
In some cases, I think that my restating their thoughts for them helped them to clarify their own spoken words and to move toward an understanding of the problems. In other cases, I may have been in too much of a hurry to assist students toward the same end. Teachers often tread a thin line where similar strategies can lead to conflicting outcomes. We want students to develop their capacity to identify good questions, but not to feel simply left to their own devices. We want them to discover that they can solve substantive mathematical problems, and assess whether a solution makes sense, but not to be overwhelmed by the challenge. We hope to establish the understanding that every idea proposed is valued, and we expect each person in the classroom to assume the responsibility for evaluating whether or not an idea makes mathematical sense. On the other hand, we do not want our students to feel that there is no educational authority in the class. We want students to engage in sensemaking, as opposed to simply following prescribed rules, but we do not want to deny them the pleasure and confidence that come from mastering techniques and mimicking experienced professionals. What tools and strategies can help teachers navigate these tricky waters? Resnick (1988) suggests that in order for students to learn mathematics, we need to teach mathematics as if it were an illstructured discipline: a domain in which multiple interpretations, argument, and debate are called for and natural. When students first start to express their mathematical thinking in words, however, they often do not use very precise language. Learning to think mathematically requires some mediating processes in order to bridge the gap between students' ordinary language and the language of mathematics.
We would like to share some of our discussions on such discourse
interventions as questioning techniques, paraphrasing, and summarizing.
We will end with some reflections on listening, particularly as this
strategy serves the dual purpose of facilitating conversation and
helping teachers figure out which strategy to use when.

