Special talents and defects

I. RELATION BETWEEN IQ AND CAPACITY FOR ARITHMETIC

Arithmetic as a psychological process has been studied analytically by psychologists more assiduously than any other of the school subjects, except reading. The psychology of arithmetic began to be investigated more than thirty years ago by laboratory workers, but so complex are the functions involved that there still remains much to be known.

Correlations show that capacity for arithmetic is closely connected with general intelligence. Most of the children who fail in the subject do so as a symptom of a general lack of competence in thinking. The great majority of those who are notably excellent arithmeticians are also superior in other performances.

The four children of more than 180 IQ, mentioned in Chapter IV as having learned to read before or during the third year of life, are also fine mathematicians, excelling at lightning calculation and at thinking in terms of numerical relations. Here, again, their marvelous skill at numbers is but symptomatic of their rare general superiority. Although the correlation between general competence and capacity for arithmetic is high and positive, it is reduced from perfection by the occurrence of discrepancies. Occasionally a very intelligent child is found, who does not readily learn arithmetic, and on the other hand there exist children whose ability at calculation far exceeds expectation from other performances.

Psychologically as well as logically, there is a distinction between arithmetic and mathematics. In both respects the former is but one phase or branch of the latter. By arithmetic is meant those functions of mathematicians which involve numerical calculation. This includes the four fundamental processes, with whole numbers and fractions, enumeration, and the solution of problems requiring choice of process to be employed.

Mathematics includes arithmetic, and also the relationships of space, time, proportion, and probability, as subsumed in algebra, geometry, trigonometry, and calculus. Psychologists find a positive intercorrelation among abilities in these various branches of mathematics, which is, however, not sufficiently close to unity so that the possibility of marked specialization in some cases is excluded. Judd has concluded that the abilities demanded by algebra, geometry, and arithmetic represent, respectively, elements not included in the others. Lightning calculators have been recorded, who could accomplish nothing, apparently, in the derivation of formulæ, or abstraction of principles.

Rogers decided as a result of experimental tests of mathematical ability, that “a marked degree of the power to analyze a complex and abstract situation, and to seize upon its implications, is the most indispensable element in mathematical proficiency.” This is the power that makes for proficiency in all life’s difficulties, and he who has it has unusual general intelligence—not mathematical proficiency only. There is certainly slight possibility that a generally stupid individual can ever deal with “higher mathematics.”

Since the processes other than the arithmetical have been very little studied, the discussion of special aptitude in mathematics will here be restricted largely to aptitude for arithmetic.

III. MENTAL FUNCTIONS IN ARITHMETICAL CALCULATION

In his recent presentation of the psychology of arithmetic, Thorndike writes as follows:

“Achievement in arithmetic depends upon a number of different abilities. For example, accuracy in copying numbers depends upon eyesight, ability to perceive visual details, and short-term memory for these. Long column addition depends chiefly upon great strength of the addition combinations, especially in higher decades, ‘carrying,’ and keeping one’s place in the column. The solution of problems framed in words requires understanding of language, the analysis of the situation described into its elements, the selection of the right elements for use at each step, and their use in the right relations.”

A great number of habits, more or less specific, must be automatized. There are all the combinations used in addition and subtraction, the multiplication tables, the reading of large numbers, the manipulation of fractions, the placing of the decimal point, and many others. These habits are of very unequal difficulty. Ranschburg has shown, for instance, that 5 + 2 is a much easier operation than is 2 + 5, and that 5 + 5 is easier than either. The difficulty of a combination is augmented by increase in the second member. The difficulty increases, also, as either or both of the members increase in value. The addition of two identical numbers, of whatever value, seems always to follow a different course from that of two unlike numbers, resembling multiplication in the time taken.

These are a few illustrations of the subtleties of habit formation in arithmetic, which are revealed only by laboratory methods. They suggest, also, the complexity and multiplicity of connections, which enter into ordinary achievement in arithmetic. Since the functions are thus highly complex and specialized, what are their interrelations? How are they organized, as regards the amounts of each found in given individuals?

