  At the University of Colorado Boulder, as part of our broader efforts to
transform middle- and upper-division physics courses, we research students'
difficulties with particular concepts, methods, and tools in classical
mechanics, electromagnetism, and quantum mechanics. Unsurprisingly, a number of
difficulties are related to students' use of mathematical tools (e.g.,
approximation methods). Previous work has documented a number of challenges
that students must overcome to use mathematical tools fluently in introductory
physics (e.g., mapping meaning onto mathematical symbols). We have developed a
theoretical framework to facilitate connecting students' difficulties to
challenges with specific mathematical and physical concepts. In this paper, we
motivate the need for this framework and demonstrate its utility for both
researchers and course instructors by applying it to frame results from
interview data on students' use of Taylor approximations.
