Decades of research show that students learn more in classes that utilize active learning than they do in traditional, lecture-only classes. Active learning also reduces the achievement gaps that are often present between various demographic groups. Given these well-established results, instructors of upper-division astronomy courses may decide to search the astronomy education research literature in hopes of finding some guidance on common student difficulties, as well as research-validated and research-based active learning curricula. Their search will be in vain. The current literature on upper-division astronomy is essentially non-existent. This is a shame, since many upper-division astronomy students will experience conceptual and problem-solving difficulties with the quantitative problems they encounter. These difficulties may exist even if students have a strong background in mathematics. In this paper, I examine one quantitative problem that is representative of those that upper-division astronomy students are expected to solve. I list many of the subtle pieces of information that students need to understand in order to advance toward a solution and I describe how such a list can be used to generate Peer Instruction (PI) questions. I also provide guidelines for instructors who wish to develop and implement their own PI questions. These PI questions can be used to increase the amount of active learning that occurs in an upper-division astronomy course. They help develop students' understandings of symbolic, mathematical representations, and they help improve students' problem-solving skills. The ideas presented in this paper can help instructors infuse their upper-division astronomy courses with active learning.

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