The novice mathematician's inquiry about new concepts: Bestowing meaning through ambivalent uses of geometrical metaphors

The doctorate project on which this paper is based, is a study of the novice mathematician's conceptual and reasoning difficulties in their encounter with mathematical abstraction. For this purpose 20 Oxford first-year mathematics undergraduates were observed and audio-recorded in their weekly tutorials and interviewed twice in two academic terms. Learning Episodes were selected and arranged in terms of mathematical topics (Foundations of Analysis, Calculus, Topology, Linear Algebra and Group Theory). Data analysis aimed at the emergence of data-grounded theory and was balanced between the topical (epistemological) and cross-topical (psychological) perspectives. The final synthesis of the findings was arranged in terms of a number of themes that characterised the novices' learning behaviour. The Learning Episode presented in this paper exemplifies one of these themes—concept-image construction through the adoption of geometrical metaphors—within the mathematical area of Group Theory and specifically the new concepts of coset and equivalence class.