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.