UNIVERSITY OF CALIFORNIA, SAN DIEGO SAN DIEGO STATE UNIVERSITY The Music of ...

572 Pages·2015·15.63 MB·English

by  David Quarfoot

UC San Diego
UC San Diego Electronic Theses and Dissertations
Title
The Music of Mathematics: Toward a New Problem Typology
Permalink
https://escholarship.org/uc/item/68f901n3
Author
Quarfoot, David
Publication Date
2015

Peer reviewed|Thesis/dissertation
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University of California
UNIVERSITY OF CALIFORNIA, SAN DIEGO
SAN DIEGO STATE UNIVERSITY

The Music of Mathematics: Toward a New Problem Typology

A dissertation submitted in partial satisfaction of the
requirements for the degree Doctor of Philosophy

in
Mathematics and Science Education
by
David Quarfoot

Committee in charge:
University of California, San Diego
Professor James Hollan
Professor Gabriele Wienhausen

San Diego State University
Professor Richard Levine, Chair
Professor Chris Rasmussen, Co-Chair
Professor Susan Nickerson

2015
Copyright
David Quarfoot, 2015
All rights reserved.
SIGNATURE PAGE

The Dissertation of David Quarfoot is approved, and it is acceptable in quality and form for
publication on microfilm and electronically:

Co-Chair

Chair

University of California, San Diego
San Diego State University
2015

iii
DEDICATION

For my parents – Beth and Jim Quarfoot – whose love and example inspired me to be the man I
am today.

For the three Matts in my life – Matt Quarfoot, Matt Madson, and Matt Ruggirello – who,
together, showed me how to live, love, and learn.

And for the two who understood me at my core – my grandmother, Reva Warren, and my best
friend, Marquis Tisdale.

