  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. But 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.
