The novice mathematician's encounter with mathematical abstraction : tensions in concept-image construction and formalisation
Mathematics is defined as an abstract way of thinking. Abstraction ranks among the least accessible mental activities. In an educational context the encounter with mathematical abstraction is the crucial step of the transition from informal school mathematics to the formalism of university mathematics. This transition is characterised by cognitive tensions. This study aimed at the identification and exploration of the tensions in the novice mathematician's encounter with mathematical abstraction. For this purpose twenty first-year mathematics undergraduates were observed in their weekly tutorials in four Oxford Colleges during Michaelmas and Hilary Term of Year 1. Tutorials were tape-recorded and fieldnotes kept during observation. The students were also interviewed at the end of each term of observation. The recordings of the observed tutorials and the interviews were transcribed and submitted to an analytical process of filtering out episodes that illuminate the novices' cognition. An analytical framework consisting of cognitive and sociocultural theories on learning was applied on sets of episodes within the mathematical areas of Foundational Analysis, Calculus, Linear Algebra and Group Theory. This topical analysis was followed by a cross-topical synthesis of themes that were found to characterise the novices' cognition. The novices' encounter with mathematical abstraction was described as a personal meaning-construction process and as an enculturation process: the new culture is Advanced Mathematics introduced by an expert, the tutor. The novices' interaction with the new concept definitions was obstructed by their unstable previous knowledge. Concept image construction was described as a construction of meaningful metaphors and an exploration of the 'raison-d'-être' of the new concepts and the new reasoning and was characterised by the tension between the Informal/Intuitive/Verbal and the Formal/Abstract/Symbolic — which was discussed in terms of semantics and reasoning. The novices were in difficulty with the mechanics of formal mathematical reasoning as well as with applying these mechanics in a contextualised manner. This decontextualised behaviour was linked to the fragility of their knowledge with regard to the nature of rigour in formal mathematics.