Orchestrating Productive Mathematical Discussions: Five Practices for Helping Teachers Move Beyond Show and Tell

Teachers who attempt to use inquiry-based, student-centered instructional tasks face challenges that go beyond identifying well-designed tasks and setting them up appropriately in the classroom. Because solution paths are usually not specified for these kinds of tasks, students tend to approach them in unique and sometimes unanticipated ways. Teachers must not only strive to understand how students are making sense of the task but also begin to align students' disparate ideas and approaches with canonical understandings about the nature of mathematics. Research suggests that this is difficult for most teachers (Ball, 1993 Ball, D. L. 1993. With an eye on the mathematical horizon: Dilemmas of teaching elementary school mathematics. Elementary School Journal, 94(4): 373–397. [Google Scholar], 2001 Ball, D. L. 2001. “Teaching, with respect to mathematics and students”. In Beyond classical pedagogy: Teaching elementary school mathematics, Edited by: Wood, T., Nelson, B. S. and Warfield, J. 11–22. Mahwah, NJ: Erlbaum. [Google Scholar]; Leinhardt & Steele, 2005 Leinhardt, G. and Steele, M. D. 2005. Seeing the complexity of standing to the side: Instructional dialogues. Cognition and Instruction, 23(1): 87–163. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]; Schoenfeld, 1998 Schoenfeld, A. S. 1998. Toward a theory of teaching-in-context. Issues in Education, 4(1): 1–95. [Crossref] , [Google Scholar]; Sherin, 2002 Sherin, M. G. 2002. When teaching becomes learning. Cognition and Instruction, 20(2): 119–150. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). In this article, we present a pedagogical model that specifies five key practices teachers can learn to use student responses to such tasks more effectively in discussions: anticipating, monitoring, selecting, sequencing, and making connections between student responses. We first define each practice, showing how a typical discussion based on a cognitively challenging task could be improved through their use. We then explain how the five practices embody current theory about how to support students' productive disciplinary engagement. Finally, we close by discussing how these practices can make discussion-based pedagogy manageable for more teachers.