Encouraging Mathematical Thinking


 Abstract
 Introduction

 Discourse
 Interventions
 Decisions

 Cylinder Problem
 Lesson Reflections
 Student Predictions

 Project Reflections
 Conclusion

 References
 Acknowledgments
 Teacher Resources



Authors'
Biographies

Table of Contents


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Discourse in
Mathematic Classroom:
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Teacher's Role in Encouraging Discourse


The rewards of effective discourse are many. Not long ago, I was observing students work out a problem together. When they finally "got" it, one of the students exclaimed, "We are so smart." Isn't that what we are striving for? Not "the teacher is so smart," not "they are so smart," but "we are." In that statement, I heard both confidence and community. Healthy discourse promotes these things.
-- Cynthia Lanius, Project teacher


Mathematics is often thought of as the discipline of "the right answer." Many of us, however, are uncomfortable with this designation because it can interfere with our efforts to help students express their mathematical thinking, learn from mistakes, experiment effectively, and pursue their mathematical interests. How does the effort to "get the right answer" fit with these goals? Can we transform the student's question "Am I right?" into "How can I develop confidence and judgment that I am on the right track when working on a problem?" and "How can I know that I am improving my mathematical problem-solving and communication skills?"

Related challenges for teachers as we work to use new materials and improve our teaching strategies include: "How do I know what students are learning? How will I judge whether the methods that I am using will result in the desired competence and knowledge? How can I learn to do this better? How can I balance the development of inquiry skills while covering content and meeting the demand for improved test scores?"

A number of us brought a strong interest in discourse to this project. In the course of our conversation we arrived at the idea that discourse could be a tool to help us and our students productively carry the question, "Am I right?" This concept further develops the role of discourse in driving our professional growth and in helping students take charge of their mathematical learning.

We see that discourse can make thinking public and create an opportunity for the negotiation of meaning and agreement (Bauersfeld, 1995). At the same time, discourse provides collective support for developing one's thinking, drawing it out through the interest, questions, probing, and ideas of others (Cobb, 1995; Krummheuer, 1995; Wood, 1995; Yackel, 1995), and discourse enables us to connect a student's own everyday language with the specialized language of mathematics. Articulating what they know allows students to clarify their own understandings. Through discourse, a teacher can better grasp the mathematical needs of the class: what the students know, misconceptions they may have, and how these might have developed (Resnick, 1988). We gain perspective on our own thoughts through the attempt to understand the thinking of others, in the process laying the foundation for a supportive learning community (Brown & Campione, 1994).

Within the mathematics education community there is strong interest in the use of discourse for teaching and learning mathematics (NCTM, 1991; Atkins, 1999; Schifter, 1996). The teacher's role is described in broad terms as facilitative, to include listening carefully to students, framing appropriate questions, and mediating competing perspectives. Students are expected to develop problem-solving skills: defining problems, formulating conjectures, and discussing the validity of solutions. Stigler and Hiebert (1998) report similar roles for teachers and students in mathematics classrooms in Japan, where mathematical discourse is an integral part of instruction.

In spite of this, we encounter many challenges when we attempt to use discourse to cultivate students' abilities to define and pursue mathematical investigations. Chazan and Ball (1995) point to an important issue in current research:

Teachers' considerations are complex, their moves subtle. Yet, conceptions of the teacher's role often seem focused on what teachers should not do. They should not tell students things; they should not be the source of knowledge. If teachers stay out of the way, the argument goes, students will construct new understandings (p. 16).
Clearly it is not enough to "stay out of the way." On the other hand, often we try to facilitate conversation through rich activities that should be ripe with engaging ideas, only to find that our strategies backfire. We end up with less conversation instead of more, or conversation that does not develop in mathematically significant directions. In such situations, we struggle to help students identify what is problematic.

We may also miss clues students give us about what they need for good conversation. Have we applied a method incorrectly? Are we withholding direct instruction at the wrong times? Do our students share with us an understanding and commitment to what we are trying to do?

What follows are ideas and thoughts that are the result of our collective reflection on and investigation into the teacher's complex role as classroom facilitator. Specifically, we begin to take a closer look at discourse interventions that teachers use to increase students' mathematical understanding -- interventions such as questioning techniques, paraphrasing, summarizing, and listening. In addition, we look at the nature of the decisions teachers face as they work to implement these interventions.

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