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Dynamic and Tangible Representations in Mathematics Education

Abstract : Dynamic geometry environments offer a new kind of representation of mathematical objects that are variable and behave "mathematically" when one of the elements of the construction is dragged. The chapter addresses three dimensions about the transformations brought by this new kind of representation in mathematics and mathematics education: an epistemological dimension, a cognitive dimension and a didactic dimension. As so often stated since the time of ancient Greece, the nature of mathematical objects is by essence abstract. Mathematical objects are only indirectly accessible through representations (D'Amore 2003, pp. 39-43) and this contributes to the paradoxical character of mathematical knowledge: "The only way of gaining access to them is using signs, words or symbols, expressions or drawings. But at the same time, mathematical objects must not be confused with the used semiotic representations" (Duval 2000, p. 60). Other researchers have stressed the importance of these semiotic systems under various names. Duval calls them registers. Bosch and Chevallard (1999) introduce the distinction between ostensive and non ostensive objects and argue that mathematicians have always considered their work as dealing with non-ostensive objects and that the treatment of ostensive objects (expressions, diagrams, formulas, graphical representations) plays just an auxiliary role for them. Moreno Armella (1999) claims that every cognitive activity is an action mediated by material or symbolic tools. Through digital technologies, new representational systems were introduced with increased capabilities in manipulation and processing. The dragging facility in dynamic geometry environments (DGE) illustrates very well the transformation technology can bring in the kind of representations offered for mathematical activity and consequently for the meaning of mathematical objects. A diagram in a DGE is no longer a static diagram, representing an instance of a geometrical
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C. Laborde, J.-M Laborde. Dynamic and Tangible Representations in Mathematics Education. Transformation - A Fundamental Idea of Mathematics Education, pp.187 - 202, 2014. ⟨hal-02381868⟩

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