• 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 and 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.

The teacher of mathematics should promote classroom discourse in which students—

• Listen to, respond to, and question the teacher and one another;

• Use a variety of tools to reason, make connections, solve problems, and communicate;

• Initiate problems and question;

• Make conjectures and present solutions;

• Explore examples and counterexamples to investigate a conjecture;

• Try to convince themselves and one another of the validity of particular representations, solutions, conjectures, and answers;

• Rely on mathematical evidence and argument to determine validity.

The teacher of mathematics, in order to enhance discourse, should encourage and accept the use of—

• Computers, calculators, and other technology;

• Concrete materials used as models;

• Pictures, diagrams, tables, and graphs;

• Invented and conventional terms and symbols

• Metaphors, analogies, and stories;

• Written hypotheses, explanations, and arguments;

• Oral presentations and dramatizations.

The teacher of mathematics should create a learning environment that fosters the development of each student’s mathematical power by—