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The utterly prosaic connection between physics and mathematics
Eugene Wigner famously argued for the "unreasonable effectiveness of
mathematics" for describing physics and other natural sciences in his 1960
essay. That essay has now led to some 55 years of (sometimes anguished) soul
searching --- responses range from "So what? Why do you think we developed
mathematics in the first place?", through to extremely speculative ruminations
on the existence of the universe (multiverse) as a purely mathematical entity
--- the Mathematical Universe Hypothesis. In the current essay I will steer an
utterly prosaic middle course: Much of the mathematics we develop is informed
by physics questions we are tying to solve; and those physics questions for
which the most utilitarian mathematics has successfully been developed are
typically those where the best physics progress has been made.Comment: 12 pages. Minor edits on an essay written for the 2015 FQXi essay
contest: "Trick or truth: The mysterious connection between physics and
mathematics
A framework for the natures of negativity in introductory physics
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
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