Studying abstract algebra to teach high school algebra [electronic resource] : investigating future teachers' development of mathematical knowledge for teaching
Keywords: Education.

A fundamental issue facing mathematics educators is developing rich, deep mathematical understanding in teacher candidates. Research suggests teachers need more than just a basic knowledge of the math they teach. They need mathematical knowledge for teaching (MKT): the ability to unpack mathematical ideas, to interpret student errors, and to flexibly use multiple representations of a single concept (Ball, Thames, & Phelps, 2008). Teachers must also be able to engage in mathematical practices, which are the tools needed to do mathematics. Though future teachers may gain general math knowledge through college courses, advanced college mathematics looks very different from the math taught in high school, creating a "vertical disconnect" (Cuoco, 2001, p. 3). To address this disconnect, math educators have called for the design of math courses for future teachers. In this study, I investigate future secondary teacher development of MKT in a math class designed for teachers. I conducted a case study of an abstract algebra course designed specifically for pre-service teachers. Abstract algebra is a ubiquitous requirement for math majors and often feels very disconnected from school mathematics. The case study provided insight into the opportunities to learn available in the content course. I investigated participants' learning through interviews and surveys that explored MKT and engagement in mathematical practices. I also collected data on pre-service teacher learning at a comparison site that did not offer math content courses for teachers. Collecting data from two sites afforded considering a wider range of teacher learning. My dissertation focuses on three key themes. In Article I, I focus on the opportunities to learn in the abstract algebra course designed for teachers. Through a novel course design, participants had opportunities to learn mathematical content (secondary and tertiary), mathematical practices, and pedagogical techniques. In Article II, I explore participant learning around engaging in mathematical practices. I found participants had different learning trajectories and all improved their ability to engage in justification on a school level algebra task. The growth was related to the opportunities to learn in the class, indicating participants were able to apply practices from abstract algebra to high school content. In Article III, I explore challenges in conceptualizing and measuring secondary MKT across participants at both sites. Performance on the survey measure revealed a ceiling effect, which illuminated the challenge of developing a secondary MKT measure with appropriate levels of difficulty. Among participants who performed similarly on the survey, task-based think aloud interviews revealed differences in engagement in mathematical practices. This suggests that survey measures may insufficiently capture practice engagement. This study has key implications for teacher preparation and research. The opportunities to learn in the abstract algebra class around math content, practices, and pedagogy suggests the value of tailoring math content courses to meet the needs of pre-service teachers. Notably, participants were able to apply their learning about practices to high school level content. In light of the emphasis on mathematical practices in the Common Core Standards, teachers must be able to engage in practices themselves and be able to teach their students to do the same. Accordingly, it is critical to better incorporate a perspective on engagement in mathematical practices into the conceptualization of secondary mathematical knowledge for teaching.