Speech and gesture provide two different access routes to a learner's mental representation of a problem. W e examined the gestures and speech produced by children learning the concept of mathematical equivalence, and found that children on the verge of acquiring the concept tended to express information in gesture which they did not express in speech. We explored what the production of such gesture-speech mismatches implies for models of concept learning. Two models of a mechanism that produces gesture-speech mismatches were tested against data from children learning the concept of mathematical equivalence. The model which best fit the data suggests that gesture and speech draw upon a single set of representations, some of which are accessible to both gesture and speech, and some of which are accessible to gesture but not speech. Thus, gesture and speech form an integrated system in the sense that they do not draw upon two distinct sets of representations. The model implies that when new representations are acquired, they are first accessible only to gesture. Over time, they are then recoded into speech.

This study examined how changes in children's problem encoding influenced their strategy selection. Fourth-grade students {N=51} solved six mathematical equivalence problems (e.g., 3+4+5=_+5) in a pretest. Children's problem encoding was then manipulated in one of two ways, or was not manipulated in a Control group. In the Subtle group, children solved four additional problems with the equal sign highlighted in red. In the Direct group, children solved the same four problems, and were directed to notice the equal sign in each problem. Children then solved six problems in a posttest, and did so again four weeks later in a follow-up test. The strategies children conveyed in their spoken and gestured explanations were assessed. Children in the Direct group considered multiple strategies for the posttest problems more often than children in the other groups, as reflected in their spoken and gestured explanations. Children in the Direct group were also most likely to generate gestured strategies and to abandon verbal strategies over the course of the study. These findings suggest that changes in problem encoding lead to changes in strategy selection.