We analyze the discourse of faculty presenting derivations in which they manipulate mathematical equations to illuminate a physical principle. Observations are interpreted through a lens of symbolic forms, conceptual and contextual meanings that are embedded in the equation. When an equation is manipulated (e.g. bringing terms to one side or another), different forms are emphasized, changing the meaning of the equation. We argue that this framework can make explicit the faculty motivations for the moves, and present two observations of manipulations that appear to have distinctly different reasons. The first manipulation brings about a change in context from a physics to a mathematical frame. In the second, a thematic manipulation --- grouping all terms of a common variable --- reveals an important conceptual point about a driven harmonic oscillator. While there is direct evidence from the observed faculty to support the inference of motivation, in neither case is the reasoning made clear to the students. The study of discourse represents a new direction in which physics education researchers can study and inform the classroom.