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: Approaches



Teachers use many approaches with students to help them develop their mathematical thinking and take ownership of their learning. These form a continuum, ranging from more direct approaches in which the teacher provides an answer, a demonstration, or a leading question, to less direct approaches that encourage students to articulate their thinking or to reflect inwardly on their questions and insights.

Some examples of less direct approaches to facilitating mathematical thinking include: non-leading questions that respond to student ideas, paraphrasing a student's answer to help him or her look carefully at what was just said; summarizing a discussion that covered many questions, connecting ideas, and problem-solving steps to be taken; and the use of wait-time, in which a teacher poses a question and provides time for the student to think through and explain his or her reasoning. Each of these interventions has the potential to help students discover that they have the capacity for logic, and that they too can think mathematically.

Although our emphasis is on helping students learn to pursue challenging mathematics questions, we must be responsive to their needs and their level of readiness. This may lead us to direct instruction (Roehler, Duffy, & Meloth, 1986). Even the most facilitation-oriented teacher introduces students to resources, although usually such a teacher does so in response to needs expressed by students. The teacher can act as an expert member of a collaborative learning community, one who has resources to bring to bear on an inquiry. In this clip, for instance, Art Mabbott explains his use of a spreadsheet to students, comparing their results to his.

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Responsiveness is a key to fostering discourse. When students are conducting an inquiry, the teacher can be a sounding board and confidence builder. In this clip, Jon Basden responds to students who are playing out the implications of being able to configure a fixed area into many rectangles that have different dimensions for their length and width.

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Another form of responsiveness involves identifying students' misconceptions based on the questions they ask. In such instances we might give an answer, but the more useful response may be a follow-up question that probes the assumptions or conclusions that led to the misguided question (Lampert, 1986). This strategy has two purposes: (a) it gives the involved students a chance to reflect on their own thinking, and (b) it refers responsibility for a question back to the person who asked it. Students need to learn to answer, "Why did you ask that question?" -- and this is quite a challenge at first, because the act is so reflexive and the assumptions are essentially taken for granted. In this type of situation, a carefully crafted question can lead students to unpack their thinking and ask themselves whether an answer or a procedure they have used makes sense. Such questioning is part of a process that serves to shift authority from the book or the teacher to the student.

One norm in the classroom that can be made explicit is the expectation that students are responsible for improving their problem-solving and we will regularly ask them questions about their decision-making. Some examples of questions that teachers might ask are:

  • What (exactly) are you doing? (Can you describe it precisely?)
  • Why are you doing it? (How does it fit into the solution?)
  • How does it help you? (What will you do with the outcome when you obtain it?) (Schoenfeld, 1987, p. 206).

Asking students to be responsible for these questions and then holding them accountable helps students begin to identify gaps and needs in their work with problems. In fact, over the course of a term, Schoenfeld (1987) reports that his students get into the habit of asking each other these questions as a regular aspect of their problem-solving.

While the process of questioning can help students recognize what they understand and what they still need to figure out, we also want our students to become responsible for identifying and posing their own questions. We want them to go beyond saying "I don't get it" and to ask about what they do not understand. We want them to talk about the resources they have explored, and where in the process of problem-solving their thinking has broken down. In posing questions, we think of ourselves as enabling students to express what they do not understand. We also think of ourselves as modeling how they might pose mathematical questions.

In thinking about questions we might ask our students, we find it useful to distinguish between leading and non-leading questions. This classification is borrowed from the legal profession because the way it is used in the courtroom clarifies our use of such questions in the classroom, with the goal of facilitating students' abilities to do their own thinking.

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