  Pre-college mathematics modeling instruction often frames mathematics as
being separated from reasoning about the real world -- and commonly treats
reasoning mathematically and reasoning about the real-world context as separate
stages of a modeling cycle. In this paper, we present evidence that helps
characterize how experts use mathematics in physics contexts while developing
graphical models. An important finding is that there was essentially no
evidence of experts reasoning in a context-free way with these tasks, but
instead they used physical reasoning -- either grounded in the context of the
task or from abstract physical models -- to guide their mathematics. The
difference in approach of physics instructors and students may lead to a
mismatch in expectations, and frustration, for both parties. This work
contributes to the body of knowledge about how mathematical reasoning appears
in physics, and can help researchers and instructors recognize the connections
to the physical world in expert mathematical reasoning. This can help
instructors and researchers be better equipped to develop materials and methods
that can help students start building those connections as well.
