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A cat springs from a walk into a run. A spiral shell takes form through growth. With suitable mathematics, what might we learn about the motion or the form? Calculus is said to help with questions about movement, growth and change. How, exactly, does it help us? What might we need to understand? How could we gain such understanding? In this paper we explore two tasks that we designed for students, one about a moving cat and one about a spiral shell. We have written extensively about these tasks (Speiser & Walter, 1994, 1996, 2004; Speiser, Walter & Maher, 2003), as well