The social and embodied nature at the heart of all knowing, doing, and learning contrasts with the images that pervade our cultural imagination of mathematical work as a solitary, cognitive activity. This article describes a playful experiment by the author group to do collective mathematics, in an extended effort to construct alternative images, instincts, and practices for ourselves. We present a pair of episodes of mathematical exploration that come from our work together and that we have seen as an early success, intimating features of a stabilized collective mathematics that we hope to continue pursuing. Coming from a single investigation of our group, these episodes offer narrative accounts of the parallel inquiries of subgroups, working to define and characterize a mathematical space we had collectively identified, and then to formulate and investigate conjectures about that space. The narratives are followed by a discussion of themes within and across them and reflections on their significance as a step toward self-organized collective mathematics
