Using Geometrical Representations as Cognitive Technologies

in Journal of Cognition and Culture
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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.

Using Geometrical Representations as Cognitive Technologies

in Journal of Cognition and Culture



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  • View in gallery
    The circles (cookies) that the teacher has drawn on the board.
  • View in gallery
    The teacher has partitioned one of the two drawn circles with perpendicular lines.
  • View in gallery
    (a) A number line with points for 0 and 1000 labeled. (b) −1000 and −1001 placed on the number line.
  • View in gallery
    (a) The teacher’s placement of –1001 on the number line; (b) the teacher’s erasure of –1001 and her query to the class about the number that should be placed at the indicated tick mark.
  • View in gallery
    (a) Jasmine’s placement of –1001; (b) Jasmine’s solution of –1006 or –1007.
  • View in gallery
    Task used by Fischbein (1993).
  • View in gallery
    Information sheet (with definitions) used in the experimental condition.
  • View in gallery
    Proportion of idealized responses for students in both treatment groups.

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