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CHAPTER VI
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.