### Abstract:

Mathematical sophistication increases rapidly as students transition from lower- to upper-division physics courses. Complex algebra is one of the mathematical tools that is not introduced or used in lower-division physics courses but is pervasive throughout upper-division courses. In this dissertation, I study middle-division physics students' developing fluency with complex number algebra as they transition from lower- to upper-division physics at Oregon State University.
Through a baseline study of students' calculation abilities with simple complex number algebra tasks, I find that there are three general categories of difficulties: performing calculations, switching between forms, and appropriately selecting forms to simplify calculations. There are varying degrees of complex number fluency among middle-division physics students, and that some complex number difficulties persist over time, even into upper-division physics courses.
Difficulties with circle trigonometry are common among middle-division physics students and contribute to students' difficulties in the three general categories. Student entering the junior year exhibit difficulties determining rectangular and polar coordinates, which suggests that distinguishing between triangle and circle trigonometry is difficult for students. Through interviews with seven students, I confirm that the simultaneous use of triangle and circle trigonometry approaches contributes to difficulties determining correct algebraic representations of complex numbers.
Students who choose to isolate approaches---triangle trigonometry, circle trigonometry, Pythagorean theorem, and norm squared---tend to be successful in determining algebraic representations from a geometric representation of a complex number. Isolating an approach requires the coordination of both algebraic and geometric representations of the complex number. The isolation of approach requires that students are able to distinguish between triangle and polar geometry, a productive strategy in translating geometric to algebraic representations.
As students translate a geometric complex number representation to several algebraic representations, I identify, describe, and demonstrate students' coordination of four fine grain-size epistemic games. Students' use of the games reflects students' development of problem solving strategies at the middle-division. The goal-oriented games demonstrate essential components of physics problem solving such as recognizing a goal to progress with the task, decisions to narrow the individual's goal, and acknowledgment that a task is complete.