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.