Abstract
In this article, we examine how it is possible, in the teaching and learning of geometry, to bridge the gap between problems involving drawings and figures, which is essential to the learning of mathematical proof. More precisely, the way students’ drawing perception has to evolve, from Iconic Visualization to Non-Iconic Visualization (Duval, Annales de Didactique et de Sciences Cognitives, 10, 5–53, 2005). We show that the Instrumental Deconstruction process is multifaceted and central in this evolution. We present a theoretical framework, in relation with an experiment based on a 3D dynamic geometry environment. Based on a case study, we show that construction tasks with specific representations make the instrumental work play a key role in the learning of geometry.
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Notes
In this case, seeing means both perceiving and interpreting what is perceived. We will keep this meaning for “visualization.”
A reviewer drew our attention to the fact that “in the United States, with DGS, ‘draw’ means to make a figure that ‘looks right.’ ‘Construct’ means to make a figure that is right due to how it is constructed and hence cannot be ‘messed up.’” In this article, we use “draw” to denote the production of a drawing in whatever way (sketching or constructing), and we use “construct” for the production of a drawing carefully using instruments and/or knowledge with the intention to be right (but sometimes being wrong).
By “intellectual proofs” we mean “[proofs] which use verbalizations of the properties of objects and of their relationships […]it requires a genuine construction of language means as an operative tool. The problem-solver must be able to use language and symbols as means to compute on statements and relations.” (Balacheff, 1988, p. 285)
Cabri 3D is a Cabrilog software. More information available here: https://cabri.com/fr/cabri3d/
The concept of “milieu”, from the Theory of Didactical Situation, refers to the “system opposing the taught system” (Brousseau, 1997, p. 57). In other words it is this part of the environment wich is meaningful from the perspective of the learning stake by the kind of feedback it provides to the student. “That is to say that the student’s answer must not be motivated by obligations related to the didactical contract but by adidactical necessities of her relationships to the milieu” (Brousseau, 1997, p. 57).
The experiment was carried out in French classes, so the transcripts in this article are our own translation. For the French original transcripts, see Mithalal (2010).
In French vertex is “sommet,” which has not only a mathematical meaning but also a meaning in everyday language, namely, “summit.”
The same truncated cube, symmetrical to the original one with respect to (ABC)
Julie described how to construct F as an intersection of two lines.
The same as the previous method, which was the intersection of two edges
For a complete presentation of the cK¢ model, the reader is invited to refer to Mithalal (2010, Appendix pp. 11 – 26).
Cabri 3D provides the user with an “intersection” tool, that constructs the intersection of two objects. It is not obvious that it does not construct the intersection of the three lines, as it only displays a message saying “this intersection point.”
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Mithalal, J., Balacheff, N. The instrumental deconstruction as a link between drawing and geometrical figure. Educ Stud Math 100, 161–176 (2019). https://doi.org/10.1007/s10649-018-9862-z
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DOI: https://doi.org/10.1007/s10649-018-9862-z