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The American Journal of Psychology XVIII April 1907 – Mathematical Prodigies – Frank D Mitchell

By a "mathematical prodigy" we shall mean a person who shows unusual ability in mental arithmetic or mental algebra, especially when this ability develops at an early age, and without external aids or special tuition. We shall use the word "calculator" in the sense of "mental calculator," as a synonym for "mathematical prodigy," and shall usually mean by "calculation" "mental calculation," unless the contrary is clearly indicated by the context. A "professional calculator" will be taken to mean a mental calculator who gives public exhibitions of his talent. "Computer," however, will be restricted to mean one who calculates on paper. All problems mentioned as solved by the mathematical prodigies will be understood to be done mentally, unless otherwise indicated.

Inaudi is a well-marked instance of the auditory memory type. When he thinks of numbers, in calculation or otherwise, he does not see them "in his mind's eye," as arrays of dots or other small objects, or as written or printed figures; numbers are for him primarily words, which he hears as if spoken by his own voice, and during his calculations he almost always pronounces at least some of these words, either with partial distinctness or in a confused murmur. Any interference with this habitual articulation embarrasses him, and prolongs his calculation. He remembers a number very much more readily after hearing it than after seeing it; in fact, if a written number is banded to him, he usually reads it aloud, in order to learn it by sound rather than by sight. Whether visual images are entirely absent is a purely theoretical question; it is at least clear that, if present at all, they play a negligible part in his mental computations. We shall later find reason to believe that this condition is by no means so rare as has been supposed. Owing to the traditions of English and French psychology, the visual theory of mental calculation has lain ready to hand, and has in the past found much apparent confirmation. But now that an unmistakably non-visual calculator is on record, it will no longer do to beg the whole question; we must insist on considering each case upon its own merits, either settling it by definite evidence or leaving it frankly in doubt. We shall see later how much of the supposed evidence for the visual theory falls before a careful examination.

One of Inaudi's most marked characteristics is his powerful memory for figures. In one experiment he was able to repeat, after a single hearing, though with an effort, 36 figures, read off to him slowly in groups of three; but in the attempt to repeat 50 figures under the same conditions he became confused, and got only 42 of them correct. This latter number, 42, Binet therefore takes as the limit of Inaudi's power of acquisition, or "mental span," under these conditions. In an experiment made to determine in what time he could learn 100 figures read off to him in groups as often as requested, he learned the first 36 in a minute and a half, the first 57 in 4 minutes, 75 in 5½ minutes, and the whole 100 (actually there were 105) in 12 minutes. On the other hand, he can repeat in order, at any time within a day or two, all the figures used in his last performance, whether in the statement of the problems, in the answers, or in the intermediate calculations. The number of these figures at times runs as high as 300, and the total duration of the performance is usually not more than 10 or 12 minutes. Each new performance, however, blots out of his memory almost entirely the figures used in the previous one; but such constants as the number of seconds in a year, etc., as well as many powers and products, and any particular numbers or results in which he for any reason takes a special interest, remain permanently with him. These facts show how important it is to take account of the conditions of such experiments if the figures established by them are to have scientific value. In an experiment lasting the same length of time as one of his regular exhibitions, but under very different conditions, Inaudi can learn only a third the number of figures he remembers with ease under his usual conditions. In these public performances, however, each number in the problem as given is repeated several times (twice by Inaudi himself and once each by his assistant and the proposer of the question), and the figures of the various calculations and the result have a logical connection in the problem. Moreover, the numbers are learned in relatively short stages, separated by intervals in which they can be assimilated.

Concerning the rapidity of Inaudi's calculations we have fairly full information, - so much fuller, in fact, than we have for any previous calculator, that no satisfactory comparisons can be made. Since the results of Binet's experiments are readily accessible, a brief summary of them will here suffice.

 In each experiment the subject was given a written column of numbers, each of which was to be mentally increased or diminished, multiplied or divided, by the same number; in other words, the addend, subtrahend, multiplier, or divisor was uniform for the whole given column of numbers. The results were called off down the column as fast as obtained, and the average time for each single operation thus determined. These tests were made on some of Binet's pupils, on Inaudi, and on four department store cashiers who were thoroughly practiced in addition, subtraction, and multiplication of small numbers, and could perform mentally 2-figure multiplications, and in some cases, though with difficulty, 3-figure multiplications. The students were of course considerably slower than Inaudi and the cashiers; but the cashiers, in dealing with the smaller numbers to which they were accustomed, were fully as rapid as Inaudi, in some cases slightly more rapid. In dealing with larger numbers, however, which exceeded the limits of their customary calculations, their inferiority to Inaudi was very marked.