• Content, representations, tasks, and materials should be chosen so as to develop all five strands of proficiency toward the big ideas of math ematics and the goals for instruction.

  • Planning for instruction should take into account what students know, and instruction should provide ways of ascertaining what students know and think as well as their interests and needs.

  • Rather than simply listing problems and exercises, teachers should plan for instruction by focusing on the learning goals for their students, keep ing in mind how the goals for each lesson fit with those of past and future lessons. Their planning should anticipate the events in the lesson, the ways in which the students will respond, and how those responses can be used to further the lesson goals.

Managing Classroom Discourse

Mathematics classrooms are more likely to be places in which mathematical proficiency develops when they are communities of learners and not collections of isolated individuals. Research on creating classrooms that function as communities of learners has identified several important features of these classrooms: ideas and methods are valued, students have autonomy in choosing and sharing solution methods, mistakes are valued as sites of learning for everyone, and the authority for correctness lies in logic and the structure of the subject, not in the teacher. In such classrooms the teacher plays a key role as the orchestrator of the discourse students engage in about mathematical ideas. Teachers are responsible for moving the mathematics along while affording students opportunities to offer solutions, make claims, answer questions, and provide explanations to their peers. Teachers need to help bring a mathematical discussion to a close, making sure that gaps have been filled and errors addressed. To develop mathematical proficiency, we believe that students require more than just the demonstration of procedures. They need experience in investigating mathematical properties, justifying solution methods, and analyzing problem situations. We recommend the following:

  • A significant amount of class time should be spent in developing math ematical ideas and methods rather than only practicing skills.

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