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?