  Mathematics is a critical part of much scientific research. Physics in
particular weaves math extensively into its instruction beginning in high
school. Despite much research on the learning of both physics and math, the
problem of how to effectively include math in physics in a way that reaches
most students remains unsolved. In this paper, we suggest that a fundamental
issue has received insufficient exploration: the fact that in science, we don't
just use math, we make meaning with it in a different way than mathematicians
do. In this reflective essay, we explore math as a language and consider the
language of math in physics through the lens of cognitive linguistics. We begin
by offering a number of examples that show how the use of math in physics
differs from the use of math as typically found in math classes. We then
explore basic concepts in cognitive semantics to show how humans make meaning
with language in general. The critical elements are the roles of embodied
cognition and interpretation in context. Then we show how a theoretical
framework commonly used in physics education research, resources, is coherent
with and extends the ideas of cognitive semantics by connecting embodiment to
phenomenological primitives and contextual interpretation to the dynamics of
meaning making with conceptual resources, epistemological resources, and
affect. We present these ideas with illustrative case studies of students
working on physics problems with math and demonstrate the dynamical nature of
student reasoning with math in physics. We conclude with some thoughts about
the implications for instruction.
