Georgia State University ScholarWorks @ Georgia State University Mathematics Dissertations Department of Mathematics and Statistics 8-7-2018 Generalizing and Transferring Mathematical Definitions from Euclidean to Taxicab Geometry Aubrey Kemp Follow this and additional works at: https://scholarworks.gsu.edu/math_diss Recommended Citation Kemp, Aubrey, "Generalizing and Transferring Mathematical Definitions from Euclidean to Taxicab Geometry." Dissertation, Georgia State University, 2018. https://scholarworks.gsu.edu/math_diss/58 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]. GENERALIZING AND TRANSFERRING MATHEMATICAL DEFINITIONS FROM EUCLIDEAN TO TAXICAB GEOMETRY by AUBREY KEMP Under the Direction of Draga Vidakovic, PhD ABSTRACT Research shows that by observing properties of figures and making conjectures in non- Euclidean geometries, students can better develop their understanding of concepts in Euclidean geometry. It is also known that definitions in mathematics are an integral part of understanding concepts and are often not used correctly in proof or logic courses by students. To further investigate student understanding of mathematical definitions, this dissertation studied students’ uses of dynamic geometry software and group work to generalize their understanding of definitions as they completed activities in Taxicab geometry. As a result of the analysis from the group work and use of Geometer’s Sketchpad by 18 students in a College Geometry class, suggestions are provided to implement cooperative learning and technology in the classroom. In addition, results are provided from the data analysis of responses to questions pertaining to the definition of circle (and its relevant concepts) of 15 students enrolled in the course who volunteered to participate in semi-structured interviews. This dissertation specifically utilizes APOS Theory (Arnon et al., 2014) and the interaction of schema framework provided by Baker et al. (2000) to determine what components of the circle schema were evoked by these participating students during these interviews. By adapting and transferring their knowledge of concepts back and forth between Euclidean and Taxicab geometry, these students provided evidence for the relationships they had formed between the components of their circle schema. Further, they demonstrated a variety of levels of schema interaction of their evoked Euclidean geometry schema and Taxicab geometry schema. As a result, a model of schema interaction and suggested pedagogical activities were developed to help facilitate student understanding of the definition of a circle and other relevant concepts. INDEX WORDS: Definitions, Geometry, Taxicab, APOS Theory, Circle, Cooperative learning, Geometer’s Sketchpad GENERALIZING AND TRANSFERRING MATHEMATICAL DEFINITIONS FROM EUCLIDEAN TO TAXICAB GEOMETRY by AUBREY KEMP 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 2018 Copyright by Aubrey Elizabeth Kemp 2018 GENERALIZING AND TRANSFERRING MATHEMATICAL DEFINITIONS FROM EUCLIDEAN TO TAXICAB GEOMETRY by AUBREY KEMP Committee Chair: Draga Vidakovic Committee: Valerie Miller Mariana Montiel Alexandra Smirnova Electronic Version Approved: Office of Graduate Studies College of Arts and Sciences Georgia State University August 2018 i In memory of Daddaddy and Grandmama (Jack and Maureen Spann) Go raibh céad míle maith agat. v ACKNOWLEDGEMENTS I would like to thank my amazing advisor, Dr. Draga Vidakovic, for her unwavering amount of encouragement, guidance, and affirmation since the day I met her. You truly have been an incredible role model, academic mom, researcher, teacher, and supporter in my life. It is not lost on me how fortunate I am to have learned as much as I have (and to continue learning) from you. I would also like to thank my committee members, Dr. Valerie Miller, Dr. Mariana Montiel, and Dr. Alexandra Smirnova, for their feedback, flexibility, and support for this dissertation and throughout the last five years of my life, as they all have had a major impact on me. You exemplify strong voices and women in my life, and I hope to influence and motivate others the way you have me. I would also like to thank the Department Chair, Dr. Guantao Chen and the rest of the Mathematics and Statistics department at GSU, including past and present instructors, graduate students, and staff members. My experiences for the last five years would not have been the same without every single one of you. Whether it be in the way you helped me study, supported, taught, or befriended me, I will never forget the effect you all have made on my life. Thank you in particular to Sandra Ahuama-Jonas