This article focuses on one common type of many different classroom discussions, open-strategy sharing (OSS), because it offers a starting point for teachers new to mathematical discussions. The goal of an OSS discussion is to explore multiple possible ways to solve a single mathematical task. Students share strategies, listen to others'…

This article highlights four routines that use visual images and open questions to invite students to discuss, justify, make arguments, and question their own thinking as well as their classmates' thinking. The authors share specific classroom examples from each routine that highlight the meaningful discourse they can inspire. Teacher prompts and…

Discourse in mathematics classrooms has been at the forefront of mathematics reform for years. Of the eight NCTM Mathematics Teaching Practices, four relate specifically to the use of discussion, questioning, and the analysis of student thinking that is based on discourse. As an answer to the demand for discourse within many mathematics classroom,…

In this article, the authors describe the Japanese term "bansho," which refers to the intentional use of board space for facilitating student learning. Bansho offers a structure for sequencing mathematics visually on the board. By purposefully organizing the board space alongside the lesson, teachers can provide students with a framework…

The inspection of mistakes can play a powerful role in an individual's learning process (Boaler 2015). The purpose of this article is to support the reader in selecting mistakes that can be leveraged to benefit the learning of all students. Specifically, the authors focus on "which" and "why": "which" mistakes to…

The first author's initial attempts at a whole-class mathematics discussion were woefully dysfunctional. Ten second graders showing their answers to a missing addend question about the number of children left riding a school bus did not result in the mathematical epiphany that the textbook promised. She did not give up on using mathematical…

This article describes how a teacher in Mexico City worked with two researchers on an instructional design project to help kindergartners become proficient in participating in whole-class mathematical conversations.

In the mathematics classroom, making sense of story problems can be a challenge for all students. Strategies that promote student discourse offer teachers one way to support their students' sense-making processes. In this article, the authors present a mathematical discourse strategy that was introduced to elementary school teachers during Project…

This article describes how one of the authors, second-grade teacher Lisa A. Brooks, challenged the raise-your-hand to-speak rule. Her desire to make this change was a result of experiences gained in a graduate class taught by the second author, Juli K. Dixon. Brooks had learned that it is possible to create a classroom community where students…

Learning through meaningful problem solving is integral in any successful mathematics program (Carpenter et al. 1999). The National Council of Teachers of Mathematics (NCTM) promotes the use of problem solving as a means to deepen understanding of all content areas within mathematics (NCTM 2000). This article describes a first-grade lesson that…

When the author recently turned her attention to the National Council of Teachers of Mathematics (NCTM) "Principles and Standards," she was startled to see communication as key. She adjusted her teaching to meet the NCTM Communication Standard and promote communication in her classroom by providing a safe environment, developing discourse and…

Many educators recommend using open-ended questions to encourage students to share their thinking. However, inexplicit open-ended questions can make it difficult for students to determine the teacher's intentions and to participate confidently in mathematical conversations. This article describes strategies teachers can use to make the language of…

This article describes how teachers can use students' conversations to assess their mathematical understanding. The authors draw on their own classroom experiences to show what they learned about planning for future lessons based on discussions in their own classrooms.

This story of a student's misconceptions about quadrilateral properties makes a case for the importance of teachers not only knowing where their students have been and where they are in terms of their learning, but also where their current learning will lead them in the future. Suggested classroom activities are included. (Contains 8 figures.)

Everyone uses language to learn mathematics. Paying close attention to the needs of students who are learning English as a second (or third) language is crucial so that teachers can modify lessons to accommodate those needs. The Equity Principle requires that they accommodate differences in their diverse student population to help everyone learn…