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

Stellenbosch University (SOUTH AFRICA)

Appears in: INTED2014 Proceedings

Publication year: 2014

Pages: 6158-6166

ISBN: 978-84-616-8412-0

ISSN: 2340-1079

Publication year: 2014

Pages: 6158-6166

ISBN: 978-84-616-8412-0

ISSN: 2340-1079

Conference name: 8th International Technology, Education and Development Conference

Dates: 10-12 March, 2014

Location: Valencia, Spain

Dates: 10-12 March, 2014

Location: Valencia, Spain

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.

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