Blending physical knowledge with mathematical form in physics problem solving

Abstract: Equations are about more than computing physical quantities or constructing formal models; they are also about understanding. The conceptual systems physicists use to think about nature are made from many different resources, formal and not, working together and inextricably linked. By blending mathematical forms and physical intuition, physicists breathe meaning into the equations they use. In contrast, in physics class, novice students often treat mathematics as only a calculational tool, isolating it from their rich knowledge of the physical world. We are interested in cases where students break that pattern by reading, manipulating, and building equations meaningfully rather than purely formally. To find examples of this and explore the diversity of ways students combine formal and intuitive resources, we conducted problem-solving interviews with students in an introductory physics for life sciences class. During the interviews, we scaffolded student use of strategies which call for both formal and intuitive reasoning, such as "examine the extreme cases". We use the analytic framework of epistemic games to model how students used the strategies and how they accessed problem-solving resources, and we present evidence that novice students using these strategies accessed more expert-like conceptual systems than those typically described in problem-solving literature. They blended physical intuition with mathematical symbolic templates, reconceptualized the nature of equations, and distinguished similar functional forms. Once introduced to a strategy, students applied it to new scenarios and found new types of uses for it, acknowledging it as a useful, general purpose problem-solving technique. Our data suggests that these strategies can potentially help novice students learn to develop and apply their physical intuition more effectively.