This research consists of teacher interviews, student interviews, and classroom observations, all based around the mathematical content area of combinatorics. Combinatorics is a part of discrete mathematics concerning the ordering and grouping of distinct elements. The data are used in four separate analyses. The first provides evidence that student interviews can be a useful source of data when considering the qualities of instruction. The case analysis shows that the teacher's instruction shifted. During interviews, the student responses showed indications of the shifts. The student interviews allowed us to see things we would not have seen through classroom observations or written assessments, and these things reflected the qualities of the instruction. The second analysis explores a framework of types of teacher knowledge in a novel way. The analysis assigns knowledge types to statements made during interviews. Not all teachers showed the same relative frequency of the different types. The implication is that with more teachers and in connection with classroom data, we may understand what these profiles suggest about a teacher's work and the types of supports that would help them. The third analysis examines the connections between students solving problems involving the multiplication principle and solving problems involving permutations. Analysis of interviews showed that on problems involving permutations, students often incorrectly overextended the multiplication principle. Students are struggling to make the transition from multiplication principle problems to permutation problems. This suggests that they need support to understand of how the two types of problems differ. The fourth analysis looks at students' representations in combinatorics. Both interviews and classroom observations showed novel student representations. The analysis shows that students generate useful non-canonical representations and that we can benefit from utilizing these. The four analyses connect to different areas of research. The first two papers consider the complex characteristics of teacher knowledge. They aim to become part of the ongoing conversation about how to prepare, evaluate, and support math teachers. The third and fourth papers focus on elements of student thinking in combinatorics. These provide examples to indicate that there is still much we do not know about this area. [The dissertation citations contained here are published with the permission of ProQuest LLC. Further reproduction is prohibited without permission. Copies of dissertations may be obtained by Telephone (800) 1-800-521-0600. Web page: http://www.proquest.com/en-US/products/dissertations/individuals.shtml.]