  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.
