Examples of how to encourage productive mathematical discourse in a classroom, using group work and discussion to develop a deeper understanding of mathematical topics and preparing them for greater learning experiences. Examples from a unit on variability show that students gain long-lasting insight into the structure and purpose of statistical formulas. The activity sheet for the discussed example is included, along with sample student work.

The differences between two kinds of discussions: sharing and collaborative. Distinguishing between these can help teachers plan and execute mathematically focused discussions to advance student understanding. The article includes suggestions for implementation including how to correct and reguide wrong answers, how to set students up for a successful exchange, and how to link concepts and ideas together between students. Sample discussions are included.

One teacher's orchestration of classroom discourse led to the development of an inductive model of teaching. In particular, this teacher’s strategic use of verbal assessment helped students generate mathematical understanding. An example of the model is discussed. Sample classroom discussions are provided that revolve around summing reciprocals of the factors of 28.

The well-planned writing workshop is an effective tool for expanding the mathematical discourse of students and helping them become more skillful writers in the discipline. This article reviews a step-by-step process for conducting a workshop and details different reader response strategies. It also offers suggestions and examples for various types of mathematical writing assignments that lend themselves to productive workshop discussion.

Mathematical discourse is the way students represent, think, talk, question, agree, and disagree in the classroom. This article illustrates how research about mathematical discourse can be translated into practice. The article shows two types of discourse, cognitive discourse and motivational discourse. Examples of student discourse and teacher-student discussions are provided.

An example of displaying cereal boxes in a triangular shape for a grocery store as a way of discussing mathematical discourse in the classroom. Efforts at implementing inquiry- and discussion-based mathematics instruction in one classroom illustrate how unexpected interpretations of one problem by students provided an opportunity to extend their mathematical investigations. Also presented are the particular demands that building on students' ideas places on curriculum and instruction. Several classroom based activities are explained along with sample student work, data, and discussions.

After frequently hearing students comment that the language used in mathematics is "like hearing a foreign language," the author investigates this perception by studying a ninth-grade mathematics classroom. Her finding suggests that students experience interference from the ways in which language is used in mathematics. The author shows the different types of interference that occurs in student understanding of mathematical language, and gives suggestions of how to block that interference.

A vision of classroom discourse that actively involves students and that uses students' explanations to develop mathematical ideas. Teachers can learn to enact this vision of classroom discourse through a process of experience, collaboration, and reflection. A classroom example of implementing this in an Algebra I course is provided.