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.