Georgia State University ScholarWorks @ Georgia State University Mathematics Dissertations Department of Mathematics and Statistics Summer 8-12-2014 The Teaching and Learning of Parametric Functions: A Baseline Study Harrison Stalvey Georgia State University Follow this and additional works at: https://scholarworks.gsu.edu/math_diss Recommended Citation Stalvey, Harrison, "The Teaching and Learning of Parametric Functions: A Baseline Study." Dissertation, Georgia State University, 2014. https://scholarworks.gsu.edu/math_diss/18 This Dissertation is brought to you for free and open access by the Department of Mathematics and Statistics at ScholarWorks @ Georgia State University. It has been accepted for inclusion in Mathematics Dissertations by an authorized administrator of ScholarWorks @ Georgia State University. For more information, please contact [email protected]. THE TEACHING AND LEARNING OF PARAMETRIC FUNCTIONS: A BASELINE STUDY by HARRISON E. STALVEY Under the Direction of Draga Vidakovic ABSTRACT This dissertation reports on an investigation of fifteen second-semester calculus students' understanding of the concept of parametric function, as a special relation from a subset of R to a subset of R2. A substantial amount of research has revealed that the concept of function, in general, is very difficult for students to understand. Furthermore, several studies have investigated students' understanding of various types of functions. However, very little is known about how students reason about parametric functions. This study aims to fill this gap in the literature. Employing Action{Process{Object{Schema (APOS) theory as the guiding theoretical perspective, this study proposes a preliminary genetic decomposition for how a student might construct the concept of parametric function. To determine whether the students in this study made the constructions called for by the preliminary genetic de- composition or other constructions not considered in the preliminary genetic decomposition, data is analyzed regarding students' reasoning about parametric functions. In particular, this study explores (1) students' personal definitions of parametric function; (2) students' reasoning about parametric functions given in the form p(t) = (f(t); g(t)); (3) students' reasoning about parametric functions on a variety of tasks, such as converting from para- metric to standard form, sketching a plane curve defined parametrically, and constructing a parametric function to describe a real-world situation; and (4) students' reasoning about the invariant relationship between two quantities varying simultaneously when described in both a graph and a real-world problem. Then the genetic decomposition is revised based on the results of the data analysis. This study concludes with implications for teaching the concept of parametric function and suggestions for further research on this topic. INDEX WORDS: Parametric function, APOS, Student understanding, Student reasoning, Sketching parametric curve, Conversion from parametric to standard, Invariance of ratio THE TEACHING AND LEARNING OF PARAMETRIC FUNCTIONS: A BASELINE STUDY by HARRISON E. STALVEY A Dissertation Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in the College of Arts and Sciences Georgia State University 2014 Copyright by Harrison E. Stalvey 2014 THE TEACHING AND LEARNING OF PARAMETRIC FUNCTIONS: A BASELINE STUDY by HARRISON E. STALVEY Committee Chair: Draga Vidakovic Committee: Mariana Montiel Valerie Miller Yongwei Yao Electronic Version Approved: Office of Graduate Studies College of Arts and Sciences Georgia State University August 2014 iv In memory of Mrs. Jean Tarpley v ACKNOWLEDGEMENTS I would like to thank my incredible advisor, Dr. Draga Vidakovic, for her guidance and phenomenal patience throughout this process. You have believed in me from the beginning, not only as my graduate advisor, but also as my undergraduate advisor. Thank you for your unwavering support and encouragement. I would also like to thank my committee members, Dr. Mariana Montiel, Dr. Valerie Miller, and Dr. Yongwei Yao, for their valuable comments on how to improve this dissertation. Thank you so much for your enthusiasm toward this project. Additionally, many thanks go to Dr. Annie Burns for her assistance in the data collection for this study. My gratitude also goes out to all of the faculty and staff at Georgia State University who have supported me over the past eleven years. I am grateful for all that you have taught me and for all of the opportunities you have given me. I would particularly like to thank Peggy Ogden and Keisha Jones for equipping me with the skills necessary to effectively communicate mathematical knowledge to a diverse community of students. I would also