Prospective Mathematics Teachers’ Perceptual and Discursive Apprehensions when Making Geometric Connections

In this paper we probe prospective mathematics teachers' thinking as displayed in their solutions to a geometry content knowledge-based task. Geometry content knowledge is examined in terms of connections made by these prospective teachers between the geometric configurations and the geometric principles established through visualization and reasoning processes. The successful completion of the task required identifying and recognizing figures, and making connections between geometry representations, properties and theorems. We focus on three different prospective mathematics teachers engaging with a written geometry task. Duval's notions of perceptual and discursive apprehensions are employed to interpret how the prospective mathematics teachers interact with a diagram. When responding to the task the prospective mathematics teachers encountered difficulties connecting the cognitive processes of visualization and reasoning. The difficulties are evidenced by weak perceptual apprehension and weak discursive apprehension. This confirms that there is a link between what is seen and what is uttered about that which is seen. Emerging from this research is a Typology of Connections in Solving geometry Tasks model for classifying and describing forms of geometric connections. The model links cognitive processes with cognitive apprehensions when making geometric connections.