Looking beyond the lecture: A qualitative examination of pedagogical decision-making and student learning in upper-level mathematics courses

Research on the teaching and learning of university mathematics has primarily focused on what happens inside the classroom. This is in spite of acknowledgment from mathematicians and mathematics education researchers that most of student learning occurs independently outside of class. The primary aim of this research is to look beyond the lecture, to gain a more holistic insight into the nature of undergraduate mathematics teaching and learning. We focus on upper-level courses, which to date have been under researched and are subject to different institutional and disciplinary demands than introductory courses. We report on two qualitative, interview-based studies. In the first, four mathematicians participated in a series of three, one-on-one interviews about their instructional design of an upper-level, lecture-based course that they had recently taught. In contrast to previous work that has focused on the lecture or individual instructional components, we examine participants’ pedagogical decision-making in relation to all instructional components of their courses, including the notes, lectures, homework, tutorial classes, and final examinations. We identify two tensions in participants’ decision-making: the content and problem-solving tensions. We explore the factors that give rise to the tensions, how participants attempted to manage them through their course design, and the implications for future research aimed at augmenting and improving lecture-based instruction. An important factor of influence on mathematicians’ instructional decision-making is their pedagogical content knowledge. Prior research on, and theoretical development of, pedagogical content knowledge has primarily occurred in the K-12 context. We combine and adapt two pedagogical content knowledge theoretical frameworks, developed in K-12 science education, for use in undergraduate mathematics. The applicability of the theoretical framework is demonstrated using one participant’s interview data. Our second empirical study focused on student learning in upper-level, lecture-based courses. Ten recent graduates were interviewed about the collaboration they undertook on their mathematics work outside of their lectures. We report on the commonalities in terms of who participants collaborated with, how their collaboration was structured, as well as the benefits attributed to collaboration and the factors that were perceived as constraining the practice. Additionally, we present a case study of two participants to demonstrate how collaborating students can be considered a community of practice and explore the impact that students’ perceptions of the nature of research mathematics can have on their learning practices. We also explore participants’ perceptions of the concept of undergraduate mathematics students being assessed on group work. Together, our findings provide further insight into the nature of teaching and learning in upper-level courses, with an emphasis on what happens outside the lecture. We conclude the thesis by outlining the implications of our work for future research and the practice of undergraduate mathematics instructors.