Bringing Understandings to Light

This excerpt is taken from Teachers' Professional Development and the Elementary Mathematics Classroom by Sophia Cohen: Bringing Understandings to Light.

Published by Lawrence Erlbaum Associates

Preface

This is a book about bringing people’s thoughts and ideas out into the open. It is about making classrooms the kinds of places where these thoughts and ideas are welcome, and more than that, where they are considered and explored. It is a book about the kind of journey it takes for a teacher to make her classroom one in which children’s ideas take center stage in this exploration. It is a book about the kind of journey it takes for a teacher to balance her attention to her own understanding of an idea, with her attention to her students’ understanding, using both in pursuit of strong subject matter knowledge. It is a book about the potential value of such a journey.

But there is no getting around that there is a journey involved in creating classrooms and teaching practices of this kind. For, while we all have thoughts and ideas, they can be fleeting, invisible, momentary events. Our own ideas can slip away. Those of others can be difficult to grasp. And, even if we manage to hold onto them, they can remain vague and ill-formed. Bringing ideas to light, carefully considering a thought’s potential takes a strong focus, a concerted effort, and initially, I will claim here, it also takes a mentor who models and supports such work.

For our children in their classrooms, this mentor can be the teacher. But the kind of teaching practice I focus on here and the subject matter knowledge it requires are far from common. In order for our children to have teachers who are mentors in this way, our teachers also need mentors; people who can guide and support them as they pursue and acquire solid subject matter knowledge and learn to teach in ways that enable the exploration of seminal ideas.

This book is built of stories about what happens when teachers have such a mentor, and engage in serious study of their own and of children’s mathematical conceptions. The stories illustrate the journeys of elementary school teachers across one year’s time, as they participated in a teacher development seminar focused on mathematics, and changed their beliefs, their knowledge, and their practices.

Two story lines run through the book, and are central to each of the individual stories told. One story line concerns psychological and social changes: teachers and students coming to listen to their own and to others’ ideas in ways they hadn’t previously done, creating the kind of classroom community in which each person treats themselves, their colleagues, and the ideas that each expresses with careful consideration. These changes relate to the quality and nature of interpersonal relationships within the classroom, including the manner in which members of a classroom community express, attend to, and grapple with one another’s thoughts and ideas.

The second story line concerns changes in mathematical thinking, among children and teachers as they learn to make mathematical conjectures, to represent these, to argue for or against them in substantive, mathematically valid ways, and to join together in considering the conjectures and arguments as they build stronger understandings of the mathematical topics in the elementary curriculum. Throughout this book we see examples of teachers in the process of learning that this kind of rigorous mathematical thinking exists, learning how to engage in it themselves, and learning to notice when children engage in it. In the last chapters we see some of the ways that teachers’ classrooms change as they build this mathematical competence, and see some evidence that their students, in turn, begin to work with more mathematical substance and rigor.

These two story lines, the psycho-social and the mathematical, are intimately related. Collectively exploring ideas in meaningful ways requires a strong, safe, respectful community. It requires trusting relationships, for who would share a thought about which he or she felt uncertain, with others from whom he or she expected disrespect, or even disinterest? At the same time, constructing new thoughts together is one way that bonds and trust are built, going deeply into ideas together is a way to build community. Just as a strong intellectual community can help to build ideas, the building of ideas is a way to build community. Supporting children as they immerse themselves in one another’s ideas requires a teacher who is able socially and emotionally, and one who is strong enough in the subject matter, mathematics in this case, to enable productive exploration. This is a tall order.

This book provides images of teachers working to fill this tall order. The stories emerge from a 2-year study of teacher learning that I conducted. The study focuses on the learning of participants in a mathematics teacher development seminar for elementary school teachers. The seminar, Developing Mathematical Ideas ((DMI) Schifter et al, 1999 a, b, c, d), is one example of intensive, domain-specific professional development: the kind of professional development so often called for by educators. DMI is a published, nationally-available teacher development curriculum in which teachers (a) study elementary mathematics to deepen their own understanding of it, (b) study the development of mathematical ideas in children, and (c) experience a teaching and learning environment focused on the joint exploration of ideas.

I was particularly interested in what teachers in this seminar learned, because I had helped to design and to write these seminar materials. While, as a team, those of us who worked on building DMI knew what we hoped the seminar might accomplish, I wanted to look carefully at the issues that actually arose for teachers taking the seminar. Further, because the DMI seminar is an example of a kind of professional development that reformers across disciplines are calling for, it seemed to me that the stories of teacher learning I was collecting and analyzing, might serve as a case that would be of interest to a broad group of educators.

The study of teacher learning that gives rise to this book followed two DMI seminars and the classroom practices of seminar participants. It involved both teachers’ own reports of what they were learning, and direct observations of participants’ classroom practices. The data include rich cases or stories about individuals, as well as check backs to data from the whole group for some sense of the representativeness of each story. The stories are used to define and illustrate 3 strands of teacher learning. Each of these strands relates to both of the 2 story lines that run through this book, although each of the strands highlights a different aspect of the story lines. Together, the strands portray the work involved for teachers who are transforming their practices. Bringing these strands to life is a central mission of this book, and the lion’s share of the text is devoted to exploring that work.

While this book focuses on elementary mathematics and the Developing Mathematical Ideas seminar, the issues it examines could be considered with respect to changing teaching practices at any grade level or any content area. This book is addressed to multiple audiences: researchers and teacher educators within universities, as well as those people in every community who have responsibility for making decisions about how teacher development time and money are spent. For all readers, this book offers an opportunity to examine – at close range – the kinds of changes in focus, knowledge, teaching practice, and opportunities for children that resulted from the work of teachers over the course of one year. For all readers the stories provide an opportunity to envision the teaching and learning towards which they would like their communities to build. But different readers will come to this book with different concerns. Below, I address a few words, separately, to different groups of readers.

To mathematics-education researchers, professional developers, and administrators:

Finally, and importantly, the teachers whose work is at the heart of this book showed enormous bravery and trust by allowing me to observe their ongoing professional development work. As professionals, it was not easy for them to make their own intellectual struggles public, to open them to study and commentary. I hope that all readers will offer, in return, the same respect for the teachers’ developing ideas that the teachers are learning to offer when they encounter ideas – their own, or students’. We will gain most from a reading of the teachers’ work that takes the stance of trying to understand what the teacher grasps, and what he or she is working at understanding or creating. The picture that emerges will inform us about the current state of teachers’ relationships to mathematics, and to teaching and learning. It will inform us about the potential of teacher seminars as vehicles for change. It will inform us about the route ahead if the national visions for reform are to come to life in classrooms across the country.

Table of Contents

Preface