On the Psychological and Physiological Foundations of Structure in Geometry: A Study in Educational Neuroscience

Perception has structure. Aspects of this structure are relevant for image-based geometrical objects and relations between them, referred to as schematic perception and inferencing, respectively. Perception of geometrical structure, is a specific cognitive function. Without direct perception of structure mathematical reasoning may be inefficient and inaccurate. It is important for mathematics educators to understand the nature of schematic perception and to identify ways in which it can be nurtured in students. The main focus of this thesis is a specific aspect of image-based geometrical reasoning, the schematic nature of geometrical diagrams. The research framework is educational neuroscience. Selected results from mathematics education research pertaining to geometrical reasoning are constrained and informed by selected results from the neurosciences pertaining to the cerebral cortex and cerebellum, and vice versa. These two epistemological domains are integrated coherently with a theoretical framework that draws on embodied cognition and the neutral monism of Spinoza. A cognitive network model of the cerebral cortex enables concepts to be understood in an extensional (i.e., generalized) sense. It may also permit an explication of the distinction between procedural reasoning and conceptual reasoning and a re-evaluation of mathematics education theories of concept formation. However, the extensional concepts of the cerebral cortex are too inexact for mathematical application. I argue that a functional role of the cerebellum is to schematize these extensional concepts of the cerebral cortex, and then these schematic concepts may be understood in an intensional (i.e., abstracted) sense. I suggest there are implications for mathematics education theories of abstraction and generalization. I present the hypothesis that decontextualization in the presentation of mathematical concepts may be a significant factor in the development of students' ability for schematic perception and inferencing from geometrical diagrams.