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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. 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. 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.