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What does it mean to understand a physics equation? The use of formal expressions in physics is not just a matter of the rigorous and routinized application of principles, followed by the formal manipulation of expressions to obtain an answer. Rather, successful students learn to understand what equations say in a fundamental sense; they have a feel for expressions, and this guides their work. More specifically, students learn to understand physics equations in terms of a vocabulary of elements that I call symbolic forms. Each symbolic form associates a simple conceptual schema with a pattern of symbols in an equation. This hypothesis has implications for how we should understand what must be taught and learned in physics classrooms. From the point of view of improving instruction, it is absolutely critical to acknowledge that physics expertise involves this more flexible and generative understanding of equations, and our instruction should be geared toward helping students to acquire this understanding. The work described here is based on an analysis of a corpus of videotapes in which university students solve physics problems.
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