IV. THE ORGANIZATION OF ARITHMETICAL ABILITIES

Thorndike and his students have shown that in general the correlation between ability in any one important feature of computation and ability in any other important feature of computation is positive and high. Thorndike holds that if enough tests were made to measure each individual fully in subtraction, multiplication with integers and decimals, division with integers and decimals, multiplication and division with common fractions, and computing with per cents, there would probably appear intercorrelations for a thousand 14-year-olds of near .90. Correlation between problem-solving and computation would doubtless be much less, probably not over .60.

Thorndike expresses the following inferences, based on interpretation of existing data.

“It should be noted that even when the correlation is as high as .90, there will be some individuals very high in one ability and very low in the other. Such disparities are to some extent, as Courtis and Cobb have argued, due to inborn characteristics of the individual in question, which predispose him to very special sorts of strength and weakness. They are often due, however, to defects in his learning, whereby he has acquired more ability than he needs in one line of work, or has failed to acquire some needed ability, which was well within his capacity.

“In general, all correlations between an individual’s divergence from the common type or average of his age for one arithmetical function, and his divergence from the average for any other arithmetical function, are positive. The correlation due to original capacity more than counterbalances the effects that robbing Peter to pay Paul may have.”

In 1910, Brown undertook to determine whether there is a special capacity for mathematics, and concluded from his correlations that there is an especially close relationship among tests involving mathematical performance. Ten years later, Collar made an effort to secure further data as to whether arithmetical ability, as a unitary combination of capacities, exists. Two hundred schoolboys were tested in the investigation. Results led to the conclusion that arithmetical ability tends to be represented in two main divisions: (1) the power to compute with ease and readiness, and (2) the power to solve problems by arithmetic, which involves the application of a higher degree of ability than is required in computation.

Arithmetical tests of various kinds correlate more closely than do arithmetical tests with non-arithmetical tests. “Hence we are compelled to interpret this relationship as evidence distinctly in favor of Burt’s suggestion, that there is an essential unity in arithmetical ability.”

All investigators have agreed in finding the correspondence between computation and problem-solving much less than that found among the various processes of computation alone. The facts are here analogous to certain facts noted in the study of reading, in Chapter IV. There it was seen that between proficiency in the mechanics of reading and comprehension in reading there may occur marked disparity; and that it is in mechanics that special discrepancies may be found between reading ability and general intelligence.

In arithmetic the same observation may be made. Marked special defects and talents are found in the mechanics of arithmetic, that is, in computation. But problem-solving in arithmetic is closely correlated with general intelligence, for it involves the capacities required for problem-solving anywhere,—response to many subtle elements, the weighing of these one against another, and choice of the procedure that will yield solution. These are the same capacities that underlie comprehension in reading, or grasp of any other situation offered by life. They are all functions measured in tests of general intelligence.

In school, arithmetical problems are usually presented as reading matter, so that reading for the comprehension of sentences is in itself of first rate importance for achievement in problem-solving.

V. PSYCHOLOGICAL STUDIES OF SPECIAL DEFICIENCY IN ARITHMETIC

Studies of children especially backward in arithmetic, with the accounts of the results of experimental teaching, have been contributed by Uhl, Smith, Schmitt, and others. Bronner has also contributed accounts of the psychological examination of such children.

Schmitt studied thirty-four pupils in the schools of Chicago, who were not feeble-minded, but were extremely retarded in arithmetic. The investigator states that tests of general intelligence were given, but does not share with the reader the exact results of such tests, saying only that the children “were not mentally defective.” The result of tabulation of circumstances involved showed that ill-health and absence were closely related to special disability in arithmetic. The inference is drawn that achievement in arithmetic calls for a hierarchy of habits, which depend on each other in a sequence. If a hiatus occurs at any essential point, as through absence, inattention, or inadequate teaching, confusion follows. (This inference seems very well justified, also, from the psychological analysis of the mental functions involved in arithmetic.) The problem of individual examination is to find out what habits have not been formed. The problem of pedagogy is to teach those habits, and to motivate the child.

