Translating Euclid: Liberating the Cognitive Potential of Collaborative Dynamic Geometry

How should one translate the classic-education approach of Euclid’s geometry into the contemporary vernacular of computer visualization, social networking and discourse-centered pedagogy? The birth of geometry in ancient Greece and its systematization by Euclid played an important role in the development of deductive reasoning and science. As it was translated and refined over the centuries, however, geometry lost some of its cognitive power and its very nature became obscured. Recently, computer-supported versions of dynamic geometry have been developed, which afford visualization, manipulation, exploration, conjectures about constraints and construction of dependencies. Particularly within a context of computer-supported collaborative learning, a dynamic-geometry environment can facilitate the gaining of mathematical insight and understanding that is the hallmark of geometry. A case study involving an intriguing geometric configuration illustrates this within a trial test of the Virtual Math Teams collaboration environment with a multi-user version of the GeoGebra dynamic-mathematics application. This indicates how significant mathematical discourse about dynamic geometry within such environments can make salient the construction of dependencies as central to deductive cognition. Translating from Classic to Post-Modern Philosophy A small community of geometers in ancient Greece established a set of discourse practices and inscription methods that defined subsequent literate rational thought in the West. A cognitive history of this accomplishment is documented in Netz (1999). Latour (2008) reviewed Netz’ analysis and suggested some of its significance. Since the golden age of Greece, geometry and rationality have gone through many transformations. To approach the question—What form should mathematics education take in the 21 Century?—it is helpful to first review the historic development of geometry. The inscriptions. Using very primitive technology—ephemeral sketches in the sand and more persistent and portable diagrams on papyrus, wood, wax or clay tablets—Euclid’s predecessors constructed intricate diagrams using just straightedge and compass. The diagrams consisted of points, line segments and arcs or circles. Construction sequences were used to establish dependencies among the components of constructed diagrams. Importantly, components were labeled with letters. The labels allowed accompanying texts to reference specific components, thus providing a clear visible connection between the elements of the inscription and specific statements in the text. The texts. The textual discourse of the early geometers consisted of a highly stylized, formulaic language. The language of geometry was derived from everyday written Greek, but required specialized training to be used. The language was geared to stating parts of propositions and proofs, such as the statement of given conditions, or well-known propositions that contributed to the proof, or steps of construction and of proof conclusions. Presentations of geometric propositions consisted primarily of proofs with their accompanying labeled diagrams. The propositions. In addition to mastering the inscription and discourse practices, a mathematician had to be very familiar with a corpus of established propositions. The knowledge of these propositions was