the total number of beans.The procedure of getting the results was freely left to the subjects, and their behaviours were very carefully observed and recorded. The experiment was repeated four times for the same subject and the results were compared.
Four types were classified of the behaviours, which the subjects showed when they dealt with these groups of beans.
I-Type: The subjects were indifferent to the groups. Each element of the group was perceived separately and directly pointed at one by one. One to one corresponding to this action of pointing, the series of number words were uttered and at last the total number was announced. This method of operation may be called counting.
II-Type: Each group was noticed and pointed as such one by one. The number of elements of each group'was added at a time and at last the total sum was aequired. This operation is called adding.
III-Tybe: The grouping of two groups, one of which being equated to the other, was a chiefc haracteristiocf this type.The number of groups pointed at one time increasedg eometricallyf,o r instance,a t firsto ne, then 2, then 4. So the number which was acquired increasdd by two fold at every time. This operation may becalled doubling, or doubling and adding.
IV-Type:In the first place all the groups were counted and the number of them was found Then the total number of elements of all groups was produced at once by the help of the multipIication-table.This opermtion is called multiplying.
These four types of behaviours are related to each other genetically and the I-Type transforms into the II-Type and the II-Type into the III and so on. (See Tab. 12 in the Japanese Text p. 173) From this fact we can conclude that adding develops from counting, doubling from adding and multiplying from doubling. The first three of these operations are successive methods and the last the simultaneous.
The simultaneous acquisition is seen evidently in the behaviour of doubling, but this behaviour is limited and applied only to a special case. The behaviour of multiplying is quite free from such a limitation. Accordingly the process of the multiplication is formed as follows;
a)to recognise each group as equal which is given as such and to treat it as an unit,
b)to count those units and to find the number of all groups,
c)to get the third number by the help of the multiplication-table to which two kinds of numbers are applied,
d)to understand this number, the product, as the total number of all elements of all groups that are given.
In this behaviqur, we can see, how great a part the multiplication-table plays. This table, to be sure, is created and constructed by an operator himself and never given from outside, though he is led by his teacher to systematize it logically and to make use of it generally.