To assess the validity of verbal reports in children's subtraction, students in Grades 1,3, and 5 were asked to solve a set of simple subtraction problems and were placed in the no-report, retrospective-report, or concurrent-report conditions. Two aspects of verbal report validity were assessed: reactivity, or whether providing a verbal report alters subsequent task performance, and veridicality, or whether the verbal reports are accurate reflections of solution strategies. Students in all grades and in both the retrospectivereport and concurrent-report conditions were able to provide veridical strategy reports, and the instruction to verbally report had few effects on task performance. Informal findings indicate that students had less difficulty reporting retrospectively than concurrently.
Problems of the form a + b -b have been used to assess conceptual understanding of the relationship between addition and subtraction. No study has investigated the same relationship between multiplication and division on problems of the form d x e ÷ e. In both types of inversion problems, no calculation is required if the inverse relationship between the operations is understood. Adult participants solved addition/subtraction and multiplication/division inversion (e.g., 9 x 22 ÷ 22) and standard (e.g., 2 + 27 -28) problems. Participants started to use the inversion strategy earlier and more frequently on addition/subtraction problems. Participants took longer to solve both types of multiplication/division problems. Overall, conceptual understanding of the relationship between multiplication and division was not as strong as that between addition and subtraction. One explanation for this difference in performance is that the operation of division is more weakly represented and understood than the other operations and that this weakness affects performance on problems of the form d x e ÷ e.
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