like to thank Dr. Alexandra Smirnova for giving me the opportunity to expand my knowledge through teaching challenging courses. Thank you to my friends for always being there, even when I couldn't be. I look forward to catching up on fun times together. See you at the lake! Last, but not least, I would like to thank my family, Richard, Sharon, Cam, Brandi, and Morgan, for their unconditional love and support. This would not have been possible without you. Thank you for encouraging me, celebrating my successes, and raising my spirits when times got tough. vi TABLE OF CONTENTS ACKNOWLEDGEMENTS ::::::::::::::::: v LIST OF TABLES :::::::::::::::::::: ix LIST OF FIGURES :::::::::::::::::::: x Chapter 1 INTRODUCTION ::::::::::::::: 1 1.1 Statement of the problem ........................ 2 1.2 Research questions ............................. 5 1.3 Theoretical perspective .......................... 6 1.4 Outline of the study ............................ 9 Chapter 2 LITERATURE REVIEW :::::::::::: 11 2.1 The general concept of function .................... 11 2.2 Covariational reasoning .......................... 17 2.3 Horizontal and vertical growth of the function concept . 23 2.4 Parametric function ............................ 27 2.5 Overview ................................... 29 Chapter 3 METHODOLOGY ::::::::::::::: 31 3.1 Role of APOS theory ........................... 31 3.2 Conceptual analysis ............................ 32 3.2.1 Survey of textbooks . 32 3.2.2 Historical considerations . 35 3.2.3 A function approach to curves defined parametrically . 37 3.3 Preliminary genetic decomposition ................... 38 3.4 Research setting .............................. 40 vii 3.4.1 Course description . 40 3.4.2 Description of instruction . 41 3.5 Subjects ................................... 41 3.6 Data collection ............................... 42 3.7 Data analysis ................................ 43 Chapter 4 DATA ANALYSIS AND RESULTS :::::::: 44 4.1 Students' personal definitions of parametric function . 45 4.2 Students' reasoning about real-valued functions and parametric functions given in the form y = f(x) and p(t) = (f(t); g(t)) . 51 4.2.1 Students' reasoning about real-valued functions given in the form y = f (x)................................ 51 4.2.1.1 Interview question 1a . 52 4.2.1.2 Interview question 1b . 56 4.2.2 Students' reasoning about parametric functions given in the form p(t) = (x(t),y(t))........................... 59 4.2.2.1 Misconceptions about parametric functions . 62 4.2.3 Relation between students' reasoning about real-valued functions and parametric functions . 73 4.3 Students' performance on tasks involving parametric functions . 75 4.3.1 Converting from parametric to standard form . 76 4.3.2 Sketching the graph of a curve given parametrically . 81 4.3.3 Constructing a parametric function to describe a real-world situation 86 4.4 Students' reasoning about the invariant relationship between two variables ................................... 98 4.4.1 Students' reasoning about the invariant relationship between two vari- ables described by a graph . 98 4.4.2 Students' reasoning about the invariant relationship between two vari- ables in a real-world problem . 102 viii 4.4.2.1 Interview questions 2a{b . 105 4.4.2.2 Interview question 2c{d . 115 4.5 Overview ................................... 125 Chapter 5 CONCLUSION :::::::::::::::: 126 5.1 Discussion of results ............................ 127 5.1.1 Research question 1 . 127 5.1.2 Research question 2 . 129 5.1.3 Research question 3 . 134 5.1.4 Research question 4 . 138 5.1.5 Brief summary . 140 5.2 Revised genetic decomposition . 141 5.3 Implications for teaching . 144 5.4 Limitations of the study . 145 5.5 Future research ............................... 145 REFERENCES ::::::::::::::::::::: 147 APPENDICES :::::::::::::::::::::: 158 Appendix A INSTRUCTIONAL TASK :::::::::::: 158 Appendix B FINAL EXAM QUESTIONS ::::::::::: 159 Appendix C INTERVIEW QUESTIONS ::::::::::: 160 Appendix D INTERVIEW PROTOCOL :::::::::::: 162 Appendix E SUGGESTED ACTIVITIES ::::::::::: 166 ix LIST OF TABLES Table 2.1 Mental actions of the covariation framework (Carlson et al., 2002). 20 Table 4.1 Students' personal parametric function definition. 50 Table 4.2 Number of student difficulties per issue on each part of interview ques- tion 2. 124 x LIST OF FIGURES Figure 4.1 Alex's work for interview question 1c. 71 Figure 4.2 Griffin’s solution to final exam question 1a. 77 Figure 4.3 Sam's solution to final exam question 1a. 78 Figure 4.4 Whitney's solution to final exam question 1a. 79 Figure 4.5 Oliver's solution to final exam question 1a. 79 Figure 4.6 Alex's solution to final exam question 1a. 80 Figure 4.7 Hannah's solution to final exam question 1a.