Philosophers and educational researchers have pondered and studied understanding for many years. From an educational perspective, illuminating how understanding is formed and improved can provide valuable theoretical and pedagogical insights. Combinatorics is an important field of research, with vast applications in mathematics and other fields of science. It is also included in many undergraduate curricula. In this thesis, I examine students' understanding of combinatorics based on two specific research questions: How do students understand a new definition? How do students improve their understanding of a concept when it is challenged? In one study, I examined students' initial understanding of a new definition and the corresponding concept image. I presented students with a new definition, trization, and a set of tasks carefully chosen to reveal different aspects of their concept image of the new definition. For this purpose I considered example generation, use of formulae, use of representations, and connections the students make to their existing knowledge and concept images. The study revealed that most students did not generate examples; however, they expected examples to be presented to them. Many students could find the appropriate formula for counting the number of possible trizations, after they were exposed to a few related tasks. Finding the formula helped them make a connection with their previous knowledge. Although a few students made this connection, they did not make any other anticipated connections. Many students' first attempt to understand trization was to create a pictorial representation, which, they did not consider significant in their understanding. However, algebraic representation was seen as a necessary and often sufficient form of understanding. In the other study, I developed a methodology, mediated successive refinement, helping students change an inappropriate concept image. This methodology was based on learner-generated examples, and it encouraged students not only to reflect on their own examples, but also to reflect and modify their peers'. This study identified the different scenarios that can occur when a student's concept image is changing. It established mediated successive refinement as a methodological tool for providing valuable research data, and a pedagogical tool for helping students improve their understanding.
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