DEVELOPING AN UNDERSTANDING OF VARIATION: AP STATISTICS TEACHERS

387 Pages·2009·2.76 MB·English

The Pennsylvania State University

The Graduate School

College of Education
DEVELOPING AN UNDERSTANDING OF VARIATION: AP STATISTICS
TEACHERS‘ PERCEPTIONS AND RECOLLECTIONS OF CRITICAL MOMENTS
A Dissertation in

Curriculum and Instruction

by

Susan A. Peters
 2009 Susan A. Peters
Submitted in Partial Fulfillment
of the Requirements
for the Degree of
Doctor of Philosophy

August 2009
The dissertation of Susan A. Peters was reviewed and approved* by the following:

Rose Mary Zbiek
Professor of Education
Dissertation Advisor
Chair of Committee

Glendon Blume
Professor of Education

M. Kathleen Heid
Distinguished Professor of Education

Thomas Hettmansperger
Professor of Statistics

Laura Simon
Lecturer in Statistics

Glendon Blume
Professor of Education
Coordinator of Graduate Program in Curriculum & Instruction

*Signatures are on file in the Graduate School
iii
ABSTRACT
This phenomenological study investigates conceptions of statistical variation that
secondary mathematics teachers who are recognized leaders in AP Statistics exhibit. This study
also investigates perceptions and recollections of activities and actions that teachers who
exhibited robust understandings of variation suggest contributed to their current understandings
of variation. The data include questionnaires, event history calendars, critical incident
descriptions, resumes, course syllabi, content-focused interviews, and two learning-context
interviews for each teacher. Constant comparative analysis (Glaser & Strauss, 1967) of content-
interview data and syllabi yielded three distinct types of teachers‘ conceptions of variation:
Expected but Explainable and Controllable (EEC), Noise in Signal and Noise (NSN), and
Expectation and Deviation from Expectation (EDE).
The teachers‘ responses to variation-related tasks were used in conjunction with the
SOLO Model, research results about students‘ learning related to variation, and expositions on
what it means to understand statistical variation to develop a framework for robust
understandings of variation. The framework consists of two cycles of levels of reasoning in the
formal mode. Robust understanding of variation is indicated from integrated reasoning about
variation across three perspectives—design, data-centric, and modeling— in the second cycle of
levels. Teachers‘ understandings of variation were assessed using the framework. Five teachers
exhibited reasoning about variation that was consistent with robust understandings of variation.
Analysis of learning experience-related data for these five teachers followed protocol for
phenomenological studies. Factors that may have contributed to these five teachers‘
developments of robust understandings include their interests in the field of statistics, their desires
to have an overarching content framework for themselves and for their students, their
foundational knowledge upon which they built deeper understandings, their propensities for
iv
critical reflection, and their acting on opportunities to engage in learning activities and rational
discourse with more knowledgeable others. The extent to which they embrace these opportunities
may distinguish them from other teachers.
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TABLE OF CONTENTS
LIST OF FIGURES ................................................................................................................. x
LIST OF TABLES ................................................................................................................... xii
ACKNOWLEDGEMENTS ..................................................................................................... xiii
Chapter 1 Rationale.................................................................................................................. 1
Status of Statistics Education Research ........................................................................... 1
Variation and Statistics .................................................................................................... 4
Research Questions .......................................................................................................... 6
Overview of the Study ..................................................................................................... 7
Chapter 2 Literature Review .................................................................................................... 9
Research About Variation and Related Concepts ............................................................ 9
Students‘ and Teachers‘ Reasoning From a Design Perspective ..................................... 10
Reasoning About Variation and Sampling Methods ................................................ 11
Reasoning About Variation and Samples ................................................................. 12
Sample Representativeness and Sample Variability ................................................ 14
Reasoning About Variability in Experimental Design ............................................. 15
Students‘ and Teachers‘ Reasoning From a Data-Centric Perspective ............................ 16
Representing Variability .......................................................................................... 17
Measuring Variability .............................................................................................. 18
Reasoning About Measures of Variation.......................................................... 19
Reasoning About Measures of Center .............................................................. 21
Reasoning About Distribution .......................................................................... 22
Describing Variability .............................................................................................. 23
Students‘ and Teachers‘ Reasoning From a Modeling Perspective ................................. 24
Variation and Binomial Models ............................................................................... 25
Variation, Models, and Statistical Inference ............................................................ 27
Research on Variation: Concluding Remarks .................................................................. 28
Research About Teacher Development ............................................................................ 29
Professional Development ........................................................................................ 30
Teacher Beliefs ......................................................................................................... 32
Professional Development, Teacher Beliefs, and Teacher Change .......................... 34
Concluding Remarks ........................................................................................................ 40
Chapter 3 Conceptual and Theoretical Grounding .................................................................. 42
Transformation Theory .................................................................................................... 42
Forms of Learning .................................................................................................... 43