Bronner’s conclusion that some children of good intelligence lack the power to form number concepts is criticized by Schmitt. When the gaps in habit formation have been located, and the child has been motivated to form the missing habits, special deficiency in arithmetic disappears.

This is, on the whole, the conclusion to be drawn from the few studies which have included experimental teaching. Uhl studied a boy who could not subtract, according to standard tests. Analysis showed that he could subtract only by multiplying. For example, to subtract 9 from 46, he first set aside 1, to get a multiple of 9. Then he disintegrated 45 into 9’s and dropped one of them. After disposing of the 9 in this devious fashion, he picked up his 1 again, and finally arrived at a correct result. It was thus found why he was so slow, and where instruction must be applied, in order to remedy the special deficiency which he showed in arithmetical calculation.

In difficult combinations, pupils invent interesting evasions. “Breaking up” larger numbers is common, so that 9 + 7 + 5 becomes 9 + 2 + 2 + 2 + 1 + 2 + 2 + 1, for instance.

Failure to form correct habits of interpreting symbols, or relations between symbols, often explains deficiency. This may be illustrated by the case of a girl who always read 40 ) 1728 as “40 divided by 1728.” Her results were thus fantastic. This error is analogous to that of writing “three dollars” as 3$.

The remedy for these conditions is to show the child what he is doing, and to give drill until the correct and rapid method is thoroughly mastered. Special deficiency in the mechanics of arithmetic is to be improved by drill, after it has been found out where the drill is needed.

VI. METHODS OF DETECTING WRONG OR INCOMPLETE HABITS

Without systematic methods of testing, it would be a very difficult task to discover just what connections might be wrongly or inadequately formed, in the case of a given child. The standardized measuring scales and practice exercises, devised during the past fifteen years, furnish a systematic means of exploration. These are constantly being extended and improved, to cover each and every kind of habit that a child must acquire, for achievement in arithmetic.

The principle of these scales and tests is to establish by experiment the speed and accuracy of typical school children, grade after grade, in the performance of the various functions separately. It thus becomes possible to discover in the case of a deficient pupil whether he needs correction and drill in every function, or in only one function. By means of the Courtis tests, for example, it may be discovered whether a child’s difficulty is in addition, multiplication, division, in speed or accuracy, or both speed and accuracy, and so forth.

The use of existing scales and tests for diagnostic purposes has been described by Courtis, Uhl, Anderson, and others. We may expect great improvement in these methods in the future. At present the standardizations are in terms of school grade norms. A better plan for diagnostic purposes would be to standardize in age norms, giving a percentile distribution for each twelve-month interval of the period of immaturity.

VII. NERVOUS INSTABILITY AND SPECIAL DEFICIENCY IN ARITHMETIC

Nervously unstable children are, as Burt has pointed out, often deficient in arithmetic, even when in general intelligence they are not deficient. This follows from the same causes of failure as were set forth under discussion of nervous instability and special difficulty in reading. To build up little by little the intricate hierarchy of arithmetical habits, each habit in its essential sequence, is a task uncongenial to the flighty, uncontrolled, or negativistic neurotic.

Individual instruction is here, again, the solution of the problem. The neurotic can learn arithmetic within the limits of his intelligence, by means of patient individual instruction, given preferably at rather brief sittings.

