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:  Paraphrasing



Paraphrasing a student's ideas can serve multiple purposes, although as Jon Basden says at the start of this discussion of teacher interventions, it is not always used to best effect. When used well, paraphrasing indicates to students that they have been heard; used poorly, however, it can distort the student's meaning and refocus attention on figuring out what the teacher is thinking. The objective of paraphrasing should be to help students hear each other's points or to organize their thoughts. In addition, using standard mathematical language to state a student's point can provide a means of introducing mathematics terms to students.

In this clip, Judith Koenig tries to understand what her students are talking about by paraphrasing their words and including an example to concretize the idea.

  [view clip]


Jon Basden also paraphrases a student's definition and uses the full mathematical terms, while providing others with an opportunity to elaborate on this definition.

  [view clip]


Paraphrasing can also provide an opportunity to link students' use of everyday language and mathematical terms. In this clip, a student explains his prediction while Art Mabbott listens. The student uses the words "shorter" and "wider" to explain his reasoning. Art then refers to "bigger diameter" in paraphrasing the student's prediction, and tries to help others focus on the explanation the student is trying to develop [view clip].

In our use of paraphrasing, we are mindful that it might derail a student's thinking. Again, we return to the idea of being responsive rather than prescriptive. If the teacher's goal is to paraphrase in order to redirect the conversation, we expect a different outcome than if we are attempting to draw attention to and encourage the student's thinking. For the latter objective, we commonly hear phrasing such as, "What I hear you saying is...," followed by a close rendering of the student's original statement.

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