University of Vermont ScholarWorks @ UVM Graduate College Dissertations and Theses Dissertations and Theses 2015 Students' Understanding Of Quadratic Functions: Learning From Students' Voices Jennifer Suzanne Stokes Parent University of Vermont Follow this and additional works at: https://scholarworks.uvm.edu/graddis Part of the Educational Administration and Supervision Commons, and the Science and Mathematics Education Commons Recommended Citation Parent, Jennifer Suzanne Stokes, "Students' Understanding Of Quadratic Functions: Learning From Students' Voices" (2015). Graduate College Dissertations and Theses. 376. https://scholarworks.uvm.edu/graddis/376 This Dissertation is brought to you for free and open access by the Dissertations and Theses at ScholarWorks @ UVM. It has been accepted for inclusion in Graduate College Dissertations and Theses by an authorized administrator of ScholarWorks @ UVM. For more information, please contact [email protected]. STUDENTS’ UNDERSTANDING OF QUADRATIC FUNCTIONS: LEARNING FROM STUDENTS’ VOICES A Dissertation Presented by Jennifer Suzanne Stokes Parent to The Faculty of the Graduate College of The University of Vermont In Partial Fulfillment of the Requirements for the Degree of Doctor of Education Specializing in Educational Leadership and Policy Studies January, 2015 Defense Date: November 5, 2014 Dissertation Examination Committee Regina Toolin, Ph.D., Advisor Susan Comerford, Ph.D., Chairperson Sean M. Hurley, Ph.D. Alan S. Tinkler, Ph.D. Cynthia J. Forehand, Ph.D., Dean of the Graduate College Abstract The objective of this multiple case study was to examine how three pairs of high school students from a northern Vermont high school approached quadratic functions through traditional and multiple representation tasks. Four research questions were examined: 1) How do students think about the quadratic function as they work on a series of tasks? 2) What mathematical strategies do students employ when they work on a series of tasks related to the quadratic function? 3) How does the type of task, traditional versus multiple representation, impact students’ understanding of the quadratic function? 4) What kinds of knowledge (procedural or conceptual) do students utilize when completing a series of tasks about the quadratic function? Qualitative research methods that utilized think-aloud protocols while students were engaged in four tasks pertaining to the quadratic function were employed in this study. Results suggested that students tend to think about isolated parts of the problem when solving quadratic problems. Early on in their learning about quadratics, students primarily relied on procedural strategies such as think-alouds, gestures, algebraic formulas, converting equation forms, process of elimination, dissecting problems, backtracking, and drawing pictures. In addition, students preferred the standard form to the vertex form when solving quadratics and often confused the y-intercept of the standard form with the y-coordinate of the vertex when the function was in vertex form. Results also indicated that students preferred to algebraically solve a problem versus tabular or graphical strategies. By exploring how students approach the quadratic function through their own voices, this study offers some insight into the conceptions and strategies that students use for solving problems that involve the quadratic function as well as possibilities for how quadratics may be taught in high school. Acknowledgements I would like to thank my advisor and committee, Dr. Regina Toolin, Dr. Sean Hurley, Dr. Alan Tinkler, and Dr. Susan Comerford, for their time and patience and for guiding me with their dense research knowledge. To Dr. Joseph Reiken, who was my much needed sounding board in the beginning, and again at the end; I have always felt blessed to have you as a friend. To Dr. Colby Kervick, who was my much needed venting and sounding board throughout it all, and to the rest of my cohort; I believe that through this program we have forged a bond that is pretty undeniable. To Salli Griggs, thank you for your supportive demeanor. You are a breath of fresh air! To my friend Jamie Bristol, who painstakingly edited this study, even though she would jokingly tell me that she had no idea mathematically what she was reading. And to many other friends who supported me throughout the process, I thank you for your many questions, concerns, and belief in me that I could do this! To my Mom, Patricia Stokes, thank you for your undying support. Mom, you have always been there to guide me, believe in me, and push me. Thank you for instilling in me the thirst for knowledge and to not be afraid to try new things. I love you and appreciate you more than words can say. To my siblings, Robert, Betsy, Mary Jo, Michael and Melissa, thank you for supporting me throughout this endeavor. ii To my Dad, Earle Bigham Stokes. I miss you Daddy, but I know that you were rooting me on from the other side and that you are proud of me. To my nephew, Captain Joseph William Schultz, U.S. Army Special Forces Green Beret. Halfway through this process we lost you in Afghanistan. Not a day goes by that I do not miss you. Among many many things, I miss how in one breath you could be teasing me about “becoming a doctor,” and in the next you seriously and compassionately asking me what my latest class was about and what it was I was learning. The world and my heart will never be the same. To my in-laws, thank you for supporting my decision to enter into the doctoral program and for letting the kids play at the farm at times so that I could work. I appreciate you and your kindness. To my beautiful children, Adaline Patricia, Michael Beauregard (Beau), and Theodore Joseph Emile. Only two of you started this journey with Mommy, yet three of you finished it with me. I hope that you will learn that hard work really does pay off, and that to have an ambition be achieved is worthy of that hard work. Mommy loves you to the moon and back. And lastly I would like to thank my husband Alan for not making too much of a fuss when I had to stay up late working and for supporting me while I took classes and did my study and research! Thank you for understanding that I have to feed that part of my brain that getting this doctorate quenched, temporarily of course. I love you very much! I am very blessed. iii Table of Contents Acknowledgements ............................................................................................................. ii List of Tables ..................................................................................................................... vi List of Figures ................................................................................................................... vii Chapter One: Introduction ................................................................................................. 1 I. Vermont’s Need for Improvement in Mathematics ..................................................... 1 II. Purpose ....................................................................................................................... 6 III. The Breakdown of the Quadratic Function ............................................................... 9 IV. Potential Significance of the Study ......................................................................... 11 V. Research Statement/Question and Sub Questions .................................................... 12 Chapter Two: Literature Review ..................................................................................... 14 I. From Learning to Understanding: The Transfer of Knowledge ................................ 16 II. Multiple Representations .......................................................................................... 20 III. The Constructivist Approach .................................................................................. 22 IV. Misconceptions about Functions ............................................................................. 24 V. Think-Aloud Method ................................................................................................26 Chapter Three: Research Methodolgy ............................................................................. 29 I. Research Design ......................................................................................................... 29 II. Participants ............................................................................................................... 31 III. Role of Researcher .................................................................................................. 33 IV. Tasks ....................................................................................................................... 33 Traditional Task Versus Multiple Representations Task – Days 1 and 2. ................ 35 Mixed Methods – Days 3 and 4. ................................................................................ 36 V. Information Collection ............................................................................................. 36 VI. Procedures ............................................................................................................... 37 VII. Information Analysis ...............................................................................................38 VIII. Trustworthiness and credibility ............................................................................ 40 Chapter Four: Results ........................................................................................................41