interested in presenting a correct solution method than in soliciting multiple methods.

A third feature of classrooms that function as communities of learners is an appreciation of the value of mistakes as sites of learning for everyone. Mistakes are not covered up; rather, they are used as opportunities to examine reasoning and to deepen everyone’s analysis. The appreciation of mistakes is a fundamental aspect of mathematical work outside the classroom; inside, it helps build the community. When Michelle makes a mistake on the grid problem, Mr. Hernandez does not tell her it is wrong and then call on someone else. He uses it instead to push her thinking.

Finally, a core feature of these classrooms is the recognition that the authority for whether something is both correct and sensible lies in the logic and structure of the subject rather than the status of the teacher or the popularity of the person making the argument. The resolution of disagreements resides in mathematical argument. Both Mr. Hernandez and Ms. Kaye have their students justify their solution strategies. Although Ms. Lawrence frequently asks her students to justify their work, when she presents the procedure for adding fractions with unlike denominators, she provides the justification. She does use mathematical properties to explain the procedure, however, rather than simply present the rule as Mr. Angelo did. Hence, in addition to selecting tasks with goals in mind and sharing essential information, the teacher’s primary role is to establish a classroom culture that supports learning with understanding, thereby serving to motivate students to learn.

Managing Discourse

An important part of classroom instruction is to manage the discourse around the mathematical tasks in which teachers and students engage. Teachers must make judgments about when to tell, when to question, and when to correct. They must decide when to guide with prompting and when to let students grapple with a mathematical issue. Their decisions do not simply rest with the mathematical task at issue. They also need to decide who should get the floor in whole-group discussions and how turns should be allocated. Teachers have responsibility for moving the mathematics along while affording students opportunities to offer solutions, make claims, answer questions, and provide explanations to their colleagues. The point of classroom discourse is to develop students’ understanding of key ideas. But it also provides opportunities to emphasize and model mathematical reasoning

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