Encouraging Mathematical Thinking


 Abstract
 Introduction

 Discourse
 Interventions
  - Approaches
  - Leading Q's
  - Non-leading Q's
  - Paraphrasing
  - Summarizing
  - Listening
 Decisions

 Cylinder Problem
 Lesson Reflections
 Student Predictions

 Project Reflections
 Conclusion

 References
 Acknowledgments
 Teacher Resources



Authors'
Biographies

Table of Contents


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Interventions: Listening



If one of our guiding concerns is to be responsive to student thinking, it follows that teachers must listen to students carefully. All of the ideas discussed above are derived from this essential element: good questioning requires good listening to what has gone before. The accuracy of paraphrasing depends on listening closely to a student's language. To summarize effectively, we must catch the rhythms and punctuation of the conversation. For students to learn how to ask good questions, they must listen to each other, to the teacher, and to themselves.

By emphasizing that everyone has something worthwhile to contribute to the class, and by actively modeling this behavior, we encourage our students to listen to themselves as well as to the teacher and each other. Many of us state explicitly to our students that we are interested in the mathematician we believe is in each one of them. Listening is key if students are to recognize that voice within themselves.

In listening, we recognize that a student's mathematical actions or explanations are reasonable from his or her point of view, even if the reason is not immediately apparent to us (Cobb, Wood, Yackel, Nicholls, Wheatley, Trigatti, & Perlwitz, 1991). As Ball (1993) points out, it is the students' sense-making that is their mathematical thinking. We need what she refers to as a "bifocal perspective," where the teacher is "perceiving the mathematics through the mind of the learner while perceiving the mind of the learner through the mathematics" (p. 159). In this way the teacher can gain insight into new approaches to concepts, while thinking about the support the student needs in order to make the next move. In this sense, listening is also part of the broader effort to learn about students' strengths and needs, and to think about when and which intervention to employ (Renninger, 1998).

Thus, although we recognize that there is much more to discuss about listening, we are led to think about decisions a teacher makes in the classroom, decisions that must take into account the particulars of the child, the interactions, the use of discourse, and the mathematics to be learned (Chazan & Ball, 1995).

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