Talking about rates conceptually, Part II: Mathematical knowledge for teaching

Part I of this report (Thompson & Thompson, 1994) described the attempts of one teacher, Bill, to teach concepts of speed to one child, Ann. We analyzed sources of their eventual dysfunctional communication, concluding that Bill had encapsulated his deep understandings of rate and proportionality within his language for numbers and operations. Part II presents the session of the researcher, Pat, with Ann, analyzing his instructional actions in light of his agenda for having Ann build a scheme of operations by which to understand distance, time, and speed. We contrast Pat's agenda with Bill's, focusing on the mathematical knowledge that guided their instructional decisions and actions and briefly discuss implications for the preparation of teachers with regard to content knowledge.