  In their study of physics beyond the first year of University -- termed
upper-division in the US, many of students' primary learning opportunities come
from working long, complex back-of-the-book style problems, and from trying to
develop an understanding of the underlying physics through solving such
problems. Some of the research at the upper-division focuses on how students
use mathematics in these problems, and what challenges students encounter along
the way. There are a number of different and diverse research studies on
students' use of mathematics in the upper-division. These typically utilize one
of two broad approaches, with some researchers primarily seeking out and
addressing challenges students face, and others working chiefly to unpack
students' in-the-moment reasoning. In this paper, we present and discuss both
approaches, and then review research efforts that strive to connect these two
approaches in order to make sense of students' use of mathematics as well as to
uncover particular challenges that students encounter. These recent efforts
represent a small step towards synthesizing the two approaches, which we argue
is necessary to more meaningfully impact student learning at the
upper-division. We close our review and discussion with suggested refinements
for future research questions for the physics education research community to
consider while it works to understand how students use math in upper-division
courses.
