In the research reported in this paper, we develop a theoretical perspective to describe and account for the growth of collective mathematical understanding. We discuss collective processes in mathematics, drawing in particular on theoretical work in the domains of improvisational jazz and theatre. Using examples of data from a study of elementary students' mathematical learning, we introduce and explore the new constructs of Collective Image Making (CIM), Collective Image Having (CIH), and Collective Property Noticing (CPN) and show how these contribute to the students' growing mathematical understanding.