# DEFINITIONAL CONFLICTS BETWEEN EUCLIDEAN GEOMETRY AND DYNAMIC GEOMETRY ENVIRONMENTS: VARIGNON THEOREM AS AN EXAMPLE

M. NdlovuStellenbosch University (SOUTH AFRICA)

The purpose of this paper is to illustrate some of the obstacles or theoretical-computational conflicts (Giraldo et al 2003) brought on by technology in the context of geometry. The question is whether we have to accept Euclidean definitions as sacrosanct or we have to accept the challenge and revise and accept some of the forbidden definitional conditions as tenably true. The traditional school mathematics quad, for example, is understood to be convex and many school syllabi present examples of these without counter-examples of alternative meanings and interpretations of the definitions. Monaghan (2000) attests that there is a connection between the perception and description of a given geometrical figure. By allowing the user to manipulate, experiment and conjecture with shapes in real-time DGEs (such as Sketchpad, Cabri, and Geogebra) have, through the dragging activities, have the potential to influence the perception of geometrical figures in a non-traditional paper and pencil way that challenges the legitimacy of some of the traditionally accepted geometrical definitions. For example, a quadrilateral with a reflex angle (concave quad) perceptually appears to be a triangle to many learners.

However learners exposed to dynamic geometry environments in which they interactively manipulate virtual shapes can readily perceive the shape as a quadrilateral they have previously constructed. Similarly, a crossed quadrilateral may be perceived not to be a quad at all by many learners until its vertices are dragged experimentally and discovered to fulfill all the properties of its convex counterpart. Varignon’s theorem on the other hand is similarly upheld when two of a quadrilateral’s sides are dragged to be co-linear surprisingly permitting us to re-define a triangle as a quad with two collinear vertices in direct contravention of this forbidden Euclidean condition. Student’s movement from a spatio-graphic geometry (of physical models, diagrams, and computer images ..) to pre-axiomatic geometry (of axioms and definitions) referred to by Parzysz (2003) can be aided by the multiple cases of the same figure that DGEs afford them. DGEs such as Sketchpad which are now ubiquitously present in many classrooms allow learners to create this so-called traditionally forbidden geometry and thus challenge mathematics educators and curriculum writers to reframe the Euclidean surface in dynamic terms to belie some of the seemingly absurd definitions in the Euclidean stable.

References:

[1] Giraldo, V., Tall, D. & Carvalho, L. M. 2003. Using theoretical computational conflicts to enrich the concept image of the derivative. Research in Mathematics Education. 5(1):63-78.

[2] Monaghan, F. 2000. What difference does it make? Children’s views of the differences between some quadrilaterals. Educational Studies in Mathematics, 42:179-196.

[3] Parzysz, B. 2003. Pre-service elementary teachers and the fundamental ambiguity of diagrams in geometry problem-solving. European Research in Mathematics Education III, Thematic Group 1.