Learning Mathematics as an Interplay Between Internal and External Representations

How do we learn mathematics when we interplay with technology? This study investigates the impact of a dynamic geometry environment, GeoGebra, in enabling grade 7 students to explore geometrical problems. Grade 7 students in India and Sweden, with no prior experience of dynamic geometry, were selected as subjects for the study and were given geometrical tasks to investigate. Their problem solving strategies, reasoning, argumentation and the process of making conjectures have been analyzed using the notion of internal and external representations related to verbal and graphical encoding. Results indicate that when students of both countries worked with the same tasks in GeoGebra, they investigated the tasks in a similar manner, and they arrived at similar conclusions. Both the groups of students perceived DGE explorations as less time consuming, more accurate and easier to attempt as compared to paper and pencil tasks. They also used external representations aside of GeoGebra, such as verbal explanations, drawings, and notes on paper-and-pencil. We argue that many dragging episodes may be seen as the development of a dynamical visual representational connection between verbal and graphical encoding.