The Professional Teaching Standards call unprecedented attention to the ‘ discourse” of mathematics classrooms, as embodied in three standards: Teacher’s Role in Discourse, Students’ Role in Discourse, and Tools for Enhancing Discourse. An unfamiliar term to many, discourse is used to highlight the ways in which  knowledge  is constructed and exchanged in classrooms. The discourse of classroom-the way of representing, thinking, talking, agreeing and disagreeing-is central to what students learn about mathematics as a domain of human inquiry with characteristic ways of knowing. Discourse is both the way ideas are exchanged and what the ideas entail:  Who talks? About what? In what ways?  What do people write, what do they record and why? What questions are important? How do ideas change? Whose ideas and ways of thinking are valued? Who determines when to end the discussion? The discourse is shaped by the tasks in which student engage and the nature of learning environment; it also influences them.

Discourse entails fundamental issues about knowledge: What makes something true or reasonable in mathematics? How can we figure out whether or not something makes sense? That something is true because the teacher or the book says so is the basis for much traditional classroom discourse. Another view, the one put forth here, centers on mathematical reasoning and evidence as the basis for the discourse. In order for students to develop the ability to formulate problems, to explore, conjecture, and reason logically, to evaluate whether something makes sense, classroom discourse must be founded on mathematical evidence.

The discourse of a classroom is formed by students and the teacher and the tools with which they work. Still, teachers play a crucial role in shaping the discourse of their classrooms through the signals they send about the knowledge and ways of thinking and knowing that are valued. For example, suppose a student claims that in tossing a penny, heads is a more likely result than tails. He explains that in ten throws, he got seven heads and three tails. He says that this out comes shows that it is “easier to get “heads. How might the teacher response to this statement? Knowing herself that heads and tails are equally likely, she might tell him that this conclusion isn’t right and explain that the outcome he noted was simply what happened that time. Or she might ask other students to respond to his assertion, for across the entire class the cumulative results of tossing a penny will probably turn up approximately half heads and half tails. Or she might observe that ten throws is not very many and suggest that he gather more data himself to see what happens. In each of these alternatives, different messages are sent about the usefulness and validity of student’s experience with coin tossing. Each conveys different implications about the role of experimentation in constructing mathematical knowledge, and each may have different  effects on the student’s view of mathematical justification. This small example illustrates how influential teacher’s interaction with students are in shaping norms of knowing and thinking. These interactions, in turn, influence students’00 ways of knowing. The norms that students and teachers come to share deeply affect the potential of the classroom as a place for learning.

Students must talk, with one another as well as in response to the teacher. When the teacher talks most, the flow of ideas and knowledge is primarily from teacher to students. When students make publics conjectures and reason with others about mathematics, ideas and knowledge are developed collaboratively, revealing mathematics as constructed by human beings within an intellectual community. Writing is another important component of  the discourse. Students learn to use, in a meaningful context, the tools of mathematical discourse- special terms, diagrams, graphs, sketches , analogies, and physical models, as well as symbols.

The teacher's role is to initiate and orchestrate this kind of discourse and to use it skillfully to foster student learning. In order to facilitate learning by all students, teacher must also be perceptive and skillful in analyzing the culture of classroom , looking out for patterns of inequality, dominance, and low expectations that are primary causes of nonparticipation by many students. Engaging every student in the discourse of the class requires considerable skill as well as an appreciation of, and respect for, students' diversity.

The teacher of mathematics should orchestrate discourse by-

v     Posing questions and tasks that elicit, engage, and challenge each student’s thinking;

v     Listening carefully to student’s ideas;

v     Asking students to clarify and justify their ideas orally and in writing;

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

v     Deciding when and how to attach mathematical notation and language to student’s ideas;

v     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;

v     Monitoring students’ participation in discussions and deciding when and how to encourage each student to participate.

Without explicit attention to the patterns of discourse in the classroom, the long-established norms of school are likely to dominate—competitiveness, an emphasis on right answers, the assumption that teachers have the answers, rejection of nonstandard ways of working or thinking, patterns reflective of gender and class biases. For example, in many mathematics classrooms, answers have traditionally been right because the teacher says so or because the teacher says so or because the teacher and the student together decipher what “they” (the textbook authors) “want”. Even with careful attention to patterns of classroom discourse, traditional norms will underlie the interactions of students and teachers. Consider the way in which right answers are treated in a mathematics class. Suppose students are solving the problem “What is two-third of nine?” and a student gives the answer, “Six.” The teacher reflex is to hear it as a “right answer” and to (a) move on; (b) praise the student; or (c) agree and repeat the answer for the benefit of the rest of the class. Even disposed to ask students to explain their answers, the teacher may ask, “How did you know that?” In listening to myself in my own classroom, I realized that using the word know seems to imply that the students’ answer is right. I heard myself saying, “How do you know?” when I agreed with what student said and asking, “Why do you think so?” or “How did you get that?” when I did not. Subsequently, I was probably giving my students clues about the “correctness” of their ideas—clues they were likely picking up.