Experiencing the multiple dimensions of mathematics with dynamic 3D geometry environments: Illustration with Cabri 3D. (English)

Electron. J. Math. Technol. 2, No. 1, 39-53, electronic only (2008).

Summary: The paper analyzes how 3D dynamic geometry environments may be used to foster the exploration of multiple dimensions of 3D geometry. The notion of dimension is twofold: it refers, one the one hand to the dimension of a geometrical object, on the other hand to the multiple types of representation and expressions used in geometry. Two kinds of processes are involved in problem solving in geometry: iconic and non iconic visualization. The non iconic visualization consists in breaking down an object into parts of same or lower dimension. This cognitive process is critical for solving problems in geometry as very often the reasoning consists in establishing relationships between elements of the figure. However this process is not spontaneous and must be learned. 3D geometry is the source of new problems regarding iconic and non iconic visualization. Iconic visualization is not always reliable as it is in 2D geometry and non iconic visualization is more complex since it deals with a larger number of kinds of objects, from dimension 0 to dimension 3. The paper examines how 3D dynamic geometry environments may enlarge the iconic visualization and assist the non iconic visualization. 3D geometry computer environments may also offer a textual description linked to the dynamic diagram. The interplay between both representations not only facilitates the construction process of figures but also may be used to move from construction tasks to proof tasks. The example of Cabri 3D is used in the paper to illustrate the argument.