Smith and Stein (2011) identify five practices that assist teachers in facilitating instruction that advances the mathematical understanding of the class: Anticipating; Monitoring; Selecting; Sequencing; Connecting. Organizing and facilitating productive mathematics discussions for the classroom take a great deal of preparation and planning. Prior to giving a task to students, the teacher should anticipate the likely responses that students will have so that they are prepared to facilitate the lesson. Students usually come up with a variety of strategies, but it is helpful if teachers have already anticipated some of the strategies when leading the discussion. The teacher then poses the problem and gives the task to the students. The teacher monitors the responses while students work individually, in pairs, or in small groups. The teacher pays attention to the different strategies that students use. To conduct the “share and summarize” portion of the lesson, the teacher selects a student to present his or her mathematical work and sequences the sharing so that the various strategies are presented in a specific order, to highlight the mathematical goal of the lesson. As the teacher conducts the discussion, he or she deliberately asks questions to connect responses to the key mathematical ideas.
Smith, Margaret S., and Mary Kay Stein. 2011. 5 Practices for Orchestration Productive Mathematics Discussions. Reston, VA: National Council of Teachers of Mathematics.