What role do gestures play in advanced mathematical thinking? We argue that the role of gestures goes beyond merely communicating thought and supporting understanding--in some cases, gestures can help generate new mathematical insights. Gestures feature prominently in a case study of two participants working on a sequence of calculus activities. One participant uses gestures to clarify the relationships between a function, its derivative and its antiderivative. We show how these gestures help create a "virtual mathematical construct," which in turn leads to a new problem-solving strategy. These results suggest that gestures are a productive, but potentially undertapped resource for generating new insights in advanced levels of mathematics. (Contains 8 figures.)