Learning and understanding in abstract algebra

435 Pages·2017·19.23 MB·English

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Doctoral Dissertations Student Scholarship
Winter 2001
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Bradford Reed Findell
University of New Hampshire, Durham
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Findell, Bradford Reed, "Learning and understanding in abstract algebra" (2001). Doctoral Dissertations.
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LEARNING AND UNDERSTANDING IN ABSTRACT ALGEBRA
BY
BRADFORD R. FINDELL
BSE, Princeton University, 1985
MA, Boston University, 1990
DISSERTATION
Submitted to the University of New Hampshire
in Partial Fulfillment of
the Requirements for the Degree of
Doctor of Philosophy
in
Mathematics Education
December, 2001
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
UMI Number: 3030602
Copyright 2001 by
Findell, Bradford Reed
All rights reserved.
___ ®
UMI
UMI Microform 3030602
Copyright 2002 by Bell & Howell Information and Learning Company.
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ALL RIGHTS RESERVED
©2001
Bradford R. Findell
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This dissertation has been examined and approved.
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Dissertation Co-directod, Karen J. Graham, Professor of Mathematics
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Disss^eerrttaattiioonn CCoo--ddiirreeccttoorr,, JJooaann FFeerrrriinnii--MMuindy, Professor of
Mathematics and Teacher Education, Michigan State University;
Former Professor of Mathematics, University of New Hampshire
Edward K. Hinson, Associate Professor of Mathematics
Donovan H. Van Osdol, Professor of Mathematics
Sharon Nodie Oja, Professor of Education
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Date
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DEDICATION
for Jenny, Elizabeth, and Randall Todd,
who exceed all hopes
and make the world grow with possibilities
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ACKNOWLEDGMENTS
Perhaps it takes several villages to raise a scholar, for if I do indeed become a scholar, it
will be largely because of the support, encouragement, and prodding I have received from
individuals of many villages in the world of mathematics education and beyond.
I thank my advisor, Joan Ferrini-Mundy, for helping me craft this study and its account
into a coherent whole, for pushing me to articulate and refme many of the premises that
were implicitly guiding my thinking, and for providing support and encouragement
throughout the process. I am indebted to Joan not only for directing my graduate
program and this dissertation but also for providing an abundance of professional
opportunities over the past eight years. In particular, I am grateful for the opportunity to
join her at the National Research Council, where she again served as my supervisor and
mentor.
Thanks go to Karen Graham, for stepping in as co-director of this study, for helping me
sharpen my ideas, for providing an ear to help me think through what needed to be done,
for merely smiling at my suggestion that I write an 80-page dissertation, and for playing
such a significant and formative role throughout my graduate program.
Thanks also go to the other members of my committee, Nodie Oja, Ed Hinson, and Don
Van Osdol, who individually provided keen critiques of drafts and who together urged
me to integrate the many disparate ideas that those drafts contained. I also thank Ann
Diller for providing helpful insights in the early phases of this study.
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I am particularly grateful to Steve Benson for allowing me to conduct this study in his
classroom and for the opportunity to help him elaborate a vision of what learning
advanced mathematics might be like. Thanks also to Carla, Diane, Lori, Robert, Wendy
and the other students for providing their time and the details of their thinking, which
formed the core of this study.
I am also grateful to the UNH Department of Mathematics and Statistics and it's chair
Ken Appel for supporting mathematics education and, in particular, for providing a
fellowship to support the completion of this study. This study was also supported in part
by a summer fellowship from the UNH graduate school.
I also appreciate the encouragement extended by the members of the Mathematical
Sciences Education Board (MSEB) and the Mathematics Learning Study Committee of
the National Research Council for whom it was a pleasure and a privilege to work. The
members allowed me to contribute to and support their work, providing an opportunity
not often available to graduate students. Among the staff at the National Research
Council, I particularly want to thank Gail Burrill and Jane Swafford for the ways that
they subtly and not-so-subtly prodded me to finish what I had started.
I especially value the advice and support provided by MSEB chairs Hyman Bass and
Joan Leitzel, vice chairs Glenda Lappan and Jere Confrey, and long-standing MSEB
member, Deborah Ball. I cannot describe in a few sentences or even a few pages the
span and depth of what they have given me through formal and informal discussions
about research, teaching, and career possibilities.
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I was particularly fortunate in my work at the National Research Council to have an
opportunity to work with Jeremy Kilpatrick, when he served masterfully as the chair of
the Mathematics Learning Study Committee. I owe him a debt that can never be repaid
for the interest he has taken in my career and for encouraging me to come to the
University of Georgia to be his colleague. Jeremy graciously responded to drafts of
several of the chapters, helped me hone my message, and provided editorial advice at all
levels.
I also thank the Mathematics Education Department at the University of Georgia,
especially Pat Wilson, for the confidence they have placed in me and the encouragement
they all have provided during the completion of this study. It is a precious opportunity to
be able to work and learn in such a vital and vibrant department.
Thanks also to my friends and colleagues in mathematics education, including Carole
Greenes, Pam Matthews, A1 Cuoco, Paul Goldenberg, Ed Dubinsky, Jim Fey, Kathy
Heid, Jim Leitzel, Rodger Bybee, Roger Howe, Hung-Hsi Wu, Kevin Miller, Ed Silver,
Steve Monk, Ann Shannon, Barbara Edwards, Dawn Berk, Tim Gutmann, and Todd
Grundmeier who provided encouragement, support, and guidance and who often told me,
in one way or another, not only that I could complete this study but that I should.
Of course, I would be nowhere without the support of my family. I owe a great debt to
my mother, Carol, for pointing me in the direction of mathematics education years ago,
for showing me what it means to engage with mathematics and to engage students in
mathematics, and, more recently, for helping me enter data, for responding to drafts, and
for letting three of us live in her house all summer. I also thank my brother Kent and his
wife Vickie for providing support and frequent meals. I am grateful to my sisters Karen,
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Leanne, and Kirsten for their encouragement and particularly to Kirsten for not gloating
that she had completed her dissertation first. A special thanks goes to my daughter
Elizabeth, who always amazes and who provides no end of joy, and to my son-to-be
Randy, whom I eagerly wait to meet.
Most of all I thank my beloved Jenny, who has single-handedly and unselfishly managed
three (almost four) lives for the past nine months, including two moves with the
anticipation of a third, and who deserves so much more than I have been able to give.
During this ever-expanding study, Jenny also graciously served as a sounding board for
my dilemmas, frustrations, and nascent ideas, often helping me formulate those ideas
more clearly. There cannot be a more generous and forgiving partner.
Since I began teaching mathematics more than sixteen years ago, there have been many
moments at which I believed and acted as though I brought some expertise to the tasks
before me, whether I was teaching, writing, editing, or talking about these activities. Not
so these past nine months. Several times, however, in moments of singular clarity, I have
caught a fleeting glimpse of what it means to be a scholar, of how it feels to be
subservient to what the data have to say, and of what is involved in building an idea that
seems new. I hope that these glimpses will become more frequent.
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