The roles of estimation and the commutativity principle in the development of third graders' mental multiplication.

In this training experiment, pretesting identified 36 third graders (mean age = 8 years 5 months, SD = 4 months) with negligible mastery of multiplication combinations involving factors from 3 to 9. Participants were randomly assigned to 2 groups, which practiced different subsets of combinations, and were then retested. The results were inconsistent with R. S. Siegler's (1988) proposal that item-specific computational practice is necessary to promote changes in error patterns and combination mastery. The results were consistent with the hypotheses that children devise increasingly flexible and accurate estimation strategies and use relational knowledge such as the commutativity principle to master combinations. Because even fast mental-arithmetic errors and correct responses may be due to strategies other than retrieval, researchers need to craft ways of distinguishing between retrieval and nonretrieval strategies.