The Psychology of Arithmetic

by Edward L. Thorndike

Fifth.—It would be still worse policy to rely exclusively on problems arising outside arithmetic. To learn arithmetic is itself a series of problems of intrinsic interest and worth to healthy-minded children. The need for ability to multiply with United States money or to add fractions or to compute percents may be as truly vital and engaging as the need for skill to make a party dress or for money to buy it or for time to play baseball. The intellectualistic needs and problems should be considered along with all others, and given whatever weight their educational value deserves.

DIFFICULTY AND SUCCESS AS STIMULI

There are certain misconceptions of the doctrine of the problem-attitude. The most noteworthy is that difficulty—temporary failure—an inadequacy of already existing bonds—is the essential and necessary stimulus to thinking and learning. Dewey himself does not, as I understand him, mean this, but he has been interpreted as meaning it by some of his followers.[22]

Difficulty—temporary failure, inadequacy of existing bonds—on the contrary does nothing whatsoever in and of itself; and what is done by the annoying lack of success which sometimes accompanies difficulty sometimes hinders thinking and learning.

Mere difficulty, mere failure, mere inadequacy of existing bonds, does nothing. It is hard for me to add three eight-place numbers at a glance; I have failed to find as effective illustrations for pages 276 to 277 as I wished; my existing sensori-motor connections are inadequate to playing a golf course in 65. But these events and conditions have done nothing to stimulate me in respect to the behavior in question. In the first of the three there is no annoying lack and no dynamic influence at all; in the second there was to some degree an annoying lack—a slight irritation at not getting just what I wanted,—and this might have impelled me to further thinking (though it did not, and getting one tiptop illustration would as a rule stimulate me to hunt for others more than failing to get such). In the third case the lack of the 65 does not annoy me or have any noteworthy dynamic effect. The lack of 90 instead of 95-100 is annoying and is at times a stimulus to further learning, though not nearly so strong a stimulus as the attainment of the 90 would be! At other times this annoying lack is distinctly inhibitory—a stimulus to ceasing to learn. In the intellectual life the inhibitory effect seems far the commoner of the two. Not getting answers seems as a rule to make us stop trying to get them. The annoying lack of success with a theoretical problem most often makes us desert it for problems to whose solution the existing bonds promise to be more adequate.

The real issue in all this concerns the relative strength, in the pupil's intellectual life, of the "negative reaction" of behavior in general. An animal whose life processes are interfered with so that an annoying state of affairs is set up, changes his behavior, making one after another responses as his instincts and learned tendencies prescribe, until the annoying state of affairs is terminated, or the animal dies, or suffers the annoyance as less than the alternatives which his responses have produced. When the annoying state of affairs is characterized by the failure of things as they are to minister to a craving—as in cases of hunger, loneliness, sex-pursuit, and the like,—we have stimulus to action by an annoying lack or need, with relief from action by the satisfaction of the need.

Such is in some measure true of man's intellectual life. In recalling a forgotten name, in solving certain puzzles, or in simplifying an algebraic complex, there is an annoying lack of the name, solution, or factor, a trial of one after another response, until the annoyance is relieved by success or made less potent by fatigue or distraction. Even here the difficulty does not do anything—but only the annoying interference with our intellectual peace by the problem. Further, although for the particular problem, the annoying lack stimulates, and the successful attainment stops thinking, the later and more important general effect on thinking is the reverse. Successful attainment stops our thinking on that problem but makes us more predisposed later to thinking in general.

Overt negative reaction, however, plays a relatively small part in man's intellectual life. Filling intellectual voids or relieving intellectual strains in this way is much less frequent than being stimulated positively by things seen, words read, and past connections acting under modified circumstances. The notion of thinking as coming to a lack, filling it, meeting an obstacle, dodging it, being held up by a difficulty and overcoming it, is so one-sided as to verge on phantasy. The overt lacks, strains, and difficulties come perhaps once in five hours of smooth straightforward use and adaptation of existing connections, and they might as truly be called hindrances to thought—barriers which past successes help the thinker to surmount. Problems themselves come more often as cherished issues which new facts reveal, and whose contemplation the thinker enjoys, than as strains or lacks or 'problems which I need to solve.' It is just as true that the thinker gets many of his problems as results from, or bonuses along with, his information, as that he gets much of his information as results of his efforts to solve problems.

As between difficulty and success, success is in the long run more productive of thinking. Necessity is not the mother of invention. Knowledge of previous inventions is the mother; original ability is the father. The solutions of previous problems are more potent in producing both new problems and their solutions than is the mere awareness of problems and desire to have them solved.

In the case of arithmetic, learning to cancel instead of getting the product of the dividends and the product of the divisors and dividing the former by the latter, is a clear case of very valuable learning, with ease emphasized rather than difficulty, with the adequacy of existing bonds (when slightly redirected) as the prime feature of the process rather than their inadequacy, and with no sense of failure or lack or conflict. It would be absurd to spend time in arousing in the pupil, before beginning cancellation, a sense of a difficulty—viz., that the full multiplying and dividing takes longer than one would like. A pupil in grade 4 or 5 might well contemplate that difficulty for years to no advantage. He should at once begin to cancel and prove by checking that errorless cancellation always gives the right answer. To emphasize before teaching cancellation the inadequacy of the old full multiplying and dividing would, moreover, not only be uneconomical as a means to teaching cancellation; it would amount to casting needless slurs on valuable past acquisitions, and it would, scientifically, be false. For, until a pupil has learned to cancel, the old full multiplying is not inadequate; it is admirable in every respect. The issue of its inadequacy does not truly appear until the new method is found. It is the best way until the better way is mastered.