The Psychology of Arithmetic

by Edward L. Thorndike

THE NEED OF STRONGER ELEMENTARY BONDS

The constituent bonds involved in the fundamental operations with numbers need to be much stronger than they now are. Inaccuracy in these operations means weakness of the constituent bonds. Inaccuracy exists, and to a degree that deprives the subject of much of its possible disciplinary value, makes the pupil's achievements of slight value for use in business or industry, and prevents the pupil from verifying his work with new processes by some previously acquired process.

The inaccuracy that exists may be seen in the measurements made by the many investigators who have used arithmetical tasks as tests of fatigue, practice, individual differences and the like, and in the special studies of arithmetical achievements for their own sake made by Courtis and others.

Burgerstein ['91], using such examples as

28704516938276546397 + 35869427359163827263

and similar long numbers to be multiplied by 2 or by 3 or by 4 or by 5 or by 6, found 851 errors in 28,267 answer-figures, or 3 per hundred answer-figures, or ³⁄₅ of an error per example. The children were 9½ to 15 years old. Laser ['94], using the same sort of addition and multiplication, found somewhat over 3 errors per hundred answer-figures in the case of boys and girls averaging 11½ years, during the period of their most accurate work. Holmes ['95], using addition of the sort just described, found 346 errors in 23,713 answer-figures or about 1½ per hundred. The children were from all grades from the third to the eighth. In Laser's work, 21, 19, 13, and 10 answer-figures were obtained per minute. Friedrich ['97] with similar examples, giving the very long time of 20 minutes for obtaining about 200 answer-figures, found from 1 to 2 per hundred wrong. King ['07] had children in grade 5 do sums, each consisting of 5 two-place numbers. In the most accurate work-period, they made 1 error per 20 columns. In multiplying a four-place by a four-place number they had less than one total answer right out of three. In New York City Courtis found ['11-'12] with his Test 7 that in 12 minutes the average achievement of fourth-grade children is 8.8 units attempted with 4.2 right. In grade 5 the facts are 10.9 attempts with 5.8 right; in grade 6, 12.5 attempts with 7.0 right; in grade 7, 15 attempts with 8.5 right; in grade 8, 15.7 attempts with 10.1 right. These results are near enough to those obtained from the country at large to serve as a text here.

The following were set as official standards, in an excellent school system, Courtis Series B being used:—

	Grade.	Speed Attempts.	Percent of Correct Answers.
Addition	8	12	80
	7	11	80
	6	10	70
	5	9	70
	4	8	70
Subtraction	8	12	90
	7	11	90
	6	10	90
	5	9	80
	4	7	80
Multiplication	8	11	80
	7	10	80
	6	9	80
	5	7	70
	4	6	60
Division	8	11	90
	7	10	90
	6	8	80
	5	6	70
	4	4	60

Kirby ['13, pp. 16 ff. and 55 ff.] found that, in adding columns like those printed below, children in grade 4 got on the average less than 80 percent of correct answers. Their average speed was about 2 columns per minute. In doing division of the sort printed below children of grades 3 B and 4 A got less than 95 percent of correct answers, the average speed being 4 divisions per minute. In both cases the slower computers were no more accurate than the faster ones. Practice improved the speed very rapidly, but the accuracy remained substantially unchanged. Brown ['11 and '12] found a similar low status of ability and notable improvement from a moderate amount of special practice.

3	5	6	2	3	8	9	7	4	9
7	9	6	5	5	6	4	5	8	2
3	4	7	8	7	3	7	9	3	7
8	8	4	8	2	6	8	2	9	8
2	2	4	7	6	9	8	5	6	2
6	9	5	7	8	5	2	3	2	4
9	6	4	2	7	2	9	4	4	5
3	3	7	9	9	9	2	8	9	7
6	8	9	6	4	7	7	9	2	4
8	4	6	9	9	2	6	9	8	9
—	—	—	—	—	—	—	—	—	—