VIII. ARITHMETICAL PRODIGIES

Extremely great ability to perform feats of mental arithmetic excites popular wonder and admiration to a degree far beyond that excited by most other manifestations of mental gifts. This may be due to the fact that in calculation each individual has a rather definite standard of performance, namely his own ability to calculate. When another goes far beyond him and his friends, in so definite a performance, he can see for himself that the typical has been phenomenally exceeded. The gifted person who exceeds the typical to an equal extent in perception of the fine shades of meaning in words, or in the detection of absurdities and contradictions in demagogy, creates no sensation among his fellow townsmen; for there is no way whereby the average man can “check up” in the performances, to show himself how phenomenally he has been exceeded in capacity for them.

Bidder, the famous English calculator, is recorded in history because he could perform mental arithmetic perhaps fifty times as well as typical persons. The facts that he also became one of the most successful civil engineers of his time, and made a large fortune, are noted as of merely incidental interest, and would not have given him a place in the history of unusual persons. A man may make fifty times as much money as the average man does, by meeting with fifty times as much acumen and energy the intricate, subtle, and difficult situations offered by modern economic life. Yet he is not so very likely to be regarded as prodigiously gifted. His fellowmen can and will explain the difference between him and themselves as due to luck or circumstance. But a gift for “lightning calculation” is obviously peculiar to the person, and makes of him an object of wonder.

The same general considerations hold in the case of children. Many children of extraordinary intelligence are found, because they have attracted attention to themselves by excellence in arithmetic; and upon examination show themselves to be equally excellent at those tests which measure IQ, excellence in which is not necessarily conspicuous except to the trained psychologist.

Accounts of prodigious calculators go back to ancient Greece, in Lucian’s reference to Nikomachos of Gerase. The word “calculation” means literally “pebbling,” coming from the Latin calculi, pebbles. Records of lightning calculators have been collected by Scripture and by Mitchell.

Jedediah Buxton (b. 1702) appears to be the first calculator on record in modern accounts. He lived at Elmton, England. “He labored hard with a spade to support a family, but seems not to have shown even usual intelligence in regard to ordinary matters of life.... In regard to matters outside of arithmetic he appeared stupid.” In 1754, when he was taken to London to be tested by the Royal Society, he went to see King Richard III performed. “During the dance he fixed his attention upon the number of steps; he attended to Mr. Garrick only to count the words he uttered. At the conclusion of the play, they asked him how he liked it.... He replied that such and such an actor went in and out so many times, and spoke so many words; another so many.... He returned to his village, and died poor and ignored.” It is said that he could give an itemized account of all the free beer he had had from the age of 12 years.

Tom Fuller, “The Virginia Calculator” (b. 1710), seems to be another case of highly specialized ability. He came from Africa as a slave when about 14 years old. He is first heard of as a calculator at the age of 70 years, when it is stated that he reduced a year and a half to seconds in about two minutes, and 70 years, 17 days, 12 hours to seconds in about a minute and a half, correcting the result of his examiner, who had not taken leap years into the reckoning. He also calculated mentally the sum of a simple geometric progression, and multiplied mentally two numbers of nine figures each. He was totally illiterate.

Other prodigious calculators, who are not known to have had superior general ability, are Zerah Colburn (b. 1804), Henri Mondeux (b. 1826), Jacques Inaudi (b. 1867), and Ugo Zaneboni (b. 1867). None of these individuals achieved eminence in any other respect, but this does not necessarily prove that they were not of superior intelligence. It would have been impossible, for instance, for the slave, Tom Fuller, to achieve intellectual eminence in a profession.

None of them was studied psychologically except Inaudi, who was examined by Binet. Inaudi was an Italian by birth. In childhood he tended sheep, as did Mondeux. His passion for numbers began at the age of about 6 years. At 7 years of age he could multiply five-place numbers by five-place numbers, “in his head.” His memory span for digits given orally was 42. He must hear them, the span being considerably reduced if he only saw them. He had little education, and did not learn to read and write until he was 20 years old. He lived by public exhibitions of his power to calculate. Binet concluded that he had no particular ability except the gift for calculation, and was not generally superior.

