the quality of teaching, little empirical confirmation of this belief can be found. Moreover, too little is known about the mathematical knowledge that teachers need and how it is used in instruction. We discuss this point more in chapter 10, but it is important to the discussion in this chapter, too. Every time we reiterate that how teachers use texts, manipulatives, and calculators makes the difference, we are hovering around questions concerning what teachers know and how they make use of that knowledge in teaching.

Finally, too little of the extant research probes the work of teaching at a sufficiently fine grain to contribute to the development of a conceptual and practical language of practice. Much of the interactive work in instruction remains unexamined, which leaves to teachers the unnecessary challenge of reinventing their practice from scratch, armed with only general advice. Suggestions that a class “discuss the solutions to a problem” provides little specificity about what constitutes a productive discussion and runs the risk of a free-for-all session that resembles sharing more than instruction. Research needs to be designed to illuminate what is entailed in a “discussion” and to probe the specific moves that teachers and students engage in that lead to productive rather than an unproductive discussions.

Instruction that develops mathematical proficiency is neither simple, common, nor well understood. It comes in many forms and can follow a variety of paths. As this chapter demonstrates, such instruction offers numerous fertile sites for research that could make a profound difference in teachers’ practice and their students’ learning.



An interactive perspective on teaching and learning has been discussed by a number of people, including Piaget, Vygotsky, Bauersfeld, Steier, Voigt, Hawkins, Gravemeijer, Easley, Cobb, and von Glaserfeld. The particular version employed here is based on the work of Cohen and Ball, 1999, 2000, in press.


Cohen and Ball, 1999, 2000, in press.


This lesson is typical of lessons observed in many U.S. classrooms during the past half-century. See, for example, the report by Fey, 1979, or the more recent TIMSS video study (Stigler and Hiebert, 1999).


Note that Mr. Angelo has avoided 10°, partly because the rule is stated in terms of moving the decimal point, and multiplying by 10°=1 leaves the number unchanged.


U.S. eighth-grade lessons from the TIMSS video study were characterized the same way. See Stigler and Hiebert, 1999.


Cohen and Ball, 2000.

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