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Understanding Quadratic Functions and Solving Quadratic Equations

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Understanding Quadratic Functions and Solving Quadratic Equations Understanding Quadratic Functions and Solving Quadratic Equations: An Analysis of Student Thinking and Reasoning Leslie Ellen Johnson Nielsen A dissertation to be submitted in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY UNIVERSITY OF WASHINGTON 2015 Reading Committee: Elham Kazemi, Chair Kara Jackson Ron Irving Program Authorized to Offer Degree: College of Education © Copyright 2015 Leslie Ellen Johnson Nielsen University of Washington Abstract Understanding Quadratic Functions and Solving Quadratic Equations: An Analysis of Student Thinking and Reasoning Leslie Ellen Johnson Nielsen Chair of the Supervisory Committee: Professor Elham Kazemi College of Education Understanding quadratic functions is critical to student success in high school mathematics and beyond, yet very little is known about what students understand about these functions. There is agreement in the field that quadratics are one of the most conceptually challenging subjects in the secondary mathematics curriculum. However, research on student learning in this area has focused on procedural aspects of solving equations, with very little known about student understanding of the behavior of quadratic functions. This study sought to learn what high school students who have completed an Algebra 2 or Precalculus class understand about quadratics. Specifically, what connections, if any, do they make between representations of quadratic functions? How do students approach solving quadratic equations, and how do they interpret the solutions? Lastly, what cognitive affordances support them in their learning and understanding of quadratic functions, and what cognitive obstacles do they encounter? This qualitative study employed cognitive interviews of 27 students in grades nine through eleven. The data included video and audio recordings as well as student work, captured on a smart pen pencast. The data was analyzed in four phases: (1) focusing on one student at a time, (2) focusing on individual problems, (3) focusing across students, and then (4) revisiting individual problems across students using a conceptual framework grounded in big ideas and essential understandings of quadratics and a children’s mathematical learning perspective. I found that students have a strong sense of the symmetry of the parent function, but are not consistently able to explain the cause of that symmetry. As students solved equations and graphed functions, they transitioned between equations set equal to constant values, expressions, and equations defining functions. At times this was a productive strategy, but for some students it reflected confusion about what they were solving. Lastly, I found that students apply their understandings from work with linear functions to solving and graphing quadratic equations. This study provides an initial framework for how students think about quadratic functions which may enable mathematics educators to better interpret how students’ prior learning influences their understanding of big ideas within the study of quadratic functions. TABLE OF CONTENTS List of Tables ............................................................................................................................ iii List of Figures ........................................................................................................................... iii Chapter 1: Introduction .......................................................................................................... 1 Chapter 2: Framing Ideas and Informing Literature .................................................... 6 Big Ideas and Essential Understandings of Quadratic Functions ...................................... 6 Conceptualizing Student Learning ............................................................................................ 20 Students’ Understandings Of Quadratics: A Review Of The Literature ........................ 28 Chapter 3: Research Methods and Study Design ........................................................ 37 Setting .................................................................................................................................................. 37 Participants ....................................................................................................................................... 37 Data Collection .................................................................................................................................. 39 Cognitive Interviews ....................................................................................................................... 39 Data Analysis ..................................................................................................................................... 43 CHAPTER 4: FINDINGS ......................................................................................................... 52 Student Performance Across Solving Problems ................................................................... 53 Demographic Differences ............................................................................................................. 55 The Problems .................................................................................................................................... 56 How Students Think About Quadratics, The Parent Function And Symmetry Of Quadratics .......................................................................................................................................... 66 The Parent Function and Symmetry of Quadratics ............................................................. 67 Affordances of Considering the Parent Function for Solving Quadratic Equations . 71 Obstacles Related to Symmetry .................................................................................................. 73 i “Solving What?” What students think they are solving or solving for when they work with quadratic equations and functions ...................................................................... 77 The Impact Of Students’ Linear Thinking On Solving Quadratic Equations And Graphing Quadratic Functions .................................................................................................... 84 Appling Linear Techniques When Solving Quadratic Equations .................................... 85 Linear Thinking When Graphing Quadratic Functions ...................................................... 91 CHAPTER 5: DISCUSSION ..................................................................................................... 97 Linear Thinking when Solving .................................................................................................... 98 Solving What? .................................................................................................................................. 101 Quality of Explanations ............................................................................................................... 105 Framework of Student Learning in Quadratics .................................................................. 108 Contributions .................................................................................................................................. 115 Limitations ....................................................................................................................................... 116 Directions for Future Research ................................................................................................ 117 Implications for Instruction ...................................................................................................... 118 Conclusion ........................................................................................................................................ 127 References .............................................................................................................................. 128 Appendices ............................................................................................................................. 135 APPENDIX A: INTERVIEW PROTOCOL .......................................................................... 136 APPENDIX B: STUDY DESIGN TABLE ............................................................................. 142 APPENDIX C: ANTICIPATED CORRECT SOLUTIONS METHODS ........................... 148 APPENDIX D: STUDENT RESPONSE FORM ................................................................... 161 APPENDIX E: CODES ............................................................................................................ 164 ii LIST OF TABLES Table 1: Participant Grade Level and Teacher by Course 38 Table 2: Scoring Values for Students’ Initial Responses to Problems 48 Table 3: Student Response Code Point Values 53 Table 4: Student performance on solving equations on a 4-point scale 54 Table 5: Foundational Understandings 109 Table 6: Framework of Student Learning in Quadratics 114 LIST OF FIGURES Figure 1. Table and Graph for the Parent Function 9 Figure 2. Solving Quadratic Functions 13 Figure 3: Completing the Square 15 Figure 4: Conceptual Understanding of Quadratics 19 Figure 5: Relationships between expressions and equations 20 Figure 6: Graph of a parabola without a visible y-intercept 31 Figure 7: Pencast of Student Work in Livescribe Player 43 Figure 8: Tabitha's Box Method for Factoring 58 Figure 9: Tracy's Work 62 Figure 10: Tabitha's Sketch of the Parent Function
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