Date of Award


Degree Type


Degree Name

Doctor of Philosophy


Three studies were conducted in order to determine the source and frequency of children's difficulties in subtraction and to examine different approaches to remediation. In Study 1, 56 third grade children were asked to solve subtraction problems and were observed and questioned regarding their solution processes. Analysis of verbal reports and written solutions suggested that the main source of difficulty involved borrowing procedures. Children who had difficulty either attempted to borrow incorrectly or made inversion errors, that is, they ignored the location of the digits and subtracted the smaller number from the larger. Study 2 examined two minimally intrusive methods of remediation. Eighty third grade children were given either instructions to borrow, promised rewards for accurate performance or no intervention and were asked to solve a series of subtraction problems requiring borrowing. Neither experimental condition resulted in a significant increase in the number of problems solved correctly. Those children who initially failed to solve any problems correctly responded to instructions with a decrease in inversion errors and an increase in borrowing errors. These results suggested that more intensive instruction was required. In Study 3, 67 third and fourth grade children were assigned to one of three conditions: Component skills Training, Criterion Training or a regular classroom control condition. The Component Skills Training condition attempted to teach the skills required for borrowing in a step by step fashion with feedback, while the Criterion Training condition simply provided feedback in the form of correctly worked solutions. Children who solved fewer than 60 percent of the problems correctly on a pretest significantly increased the number of problems solved correctly on posttests conducted 1-2 days and two weeks following training. These increases were reflected by a reduction of errors involving borrowing. Those children who did not complete the training programs successfully showed patterns of performance across days which were suggestive of fatigue, boredom, or the presence of some other interfering factor. In all, the results suggested that intensive programming through careful attention to contingencies, as well as an appreciation of how program variables interact with learner characteristics and development may be necessary for effective remedial intervention in subtraction.



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