Encouraging Mathematical Thinking


 Abstract
 Introduction

 Discourse
 Interventions
 Decisions

 Cylinder Problem
 Lesson Reflections
 Student Predictions

 Project Reflections
 Conclusion

 References
 Acknowledgments
 Teacher Resources



Authors'
Biographies

Table of Contents


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Reflections

John McKinstry       [Biography]



I have greatly enjoyed and profited from my participation in the group known as WRAP. I have profited in several ways. First, it has been great discussing our ideas of teaching math with such a thoughtful group of teachers. Second, it has been helpful to think about how conversation works in the classroom.

The purpose of this reflection was originally for us to focus on one concrete part of our teaching that we found through our work together. I find that the one concrete thing I got out of this was a clearer understanding of my beliefs about teaching, for there is nothing quite as practical as a good theory. Susan Stein and I come from similar schools and thus found we shared certain language when describing classrooms and discourse. After talking with Ann Renninger, I realized that much of the university-based research coincides with the kind of teaching practiced in Friends (Quaker) schools. The big differences in my mind between that research and the teaching among Friends are 1) the vocabulary sometimes used and 2) the underlying basis. Hence I have come to realize how much of the educational philosophy of Friends shapes my beliefs about teaching, and how much that philosophy parallels the research in good pedagogy.

The Friends philosophy of education rests on a basic religious belief of Quakers that each person can know personal religious truth, without any intermediary. Quakers express it in many ways, the most common of which is that each of us has "that of God." One can find religious truth, not individually, but by coming together, in a meeting, where the sense of Truth is discerned together, and there is an understanding that Truth is continually being revealed. This belief is radically democratic, since all authority is discerned by the group, but no one person has complete authority on the Truth. The source of Truth is mainly a person's personal experience, although Friends recognize that when we share others' experiences, including those of the past, our understanding is deeper.

Since Quakers make no distinction between the religious and non-religious, these beliefs work in the classroom. Thus, a classroom is a place where there is a corporate search for truth, with the teacher as one authority among others. The understanding of Truth is based upon people's personal experience, and listening to one another, to come to a collective understanding. A math class is a place where knowledge is shared, not handed down. By insisting on relying on personal experience, we limit ourselves as teachers, and this makes us more likely to listen, and less likely to talk, for how much of what we say has come from personal experience, as opposed to what we have read or were told somewhere?

In our discussion of discourse in the classroom, there is also an implicit trust in our students, not in isolation, but as a classroom community. If the trust is placed in that classroom community, our role as teachers is lessened, and it rejects the image of the sole mathematician in place of the group working together. It is our initial role to create this atmosphere of trust. The part of our BRAP experience that I found most interesting was the ways in which we created this trusting classroom atmosphere. But I believe that the best way to create such an atmosphere and nurture it is through our own example: How well do we listen? How do we acknowledge and learn from our students' insights? Are we prepared to find ourselves in a place at the end of class or the end of a week that is far different from where we initially thought we would be? Are our classrooms joyful, kind, and fun places to be? Are we comfortable with silence, understanding that silence means that there is time for reflecting, thinking, and probing the depths of one's mind and allowing the reticent to come forward with a sense of safety?

Thus, the concrete response I have taken from this, aside from better understanding my overall philosophy, has been my ability to structure classes and lessons to do several things. 1. help students think about basic mathematical ideas, 2. listen to my students, 3. wait for them, 4. ask questions that help them think about math in different ways, and 5. encourage my students to ask good questions themselves.

From this group I have profited so from the great respect we all have shown one another, and the excitement we have had over our teaching. It has been a terrific experience and I am grateful to all of you.

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