Encouraging Mathematical Thinking



 Cylinder Problem
 Lesson Reflections
 Student Predictions

 Project Reflections

 Teacher Resources


Table of Contents

VIDEO CLIPS: Internet access via modem may mean very long download times for video clips. If you are not on a fast line, you may want to read this paper without viewing the clips.

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Being a professional educator takes time -- time to plan, time to practice, time to grade, time to communicate -- and I never have enough time. However, I now realize that adding reflection and research to my agenda have made my life as a teacher easier, not more difficult.
-- Judith Koenig, Project teacher

This paper reflects our working knowledge about how teachers engage students in mathematical thinking, and why this is important. We focus in particular on discourse in the classroom. By this we mean the use of questioning, listening, writing, and reflection as a means of encouraging reciprocal conversation -- the kind of teaching that allows every person to have a voice in creating mathematical understanding.

We are a group of mathematics teachers and Math Forum staff who differ in our perspectives and backgrounds. We work with students who vary in the level of math they are learning, in their prior experiences with math, and in their school contexts and demographics. We began talking and working together as part of a National Science Foundation project hosted by TERC, called Bridging Research and Practice (BRAP). The goal of the Math Forum partnership was the preparation of a practitioner videopaper that would bridge research and practice. A brief way into the project we imagined that instead of bridging research and practice, we were actually weaving them together.

In our work together we have read and discussed research in mathematics education, attempted to think about mathematics and teaching in a reflective way, and shared our ideas. We do not consider this paper a finished product; rather, we see it as the basis of a conversation still being developed, one in which you, the reader, are invited to participate. We are posting this article on the Internet as a conversation in progress, inviting on-line exchanges. We hope to make it easy for others to join in, or to include us in discussions taking place elsewhere.

During the past nine months we have spent time together in face-to-face workshops, participated in conversations through email and virtual meetings, read articles and chapters, shared videotapes of our classrooms, and talked about our teaching. The process of collaborating in this research "from the inside" (Ball, 1993) has been exciting and empowering for us. We have developed a common vocabulary, shared reflections, encountered new ideas, and built trusting relationships. Our conversations have deepened our understanding of mathematics, teaching, and student learning, and have provided us with ways to improve what we do.

We include in this videopaper the presentation of a stimulating mathematics problem about cylinders as it was taught in six classrooms, video clips of students in these classrooms, and an exploration of teaching practices and discussions about how to encourage mathematical thinking. We reflect on our own experience in relation to research on mathematics classroom discourse (Romberg, 1994), and describe the challenges involved in using questions, observations, explanation, and the like, as ways of helping students learn mathematics.

Since mathematics education research sometimes seems distant and disconnected from our work as mathematics teachers, we began our investigation by grounding it in the following focused questions about teaching practices:

  • How can teachers guide students' oral communication, for example, through active listening and careful paraphrasing of their language?

  • What kinds of questions promote good discourse, and at what stage of learning should they be used?

  • How can we make our own and our students' self-assessments and reflections explicit, deliberate, and systematic?

  • What are the characteristics of problems that motivate discourse in a mathematics class?

  • How can we give students ownership of the problem-solving process, help them ask questions that show their understandings and misunderstandings, and encourage them to describe their ideas in writing? How can we model this behavior in our own reflective teaching?

We reject the extremes that plague much of the ongoing debate regarding mathematics education. Whether the issue be algorithms vs. student understanding, back-to-basics vs. problem-solving, or teaching through lectures vs. cooperative group work, we do not find the controversy that comes with the current debates over mathematics education to help our practice. Our experience leads us to believe that for each of the above questions, the "conflicting" approaches need to be appreciated as complementary. They need to be used together, dynamically and creatively.

Resnick (1988) notes that there are numerous ways to set up classes in which students learn. For example, Lampert (1986) conducts full-class discussions in which her students justify solutions to mathematical problems. Schoenfeld (1987) works with small groups of students, modeling thinking aloud for them when necessary. Lesh (1985) has students work in small groups, but without a teacher present. Countryman (1992) encourages students to write in journals.

Clearly, there are many ways in which teachers can work effectively with their students. The present discussion, however, focuses specifically on the role of discourse in the mathematics classroom. While issues such as assessment, class norms and structure, student cognition and learning, and the development of mathematics are also important to us, our goal here has been to work with our students to describe and question what they understand. This raises issues of how we can facilitate their ability to do this. It leads us to ask, "How do I know I am right?" -- especially when our students ask us this question.

In this paper we share with you the challenges we face in discovering how to encourage our students to think mathematically. We invite colleagues who are interested in using collaboration, communication, and reflection as important tools for professional growth to join us.

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