TEACHING AND LEARNING UNDERGRADUATE ABSTRACT ALGEBRA Timothy Patrick Fu

303 Pages·2007·1.51 MB·English

ABSTRACT

Title of Dissertation: A TALE OF TWO COURSES; TEACHING
AND LEARNING UNDERGRADUATE
ABSTRACT ALGEBRA

Timothy Patrick Fukawa-Connelly, Doctor of
Philosophy, 2007

Dissertation Directed By: Associate Professor Patricia F. Campbell and
Professor James T. Fey
Education, Curriculum and Instruction

The abstract algebra course is an important point in the education of
undergraduate mathematics majors and secondary mathematics teachers. Abstract
algebra teachers have multiple goals for student learning, and the literature suggests
that students have difficulty meeting these goals. Advisory reports have called for a
move away from lecture toward investigation-based class sessions as a means of
improving student understanding. Thus, it is appropriate to understand what is
happening in the current teaching and associated learning of abstract algebra.
The present study examined teaching and learning in two abstract algebra
classrooms, one consciously using a lecture-based (i.e., deduction-theory-proof, or
DTP) mode of instruction and the other an investigative approach. Instructional data
was collected in classroom observations, and multiple written instruments and a set of
interviews were used to evaluate student learning.
Each instructor hoped students would develop a deep and connected
knowledge base and attempted to create classroom environments where students were
constantly engaged as a means of doing so. In the lecture class, writing proofs was
the central activity of class meetings; nearly every class period included at least one
proof. In the investigative class, the processes of computing and searching for
patterns in various structures were emphasized.
At the end of the semester, students demonstrated mixed levels of proficiency.
Generally, students did well on items that were relatively familiar, and poorly when
the content or context was unfamiliar. In the DTP course, two students demonstrated
significant proficiency with analytical argument; the remainder demonstrated mixed
proficiency with proof and very little proficiency with other content. The students in
the investigative class all seemed to develop similar levels of proficiency with the
content, and demonstrated more willingness to explore unknown structures.
This study may prompt discussions about the relative importance of
developing proof-proficiency, students’ ability to formulate and investigate
hypotheses, developing students’ content knowledge, and students’ ability to operate
in and analyze novel structures.
A TALE OF TWO COURSES; TEACHING AND LEARNING
UNDERGRADUATE ABSTRACT ALGEBRA

By

Timothy Patrick Fukawa-Connelly

Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park, in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
2007

Dissertation Committee:
Dr. Patricia F. Campbell, Chair
Dr. James T. Fey, Chair
Dr. Kenneth Berg
Dr. Steven Selden
Dr. Lawrence Clark
© Copyright by
Timothy Patrick Fukawa-Connelly
2007
DEDICATION
To my wife Kate

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ACKNOWLEDGEMENTS
I want to thank:
• Dr. James Fey and Dr. Patricia Campbell for their guidance and support.
• Dr. Nadine Connell’s friendship, experience and occasional snacks made my
time as a graduate student far richer than I had hoped for.
• My parents, for everything.