None of these calculators showed any gift for mathematics beyond arithmetic. Many others are on record who are known to have had great all-round superiority, and mathematical genius of the highest order, as is proven by their achievements. Bidder (b. 1806), Bidder, Jr. (b. 1837), Safford (b. 1836), Gauss (b. 1777), Ampère (b. 1775), Hamilton (b. 1788), and Whatley (b. 1787), all were lightning calculators.

George Parker Bidder was the son of a stonemason, of Devonshire. His family history is on record, and is quite interesting in connection with his gifts. His eldest brother, a Unitarian minister, had an extraordinary memory for Bible texts, but took no special interest in arithmetic. Another brother was an excellent mathematician and insurance actuary. Still other members of the family were distinguished in non-mathematical pursuits. Bidder’s ability was first noticed when he was 6 years old. In 1822, at the age of 16 years, he took a prize in mathematics at the University of Edinburgh. He became a distinguished engineer, and accumulated wealth, as before stated. His son, the younger Bidder, was wrangler at Cambridge, and became barrister and Queen’s counsel. He could multiply fifteen-place numbers by fifteen-place numbers, and could play two games of chess simultaneously, blindfolded. Two of his daughters “showed more than average ability in mental arithmetic.”

Truman Henry Safford was the son of a Vermont farmer, both parents having been school teachers. His power in calculation was noticed when he was 3 years old. At about 7 years of age, he began to study algebra and geometry, and soon thereafter, astronomy. In his tenth year he published an almanac, computed entirely by himself. His interests included chemistry, botany, philosophy, geography, and history in addition to astronomy and mathematics. He took his degree at Harvard in 1854, at the age of 18 years, and became an astronomer. He was professor of astronomy in Williams College for many years, until his death, and made many important astronomical calculations and discoveries.

Carl Frederick Gauss, the great mathematician, was a lightning calculator, the marvels of his performance exceeding those of nearly all others. Gauss entered the gymnasium when he was 11 years old, and in mathematics soon surpassed his teachers. He began the study of higher analysis at 10, and at 14 could read Newton with understanding. At 24 he published Disquisitiones Arithmeticæ, which is a fundamental contribution to mathematics. He himself has related that he remembers having followed by mental arithmetic a calculation concerning the wages of his father’s workmen, and of having thus detected an error in the reckoning, at the age of 3 years. He could use from memory the first decimals of logarithms, and was especially ingenious at discovering new methods. Gauss was unquestionably a person of very extraordinary general intelligence. As a child he mastered not only mathematics, but also the classical languages with wonderful ease. It is quite possible, however, that his gift for mathematics exceeded his general capacity in other respects.

The renown of André Ampère’s achievements in science is commemorated in the ampère. As a child, he showed all-round ability, and encyclopedic interests. He learned counting at 3 or 4 years of age, by means of pebbles, “and was so fond of this diversion that he used for purposes of calculation pieces of a biscuit, given him after three days’ strict diet.” There is no question that Ampère was a child of extremely high IQ, the ability at calculation being but one manifestation of his great genius. He was a chemist, a metaphysician, and a mathematician. He became professor of mathematics, and wrote on probabilities, the unity of structure in organisms, and electrodynamics. In this last field he discovered fundamental truths, and immortalized his name. He was elected to the Academy of Sciences in Paris, and is recognized as one of the world’s great thinkers, not as a calculator merely.

Richard Whatley, Archbishop of Dublin, was a prodigious calculator as a child. From 5 to 9 years of age he astonished onlookers by his feats. He afterwards ceased to interest himself in calculation, but used his intellectual capacity for achievement in other fields.

The greatest calculator on record, according to the researches of Scripture, is Johann Dase, born in Hamburg, in 1824. He could count objects with extreme rapidity. “With a single glance he could give the number, up to 30 and thereabouts, of peas in a handful, scattered on the table”; could give the number of sheep in a herd, or books in a case so quickly that his record remains unequaled. He could carry on enormous and protracted calculations, without recording figures, but seemed not to comprehend mathematical principles. He attended school when 2 to 3 years old, and began public exhibitions at 15 years of age. From the records it is not possible to prove or disprove superior general intelligence.

