In this study, I will examine the beliefs and conceptual understanding of instructors and students from two abstract algebra classes. This research takes the form of a case study in which I answer four research questions, each addressing a relationship between instruction and beliefs or conceptual understanding. Specifically, these research questions are:

1. What beliefs do the instructors have about math, teaching, and learning and what relationship exists between these beliefs and instructional practice?
2. What is the relationship between instructional practice and students' beliefs about math, teaching, and learning?
3. What conceptual metaphors do the professors use to describe isomorphisms and homomorphisms and what relationship exists between these metaphors and the mathematical content in instruction?
4. What is the relationship between the mathematical content in instruction and conceptual metaphors the students use to describe isomorphisms and homomorphisms?

In terms of beliefs, the instructors articulated considered positions on the nature of math, math learning, and math teaching. These beliefs were clearly reflected in their overall approaches to teaching. However, their instruction shifted in practice over the course of the semester. Students' beliefs seemed to shift slightly as a result of the ways their instructors taught. However, their core beliefs about math seemed unchanged and some lessons students took away were similar in the two classes.

In terms of conceptual understanding, the instructors provided many conceptual metaphors that related to how they understood isomorphism. They struggled more to provide an image for homomorphism, which requires thinking about a more complicated mathematical object. Their understandings of isomorphism and homomorphism were largely reflected in their instruction with some notable differences. Students took away similar understandings of isomorphism to the instructors, but did not all take away the same level of structural understanding of homomorphism.

In short, relationships between instructors' beliefs and instruction and between instructors' conceptual understanding and instruction were evident. However, certain elements were not made as clear as they perhaps intended. Relationships between instruction and students' beliefs and between instruction and students' conceptual understanding were also evident. However, relationships between instruction and beliefs were subtler than between instruction and conceptual understanding.