For physicists, 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, and this process is fundamental to what it means for an expert to understand something. 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" and "think about the dimensions". 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 variables and equations, and distinguished superficially-similar functional forms. Once introduced to a strategy, students sometimes applied it to new scenarios spontaneously or applied it in new ways to the present scenario, 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.

Students often get stuck in problem solving, ignoring their physical intuition in favor of plug-and-chug or pattern-matching approaches. We suggest that examining the extreme cases is a useful way of moving students towards more expert-like problem solving. Based on a case study, we show that novice students can quickly learn to use extreme cases productively in problem-solving. In reasoning about extreme cases, students blend conceptual and mathematical cognitive resources. At the same time, they can generate new and creative uses for extreme-case reasoning, here recasting it from a tool for evaluating answers to one for generating them. Extreme case reasoning may prove a valuable instructional goal at the introductory level.

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