Learning Through Meaning Schemes .............................................................. 43
vi
Learning New Meaning Schemes ..................................................................... 45
Learning by Transforming Meaning Schemes ................................................. 45
Learning by Perspective Transformation ......................................................... 46
Types of Meaning Perspectives ................................................................................ 48
Epistemic Perspectives ..................................................................................... 48
Sociolinguistic Perspectives ............................................................................. 52
Psychological Perspectives ............................................................................... 53
Perspectives and Learning ................................................................................ 53
Domains of Learning ................................................................................................ 54
Instrumental Domain ........................................................................................ 55
Communicative Domain ................................................................................... 56
Elements of Transformational Learning................................................................... 57
Critical Reflection ............................................................................................ 61
Rational Discourse ........................................................................................... 62
Action ............................................................................................................... 62
Conditions for Perspective Transformation ............................................................. 63
The SOLO Model ............................................................................................................. 66
Levels of Response................................................................................................... 69
The SOLO Model as a Tool for Eliciting Understanding of Variation .................... 70
Chapter 4 Research Methods ................................................................................................... 73
Participant Selection ........................................................................................................ 73
Background Information .......................................................................................... 74
Selection Criteria ...................................................................................................... 75
Selection Process ...................................................................................................... 78
Data Collection to Address Research Question One ........................................................ 81
The SOLO Model in Relation to Data Collection .................................................... 82
The Consultant Task................................................................................................. 84
Design Perspective ........................................................................................... 85
Data-Centric Perspective .................................................................................. 86
Modeling Perspective ....................................................................................... 89
The Caliper Task ...................................................................................................... 89
DesignPerspective ............................................................................................ 90
Data-Centric Perspective .................................................................................. 91
Modeling Perspective ....................................................................................... 91
The Handwriting Task .............................................................................................. 93
Data Analysis to Address Research Question One .......................................................... 95
Pre-Pilot and Pilot Study Analysis ........................................................................... 95
In-Depth Analysis of Data in Response to Research Question One ........................ 101
Data Collection to Address Research Question Two ....................................................... 108
Self-Report Methods ................................................................................................ 108
Recall Effects ................................................................................................... 109
Event History Calendars ................................................................................... 110
Critical Incidents .............................................................................................. 112
Teacher Interviews ................................................................................................... 115
Additional Data Sources ........................................................................................... 119
Data Analysis to Address Research Question Two .......................................................... 122
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Pre-Pilot and Pilot Study Analysis ........................................................................... 122
In-Depth Analysis of Data in Response to Research Question Two ........................ 124
Concluding Comments ..................................................................................................... 126
Chapter 5 Conceptions of Variation......................................................................................... 127
Conception: Expected but Explainable and Controllable (EEC) ..................................... 128
EEC and the Design Perspective .............................................................................. 129
Expected ........................................................................................................... 129
Explainable ....................................................................................................... 130
Controllable ...................................................................................................... 134
Summary of Design Perspective and EEC Conceptions .................................. 138
EEC and the Data-Centric Perspective ..................................................................... 138
EEC and the Modeling Perspective .......................................................................... 142
Summary .................................................................................................................. 145
Conception: Noise in Signal and Noise (NSN) ................................................................ 146
NSN and the Data-Centric Perspective .................................................................... 147
Noise With Measure of Center as Signal ......................................................... 147
Noise With Data Patterns/Distributions as Signal ............................................ 153
Noise With Relationship between Variables as Signal .................................... 155