iv
TABLE OF CONTENTS

Signature Page ............................................................................................................................. iii
Dedication ..................................................................................................................................... iv
Table of Contents........................................................................................................................... v
List of Abbreviations .................................................................................................................... xi
List of Figures ............................................................................................................................. xii
List of Tables ............................................................................................................................... xv
Acknowledgements ................................................................................................................. xviii
Vita ............................................................................................................................................. xix
Abstract of the Dissertation ......................................................................................................... xx
Chapter 1: The Problem ................................................................................................................ 1
A Thought Experiment ..................................................................................................... 1
Why Mathematical Problems? .......................................................................................... 3
Why Mathematical Problems Now? ................................................................................. 6
Overview of the Research Plan ...................................................................................... 12
Research Questions ........................................................................................................ 16
Conclusion ...................................................................................................................... 17
Chapter 2: Literature Review ..................................................................................................... 19
Overview ........................................................................................................................ 19
Mathematical Problem Solving ...................................................................................... 19
Cognition ........................................................................................................... 20
Metacognition .................................................................................................... 29
Affect, Beliefs, and Society ............................................................................... 35
Conclusion ......................................................................................................... 41
v
Mathematical Problem Features ..................................................................................... 42
Cognition/Metacognition ................................................................................... 45
Affect/Beliefs/Society ....................................................................................... 62
Conclusion ......................................................................................................... 71
Typologies ...................................................................................................................... 71
Binary MPTs ..................................................................................................... 73
Multilevel MPTs ................................................................................................ 74
Two-Dimensional MPTs ................................................................................... 76
The Jonassen Typology ..................................................................................... 79
Conclusion ......................................................................................................... 82
Educational Data Mining ................................................................................................ 83
A Description of EDM ...................................................................................... 83
Organization of EDM ........................................................................................ 84
Conclusion ......................................................................................................... 87
Conclusion ...................................................................................................................... 87
Chapter 3: Methodology ............................................................................................................. 88
Thread 1: From MPFs to LAMs and PSMs ................................................................... 88
Number of Entry/Exit Points ............................................................................. 91
Number of Solutions .......................................................................................... 91
Number of Steps ................................................................................................ 92
Difficulty ........................................................................................................... 92
Presence of Misconceptions .............................................................................. 93
Resource Creation ............................................................................................. 93
Development of Metacognition ......................................................................... 94
Variety of Representational Media .................................................................... 94
vi
Appropriateness ................................................................................................. 95
Internal Resource Collaboration ........................................................................ 95
Insufficient/Extraneous Information .................................................................. 96
Amount Learned ................................................................................................ 96
Cognitive Engagement ...................................................................................... 96
Novelty .............................................................................................................. 98
Cognitive Sophistication ................................................................................... 99
Creativity ........................................................................................................... 99
Elegance ............................................................................................................ 99
Productive Dispositions ................................................................................... 100
Degree of Encouragement ............................................................................... 100
Lifelong/Self-Directed Learning ..................................................................... 101
External Collaboration ..................................................................................... 101
Affective Engagement ..................................................................................... 101
Focus on Inequity ............................................................................................ 102
Authenticity ..................................................................................................... 102
Concluding Thoughts ...................................................................................... 102
Thread 2: From Data Set to LAM-Side MST .............................................................. 104
Alcumus Data Set Overview ........................................................................... 104
Munging .......................................................................................................... 111
New Columns .................................................................................................. 114
LAM Construction ........................................................................................... 121
Thread 3: Metric Creation to the Full MST ................................................................. 123
Metric Parameter Exploration ......................................................................... 124
vii
The Expert Panel ............................................................................................. 127
Inter-Rater Reliability ...................................................................................... 131
Video Analysis ................................................................................................ 135
CDW/DQ Scoring ........................................................................................... 137
Thread 4: From Contexts to Correlations .................................................................... 142
Figure 3.23, Stage 1 ......................................................................................... 144
Figure 3.23, Stage 2 ......................................................................................... 146
Figure 3.23, Stage 3 ......................................................................................... 152
Conclusion .................................................................................................................... 155
Chapter 4: Contextualized Analysis ......................................................................................... 156
Continuing Thread 2: The LAM Side of the Full MST ............................................... 156
Continuing Thread 4: Analysis of the Contextualized Results .................................... 166
Appendix E Findings ....................................................................................... 170
Appendix F Findings ....................................................................................... 190
Conclusion .................................................................................................................... 202
Macro Findings ................................................................................................ 202
Micro Findings ................................................................................................ 202
Chapter 5: Decontextualized Analysis ...................................................................................... 206
Metric Analysis ............................................................................................................ 207
Predicting the PSMs from the LAMs ........................................................................... 217
New Dimensions for Mathematical Problems .............................................................. 223
Problem Analysis .......................................................................................................... 241
A New Mathematical Problem Typology ..................................................................... 247
Conclusion .................................................................................................................... 261
On Metrics ....................................................................................................... 261
viii
On Problems and Typologies .......................................................................... 262
Chapter 6: Conclusion .............................................................................................................. 265
Overview of Results ..................................................................................................... 265
Implications for Teachers and Students ........................................................................ 268
Implications for Research ............................................................................................. 271
Limitations .................................................................................................................... 274
Future Directions .......................................................................................................... 276
Final Thoughts .............................................................................................................. 280
Appendices ................................................................................................................................ 282
Appendix A: Final Metric Scoring Instruments (PSMs) ............................................. 282
Metric 1: Difficulty ......................................................................................... 282
Metric 2: Elegance .......................................................................................... 283
Metric 3: Novelty ........................................................................................... 284
Metric 4: Cognitive Sophistication ................................................................. 285
Metric 5: Internal Resource Collaboration ..................................................... 286
Metric 6: Number of Steps ............................................................................. 287
Metric 7: Creativity ........................................................................................ 288
Metric 8: Representational Media .................................................................. 289
Metric 9: Resource Creation ........................................................................... 290
Metric 10: Misconceptions ............................................................................. 291
Metric 11: Number of Solutions ..................................................................... 292
Metric 12: Productive Dispositions ................................................................ 293
Metric 13: Affective Engagement .................................................................. 294
Metric 14: Authenticity .................................................................................. 295
Appendix B: Complete R Code ................................................................................... 296
ix