Table Of ContentUniversity of South Florida
Scholar Commons
Graduate Teses and Dissertations Graduate School
2011
Geometric Transformations in Middle School
Mathematics Textbooks
Barbara Zorin
University of South Florida,
Geometric Transformations in Middle School Mathematics Textbooks
by
Barbara Zorin
A dissertation submitted in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
Department of Secondary Education
College of Education
University of South Florida
Major Professor: Denisse R. Thompson, Ph.D.
Richard Austin, Ph.D.
Catherine A. Beneteau, Ph.D.
Helen Gerretson, Ph.D.
Gladis Kersaint, Ph.D.
Date of Approval:
March 10, 2011
Keywords: Geometry, Transformations, Middle School, Mathematics Textbooks, Content
Analysis
Copyright © 2011, Barbara Zorin
1
Dedication
This dissertation is dedicated to my father, Arthur Robert, who provided me with
love and support, and who taught me, from a very early age, that I could accomplish
anything that I put my mind to. To my son, Rick, for always believing that I was the
smartest person he had ever met and showing me that he knew that I would complete this
endeavor. To my departed husband, Glenn, for all of the love, faith, and understanding
that he brought into my life. And to my partner, Michael, who has provided me with love,
respect, understanding, and the support that I needed to finish what I started.
Acknowledgments
I want to thank the following people for the assistance and encouragement they
gave me during my program of study and throughout the writing of this dissertation. Dr.
Denisse R. Thompson, my major professor, for the support and consistent guidance, as
well as the considerable time spent reviewing and critiquing my writing. Dr. Gladis
Kersaint, Dr. Helen Gerretson, Dr. Rick Austin, and Dr. Catherine Beneteau, my
dissertation committee members, for their encouragement and numerous reviews of my
manuscript; and, to Michael for his understanding and willingness to allow me to
dedicate myself to the completion of this project.
I also want to extend a thank you to the doctoral students of USF whom I have
met during my course work and the writing of this dissertation. Especially, to Suzie
Pickle and Matt Kellogg who have provided enduring support; and to Gabriel Cal and
Sarah Bleiler who generously shared their valuable time in the coding procedures for this
study.
I offer a very sincere thank you…
Table of Contents
List of Tables _______________________________________________________ viii
List of Figures _________________________________________________________ x
Abstract _____________________________________________________________ xii
Chapter 1: Introduction and Rationale for the Study ___________________________ 1
Opportunity to Learn and Levels of Cognitive Demand __________________ 6
Statement of the Problem _________________________________________ 10
The Purpose of the Study _________________________________________ 11
Research Questions ______________________________________________ 12
Significance of the Study _________________________________________ 13
Conceptual Issues and Definitions __________________________________ 15
Chapter 2: Literature Review ____________________________________________ 21
Literature Selection ________________________________________ 21
The Curriculum and the Textbook __________________________________ 22
Types of Curriculum _______________________________________ 22
The Mathematics Textbook and the Curriculum _________________ 23
The Textbook and its Use in the Classroom _____________________ 25
Curriculum Analysis _______________________________________ 28
Related Textbook Content Analysis _________________________________ 29
Types of Textbook Content Analysis __________________________ 30
Curriculum Content Analysis for Textbook Selection _____________ 33
i
Curriculum Content Analysis for Comparison to
International Tests _______________________________________ 35
Content Analysis on Textbook Presentations and Student
Expectations __________________________________________________ 38
Analyses of Levels of Cognitive Demand Required in Student
Exercises ____________________________________________________ 43
Research on Transformation Tasks and Common Student Errors __________ 43
Transformations __________________________________________ 44
Issues Students Experience with Transformation
Concepts _________________________________________ 46
Translations __________________________________ 46
Reflections __________________________________ 47
Rotations ____________________________________ 48
Dilations ____________________________________ 50
Composite Transformations _____________________ 51
Conceptual Framework for Content Analysis of Geometric
Transformations _______________________________________________ 52
Summary of Literature Review _____________________________________ 53
Chapter 3: Research Design and Methodology 54
Research Questions 54
Sample 55
Development of the Coding Instrument for Analysis of
Transformations 59
Global Content Analysis Conceptual Framework ______________________ 61
Sample Application of the Coding Instrument 69
ii
Reliability Measures 71
Summary of Research Design and Methodology 75
Chapter 4: Findings ____________________________________________________ 77
Research Questions _____________________________________________ 77
Analysis Procedures _____________________________________________ 78
Organization of the Chapter _______________________________________ 79
Physical Characteristics of Transformation Lessons in Each
Series _________________________________________________ 80
