  Mathematical reasoning skills are a desired outcome of many introductory
physics courses, particularly calculus-based physics courses. Positive and
negative quantities are ubiquitous in physics, and the sign carries important
and varied meanings. Novices can struggle to understand the many roles signed
numbers play in physics contexts, and recent evidence shows that unresolved
struggle can carry over to subsequent physics courses. The mathematics
education research literature documents the cognitive challenge of
conceptualizing negative numbers as mathematical objects--both for experts,
historically, and for novices as they learn. We contribute to the small but
growing body of research in physics contexts that examines student reasoning
about signed quantities and reasoning about the use and interpretation of signs
in mathematical models. In this paper we present a framework for categorizing
various meanings and interpretations of the negative sign in physics contexts,
inspired by established work in algebra contexts from the mathematics education
research community. Such a framework can support innovation that can catalyze
deeper mathematical conceptualizations of signed quantities in the introductory
courses and beyond.