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TABLE OF CONTENTS

DEDICATION..........................................................................................................ii
ACKNOWLEDGEMENTS......................................................................................iii
TABLE OF CONTENTS..........................................................................................iv
List of Tables...........................................................................................................vii
List of Figures........................................................................................................viii
CHAPTER 1: RATIONALE AND SIGNIFICANCE.................................................1
Rationale...............................................................................................................1
Abstract Algebra and the Undergraduate Mathematics Curriculum....................1
Teaching and Learning Advanced Mathematics.....................................................5
Teaching and Learning Algebra.........................................................................8
Research Questions and Overall Study Design.....................................................10
Teaching and Learning in the DTP Class.........................................................11
Teaching and Learning in the Investigative Class.............................................11
Comparison of DTP and Investigative Class Teaching and Learning................12
Developed Mathematical Proficiency in DTP and Investigative Abstract Algebra
Classes.............................................................................................................12
Limitations of the Study.......................................................................................13
CHAPTER 2: LITERATURE REVIEW..................................................................15
Teaching Advanced Mathematics........................................................................16
DTP Teaching..................................................................................................16
Investigative Teaching.....................................................................................20
Teachers’ pedagogical decisions......................................................................33
Summary.........................................................................................................38
Student learning...................................................................................................39
Conceptual Understanding...............................................................................40
Procedural fluency...........................................................................................47
Strategic competence.......................................................................................48
Adaptive reasoning..........................................................................................50
Productive disposition......................................................................................54
Summary.........................................................................................................57
CHAPTER 3: CONTEXT AND METHODOLOGY...............................................59
Introduction.........................................................................................................59
Context................................................................................................................59
The Mathematics Department..........................................................................60
The Role of the Researcher..................................................................................66
Data Sources........................................................................................................67
Paper-and-pencil instruments...........................................................................68
Data Analysis ......................................................................................................77
Analysis of abstract algebra instruction............................................................78
Analysis of student learning.............................................................................80
CHAPTER 4: LEARNING ABOUT TEACHING...................................................82
Methodology.......................................................................................................83
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Proof in the DTP class.....................................................................................84
Characterizing DTP Teaching..............................................................................98
Teaching Scripts..............................................................................................99
Salient characteristics of the observed teaching scripts...................................108
Student Scripts...............................................................................................112
Summary of DTP teaching.................................................................................113
Analysis of Teaching in the Investigative Class .................................................115
Teaching scripts.............................................................................................116
Salient characteristics of the observed teaching scripts...................................134
Student Scripts...............................................................................................138
A Comparison of the Observed Teaching Scripts...............................................139
Teaching scripts which differentiate the sections............................................141
Shared teaching scripts...................................................................................143
Summary...........................................................................................................148
CHAPTER 5: STUDENT PROFICIENCY WITH ABSTRACT ALGEBRA........152
Introduction and Methodology...........................................................................153
Identities, Inverses, and Units............................................................................156
Understanding Identities, Inverses, and Units.................................................156
Opportunities to Learn...................................................................................157
Assessment of Student Understanding............................................................158
Summary of Demonstrated Proficiency with Identity, Inverse, and Unit.........176
Polynomials.......................................................................................................180
Understanding polynomials............................................................................180
Idealized Student Proficiency with Polynomials.............................................181
Opportunities to Learn...................................................................................182
Assessment of Student Understanding............................................................184
Evidence of Student Proficiency....................................................................185
Summary of Demonstrated Proficiency with Polynomials..............................205
Structure/substructure and what is inherited.......................................................207
Understanding groups, rings and fields...........................................................208
Idealized Student Proficiency with Structure..................................................209
Opportunities to Learn...................................................................................210
Assessment of Student Understanding............................................................211
Evidence of Student Proficiency....................................................................213
Summary of Demonstrated Proficiency with Structure...................................223
Proof proficiency...............................................................................................224
Idealized Student Proficiency with Proof.......................................................225
The local situation..........................................................................................226
Opportunities to Learn...................................................................................227
Assessment of Student Understanding............................................................228
Evidence of Student Proficiency....................................................................228
Polynomial Proofs .........................................................................................237
Summary of Demonstrated Proof Proficiency................................................248
Comparing the Demonstrated Proficiencies of the Two Classes.........................251
Conclusion.........................................................................................................252
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CHAPTER 6: SUMMARY, IMPLICATIONS AND DIRECTIONS FOR THE
FUTURE...............................................................................................................262
Teaching............................................................................................................262
Students’ Demonstrated Proficiencies................................................................265
Limitations of the Current Study........................................................................269
Implications for the Field...................................................................................271
Directions for Future Study................................................................................274
APPENDIX A: STUDENT BACKGROUND ASSESSMENT.............................277
APPENDIX D: END-OF-SEMESTER WRITTEN ASSESSMENT.....................280
REFERENCES......................................................................................................283

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List of Tables

Table 1. Data sources………………………………………………………………..69

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