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Geometric Transformations in Middle School Mathematics Textbooks University of South Florida Scholar Commons Graduate Theses and Dissertations Graduate School 2011 Geometric Transformations in Middle School Mathematics Textbooks Barbara Zorin University of South Florida, [email protected] Follow this and additional works at: https://scholarcommons.usf.edu/etd Part of the American Studies Commons, and the Science and Mathematics Education Commons Scholar Commons Citation Zorin, Barbara, "Geometric Transformations in Middle School Mathematics Textbooks" (2011). Graduate Theses and Dissertations. https://scholarcommons.usf.edu/etd/3421 This Dissertation is brought to you for free and open access by the Graduate School at Scholar Commons. It has been accepted for inclusion in Graduate Theses and Dissertations by an authorized administrator of Scholar Commons. For more information, please contact [email protected]. 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 ____________________________________
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