There are on record but three calculators, who were personally examined by psychologists, so far as the present writer can learn. Inaudi, already mentioned, and Pericles Diamandi, a Greek grain merchant, born in 1868, were examined by Binet. Arthur Griffith, son of a stonemason, born in 1880, was examined by Lindley and Bryan, in the laboratory at the University of Indiana, in 1899.

Binet concluded that Inaudi had no unusual ability except for mental calculation, and that his auditory memory for digits was a special gift. Diamandi, on the other hand, in addition to his ability in calculation, knew five languages, was an incessant reader, and wrote both novels and poetry. He entered school at 7, and remained until he was 16, always heading his class in mathematics. His methods in calculation were visual. “He has a number-form of a common variety, running zigzag from left to right, and giving most space to the smaller numbers. This number-form he sees as localized within a peculiar grayish figure, which also serves as a framework for any particular number or other object, which he visualizes.”

Griffith had, from the age of 3, a passion for counting and made fair records in all studies. He entered school at 10, and attended school seven years. In scope and tenacity of memory, and in rapidity at calculation, he ranked with the best recorded cases, according to the investigators who examined him. Memory was described as very systematic; and rapidity was seen to depend on the great number of numerical relations committed to memory, and upon reduction in number of operations through short-cut methods.

These three examinations were all conducted more than twenty years ago, before standardized methods of measurement had been developed. It is difficult to glean from them, and from the biographical material compiled by Scripture and by Mitchell, what the truth is, as regards the extent to which this gift for calculation was special in these persons. Many of them, as we have seen, were certainly men of genius, with general capacity for selective thinking. Several others probably were not of superior general intelligence, but in no case can we be certain, on the basis of anecdotal evidence alone. Some of them were peasants or slaves, born to manual toil, in the absence of free schools, and in the presence of rigid class distinctions. It is not inconceivable that a child of IQ over 170, condemned by unavoidable environment to herd sheep or pick cotton through his youth, might find relief from the monotony of his work by calculating. As Mitchell, himself a lightning calculator, says, “Given a knowledge of how to count, and later a few definitions, and any child of average ability can go on, once his interest is accidentally aroused, and construct, unaided, practically the whole science of arithmetic, no matter how much or how little he knows of other things.” This statement is probably true, if we change one word, and substitute for “child of average ability,” “child of great ability.”

All who have examined lightning calculators, or searched their biographical records, are agreed that the secret of their power lies in highly developed mechanics. Special habits of combining and recognizing numbers are formed, which differ from ordinary calculation comparatively in somewhat the same way as the method of the child who added 7 + 5 by adding 7 + 2 + 2 + 1, the latter being analogous to the usual method.

The lightning calculator memorizes combinations far beyond those ordinarily memorized, so that he is, for instance, able to add 2581 + 1763 as quickly as an ordinary person can add 15 + 8. He learns multiplication tables up to 100 × 100, whereas we learn only through 12 × 12. He devises and uses many “short cuts,” e.g. multiplying by two easy numbers and taking the difference, instead of multiplying by an awkward number. Multiplication is probably used as the fundamental operation.

This specialization in and perfection of arithmetical connections, by a person of original aptitude for and interest in numbers, results in the prodigious calculator. As Scripture concludes, “These persons had enormous ability to learn calculation, not to calculate without learning.” The rôle played by practice is seen in the fact that if interest in counting wanes, and practice at calculation ceases, the skill acquired deteriorates through disuse. Whatley, and others, who became distracted from calculation by other interests as they grew up, lost the power they had possessed. However, by resuming practice, the skill can be regained by those who have acquired it, as is the case with skills in general.

