How children change their minds: strategy change can be gradual or abrupt.

This study investigated patterns of change in children's strategies for solving mathematical equivalence problems. The strategies children expressed in speech and in gesture were assessed both before and after an instructional intervention. In the intervention, children received either no input, accuracy feedback, or feedback plus instruction about a principle, an analogy, or a procedure. From pretest to posttest, many children changed both the variability of their strategy use and the content of their strategy repertoires. Patterns of change depended on type of instruction and on children's initial level of variability. Children who received instruction were especially likely to generate new strategies, and children with high variability were especially likely to abandon prior strategies. Gradual change was most common; however, many children modified their repertoires abruptly. Abrupt strategy change was especially prevalent among children who received procedure-based instruction and among children with low initial variability.

Knowledge conveyed in gesture is not tied to the hands.

Children frequently gesture when they explain what they know, and their gestures sometimes convey different information than their speech does. In this study, we investigate whether children's gestures convey knowledge that the children themselves can recognize in another context. We asked fourth-grade children to explain their solutions to a set of math problems and identified the solution procedures each child conveyed only in gesture (and not in speech) during the explanations. We then examined whether those procedures could be accessed by the same child on a rating task that did not involve gesture at all. Children rated solutions derived from procedures they conveyed uniquely in gesture higher than solutions derived from procedures they did not convey at all. Thus, gesture is indeed a vehicle through which children express their knowledge. The knowledge children express uniquely in gesture is accessible on other tasks, and in this sense, is not tied to the hands.

Gesture-speech mismatch and mechanisms of learning: what the hands reveal about a child's state of mind.

Previous work has shown that, when asked to explain a concept they are acquiring, children often convey one procedure in speech and a different procedure in gesture. Such children, whom we label "discordant," have been shown to be in a transitional state in the sense that they are particularly receptive to instruction--indeed more receptive to instruction than "concordant" children, who convey the same procedure in speech and gesture. This study asks whether the discordant state is transitional, not only in the sense that it predicts receptivity to instruction, but also in the sense that it is both preceded and followed by a concordant state. To address this question, children were asked to solve and explain a series of problems instantiating the concept of mathematical equivalence. The relationship between gesture and speech in each explanation was monitored over the series. We found that the majority of children who learned to correctly solve equivalence problems did so by adhering to the hypothesized path: They first produced a single, incorrect procedure. They then entered a discordant state in which they produced different procedures--one in speech and another in gesture. Finally, they again produced a single procedure, but this time a correct one. These data support the notion that the transitional state is characterized by the concurrent activation of more than one procedure, and provide further evidence that gesture can be a powerful source of insight into the processes involved in cognitive development.

Transitions in concept acquisition: using the hand to read the mind.

Thoughts conveyed through gesture often differ from thoughts conveyed through speech. In this article, a model of the sources and consequences of such gesture-speech mismatches and their role during transitional periods in the acquisition of concepts is proposed. The model makes 2 major claims: (a) The transitional state is the source of gesture-speech mismatch. In gesture-speech mismatch, 2 beliefs are simultaneously expressed on the same problem--one in gesture and another in speech. This simultaneous activation of multiple beliefs characterizes the transitional knowledge state and creates gesture-speech mismatch. (b) Gesture-speech mismatch signals to the social world that a child is in a transitional state and is ready to learn. The child's spontaneous gestures index the zone of proximal development, thus providing a mechanism by which adults can calibrate their input to that child's level of understanding.