• 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.

Standard 3. Students’ Role in Discourse

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.

Standard 4. Tools for Enhancing Discourse

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.

Standard 5. Learning Environment

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

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