Satisfaction in mental activity for its own sake is expressed by those calculators who have given introspections. After Safford had lost the power of lightning calculation through disuse, he continued to take pleasure in factoring large numbers, or in satisfying himself that they were prime. The younger Bidder said, “With my father, as with myself, the handling of numbers or playing with figures afforded a positive pleasure, and constant occupation of leisure moments. Even up to the last year of his life,^[16] my father took delight in working out long and difficult arithmetical and geometrical problems.”

All who have studied material relating to prodigious calculators have especially stressed the very early age at which the gift has shown itself. This is especially true of those who achieved greatness in science, as adults. Gauss, Whatley, and Ampère were all first noted at the age of 3 years, and Safford and Bidder at the age of 6 years. It appears to the present writer to be probable that any child of IQ over 180 could be taught to be a lightning calculator. This inference comes from observing such children, as they master numbers.

To illustrate mathematical aptitude in children of high IQ, a brief account is herewith given of two boys, both known professionally to the present writer since early childhood. These children are both of a degree of general intelligence so rare as to be scarcely ever found, and both are especially interested in mathematics.

The boy D, of IQ 184, was described first by Terman, in The Intelligence of School Children. His achievements are most remarkable in every kind of intellectual activity, including music and drawing. Among his favorite pastimes since infancy has been the manipulation of numbers. His calculations, dating from the time his hand could wield a pencil, have covered hundreds of pages. As a child of 7, 8, and 9 years, D found the keenest satisfaction in deriving formulæ to render himself unbeatable at family games based on number. At the age of 12 years he has completed the mathematical curriculum of the elementary and secondary schools, through arithmetic, algebra, geometry, and trigonometry. (It should be added that he has also completed the curriculum of the elementary and secondary schools in all other respects, and is ready at 12 years to enter college.)

Figure 15 shows D’s calculations on Test 2, of Army Alpha, Form 5, five minutes being allowed for the performance. Figure 16 shows his calculations on Test 6, of the same form of Alpha, three minutes being allowed. D was 10 years 11 months old on the date of these calculations. He had never previously seen either of these tests.

The second child to whom we wish to refer briefly is R, of IQ 187. He, too, has delighted in number from about the third year of life. When first seen by the present writer, at the age of 6 years 6 months, R’s memory span for digits was at least eight (beyond this he was not tested), and he could easily reverse seven digits at least (beyond this the test did not go). He has been taught short cuts and other mechanics of lightning calculation till now, at the age of 8, he can with great speed calculate the answer to such a series as “2 × 2 × 2 × 2 multiplied by twice the square of 2; square it,” or “2255² − 2245².”

In Figure 17 is shown R’s calculation on Test 2, Army Alpha, Form 5, and in Figure 18, his performance in Test 6, the time limits being the same as indicated for D. R was 7 years 6 months old on the date of these performances. The ordinary child of that age can, of course, make no score whatever. R had never previously seen either of the tests.

R’s teacher^[17] writes of him, “His ability in academic work seems well distributed, though strongest in mathematics. For this grade he is remarkably low in art and industrial work, but he would be average in the second grade, where his age would usually place him. His artistic feeling is all for music and literature.... I think he is rather clumsy with his hands even for his age, though not much below the average child. With his mental ability he can learn to do anything in which his interest is aroused.... As he goes on, I hope that we can arrange for him to work with more advanced groups in mathematics and science, though remaining in the present group for most of the day.... In mathematics it is noticeable that although he can use short cuts which are Greek to the class, he is quite as apt to make an error in concrete problems as the other bright children. This is not lack of attention or interest, for he is always keenly alive in any lesson in mathematics. For example, in shop where he was making a table with a top 24 inches square, he was shown the lumber (12 inches wide) and asked how many pieces he must prepare for the table. He replied ‘three,’ and it was some time before he was led to recognize his mistake.”

