|dc.description.abstract||As an experienced and passionate problem solver for years, I wanted to better understand the metacognition that students exhibit when solving nonroutine geometry problems in a dynamic geometry environment. In this study, dynamic tool software—namely, the Geometer’s Sketchpad—was used by the participants. My intention was to focus on participants’ decision making, reflection, reasoning, and problem solving as well as to understand what situations and interactions in a dynamic geometry environment promote metacognitive behavior.
Case studies were conducted of two mathematics education preservice teachers who had previously completed a semester of college geometry and had prior experience working in Geometer’s Sketchpad. Artigue’s (2002) instrumental approach and Schoenfeld’s (1981) model of episodes and executive decisions in mathematics problem solving were used to identify patterns of metacognitive processes in a dynamic geometry environment. Data sources for this study consisted of think-aloud protocols, individual interviews after each problem-solving session, students’ written solutions, researcher’s observation notes, video files of problem solving sessions and a final interview. All collected data were analyzed using the constant comparative method for both the within-case and the cross-case analysis.
Problem solving of the two participants was described through identifying the metacognitive processes within each problem-solving episode, and associating them with the Geometer’s Sketchpad use. During the reading, understanding, and analysis episodes, the participants engaged in monitoring behaviors such as sense making, drawing a diagram, and allocating potential resources and approaches that helped make productive decisions. During the exploring, planning, implementation, and verification episodes, the participants made decisions to access and consider knowledge and strategies, make and test conjectures, monitor the progress, and assess the productivity of activities and strategies and the correctness of an answer. Geometer’s Sketchpad played an important role in supporting these metacognitive processes. Their use of metacognitive questions helped prompt a metacognitive activity. The effectiveness of solution approaches was dependent on the presence of managerial decisions. Cognitive problem-solving actions not accompanied by appropriate metacognitive monitoring actions appeared to lead to unproductive efforts. Redirection and reorganizing of thinking in productive directions occurred when metacognitive actions guided the thinking and when affective behaviors were controlled.||