Summary of Data-Centric Perspective and NSN Conceptions ........................ 157
NSN and the Design Perspective ............................................................................. 158
NSN and the Modeling Perspective ......................................................................... 160
Summary .................................................................................................................. 165
Conception: Expectation and Deviation from Expectation (EDE) .................................. 165
EDE and the Modeling Perspective.......................................................................... 167
Deviation of Statistics From Parameters .......................................................... 167
Deviation of Observed Data From Expected Patterns ...................................... 171
Deviation of Observed Data From Expected Relationships ............................. 173
Summary of Modeling Perspective and EDE Conceptions .............................. 178
EDE and the Design Perspective .............................................................................. 179
NSN and the Data-Centric Perspective .................................................................... 181
Summary .................................................................................................................. 182
Comparison of Conceptions ............................................................................................. 182
Summary of the Three Conceptions ......................................................................... 183
Design ...................................................................................................................... 186
Data. ......................................................................................................................... 188
Models ...................................................................................................................... 191
Context ..................................................................................................................... 194
Conceptions and Teachers in This Study ......................................................................... 195
Concluding Comments ..................................................................................................... 199
Chapter 6 Robust Understandings of Variation ....................................................................... 200
Framework Based on SOLO Model ................................................................................. 201
Development of Detailed Framework ...................................................................... 203
Elements and Reasoning Indicative of Robust Understandings of Variation .......... 207
Design Perspective ................................................................................................... 209
viii
Variational Disposition (DP1) .......................................................................... 210
Variability in Data for Contextual Variables (DP2) ......................................... 213
Variability and Relationships Among Data and Variables (DP3) .................... 214
Effects of Sample Size on Variability (DP4).................................................... 218
Relational Reasoning About Variation Within the Design Perspective ........... 219
Data-Centric Perspective .......................................................................................... 220
Variational Disposition (DCP1) ....................................................................... 223
Variability in Data for Contextual Variables (DCP2) ...................................... 225
Variability and Relationships Among Data and Variables (DCP3) ................. 227
Effects of Sample Size on Variability (DCP4) ................................................. 230
Relational Reasoning About Variation Within the Data-Centric
Perspective ................................................................................................ 231
Modeling Perspective ............................................................................................... 232
Variational Disposition (MP1) ......................................................................... 233
Variability in Data for Contextual Variables (MP2) ........................................ 236
Variability and Relationships Among Data and Variables (MP3) ................... 241
Effects of Sample Size on Variability (MP4) ................................................... 243
Relational Reasoning about Variation Within the Modeling Perspective ........ 244
Relational Reasoning in the Second SOLO Level ................................................... 245
Integrated Reasoning Across Multiple Perspectives for a Single Element ...... 246
Integrated Reasoning Across Multiple Perspectives for Multiple Elements .... 248
Robust Understandings of Variation ................................................................ 249
Conceptions and Robust Understanding .......................................................................... 251
Relationship Between Conceptions and Robust Understanding .............................. 251
Robust Understanding, Conceptions, and Teachers in This Study .......................... 252
Chapter 7 Influential Factors for Learning .............................................................................. 256
Triggers ............................................................................................................................ 256
Triggers of Epochal Dilemmas ................................................................................ 259
Triggers of Incremental Dilemmas .......................................................................... 260
Triggers From Workshop or Conference Activities ......................................... 260
Triggers From Dialogue With Colleagues and Statisticians ............................ 262
Triggers From Classroom Teaching ................................................................. 263
Triggers From Planning for AP Statistics Instruction ...................................... 264
Reactions to Triggers ............................................................................................... 266
Recursive Nature of Learning .................................................................................. 266
Personal Factors Related to Learning .............................................................................. 267
Interest in the Field of Statistics ............................................................................... 267
Motivation to Encounter and to Resolve Dilemmas ................................................ 269
Confidence in Ability to Learn ................................................................................. 271
Reflection on Content............................................................................................... 272
Commitment to Students and Teaching ................................................................... 276
Knowledge Base ....................................................................................................... 278
