What is mathematics? : an examination of the perceptions of mathematics, mathematical expertise and its relationship to reasoning and language

The nature of mathematical abilities is poorly understood in comparison to most other abilities, such as language and reasoning. Most research on the psychology of mathematics has focused on simple mathematical abilities such as numerical calculation. This thesis contributes to foundations for research examining high level mathematical abilities, particularly expert level abilities. These contributions are twofold. The first contribution is ascertaining how mathematics is defined and perceived by individuals currently working in high-level mathematics. The studies in chapters 3 and 4 examined mathematicians' definitions of mathematics, perceptions of how expertise in mathematics may be defined, the skills important for mathematics, and the barriers to studying mathematics. The results suggest that mathematicians view criteria commonly used in psychological research on expertise as less important than performance-based measures, and differentiate between the skills required by a successful student of mathematics and a mathematical expert. The second contribution concerns the relationships between mathematical abilities and abilities in other appropriate domains. Results reported in chapters 1 and 2 and a review of the relevant literature indicate that the two most relevant domains for such investigations are deductive reasoning and language processing. Chapter 5 examined the relationship between performance on mathematical and deductive reasoning tasks, specifically syllogisms and the Wason Card Task. Performance on neither reasoning task was found to be related to mathematical performance, suggesting that performance on mathematical tasks and simple reasoning may be unrelated. The final three empirical chapters examined the relationship between mathematical and linguistic abilities from three perspectives. Chapter 6 reports a survey of high school teachers' perceptions of the skills required for success in mathematics, English and science. In general the skills perceived to be important for the three subjects differed and, importantly, not just the skills traditionally linked to the different subjects but also general cognitive abilities such as long term memory and creativity. Chapters 7 and 8 examined the link between mathematical processing and linguistic processing abilities which has been proposed by previous research. The study in chapter 7 found that reaction times to mathematical stimuli generally showed the same pattern as the linguistic stimuli they were modelled on. The type of stimulus influenced the way participants interacted with the stimuli so the relationship and the degree of similarity may depend on the area of mathematics, as well as how the stimuli are presented. Chapter 8 continued this examination using eye tracking to more closely examine how stimuli are being parsed. Influences on parsing included expertise and the mathematical stimulus format, (for example numerical as oppose to worded problems). Findings are not consistent with predictions from previous linguistic research, even with linguistic stimuli used in earlier studies, thereby raising new questions regarding the effects of stimulus format and task demands. The thesis concludes with a discussion of methodological issues in experimental studies of mathematical and other cognitive abilities, and research on expertise