Categories, definitions and mathematics : student reasoning about objects in analysis
This thesis has two integrated components, one theoretical and one investigative. The theoretical component considers human reason about categories of objects. First, it proposes that the standards of argumentation in everyday life are variable, with emphasis on direct generalisation, whereas standards in mathematics are more fixed and require abstraction of properties. Second, it accounts for the difficulty of the transition to university mathematics by considering the impact of choosing formal definitions upon the nature of categories and argumentation. Through this it unifies established theories and observations regarding student behaviours at this level. Finally, it addresses the question of why Analysis seems particularly difficult, by considering the relative accessibility of its visual representations and its formal definitions. The investigative component is centred on a qualitative study, the main element of which is a series of interviews with students attending two different first courses in Real Analysis. One of these courses is a standard lecture course, the other involves a classroom-based, problem-solving approach. Grounded theory data analysis methods are used to interpret the data, identifying behaviours exhibited when students reason about specific objects and whole categories. These behaviours are linked to types of understanding as distinguished in the mathematics education literature. The student's visual or nonvisual reasoning style and their sense of authority, whether "internal" or "external" are identified as causal factors in the types of understanding a student develops. The course attended appears as an intervening factor. A substantive theory is developed to explain the contributions of these factors. This leads to improvement of the theory developed in the theoretical component. Finally, the study is reviewed and the implications of its findings for the teaching and learning of mathematics at this level are considered.