Mathematical argumentation, justification, and proof are practices at the heart of mathematics. Yet mathematics teachers generally operate without clear definitions of these practices making it difficult to communicate expectations, decide what to accept or expect from students at different grade levels, and distinguish these activities from similar practices used in other disciplines. In this paper, we advance a teacher-centered view of these practices by articulating a set of criteria to guide term usage for communities of teachers. We offer examples to demonstrate how this approach centralizes teachers' perspectives and helps reveal the different support-for-claims activities we value in math classrooms.