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: Leading Questions



Just as a lawyer can use leading questions to induce a witness to change the subject, and on cross-examination to test the knowledge, understanding, and experience of a witness, so too a teacher may use leading questions to begin a lesson, to probe the depth of student understanding, to elicit content, to help a student clarify and extend his or her thinking, and to provide focus.

Leading questions can be used effectively in ways that range from the very didactic to strategies that are more simply facilitative. At the instructional end of the spectrum, some teachers use leading questions to focus a class or to model a line of thinking they would like their students to develop. In such an instance, the teacher has a clear idea of where the discussion needs to lead, and does not use open-ended questions such as "What happened?" or "What did you think?" For example, here Susan Boone uses questions to review key concepts and to prepare her students for the investigation they are about to start.

  [view clip]


In another clip, during student presentations Judy Koenig asks students how their graphs are similar or different, and why. When the class has difficulty answering the question, she uses leading questions to guide them step by step [view clip]. When her class is unresponsive, she asks about the two graphs. Are they linear? Do they increase? Are they curved? etc.

These leading questions differ from questions that enable students to pursue their own line of thinking. Such questions are more open-ended, in the sense that they support student thinking without suggesting an answer. They are responsive to the student, rather than being prescriptive. Typically, they pick up on the particular direction of a thought and attempt to focus attention on it. For example, we might hear a teacher asking, "Why is knowing the radius important?" or, "How did we get from circumference to area?" Such questions are not "leading" in the legal sense, but are leading in the sense that they help direct the flow of conversation and lead a class along a line of thought.

In the following clip, for example, we see Susan Stein working with students to help them understand why increasing the area of the base has a greater effect on the volume of the cylinder than does increasing the height of the cylinder.

  [view clip]


One student suggested that increasing the area of the base of the cylinder has a "bigger effect" on the volume than the height, but Susan wants her to dig deeper and articulate what it is about the area that makes this true. Susan's questions confirm the student's intuition and try to give her further support for translating her insight into a mathematical explanation.

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