This thesis studies the reasoning behaviour of successful mathematicians. It is based on the philosophy that, if the goal of an advanced education in mathematics is to develop talented mathematicians, it is important to have a thorough understanding of their reasoning behaviour. In particular, one needs to know the processes which mathematicians use to accomplish mathematical tasks. However, Rav (1999) has noted that there is currently no adequate theory of the role that logic plays in informal mathematical reasoning. The goal of this thesis is to begin to answer this specific criticism of the literature by developing a model of how conditional “if…then” statements are evaluated by successful mathematics students. Two stages of empirical work are reported. In the first the various theories of reasoning are empirically evaluated to see how they account for mathematicians’ responses to the Wason Selection Task, an apparently straightforward logic problem (Wason, 1968). Mathematics undergraduates are shown to have a different range of responses to the task than the general well-educated population. This finding is followed up by an eve-tracker inspection time experiment which measured which parts of the task participants attended to. It is argued that Evans’s (1984, 1989, 1996, 2006) heuristic-analytic theory provides the best account of these data. In the second stage of empirical work an in-depth qualitative interview study is reported. Mathematics research students were asked to evaluate and prove (or disprove) a series of conjectures in a realistic mathematical context. It is argued that preconscious heuristics play an important role in determining where participants allocate their attention whilst working with mathematical conditionals. Participants’ arguments are modelled using Toulmin’s (1958) argumentation scheme, and it is suggested that to accurately account for their reasoning it is necessary to use Toulmin’s full scheme, contrary to the practice of earlier researchers. The importance of recognising that arguments may sometimes only reduce uncertainty in the conditional statement’s truth/falsity, rather than remove uncertainty, is emphasised. In the final section of the thesis, these two stages are brought together. A model is developed which attempts to account for how mathematicians evaluate conditional statements. The model proposes that when encountering a mathematical conditional “if P then Q”, the mathematician hypothetically adds P to their stock of knowledge and looks for a warrant with which to conclude Q. The level of belief that the reasoner has in the conditional statement is given by a modal qualifier which they are prepared to pair with their warrant. It is argued that this level of belief is fixed by conducting a modified version of the so-called Ramsey Test (Evans & Over, 2004). Finally the differences between the proposed model and both formal logic and everyday reasoning are discussed.

The aim of this study was to generate a model (or theory) that explains students’ concerns as they tackle non-routine mathematical problems. This was achieved by using the grounded theory approach as suggested by Glaser and Strauss (1967) and further developed by Glaser (1978; 1992; 1998; 2001; 2003). The study took place in the context of a problem-solving course offered at the undergraduate level. As methods of data collection, the study made use of the problem-solving rubrics (or scripts) that students generated during the course. Other sources of data included interviews with the students and observations in class. The model generated as a result of this study suggests that problem solving can be seen as a four-stage process. The process was labelled ‘solutioning’ and is characterised by students trying to resolve the following concerns: Generating knowledge; Generating solutions; Validating the results, and Improving the results. The model also makes reference to pseudo-solutioning as an alternative approach to solutioning. During pseudo-solutioning, instead of trying to resolve the concerns listed above, students focus on trying to satisfy the academic requirement to submit an acceptable piece of work. Thus, pseudo-solutioning can be seen as an important variation to solutioning. After presenting the model of ‘solutioning’, the study provides an illustration of how it can be used to describe students’ processes. This is done in set of case studies in which three problem-solving processes are considered. The case studies provide a view of how the model developed fits the data and serves to highlight relevant patterns of behaviour observable as students solve problems. The case studies illustrate how the concepts suggested by the model can be used for explaining success and failure in the processes considered. This study contributes to the study of problem solving in mathematics education by providing a conceptualisation of what students do as they try to solve problems. The concepts that the model suggests are relevant for explaining how students resolve their main concerns as they tackle problems during the course. However, some of these concepts (e.g., ‘reducing complexity’, ‘blinding activities’, ‘transferring’) may also be of relevance to problem solving in other areas.