Many educators and researchers advocate for student engagement in disciplinary activity. This is especially the case in advanced undergraduate courses taken by mathematics majors. In our respective design-based research projects, we found a need to better operationalize the activity of mathematicians in order to both plan for and document student engagement in disciplinary activity. In this report, we share our literature-based efforts to identify the tools and objects used by pure mathematicians in their work. We share the overarching framework we developed, Authentic Mathematical Proof Activities (AMPA), and illustrate the ways we have used this framework to analyze teacher-student activity using an activity theory lens. We conclude with reflections on how tensions between authenticity-to-the-discipline and authenticity-to-the-students shape the teacher-student activity system. [For the complete proceedings, see ED630060.]