Understanding abstract algebra concepts

238 Pages·2017·4.33 MB·English

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Doctoral Dissertations Student Scholarship
Spring 2007
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Anna S. Titova
University of New Hampshire, Durham
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Titova, Anna S., "Understanding abstract algebra concepts" (2007). Doctoral Dissertations. 362.
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UNDERSTANDING ABSTRACT ALGEBRA CONCEPTS
BY
ANNA S. TITOVA
BS, Russian State Pedagogical University, St. Petersburg, Russia, 1998
MS, University of New Hampshire, 2005
DISSERTATION
Submitted to the University of New Hampshire
In Partial Fulfillment of
The Requirements for the Degree of
Doctor of Philosophy
In
Mathematics Education
May, 2007
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UMI Number: 3260585
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This dissertation has been examined and approved.
1(PJj
Dissertation Director, Dr. Sonia Hristovitch
Assistant Professor of Mathematics
vs
Dr. Karen J. Graham
Professor of Mathematics
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Dr. Dmitri Nikshych
Associate Professor of Mathematics
Dr. Edward Hinson
Associate Professor of Mathematics
Dr. Maria Basterra
Assistant Professor of Mathematics
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Date
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DEDICATION
for Dasha and Andrei Smuk;
Ludmila Titova, Sergei Titov;
Nina Nenasheva, Ivan Nenashev
who make me and my life possible,
for their continued encouragement and support
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ACKNOWLEDGEMENTS
Following the thesis manual, I only write my name on the front page of this
dissertation. However, it would not be possible without help, support and
guidance of many people. Writing the dissertation was a great challenge and now
I owe my gratitude to all of those who intentionally or unintentionally assisted me
during my graduate research and those who inflamed me for this study.
I would like to express my sincere gratitude to Professor Sonia Hristovitch
who was advising my study and me during all these years. She was able to teach
me what research is, how to conduct it, and what it means to write a dissertation.
Her knowledge and positive attitude motivated and inspired me.
Special thanks go to Professor Karen Graham for her support during my
doctoral study, especially during the most difficult first years. I am grateful to her
for giving me the feedbacks and helping me to organize the thesis.
I would like to thank Professor Edward Hinson for all his help with the data
collection for this study. It would not be possible for me to have a newborn, to
commute between Rhode Island and New Hampshire, and collect the data
without his help. Thank you for your optimism, your interest in my research and
your support.
I am also greatly indebted to Professor Dmitri Nikshych for providing me
with great quality instructions and teaching me Abstract Algebra. His teaching
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and interest in Abstract Algebra structures allowed me to take the responsibility
and study how other people understand Abstract Algebra concepts.
I would like to thank Professor Maria Basterra for increasing my interest in
mathematics and mathematics education, for “being-a new-mom-and-researcher”
example, and agreeing to be a member of my dissertation committee.
Again, I would like to thank all my dissertation committee members for
their help and support.
I cannot overstate the role that Dr. Melissa Mitcheltree played in my life
during my graduate study. I thank her for helping me to assimilate in a new
learning environment, for teaching me how to be a graduate student, for
supporting me and helping me with my study. I am very lucky to have a friend like
her.
I would like to thank the Graduate School of University of the New
Hampshire for giving me this wonderful opportunity. Special thanks to the
Department of Mathematics and Statistics for providing me with teaching
assistantship that gave me a unique teaching experience and financial support.
This study was in part supported by the Graduate School Summer Fellowship. I
thank all professors and graduate students of the department, and also all the
students who agreed to participate in my study.
I am very grateful to my family. Without them I would have never start my
graduate study and would have never completed this dissertation. Thank goes to
my husband Andrei Smuk for his support, help, and inspiration for learning he
gave me. My daughter Dasha who, being born before the dissertation was
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complete, gave me a chance to graduate. Special thanks to my parents, Ludmila
Titova and Sergey Titov, and grandparents, Nina Nenasheva and Ivan Nenashev
for their life-long commitment to my personal and scholarly development.
Also I would like to thank all my friends for believing in me, for their
personal support and help.
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TABLE OF CONTENTS
SIGNATURE PAGE.............................................................................................................................ii
DEDICATION........................................................................................................................................iii
ACKNOWLEDGEMENTS..................................................................................................................iv
LIST OF TABLES.................................................................................................................................x
LIST OF FIGURES..............................................................................................................................xi
ABSTRACT.........................................................................................................................................xiv
CHAPTER PAGE
I. INTRODUCTION.........................................................................................................................1
II. LITERATURE REVIEW.............................................................................................................4
Philosophical View.................................................................................................................4
Classical View by J. Piaget..................................................................................................5
Theories Based on Piagetian Idea of Reflective Abstraction....................................10
Alternative views on the notion of abstraction..............................................................14
Mathematical Knowledge Acquisition: Learning Abstract Algebra Concepts 24
APOS Theory and Researches Based on It...............................................................25
Studies framed in different perspectives.....................................................................31
III. THEORETICAL FRAMEWORK............................................................................................36
IV. RESEARCH QUESTIONS....................................................................................................41
V. METHODOLOGY.....................................................................................................................43
Introduction............................................................................................................................43
Settings and Instructional Context..................................................................................44
Participants............................................................................................................................44
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Data Collection Procedure 45
Students Artifacts..............................................................................................................45
Students Interviews..........................................................................................................45
Observations......................................................................................................................47
Instruments............................................................................................................................47
Data Analysis.........................................................................................................................47
VI. DATA ANALYSIS...................................................................................................................49
Quiz 1......................................................................................................................................49
Set - operation relations: universal quantifications..................................................51
Set - operation relations: closure.................................................................................56
Interesting response.........................................................................................................62
Quiz 2......................................................................................................................................63
Set - operation relations. Groups and their subgroups...........................................63
Exam 1...................................................................................................................................72
Identity - quantifiers.........................................................................................................73
Identity - uniqueness........................................................................................................78
Set - operation relations. Closure.................................................................................85
Concrete examples..........................................................................................................87
Identity element - group axioms....................................................................................90
Exam 2 ...................................................................................................................................93
Analysis of Problems 1 and 2 ........................................................................................94
Analysis of Problem 3.....................................................................................................106
Analysis of Problem 4.....................................................................................................115
Analysis of Problem 5.....................................................................................................124
Interview 1.............................................................................................................................128
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