INVESTIGATING HIGH SCHOOL STUDENTS' UNDERSTANDINGS of ANGLE MEASURE by HAMILTON L. HARDISON (Under the Direction of Leslie P
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INVESTIGATING HIGH SCHOOL STUDENTS’ UNDERSTANDINGS OF ANGLE MEASURE by HAMILTON L. HARDISON (Under the Direction of Leslie P. Steffe) ABSTRACT The purpose of this study was to understand how ninth-grade students quantify angularity. Angle and angle measure are critical topics in mathematics; however, students’ difficulties with these topics are well documented in research literature. In contrast, little is known about how students develop propitious quantifications of angularity, and researchers have critiqued previous curricular approaches. Although some scholars have identified quantifications of angularity beneficial for the study trigonometry, these quantifications leverage multiplicative comparisons of circular quantities (e.g. arc length and circumference) and, as such, are an unlikely starting point for students’ initial quantifications. Previous studies examining how students reason with angles have obfuscated students’ quantifications by failing to emphasize the attribute to be measured or providing measurement tools like protractors. This dissertation research is a response to calls for studies investigating students’ quantifications of angularity, particularly at the high school level. As teacher-researcher, I taught a pair of ninth-grade students at a rural high school in the southeastern U.S. in a constructivist teaching experiment from October 2015 to April 2016. Teaching practices included creating and posing problems involving angle models, interacting with students to understand their ways of reasoning, and reflecting on students’ mathematical activities. From video records of these sessions, I constructed second-order models accounting for students’ initial quantifications of angularity and modifications to these quantifications occurring throughout the teaching experiment. From the students’ activities, I abstracted three motions involved in the construction of angularity and five operations used to alter the side lengths of angle models while preserving angularity. During the teaching experiment, both students constructed quantifications of angularity involving extensive quantitative operations (e.g., segmenting, iterating); these extensive quantifications have not been discussed in previous empirical literature. Students’ constructions of right-angle and full-angle templates were a significant resource for students’ understandings of the degree as a standard unit of angular measure. Students developed conceptions of degrees that involved positing familiar angular templates as composite units (e.g., a right angle as a 90-unit composite or a full angle as a 360-unit composite), and their quantifications differed across rotational and non-rotational angle contexts. INDEX WORDS: Angle, Angle Measure, Angularity, Degree Measure, Quantification, Quantity, Quantitative Reasoning, Radical Constructivism, Scheme Theory, Teaching Experiment INVESTIGATING HIGH SCHOOL STUDENTS’ UNDERSTANDINGS OF ANGLE MEASURE by HAMILTON L. HARDISON B.S.Ed., University of Georgia, 2006 M.Ed., University of Georgia, 2008 A Dissertation Submitted to the Graduate Faculty of The University of Georgia in Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY ATHENS, GEORGIA 2018 © 2018 Hamilton L. Hardison All Rights Reserved INVESTIGATING HIGH SCHOOL STUDENTS’ UNDERSTANDINGS OF ANGLE MEASURE by HAMILTON L. HARDISON Major Professor: Leslie P. Steffe Committee: Kevin C. Moore James W. Wilson Electronic Version Approved: Suzanne Barbour Dean of the Graduate School The University of Georgia May 2018 ACKNOWLEDGEMENTS I could not have completed this degree without the help of others. In this section, I acknowledge the numerous individuals who supported me in writing this dissertation or in my pursuit of higher education more broadly. An investigation of students’ mathematical thinking necessarily involves one or more students. As such, I first express my gratitude to the four students who volunteered to work with me on mathematics tasks on many afternoons during their ninth-grade year. I thank Camille, Kacie, Bertin, and Mac for sharing their time with me and describing their thinking; obviously, this dissertation would not have been possible without these students. Additionally, I thank Daniel and Hillary, who were participants in a previous study that informed my dissertation research. I have tried to convey some of what I have learned from these six students in this manuscript; I learned much more than I was able to write here. In the future, I will share more of what I have learned from these participants. The students mentioned above were able to participate in this research because of the efforts of many other individuals. I