Placenticeras: evolution of a task

We study a sequence of concrete case examples, anchored to a single image (of a fossil shell) to explore the complexity of building mathematical models in compelling realistic situations. The case examples are drawn from work with calculus undergraduates, and also dance majors from an experimental mathematics course who built evolving spiral forms in choreographic projects. The exposition is narrative, in the sense that it emphasizes, case by case, the evolution of the authors’ thinking in the course of extended conversations and collaborations with students over several years.