Abstract

Although standards of rigor in mathematics are subject to debate among philosophers, mathematicians and educators, proof remains fundamental to mathematics and distinguishes mathematics from other sciences. There is no doubt that the ability to appreciate, understand and construct proofs is necessary for students at all levels, in particular for students in advanced undergraduate and graduate mathematics courses. However, studies show that learning and teaching proof may be problematic and students experience difficulties in mathematical reasoning and proving.
This thesis is influenced by Lakatos’ (1976) view of mathematics as a ‘quasi-empirical’ science and the role of experimentation in mathematicians’ practice. The purpose of this thesis was to gain insight into undergraduate students’ ways of validating the results of their mathematical thinking. How do they know that they are right? While working on my research, I also faced methodological difficulties. In the thesis, I included my earliest experiences as a novice researcher in mathematics education and described the process of choosing, testing and adapting a theoretical framework for analyzing a set of MAST 217 (Introduction to Mathematical Thinking) students’ solutions of a problem involving investigation. The adjusted CPiMI (Cognitive Processes in Mathematical Investigation, Yeo, 2017) model allowed me to analyze students’ solutions and draw conclusions about the ways they solve the problem and justify their results. Also I placed the result of this study in the context of previous research.