Presents the latest research on how reasoning with analogies, metaphors, metonymies, and images can facilitate mathematical understanding. For math education, educational psychology, and cognitive science scholars.

This article reports on a study of children's deductive reasoning in solving novel relational problems. Detailed protocols were obtained from 264 children (aged 9- 12 years) who verbalised their thinking as they solved the problems. The study included the development of a three-phase theory based on Johnson-Laird and Byrne's mental models perspective, but with some distinct modifications. These include a focus on the relational complexity entailed in model construction and in premise integration, and the advancement of four reasoning principles that (...) are applied throughout problem solution (in contrast to Johnson-Laird's falsification processes as the hallmark of deductive reasoning). The reported case studies and the results of statistical analyses supported predictions arising from the proposed theory, including the key role of the reasoning principles. The results also showed that problem difficulty is a function of relational complexity, not of the number of models to be constructed, as argued by Johnson-Laird and Byrne. (shrink)