A productive way to think about imagistic mental models of physical systems is as though they were sources of quasi‐empirical evidence. People depict or imagine events at those points in time when they would experiment with the world if possible. Moreover, just as they would do when observing the world, people induce patterns of behavior from the results depicted in their imaginations. These resulting patterns of behavior can then be cast into symbolic rules to simplify thinking about future problems and (...) to reveal higher order relationships. Using simple gear problems, three experiments explored the occasions of use for, and the inductive transitions between, depictive models and number‐based rules. The first two experiments used the convergent evidence of problem‐solving latencies, hand motions, referential language and error data to document the initial use of a model, the induction of rules from the modeling results, and the fallback to a model when a rule fails. The third experiment explored the intermediate representations that facilitate the induction of rules from depictive models. The strengths and weaknesses of depictive modeling and more analytic systems of reasoning are delineated to motivate the reasons for these transitions. (shrink)

This study investigated the roles of problem structure and strategy use in problem encoding. Fourth‐grade students solved and explained a set of typical addition problems (e.g., 5 + 4 + 9 + 5 = _) and mathematical equivalence problems (e.g., 4 + 3 + 6 = 4 + _ or 6 + 4 + 5 = _ + 5). Next, they completed an encoding task in which they reconstructed addition and equivalence problems after viewing each for 5 s. Equivalence problems (...) of the form 4 + 3 + 6 = 4 + _ overlap with a perceptual pattern found in traditional arithmetic problems (i.e., answer blank in final position), and students' encoding was poorest on problems of this type. Individual differences in encoding the equivalence problems were related to variations in strategy use. Some students solved blank‐final equivalence problems using the standard arithmetic strategy of performing all given operations on all given numbers. These students made more errors in encoding problem structure, but fewer errors in encoding the numbers, than did students who solved the problems using correct or other incorrect strategies. Moreover, students who expressed many strategies for solving the blank‐final equivalence problems made fewer errors in encoding problem structure, but more errors in encoding the numbers, than did students who expressed only a single strategy. Results highlight that encoding is intended to guide action and that prior experience can simultaneously facilitate and interfere with accurate encoding. (shrink)

Word problems are notoriously difficult to solve. We suggest that much of the difficulty children experience with word problems can be attributed to difficulty in comprehending abstract or ambiguous language. We tested this hypothesis by (1) requiring children to recall problems either before or after solving them, (2) requiring them to generate f'mal questions to incomplete word problems, and (3) modeling performance pattems using a computer simulation. Solution performance was found to be systematically related to recall and question generation performance. (...) Correct solutions were associated with accurate recall of the problem structure and with appropriate question generation. Solution "errors" were found to be correct solutions to miscomprehended problems. Word problems that contained abstract or ambiguous language tended to be miscomprehended more often than those using simpler language, and there was a great deal of systematicity in the way these problems were miscomprehended. Solution error pattems were successfully simulated by manipulating a computer model’s language comprehension strategies, as opposed to its knowledge of logical set relations. o was.. (shrink)