# The American Journal of Psychology/Volume 21/Practice in the Case of Addition

The American Journal of Psychology, Vol. 21, No. 3 (July, 1910), pp. 483-486.
PRACTICE IN THE CASE OF ADDITION.
By Edward L. Thorndike, Teachers College, Columbia University.
The experiment reported here was designed to secure information concerning only the amount and rate of improvement and the value of the practice experiment as a method for school work. The practice was not continued long enough nor taken under uniform enough conditions to justify inferences concerning changes in the rate of improvement; and I shall make no attempt to analyze out the factors producing the improvement.
The experiment consisted in adding daily for seven days fortyeight columns each of ten numbers (no I's or o's being included). Seven printed blanks had been arranged of equal difficulty. The forty-eight sums were written. The time required was recorded in seconds. The subjects were nineteen university students eight men and eleven women.
The time taken and the number of examples wrong for each set for each of the nineteen subjects are recorded in Table I. Table II repeats Table I with an addition of one per cent, of the time for fortyeight examples for each example wrong. That is, I estimate that half the time for one example is a just allowance to balance its inaccuracy. This system of allowance is, of course, arbitrary, but it will not prejudice any of the conclusions which I shall draw. They would be the same by any reasonable allowance. Table III summarizes conviently the facts as to the amount and rate of improvement, and its relation to initial ability.
Taking the whole group together, improvement in speed and in accuracy are about equal, the median reduction in time regardless of errors being 31 per cent, and the median reduction in errors regardless of time being 29 per cent. By the scores with allowance for errors the median improvement in general efficiency in addition is 33 per cent. The average improvement is 29 per cent. This is for less than one hour of practice (about fifty-three-minutes).
The individuals vary from cases making no improvement (F and G) to a case of nearly fifty percent, improvement (K).
That the practice represented by only 2,592 additions made by an educated adult whose addition associations have been long established and often used should produce an improvement of threetenths, bears witness to the continued plasticity or educability of the synapses involved. It also supports the contention that the degree of efficiency shown by persons in any intellectual function is a result chiefly of specific training in it or the elements of it and only slightly of the transfer to it of the effects of training other functions. If the general training of from -fifteen to twenty-five years of a scholarly life were responsible for a large fraction of one's efficiency in "quickness of association" or "accuracy in response," one would not by so little specific training be able to improve so much.
The amount of improvement in this experiment may also add to our confidence that the method of the practice experiment wherein ​one works at one's limit and competes with one's own past record may well be made a regular feature in many school drills. Even if the same length of time produced in children a percentile improvement only half as great as here, the gain would still probably be far greater than the gain by any of the customary forms of drill.
Table I
Gives Scores in the Seven Successive Practice Periods :
| | | | | First | First | Second | Second | Third | Third | Fourth | Fourth | Fifth | Fifth | Sixth | Sixth | Seventh | Seventh |
| Individual | Sex | Day of beginning | Interval between tests Hours | Time | Examples wrong | Time | Examples wrong | Time | Examples wrong | Time | Examples wrong | Time | Examples wrong | Time | Examples wrong | Time | Examples wrong |
| A | m | M | 24 | 550 | 3 | 530 | 5 | 420 | 3 | 420 | 4 | 430 | 4 | 402 | 2 | 382 | 4 |
Table II
Scores Reduced to Single Variables by Allowance for Examples wrong :
| Individuals | Sex | First | Second | Third | Fourth | Fifth | Sixth | Seventh |
| A | m | 565 | 555 | 432 | 436 | 446 | 410 | 398 |
The relation of the amount of improvement to initial ability in any practice experiment is of great interest because it gives evidence bearing upon the fundamental problem of the relative shares of original nature and environment in determining the achievements of men.
It has been shown that in the case of educated adults the relative (that is, percentile) differences amongst educated adults in the ability to multiply mentally a three-place number by a three-place number are left unreduced by submitting all the individuals to equal practice.
The differences amongst individuals in the ability to add seem to be due in larger measure to differences in environmental influence. For equal practice does here reduce a little the relative or percentile differences within our group. This will be seen by comparing the relative variability of the group in the seventh practice period with that in the first, or by calculating the co-efficient of correlation between initial ability and percentile improvement. The proportions for highest and lowest individuals, next to highest and next to lowest are:
In first practice period In second practice period ist 19th .45 3.36 nd 18th .25 1.99 rd I7th .92 1.87 th 16th th I5th .82 1.46 1.62 1.46
The correlation between initial ability and percentile improvement is negative, roughly #
There is, of course, no essential conflict between this result for addition and the opposite result for mental multiplication with two ​three-place numbers. The same theoretical view which would expect mental span and ability to manage very complex relationships in a given field to be increased by practice in close dependence upon original capacity, would expect particular associative habits such as thinking of thirteen upon seeing 4, 7 and 2 in a column, to be increased by practice in less close dependence upon original capacity.
The improvements recorded are of the seventh set of 48 ten-figure examples over the first such set. They represent approximately the practice effect of 2,192 additions, or of from 30 to 75 minutes work.
This work is in the public domain in the United States because it was published before January 1, 1929.
The longest-living author of this work died in 1949, so this work is in the public domain in countries and areas where the copyright term is the author's life plus 74 years or less. This work may be in the public domain in countries and areas with longer native copyright terms that apply the rule of the shorter term to foreign works.
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