Abstract 
 Policy and research have increased attention in K12 mathematics education on core mathematical practices such as argumentation, justification, and proof, that are central to students learning and doing mathematics. One reason for this increased focus is the empirical evidence that engaging students in mathematical practices provide more equitable opportunities for student learning. Although mathematics disciplinary practices (MDPs) are found in postsecondary educational literature, they are often described in context to particular mathematical content areas in undergraduate mathematics. Research in undergraduate mathematics rarely takes mathematical practices as a focal point of research, even though these practices are essential to doing mathematics across multiple content areas. Because students in advanced mathematics courses are more explicitly engaged in the practices of the discipline, understanding how mathematicians report on using and teaching MDPs is an essential next step toward understanding the role that MDPs play across mathematical content domains.
My qualitative dissertation study presented in two manuscripts collectively examines mathematics disciplinary practices (MDPs) and their relation to teaching and learning in advanced undergraduate mathematics courses. Across two phases, I initially conducted an interview study with eight mathematicians that teach upperdivision undergraduate mathematics courses to understand the ways that mathematicians use MDPs in their professional research work and how they approach teaching MDPs in their advanced mathematics courses. In the second phase, I identified two case studies from the first phase of the research of mathematicians’ advanced
undergraduate instruction that I investigated with the goal of better understanding the ways that MDPs might emerge in classroom settings. Across both manuscripts of the study, I provide a descriptive analysis of the MDPs that mathematicians use in research and teaching.
In the first manuscript, I highlight three themes that I identified through the process of my analysis of interviews with eight mathematicians. To situate the study within the educational literature, I trace the ways that mathematical practices were developed across policy documents. I offer a conceptualization of MDPs grounded in the research literature that provides the foundation for discussing MDPs in the professional work of mathematicians and how such practices are taught in advanced mathematics. The first theme outlines the MDPs that were reported by mathematicians and describes how these practices coalesced. The second theme reports on are the ways that MDPs are used in mathematicians' research. The final theme examines mathematicians' reflections on teaching students MDPs in their advanced undergraduate mathematics courses. Within this theme, I explicate the social and analytical scaffolding mathematicians' reported to teach students MDPs in their advanced courses. When I pressed mathematicians for teaching practices that support students’ learning MDPs, they often drew on general teaching practices, including demonstrations with a variety of examples, employing group work, or assigning homework problems. Collectively, my analysis of the interview data highlights the difficulty of teaching towards MDPs and I discuss the complexity of MDPs in mathematicians' professional work and the ways MDPs might be taught in advanced mathematics settings. From my analysis, I suggest that teaching mathematical practices is a difficult endeavor.
In my second manuscript, I focus on an important MDP that I identified from my analysis of mathematicians' interviews. I describe how mathematicians attended tomathematical conditions, assumptions, and properties as they engage in mathematical work across content domains (which I call the CAPs practice). Although the CAPs practice appears in literature around mathematical modeling and to some extent in mathematical proof, the CAPs practice is treated by many researchers as a smaller component of these larger practices (e.g., creating mathematical models, proving mathematical statements). In this manuscript, I argue that attending to conditions, assumptions, and properties is an importantcrosscutting practice of the discipline that mathematicians described as an important part of their research and in their teaching. This manuscript offers a framework for understanding four ways that the CAPs practice is used in the
discipline of mathematics, as well as how the CAPs practice extends to other STEM disciplines. The findings offer mathematician perspectives on the ways they learned to engage in the CAPs practice, how they currently engage in the practice professionally, and instances in which they discussed CAPs within their teaching. Implications from this work call attention to a disciplinary practice that is largely unexplored as an entity and offers an empirically grounded definition.
An implication of this overall study is for reframing the mathematics students should learn and ways to improve students' undergraduate experiences in advanced mathematics courses to support their learning. This implication reflects similar calls in K12 mathematics education for greater attention to mathematical practices. In particular, this study advances the importance of teaching towards mathematical practices across all content domains in postsecondary mathematics courses to support students’ explicit engagement in MDPs in undergraduate mathematics classrooms. There is a real need to explore the relationships between the variety of professional mathematical practices and how mathematicians can employ specific instructional strategies to support students’ engagement in these MDPs in their teaching. The second implication of this dissertation study is for the community of mathematics education researchers whose interests are in undergraduate mathematics. The results of this study indicate a need for more research on instructional approaches that support students in advanced mathematics to learn the constellation of MDPs that mathematicians use in their research. These two manuscripts describe the importance of MDPs in learning and doing mathematics. MDPs should not be treated implicitly as they are often subsumed in educational research focused on specific mathematical content areas. I look forward to advancing this work to explore MDPs in developmental courses and lowerdivision mathematics courses. A goal of future research will study specific classroom teaching practices that promote students' engagement in MDPs, as well as focus on student experiences as they learn to engage in MDPs in classroom settings.