With his love of mathematics, R combines a passion for classifying. As early as his first year of life, he would classify his playing blocks according to the shape of the letters on them,—O, Q, P, and the like together, and A, V, W, N, M, and the like in another group, and so forth. This delight in classifying is also one of D’s most conspicuous characteristics.

X. THE INHERITANCE OF ARITHMETICAL ABILITIES

From his search through the literature pertaining to arithmetical prodigies, Mitchell concluded that he could not find sufficient data from which to generalize concerning heredity. This conclusion is no doubt justified. We must wait upon modern studies, in order to gain knowledge of the extent to which such tendencies may be inherited. We may note, however, that many relatives, gifted in some way, are reported among the lightning calculators of history. Diamandi’s mother “had an excellent memory for all sorts of things,” and a brother and a sister out of a family of fourteen siblings shared his aptitude for mental arithmetic; the family history of the Bidders has been referred to already; Safford’s father and mother were both teachers; Gauss had a maternal uncle of known mechanical and mathematical talent; Mitchell’s younger brother could play chess blindfolded. Of the two children, D and R, herein described, both have many adult relatives who are or were writers, money makers, inventors, or organizers. Of this generation, D is an only child, but he has several cousins. Of these, three who have been measured show IQ’s of 150, 156, and 157, respectively. R’s only brother has an IQ of 150, and of his two cousins, both girls, the only one yet measured has an IQ of 170. These are suggestive fragments of facts concerning family resemblances.

Cobb has made a quantitative study of resemblance between parents and children, in the various fundamental processes, using five of Courtis’ standard tests. She finds that the coefficient of correlation between child and like parent is .60, between child and unlike parent, .01, between child and mid-parent, .49. By “mid-parent” is meant the ability that falls midway between the abilities of the two parents. Twenty persons were studied in eight families. No sex differences were noted. A child of either sex may resemble either parent, and not all children of the same family do resemble the same parent. Cobb concludes that the likeness found is due to heredity.

In the matter of sex differences, it is notable that of all the lightning calculators recorded only one, and she of minor importance, was of the female sex. It is possible that this difference may be due to native sex differences in the inheritance of endowment. It is much more probably due, however, to those differential pressures—social, educational, and economic—which cast up to public notice more deviates of all kinds among the male sex. During the periods from which the records of lightning calculators have been gathered, this differential pressure was much more forceful than it is now. Because of the differential action upon the sexes of social pressures, it is never possible to make valid comparisons of the sexes in respect to mental deviation, unless the sampling has been rigidly made in some manner absolutely indifferent to selection, and unless the measurements have been objectively taken.

Studies thus far made would convince us that arithmetical skill consists in the automatization and integration of a hierarchy of habits, which can be acquired to a passable degree by all children of average intelligence. Lightning calculation results from building up and rendering automatic still further habits, and can be achieved by persons of great general intelligence. It remains an open question whether a generally stupid person can ever become a prodigious calculator, but it seems certain that interest in and aptitude for arithmetic may be especially marked in generally superior children.

Arithmetical ability may develop, without simultaneous development of ability in other branches of mathematics. One may calculate prodigiously, without comprehending algebraic and geometric principles, or being interested in them. Also one may be more or less adept, either by nature or by training, in one kind of arithmetical function than in others.

Drill is the means for improving arithmetical ability, so far as speed and accuracy of calculation are concerned. Ability in problem solving can probably not be much affected by drill, since “a problem” is, by definition, something that requires independent adjustment, and not the response of automatic habit. It therefore calls on general intelligence, and cannot be improved after the mechanics of reading and calculating have been mastered up to the limits of capacity.

REFERENCES

Anderson, C. I.—“The Use of the Woody Scale for Diagnostic Purposes”; Elementary School Journal, 1918.

Binet, A.—Psychologie des grands calculateurs et joueurs d’échecs; Paris, 1894.

Brown, W.—“An Objective Study of Mathematical Intelligence”; Biometrika, 1910.

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