Location of Pages Related to Transformations __________________ 80
Relative Position of Transformation Lessons ____________________ 83
Lesson Pages Related to Each Type of Transformation ____________ 86
Characteristics and Structure of Transformation Lessons ________________ 88
Components of Transformation Lessons _______________________ 88
Characteristics of Transformation Constructs in Each
Textbook Series _________________________________________ 91
Prentice Hall Textbook Series _________________________ 91
Symmetry, Line of Symmetry, and Reflection _______ 91
Translations __________________________________ 92
Rotations ____________________________________ 93
Dilations ____________________________________ 94
Glencoe Textbook Series _____________________________ 94
Symmetry ___________________________________ 95
Reflection and Translations _____________________ 95
Rotations ____________________________________ 96
iii
Dilations ____________________________________ 97
Connected Mathematics 2 Textbook Series _______________ 97
Symmetry and Line of Symmetry _________________ 98
Reflections, Translations, and Rotations ___________ 98
Dilations ____________________________________ 99
UCSMP Textbook Series _____________________________ 99
Symmetry and Reflections ______________________ 99
Translations _________________________________ 100
Rotations ___________________________________ 101
Dilations ___________________________________ 101
Summary of Textbook Series _________________________ 102
Number of Transformation Tasks __________________________________ 102
Number of Tasks in Each Series _____________________________ 103
Number of Each Type of Transformation Task in
Student Exercises _______________________________________ 105
Characteristics of the Transformation Tasks in the Student
Exercises _____________________________________________ 110
Translations _______________________________________ 111
Reflections _______________________________________ 113
Rotations _________________________________________ 118
Dilations _________________________________________ 120
Composite Transformations __________________________ 122
Student Exercises Analyzed by the Characteristics of
Performance Expectations ___________________________ 124
iv
Suggestions for Instructional Aids and Real-World
Connections ___________________________________________ 126
Student Exercises Summarized by Textbook Series ______________ 129
Prentice Hall ______________________________________ 129
Glencoe __________________________________________ 129
Connected Mathematics 2 ____________________________ 130
UCSMP __________________________________________ 130
Level of Cognitive Demand Expected by Students in the
Transformation Exercises ______________________________________ 130
Summary of Findings ___________________________________________ 134
“Where” the Content of Transformation Lessons are
Located in the Textbooks _________________________________ 135
“What is Included in the Transformation Lessons of each
Textbook Series ________________________________________ 137
“How” Transformation Exercises are Presented in the
Lessons _______________________________________________ 138
Level of Cognitive Demand Required by Student Exercises _______ 141
Chapter 5: Summary and Conclusions ____________________________________ 145
Overview of the Study __________________________________________ 145
Research Questions _____________________________________________ 146
Purpose of the Study ____________________________________________ 147
Summary of Results ____________________________________________ 148
Opportunity to Learn Transformation Concepts in the
Prentice Hall Textbook Series _____________________________ 149
Opportunity to Learn Transformation Concepts in the
Glencoe Textbook Series _________________________________ 151
Opportunity to Learn Transformation Concepts in the
v
Connected Mathematics 2 Textbook Series ___________________ 153
Opportunity to learn transformation concepts in the
University of Chicago School Mathematics Project
Textbook series ________________________________________ 156
Discussion ____________________________________________________ 158
Limitations of the Study _________________________________________ 163
Significance of the Study ________________________________________ 165
Implications for Future Research __________________________________ 168
References __________________________________________________________ 174
Appendices _________________________________________________________ 212
Appendix A: Pilot Study ________________________________________ 213
Appendix B: Composite Transformation Sample Conversions and
Properties List ___________________________________ 233
Appendix C: Properties of Geometric Transformations Expected
to be Present in Lessons ___________________________ 235
Appendix D: Aspects of Transformations and Student Issues __________ 236
Appendix E: Examples of Student Performance Expectations in
Exercises ______________________________________ 238
Appendix F: Coding Instrument __________________________________ 243
Appendix G: Instrument Codes for Recording Characteristics of
Student Exercises ________________________________ 248
Appendix H: Transformation Type Sub-grouped Categories and
Exercises _______________________________________ 250
Appendix I: Examples of Tasks Characterized by Levels of
Cognitive Demand in Exercises _____________________ 256
Appendix J: Background for Content Analysis and Related
Research Studies _________________________________ 259
vi
