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Using Geometrical Representations as Cognitive Technologies

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In this article we provide a treatment of geometric diagrams as a semiotic ‘technology’ in mathematics education. To this end, we review two classroom observations and one experimental study that illuminate the potential for communicative breakdown in the use of this technology, as well as possibilities for mitigating the breakdown. In the classroom observations, we show that rather than ‘seeing through’ the diagrams to the idealized mathematical objects (a pervasive semiotic practice in academic mathematics), upper-elementary grade students may treat the misleading appearances of the diagrams as mathematically significant. We argue that one source of such communicative breakdowns is that the ‘rules’ for using the semiotic technology are rarely made explicit in classrooms. In the experimental study, we review findings showing that when students’ have access to mathematical definitions that distinguish between the diagram and the idealized mathematical object (e.g., a diagram of a mathematical point is the size of a small dot, but a true mathematical point has no size), they are less likely to rely on the misleading appearances of the diagrams; we also found that this effect increases over grade level. The issues explored in this article suggest that upper elementary students could benefit from explicit treatment of definitional practices in academic mathematics. At a broader theoretical level, the findings serve to emphasize the potential for any semiotic technology to have multiple meanings and the role of active sense making in communications with and about technologies.

Affiliations: 1: * Graduate School of Education1501 Tolman HallUniversity of California BerkeleyBerkeley, ca 94720-1670USA; 2: Graduate School of Education1501 Tolman HallUniversity of California BerkeleyBerkeley, ca 94720-1670USA


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