The teacher of mathematics should orchestrate discourse by-

posing questions and tasks that elicit, engage, and challenge each student's thinking;

listening carefully to students' ideas;

asking students to clarify and justify their ideas orally and in writing;

deciding what to pursue in depth from among the ideas that students bring up during a discussion;

deciding when and how to attach mathematical notation and language to students' ideas;

deciding when to provide information, when to clarify an issue, when to model, when to lead, and when to let a student struggle with a difficulty;

monitoring students' participation in discussions and deciding when and how to encourage each student to participate.

Elaboration

Like a piece of music, the classroom discourse has themes that pull together to create a whole that has meaning. The teacher has a central role in orchestrating the oral and written discourse in ways that contribute to students' understanding of mathematics.

The kind of mathematical discourse described above does not occur spontaneously in most classrooms. It requires an environment in which everyone's thinking is respected and in which reasoning and arguing about mathematical meanings is the norm. Students, used to the teacher doing most of the talking while they remain passive, need guidance and encouragement in order to participate actively in the discourse of a collaborative community. Some students, particularly those who have been successful in more traditional mathematics classrooms, may be resistant to talking, writing, and reasoning together about mathematics.

One aspect of the teacher's role is to provoke students' reasoning about mathematics. Teachers must do this through the tasks they provide and the questions they ask. For example, teachers should regularly follow students' statements with, "Why?" or by asking them to explain. Doing this consistently, irrespective of the correctness of students' statements, is an important part of establishing a discourse centered on mathematical reasoning. Cultivating a tone of interest when asking a student to explain or elaborate on an idea helps to establish norms of civility and respect rather than criticism and doubt. Teachers also stimulate discourse by asking students to write explanations for their solutions and provide justifications for their ideas.

Emphasizing tasks that focus on thinking and reasoning serves to provide the teacher with ongoing assessment information. Well-posed questions can simultaneously elicit and extend students' thinking. The teacher's skill at formulating questions to orchestrate the oral and written discourse in the direction of mathematical reasoning is crucial.

A second feature of the teacher's role is to be active in a different way from that in traditional classroom discourse. Instead of doing virtually all the talking, modeling, and explaining themselves, teachers must encourage and expect students to do so. Teachers must do more listening, students more reasoning. For the discourse to promote students' learning, teachers must orchestrate it carefully. Because many more ideas will come up than are fruitful to pursue at the moment, teachers must filter and direct the students' explorations by picking up on some points, and by leaving others behind. Doing this prevents student activity and talk from becoming too diffuse and unfocused. Knowledge of mathematics, the curriculum, and of students should guide the teacher's decisions about the path of the discourse. Other key decisions concern the teacher's role in contributing to the discourse. Beyond asking clarifying or provocative questions, teachers should also, at times, provide information and lead students. Decisions about when to let students struggle to make sense of an idea or a problem without direct teacher input, when to ask leading questions, and when to tell students something directly are crucial to orchestrating productive mathematical discourse the classroom. Such decisions depend on teachers' understandings of mathematics and of their students-on judgments about the things that, students can figure out on their own or collectively and those for which they will need input.

A third aspect of the teacher's role in orchestrating classroom discourse is to monitor and organize students' participation. Who is volunteering comments and who is not? How are students responding to one another? What are different students able to record or represent on paper about their thinking? What are they able to put into words, in what kinds of contexts? Teachers must be committed to engaging every student in contributing to the thinking of the class. Teachers must judge when students should work and talk in small groups and when the whole group is the most useful context. They must make sensitive decisions about how turns to speak are shared in the large group-for example, whom to call on when and whether to call on particular students who do not volunteer. Substantively, if the discourse is to focus on making sensed mathematics, on learning to reason mathematically, teachers must refrain from calling only on students who seem to have right answers a valid ideas to allow a broader spectrum of thinking to be explored in the discourse. By modeling respect for students' thinking and conveying the assumption that students make sense, teachers can encourage students to participate within a norm that expects group members to justify their ideas. Teachers must think broadly about a variety of ways for students to contribute to the class's thinking-using means that are written or pictorial, concrete or representational, as well as oral.

Vignettes