ARITHMETIC
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.
II. DISTINCTION BETWEEN ARITHMETIC AND MATHEMATICS
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.
IX. ARITHMETICAL ABILITY OF TWO CHILDREN OF IQ 184 AND IQ 187
(STANFORD-BINET)
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.
[Illustration:
FIG. 15.—Showing D’s calculations on Test 2, Army Alpha, Form 5, at
the age of 10 years 11 months, five minutes being allowed for the
performance. The only figuring done on paper appears in the margin.
]
[Illustration:
FIG. 16.—Showing D’s calculations on Test 6, Army Alpha, Form 5, at
the age of 10 years 11 months, three minutes being allowed for the
performance.
]
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 “2255^2 − 2245^2.”
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.”
[Illustration:
FIG. 17.—Showing R’s calculation on Test 2, Army Alpha, Form 5, at the
age of 7 years 6 months, five minutes being allowed for the
performance. Note immature formation of the numerals. The only part
of the figuring done on paper appears in the margin.
]
[Illustration:
FIG. 18.—Showing R’s calculation on Test 6, Army Alpha, Form 5, at the
age of 7 years 6 months, three minutes being allowed for the
performance. Note immature formation of the numerals.
]
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.
XI. IMPLICATIONS FOR EDUCATION
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.
COBB, M.—“The Inheritance of Arithmetical Abilities”; _Journal of
Educational Psychology_, 1917.
COLLAR, D. J.—“A Statistical Survey of Arithmetical Ability”; _British
Journal of Psychology_, 1920.
GILLINGHAM, A.—_One Child’s Struggle in the Preparation for Life_;
Pedagogical Seminary, 1913.
LANTERNE, S.—_Psychologie du nombre et des opérations élémentaires de
l’arithmétique_; Paris, 1907.
LAZAR, E., and PETERS, W.—“Rechenbegabung und Rechendefekte bei
abnormen Kindern”; _Fortschritte der Psychologie_, 1915.
LINDLEY, E. H., and BRYAN, W. L.—“An Arithmetical Prodigy”;
_Psychological Review_, 1900.
MÁDAY, H. V.—“Die Fähigkeit des Rechnens beim Menschen und beim
Tiere”; _Zeitschrift für angewandte Psychologie_, 1913.
MILLER, G. A.—“Mathematical Prodigies”; _Science_, 1907.
MITCHELL, F. B.—“Mathematical Prodigies”; _American Journal of
Psychology_, 1907.
MÖBIUS, P. J.—_Ueber die Anlage zu Mathematik_, 2nd edition; Barth,
Leipzig, 1907.
RANSCHBURG, P.—_Die Rechenschwäche (Arithmasthenie) der Schulkinder im
Lichte des Experiments_; J. Springer, Berlin, 1916.
ROGERS, A. L.—_Experimental Tests of Mathematical Ability and Their
Prognostic Value_; Teachers College, Columbia University, 1918.
SCRIPTURE, E. W.—“Arithmetical Prodigies”; _American Journal of
Psychology_, 1891.
SCHMITT, C.—“Extreme Retardation in Arithmetic”; _Elementary School
Journal_, 1921.
SMITH, J. H.—“Individual Variations in Arithmetic”; _Elementary School
Journal_, 1916.
TERRY, P. W.—“The Reading Problem in Arithmetic”; _Journal of
Educational Psychology_, 1921.
THORNDIKE, E. L.—_The Psychology of Arithmetic_; The Macmillan Co.,
New York, 1921.
UHL, W. L.—“The Use of Standardized Material in Arithmetic for
Diagnosing Pupils’ Methods of Work”; _Elementary School Journal_,
1917.