thank the students’ parents for allowing their children to participate in my study. Additionally, I thank all the school personnel—office staff, mathematics teachers, and administrators—who supported this study in so many ways. This dissertation marks the end of my studies at the University of Georgia, where I also obtained my Bachelor of Science in Education and Master of Science in Education degrees. Now as a “triple dawg,” I thank the UGA community at large for my lengthy iv and rewarding collegiate educational experiences. In the Department of Mathematics, I extend special thanks to Ted Shifrin, Mo Hendon, Clint McCrory, and Neil Lyall who demonstrated passion for mathematics and appreciation for their students’ mathematical thinking. The same is true of Sybilla Beckmann, who also helped to pique my interest in working with elementary and middle grades teachers by inviting me to participate in her content courses. In the Department of Statistics, I thank Chris Franklin for preparing me to teach high school students the art and science of learning from data. When I elected to attend the University of Georgia for my undergraduate studies, I was entirely unaware the Department of Mathematics and Science Education housed nationally (and internationally) renowned programs for mathematics teacher education and research in mathematics education. I thank the mathematics education faculty who comprised the department during my years of study, and I consider myself extraordinarily fortunate to have studied under a multitude of headliners in the field. In the following paragraph, I thank some individual faculty members for a few salient memories from my years as a student of the department. I thank John Olive and Nicholas Oppong for designing impactful undergraduate courses and for introducing me to the pedagogical power of mathematics software. From my master’s program, I will never forget sketching the moon in the night sky as homework for Andrew Izsák’s course on conic sections, nor will I forget his sense of humor. I learned much about the history of mathematics from Larry Hatfield, whose enthusiasm for the subject was unparalleled. I thank Denise Spangler for supporting me to be more knowledgeable about mathematics teacher education and for her invaluable advice on navigating the job market. What I have learned about supervising prospective v teachers during field experiences is largely due to the efforts of Ryan Smith and AnnaMarie Conner. I thank Dorothy White for introducing me to research on critical issues in mathematics education, and I thank Jeremy Kilpatrick for teaching me to read mathematics education research critically. Much gratitude is due to Jessica Bishop, who not only introduced me to working with practicing teachers in research and professional development contexts but also encouraged me to teach prospective elementary teachers. Finally, I appreciate Pat Wilson for supporting me as I prepared the prospectus for this dissertation study. I am particularly grateful to the three faculty members who served on my dissertation committee: Jim Wilson, Kevin Moore, and Les Steffe. I thank each of you in the paragraphs below. To Jim Wilson: As a master’s student, I learned much mathematics from your courses, particularly the course on problem solving. Thank you for serving as my advisor during my initial years as a PhD student and for your sound advice on navigating the doctoral program. Regarding the dissertation, I am particularly grateful for your comments and painstaking edits, which immensely improved this manuscript. Additionally, I thank you and Corene for regularly opening your home to graduate students over the years; your hospitality is something I hope to emulate with my future students. To Kevin Moore: I would likely still be searching for a dissertation topic were it not for taking your courses on quantitative reasoning and preparing prospective secondary mathematics teachers to teach precalculus topics. Thank you for listening to vi my nascent research ideas, offering discussion-based seminars, and inviting me to participate on your research projects. I am fortunate to have you as a mentor and friend. To Les Steffe: Thank you for agreeing to supervise this dissertation and for your tremendous support and unwavering patience throughout my writing process. Thank you for introducing me to radical constructivism, which has helped me to model how and what individuals know. Although I am certain I will never understand things exactly the way you do, I will continue to work to understand the distinctions you have made in your research; I look forward to learning more from you. I will always admire your approach to mentoring, particularly how you treat your students as knowledgeable others and how you always seem to act in ways that promote tremendous progress in each of them.