Abstract

This study examined children's justifications and evaluations of derived-fact strategies-strategies for solving addition and subtraction problems by using known combinations-to explore their knowledge of underlying part-whole relationships. Twenty-two normally achieving third graders and 19 learning-disabled (LD) students were asked to evaluate, justify, and complete derived fact strategies modeled correctly and incorrectly by puppets. About half of the students gave adequate explanations of the derived-fact strategies for addition-explanations that included the appropriate transformations to parts and wholes. Only 10% to 20% of students gave adequate explanations of subtraction strategies. Students, especially LD students, often solved subtraction problems by rigidly applying previously learned or invented methods, rather than explaining the derived-fact strategies modeled by the puppets. Although the performance of LD students lagged considerably behind that of their peers, their successful explanations were essentially indistinguishable from those of normally achieving students. The students' explanations were used to develop a partial model of the knowledge underlying the derived-fact strategies. The model has two major components, corresponding to two sets of operations involved in the justification task: mapping a problem onto the part-whole schema and making transformations to elements of the part-whole schema.