The Electronic Journal of Mathematics and Technology, Volume 3, Number 3, ISSN 1933-2823 The Impact of Structural Algebraic Units on Students’ Algebraic Thinking in a DGS Environment Stavroula Patsiomitou spatsiomitou@sch.gr Department of Primary Education University of Ioannina, Greece Abstract: The present paper attempts to bridge the world of digital technology and the world Euclid bequeathed us in his "Elements". The role of the design process of activities in an dynamic geometry interactive environment such as that of Geometer’s Sketchpad v4 is examined, along with ways in which students can be assisted to understand algebraic concepts through geometrical reconstructions. The ways in which it can facilitate the understanding of geometrical concepts are examined, along with the bridging, between the fields of algebra and geometry, and strategies for overcoming obstacles. Two volunteer teams (one control, one experimental) were evaluated with regard to their ability to represent concrete algebraic expressions using geometrical representations on cardboard. From the results, it can be concluded that the experimental team managed not only to construct the concrete identities, but also to connect them with formal reasoning. 1. Introduction The geometrical representations of identities and algebraic expressions in general were initially developed in Euclid‘s ―Elements‖ (325-265). Stamatis [14] declares: ―The second book of Euclid‘s ―Elements‖… includes the application of Geometry to Algebra and is ascribed for the most to Pythagoreans. The first 10 theorems relate to algebraic identities, which we are able to represent in the following way: with the letters a, b, c assigned to represent straight segments.‖ (my translation from the Greek text). Netz [28] also shows that the Greeks considered diagrams essential to geometrical proofs and they allowed properties to be conceptualized from the diagram. Fowler [7] makes it clear Greek geometry was non-arithmetical and did not use fixed units for measurement [10]. Herbst‘s opinion is [12] that Netz‘s [28] study of lettering practices in Greek geometry permits the observation that Greek geometers produced their diagrams at the same time that they conceived their proofs: i.e. the diagram was not drawn at the end to illustrate the written proof, and was not drawn in its entirety before the production of the argument‖. Representations were the first empirical mode leading to the proving process in Ancient Greece, (see, for example, Socrates and Meno) although the process observed in Euclid‘s ―Elements‖ does not display a transition from visual representation to rigorous reasoning. The visual representations can prove only specialized cases, while the Euclidean proof can empower every case by reinforcing the initial visual proof. In the Platonic dialogue (Socrates: slave), the slave‘s incorrect answers are restructured with a shape: a concrete representation, with the particular shape functioning as the visual proof of the accuracy of Socrates‘ proposition to the slave. The phrase ―ei mi voulei arithmein alla deixon‖ which means ―if you don‘t want to measure, just prove‖ (my translation from the ancient Greek text) can be considered an interpretation of what the ancient Greeks meant by ‗proof‘, making use of the concrete expression of the ancient verb ―deiknymi‖. Furinghetti and Paola [16] argue that the didactical suggestion implicit in Lakatos‘ work is that a return to the spirit of the Greek geometers would be advisable. Referencing Szabo, they call for ‗deiknymi‘ to be developed both analytically and synthetically [16]. How meaningful can the act of ―deiknimi‖ be nowadays, and how do students understand it? How can this be achieved through