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Algebraic manipulation as motion within a landscape

Abstract

We show that students rearranging the terms of a mathematical equation in order to separate variables prior to integration use gestures and speech to manipulate the mathematical terms on the page. They treat the terms of the equation as physical objects in a landscape, capable of being moved around. We analyze our results within the tradition of embodied cognition and use conceptual metaphors such as the path-source-goal schema and the idea of fictive motion. We find that students solving the problem correctly and efficiently do not use overt mathematical language like multiplication or division. Instead, their gestures and ambiguous speech of moving are the only algebra used at that moment.

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Acknowledgements

The work described in this paper was supported in part by National Science Foundation grants DUE-9455561, DUE-0442388, DRL-0633951, and DUE-0941191. The authors thank Rachel E. Scherr, Eric Brewe, and Evan Chase for their valuable input and feedback during preparation of this manuscript. We further thank three anonymous reviewers for their very careful reading and helpful suggestions. Data were gathered by Katrina E. Black at the University of Maine and form part of her Ph.D. dissertation (Black, 2010).

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Correspondence to Michael C. Wittmann.

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Wittmann, M.C., Flood, V.J. & Black, K.E. Algebraic manipulation as motion within a landscape. Educ Stud Math 82, 169–181 (2013). https://doi.org/10.1007/s10649-012-9428-4

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  • DOI: https://doi.org/10.1007/s10649-012-9428-4

Keywords

  • Gesture analysis
  • Algebra
  • Embodied cognition
  • Physics
  • Differential equations
  • Separation of variables