Prior Knowledge ............................................................................................... 278
Overarching View of or Framework for Statistics ........................................... 279
Summary of Personal Factors .................................................................................. 281
Environmental Influences Related to Learning ................................................................ 282
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Comfortable Learning Environment ........................................................................ 282
Rational Discourse ................................................................................................... 283
Collaboration and Interactions With Teachers ................................................. 283
Interactions With Statisticians and More Knowledgeable Others .................... 285
Consideration of Alternative Views ................................................................. 286
Community ............................................................................................................... 290
Resources ................................................................................................................. 292
Characteristics of Learning Experiences .......................................................................... 292
Role of Theory ......................................................................................................... 293
Role of Data and Activity ......................................................................................... 295
Key Examples and Problems Targeting Key Ideas .................................................. 300
Exploring New Roles ....................................................................................................... 301
Conducting Professional Development .................................................................... 302
Assuming Leadership Positions and Authoring Statistics Publications ................... 303
Summary of Teachers‘ Transformations .......................................................................... 304
Nature of Teachers‘ Transformations .............................................................................. 306
Chapter 8 Implications, Limitations, and Future Directions .................................................... 314
How do interactions with more knowledgeable others, particularly in terms of
how they support critical reflection and rational discourse, affect how
teachers (and students) construct robust understandings of statistical
concepts? .......................................................................................................... 315
What professional development factors or experiences trigger dilemmas and
facilitate resolution of dilemmas in ways that contribute to the
development of teachers‘ robust conceptions and understandings of
statistical concepts? .......................................................................................... 317
What strategies and tools are useful in investigating how teachers develop
statistical and pedagogical knowledge needed for teaching statistical
concepts? .......................................................................................................... 319
What characterizes prospective and inservice teacher education courses and
programs that facilitate teachers‘ development of robust statistical
conceptions and pedagogical strategies for successfully teaching statistics? ... 320
Concluding Thoughts ....................................................................................................... 322
References ................................................................................................................................ 324
Appendix A Initial Questionnaire ............................................................................................ 348
Appendix B Sample Event History Calendar .......................................................................... 350
Appendix C Event History Calendar ....................................................................................... 356
Appendix D Critical Incident Description ............................................................................... 366
Appendix E Abbreviated Context I Interview Schedule .......................................................... 367
Appendix F Abbreviated Context II Interview Schedule ......................................................... 370
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LIST OF FIGURES
Figure 2-1: Model for Addressing Teachers‘ Feelings and Beliefs. ........................................ 37
Figure 2-2: Adaptation of a Model of Professional Growth. ................................................... 38
Figure 3-1: SOLO and the Cycle of Levels for Each Perspective. .......................................... 71
Figure 3-2: The SOLO Model and Robust Understanding of Variation. ................................. 72
Figure 4-1: Elements and Indicators of Robust Understanding as Two Cycles of Levels
in the SOLO Model. ......................................................................................................... 83
Figure 4-2: The Consultant Task. ............................................................................................ 84
Figure 4-3: Summary Values and Dotplots for Sample Exam Scores ..................................... 87
Figure 4-4: Corrected Summary Values and Dotplots for Sample Exam Scores. ................... 87
Figure 4-5: Initial Graph for the Caliper Task. ........................................................................ 90
Figure 4-6: Initial Question for the Caliper Task. .................................................................... 90
Figure 4-7: Scatterplot Resulting From a Larger Data Set. ..................................................... 92
Figure 4-8: Regression Output for the Data Displayed in Figure 4-7 ...................................... 93
Figure 4-9: Excerpt to Critique Design. ................................................................................... 94
Figure 4-10: Excerpt to Create Design. ................................................................................... 94
Figure 4-11: Original Presentation of Size-15 Samples in the Consultant Task. ..................... 98
Figure 4-12(a): Example of the Matrix Layout. ...................................................................... 103
Figure 4-12(b): Example of the Matrix Entry. ......................................................................... 103
Figure 4-13: Sample Summary Statements of Everett‘s Data-Centric Reasoning. .................. 104
Figure 5-1: Haley‘s Sketch of a Nonrandom Residual Plot. .................................................... 140
Figure 5-2: Isaac‘s Recommendation for the Student. ............................................................. 143
Figure 5-3: Scatterplot With the Lines of Best Fit Suggested by Isaac and by the
Researcher. ....................................................................................................................... 145
Figure 5-4: Everett‘s Linear Pattern......................................................................................... 157