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Discernment of invariants in dynamic geometry environments

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Abstract

In this paper, we discuss discernment of invariants in dynamic geometry environments (DGE) based on a combined perspective that puts together the lens of variation and the maintaining dragging strategy developed previously by the authors. We interpret and describe a model of discerning invariants in DGE through types of variation awareness and simultaneity, and sensorimotor perception leading to awareness of dragging control. In this model, level-1 invariants and level-2 invariants are distinguished. We discuss the connection between these two levels of invariants through the concept of path that can play an important role during explorations in DGE, leading from discernment of level-1 invariants to discernment of level-2 invariants. The emergence of a path and the usefulness of the model will be illustrated by analysing two students’ DGE exploration episodes. We end the paper by discussing a possible pathway between the phenomenal world of DGE and the axiomatic world of Euclidean geometry by introducing a dragging exploration principle.

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Notes

  1. Since dragging is the fundamental activity, we will be referring to within DGE, we refer to the student, or, in general, the person dragging, as dragger.

  2. We remark that, in DGE like Cabri, dependent elements cannot be moved by dragging them directly but only indirectly by dragging the elements they depend upon. However, in some other DGE for the same construction, r and s could be moved by directly dragging them.

  3. In some other DGE, they can be acted upon directly but in this case they lose their defining properties.

  4. It is beyond the scope of this paper to compare our model with other models. In particular, it should not be thought of as “developmental”: The transition it describes is given in terms of a set of means of discernment that we do not relate to any particular stage of cognitive development of the individual. Being able to discern a level-2 invariant and interpret it geometrically, per se, with respect to a model like Van Hiele’s certainly implies being able to operate at least at level 2 (Van Hiele, 1986, p.53); however, what we are concerned with is describing, from a cognitive point of view, what might be involved in discerning such an invariant during an (even single) experience in a dynamic geometry software.

  5. The notion is consistent with Leung and Lopez-Real’s (2002) notion of locus of validity.

  6. This is why a path can become the key element in a process of conjecture–generation. So analyzing how students discern paths and elaborate them can provide insights on their processes of conjecture–generation.

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Correspondence to Allen Leung.

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Leung, A., Baccaglini-Frank, A. & Mariotti, M.A. Discernment of invariants in dynamic geometry environments. Educ Stud Math 84, 439–460 (2013). https://doi.org/10.1007/s10649-013-9492-4

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