We argue that progress in the area of research on mathematics teacher responses to student thinking could be enhanced were the field to attend more explicitly to important facets of those responses, as well as to related units of analysis. We describe the Teacher Response Coding scheme (TRC) to illustrate how such attention might play out, and then apply the TRC to an excerpt of classroom mathematics discourse to demonstrate the affordances of this approach. We conclude by making several further observations about the potential versatility and power in articulating units of analysis and developing and applying tools that attend to these facets when conducting research on teacher responses.