The Development of Students Geometrical Thinking through Transformational Processes and Interaction Techniques in a Dynamic Geometry Environment

Abstract

The paper draws on a didactic experiment conducted in a secondary school mathematics classroom in Greece which aimed to explore a) ways in which students develop problem representations, reasoning and problem-solving, making decisions and receiving feedback about their ideas and strategies in a DGS-supported environment b) ways in which students develop rigourous proof through building linking visual active representations and c) ways to develop students’ van Hiele level. The mathematical problem the students engaged with - either in the Geometer’s Sketchpad dynamic geometry enviroment (Jackiw, 1988) or in the static environment - generated potentially insightful data on the issues focused on the comparison between the experimental and control groups. Initially, three pairs from the experimental group explored the treasure problem within a dynamic geometry environment. The discussions and results of the discussion were videotaped. The problem was then reformulated by the researcher taking into account the research group’s retroaction, and re-explored by both the control and experimental groups in a paper-pencil test. The researcher then (semi) pre-designed multiple-page sketches detailing the sequential phases of the solution to the problem using rigorous proof, and in so doing transferring her classroom reaching style into the software design, drawing on the chain questioning method of Socrates, which aim to stimulate interaction. For this reason, she linked all the software func-tions/actions using the interaction techniques supported /facilitated by the Geometer’s Sketchpad v4 (DGS) environment (Jackiw, 1988) to better allow students to discover solution paths and to reason by rigorous proof. This mode of design and the results of the experimental use of the software with students led to the need to define two new concepts: the meanings of Linking Visual Active Representations (LVAR) and Reflective Visual Reaction (RVR). The researcher observed the students’ actions and thinking processes during the research process and offers a description and analysis of these processes. An analysis of the results of the experimental procedure revealed