and Beth Connor for the hallway chats and answering my countless emails. Also, to Harrison Stalvey, Annie Burns-Childers, and Leslie Meadows – I thank you for four plus years of great friendship and collaboration in research. I would also like every student that I have had the opportunity to teach, with a special thank you to the participants in this study. You all have taught me about myself as an educator and a person more so than I ever thought possible. I would also like to thank running, yoga, dry shampoo, and red wine for helping me handle graduate school. In addition, I want to thank my dogs, Parker and Quinn, who are so much more to me than just dogs. The cuddles and licks have been a necessity in keeping my sanity. Finally, I would like to thank my entire family, of which I have three types. vi My academic family – Darryl Chamberlain Jr., Russell Jeter, and Robin Baidya, among so many others – I can’t thank you enough for the constant support you all have provided me. Darryl – for your amazing mentorship, for approaching me about pursuing this degree so many years ago, and for showing me how to perfectly merge friendship and research. Russell – for your overall inspiring nature and pushing me to accomplish so many things that I never thought would be possible. You are the only person I know who would have suggested that we train for a marathon while we were both writing our dissertations. Robin – not only for continuing to inspire me mathematically, but for the incredible openness and genuineness you bring to our friendship. You were the first person that befriended me in graduate school which helped me to feel comfortable, and I know I would not have made it to this point without you and your encouragement. My non-academic family – Nicole Bardizbanian, Kristen Owen, Samantha Skadan, Hannah Warner, Alicia Fortino, Tariq Talley, Alex Coble, Robert Vogel, Daniel Owen, Dany Bardizbanian, Sean Skadan, Daniel Evans, Aiken Davis, Chad Dobler, and so many more. There are too many of you to individually and personally thank here, but I want you to know how much I appreciate all of you and the tolerance you have shown me over the last five years by listening to me stress and freak out about the program. I love each and every one of you so uniquely, and I cannot express how much I value the moments we have had together throughout our friendships. Finally, but certainly not least, my family family – Mama, Daddy, Amy, Alex, Felix, Aunt Mare, Brianna, Jason, Uncle Mike, Colin, Uncle Bobby, Aunt Jane, Uncle Steve, Paul, Ally, Aunt Juju, Uncle Pat, John Henry, Angie, Uncle Tommy, Aunt Mary Anne, and Grandma. You all have also supported me and listened to me vent about my stress over the last five years, so I thank you for not throwing food at me across the table at one of our family birthday parties or holiday functions. Also, for continuing to tell me that I could do it – I can’t thank you enough. vii I especially want to thank my immediate family. Mama – for the unending love and compassion you give me and everyone else in your life. Grandmama’s saint-ness was passed down to you (and Aunt Mare), and anyone who has met you knows that you legitimately would do anything for anyone without expecting anything in return. Everything that you have done for me in the last 27 years that I am thankful for is inexplicable. You really are the best Mama and friend in the world. Daddy – for your strength and leadership of our family and for giving me such a wonderful person to look up to. I know you are my Daddy, but you are also one of my closest friends and I have learned so much about life from you. From hiking/camping trips or dancing with you as I stood on your feet when I was younger, to our more recent pastimes like running, sitting in a cove at the lake, or grabbing a beer when we have time…I always have and always will be your little girl. Amy and Alex – I am so proud you both and of the relationship we have with one another. I can’t thank you enough for being such great people to grow up with. Mama always told us that she wanted us to not only be siblings but become each other’s best friends, and I know that we have done this. You two bring such love and laughter to my life, and I would not be the person I am today if it weren’t for you helping to shape me along the way. I can’t thank God enough for the entire family (of all types) he has blessed me with. To Daddaddy and Grandmama, for whom this dissertation is dedicated, I want to thank you for showing me what unconditional love looks like. The family you raised and have left behind is an amazing one and has provided so much support for one another.