Substantiating the Dynamic Views in Learning Geometry : From the Viewpoints of Image Schemata

Abstract

This research aims at clarifying the transition process from elementary mathematics to secondary one through the design experiment methodology. In this paper, we sorted out students' dynamic views which enabled them to enhance their understanding of inclusion relations between figures through analyzing our data of classroom lessons "constructing the figures using the operative sheets" based on the grounded theory approach. We could find the follwoing seven dynamic views as a result of our analysis. (A) Concrete manipulation, (B) Idealization and mental operation, (C) Gesture, (D) Grasping the movement of some point during the whole figure change, (E) Recognition of invariants, (F) Reverse operation, and (G) Simultaneous identification of invariants and variables. Then, we characterized the dynamic views of (A) concrete manipulation and (B) idealization and mental operation as the figurative image schema, the dynamic views of (C) gesture, (D) grasping the movement of some point during the whole figure change and (E) recognition of invariants as the opeartive image schema, and (E) reverse operation and (G) simultaneous identification of invariants and variables as the relational image schema. We consider that these characterizations may suggest the possible categories of the image schemata for the recognition of geometry, although we don't claim that these are all cases. Also, we may know why some students could learn the inclusion relations between figures and the others not, if we observe children's thinking in geometry in terms of the framework.