This study examines how collaborative activity among students and the teacher to investigate disciplinary questions, which we term 'joint exploration', is established and maintained in a secondary mathematics classroom. Although collaborative and active learning is increasingly sought after in mathematics classrooms, studies of instances of joint exploration remain relatively rare. In this study, we use the theoretical perspective of positioning to conceptualize joint exploration as involving the negotiation among participants to position students with epistemic authority and agency. Using a constant comparative method, we use classroom video data of two episodes containing joint exploration and closely analyse the shifts in epistemic positioning within them. We find that shifts in epistemic positioning, especially with respect to students positioning one another with epistemic authority and exercising epistemic agency, help to support continued joint exploration. We also find that the teacher can play an important role in decentring themselves as the epistemic authority. In addition to these findings, this study contributes a distinction in epistemic authority and agency, as we explain how the two concepts are related and involved in establishing and maintaining joint exploration.