The Psychology of Arithmetic

by Edward L. Thorndike

CHAPTER VI

THE PSYCHOLOGY OF DRILL IN ARITHMETIC: THE AMOUNT OF PRACTICE AND THE ORGANIZATION OF ABILITIES

THE AMOUNT OF PRACTICE

It will be instructive if the reader will perform the following experiment as an introduction to the discussion of this chapter, before reading any of the discussion.

Suppose that a pupil does all the work, oral and written, computation and problem-solving, presented for grades 1 to 6 inclusive (that is, in the first two books of a three-book series) in the average textbook now used in the elementary school. How many times will he have exercised each of the various bonds involved in the four operations with integers shown below? That is, how many times will he have thought, "1 and 1 are 2," "1 and 2 are 3," etc.? Every case of the action of each bond is to be counted.

THE FUNDAMENTAL BONDS

1 + 1	2 − 1	1 × 1	2 ÷ 1
1 + 2	2 − 2	2 × 1	2 ÷ 2
1 + 3		3 × 1
1 + 4		4 × 1
1 + 5	3 − 1	5 × 1	3 ÷ 1
1 + 6	3 − 2	6 × 1	3 ÷ 2
1 + 7	3 − 3	7 × 1	3 ÷ 3
1 + 8		8 × 1
1 + 9		9 × 1
	4 − 1		4 ÷ 1
	4 − 2		4 ÷ 2
11 (or 21 or 31, etc.) + 1	4 − 3	1 × 2	4 ÷ 3
11 " + 2	4 − 4	2 × 2	4 ÷ 4
11 " + 3		3 × 2
11 " + 4		4 × 2
11 " + 5	5 − 1	5 × 2	5 ÷ 1
11 " + 6	5 − 2	6 × 2	5 ÷ 2
11 " + 7	5 − 3	7 × 2	5 ÷ 3
11 " + 8	5 − 4	8 × 2	5 ÷ 4
11 " + 9	5 − 5	9 × 2	5 ÷ 5
	6 − 1	1 × 3	6 ÷ 1
2 + 1	6 − 2	2 × 3	6 ÷ 2
2 + 2	6 − 3	3 × 3	6 ÷ 3
2 + 3	6 − 4	4 × 3	6 ÷ 4
2 + 4	6 − 5	5 × 3	6 ÷ 5
2 + 5	6 − 6	6 × 3	6 ÷ 6
2 + 6		7 × 3
2 + 7		8 × 3
2 + 8	7 − 1	9 × 3	7 ÷ 1
2 + 9	7 − 2		7 ÷ 2
	7 − 3		7 ÷ 3
	7 − 4	1 × 4	7 ÷ 4
12 (or 22 or 32, etc.) + 1	7 − 5	2 × 4	7 ÷ 5
12 " + 2	7 − 6	and so on	7 ÷ 6
	7 − 7	to 9 × 9	7 ÷ 7
and so on to	and so on		and so on to
9 + 9	to 18 − 9		82 ÷ 9 83 ÷ 9, etc.
19 (or 29 or 39, etc.) + 9

If estimating for the entire series is too long a task, it will be sufficient to use eight or ten from each, say:—

3 + 2	13, 23, etc. + 2	7 + 2	17, 27, etc. + 2
" 3	" 3	" 3	" 3
" 4	" 4	" 4	" 4
" 5	" 5	" 5	" 5
" 6	" 6	" 6	" 6
" 7	" 7	" 7	" 7
" 8	" 8	" 8	" 8
" 9	" 9	" 9	" 9