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The Case of Infinity

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The Case of Infinity University of Texas at El Paso DigitalCommons@UTEP Open Access Theses & Dissertations 2015-01-01 Challenges in Assessing College Students' Conception of Duality: The aC se of Infinity Grace Olutayo Babarinsa-Ochiedike University of Texas at El Paso, [email protected] Follow this and additional works at: https://digitalcommons.utep.edu/open_etd Part of the Higher Education Administration Commons, Higher Education and Teaching Commons, Mathematics Commons, and the Science and Mathematics Education Commons Recommended Citation Babarinsa-Ochiedike, Grace Olutayo, "Challenges in Assessing College Students' Conception of Duality: The asC e of Infinity" (2015). Open Access Theses & Dissertations. 997. https://digitalcommons.utep.edu/open_etd/997 This is brought to you for free and open access by DigitalCommons@UTEP. It has been accepted for inclusion in Open Access Theses & Dissertations by an authorized administrator of DigitalCommons@UTEP. For more information, please contact [email protected]. CHALLENGES IN ASSESSING COLLEGE STUDENTS’ CONCEPTION OF DUALITY: THE CASE OF INFINITY GRACE OLUTAYO BABARINSA-OCHIEDIKE Department of Teacher Education APPROVED: ________________________________ Mourat Tchoshanov, Ph.D., Chair ________________________________ Olga Kosheleva, Ph.D. ________________________________ Helmut Knaust, Ph.D. Charles Ambler, Ph.D. Dean of the Graduate School Copyright © by Grace Olutayo Babarinsa-Ochiedike 2015 Dedication I dedicate this dissertation to the Ancient of Days; the author and the finisher of my faith. When my heart was overwhelmed, you led me beside the still waters and you restored my soul. To my husband, Uzoma, with whom I am blessed to share my life. This journey begun 7 ½ years could not have been possible without your inspiration, understanding, and support. Thank you for bearing with me for eleven years as a wife, mother, instructor and a full time student. To my children Emmanuela, Valerie & Joel, your understanding motivated me to complete the program. To my parents, Late Chief Isaac Olusanya Babarinsa and Late Chief (Mrs) Glory Eyo Babarinsa, who are my greatest role models and more than any other people, shaped who I am. Mom, I’m eternally grateful for your endurance for eleven years. I really wish you were still here with us to rejoice at the completion of my doctoral degree. CHALLENGES IN ASSESSING COLLEGE STUDENTS’ CONCEPTION OF DUALITY: THE CASE OF INFINITY by GRACE OLUTAYO BABARINSA-OCHIEDIKE, B.Sc., PG. Dip., M.S., M.Ed. DISSERTATION Presented to the Faculty of the Graduate School of The University of Texas at El Paso in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY Department of Teacher Education THE UNIVERSITY OF TEXAS AT EL PASO May 2015 Acknowledgements With the completion of this dissertation, I want to express my sincere thanks to all the people who have supported me through the completion of this study. I deeply appreciate Dr. Mourat Tchoshanov for his role in my career. He has overseen this study in its entirety, and provided exceptional mentorship. From him I have learned much. As my dissertation chair and advisor, he guided me patiently with understanding, encouraged me, supported me, and helped me clarify my thoughts about my own understandings of the duality of infinity. Many thanks go to Dr. Olga Kosheleva and Dr. Helmut Knaust, for serving on my dissertation committee and giving me feedback throughout the course of this study. This study would not have been possible without your guidance and support. Besides the committee members, a lot of people have contributed to my academic and professional growth. In particular, I want to thank Dr. Brian Giza who offered support when I encountered valleys during this journey, and Dr. Maria de la Piedra, for the assistantship offered me, to support me throughout my doctoral studies. I have learned a lot from presenting cross-departmental lecture/workshop series in qualitative methods for graduate students and faculty at the Research & Evaluation Lab. In addition, I want to appreciate the office staff in the Teacher Education Department for being very supportive of me throughout my graduate studies, and the cleaners at the College of Education who made staying back after work hours to study conducive for me. Thank you! I would also like to thank my research participants’ for their willingness and time sacrifice, as well as the cooperation of their professors in sparing their class time for questionnaire administration. My doctoral cohort colleague Ben McDermott, who played a significant role during my data collection. Thank you! I want to especially thank the beloved brethren of RCCG Living Word Center for their ceaseless prayers, supports and words of encouragement throughout my course of study. The Lord continue to bind us together with his unbroken cord. I am grateful to all families and friends in El Paso, who showed genuine love and have assisted in one way or the other, especially with my kids. The Lord reward your labor of love. v My deepest gratitude goes to my dear father in the Lord, Bishop Fidelis Ugbong, for his consistent prayers, counsel and encouragement all the way through my studies. You were just a phone call away, anytime! Thank you for all the sacrifices. The Lord honor you sir! Finally, special thanks go to my precious sisters, brother, and in-laws for their stanch support and encouragement throughout my studies. You believed in me, encouraged me and gave so much. Thank you and God bless you richly! vi Abstract Interpreting students’ views of infinity posits a challenge for researchers due to the dynamic nature of the conception. There is diversity and variation among students’ process-object perceptions. The fluctuations between students’ views however reveal an undeveloped duality conception. This study examined college students’ conception of duality in understanding and representing infinity with the intent to design strategies that could guide researchers in categorizing students’ views of infinity into different levels. Data for the study were collected from N=238 college students enrolled in Calculus sequence courses (Pre-Calculus, Calculus I through Calculus III) at one of the southwestern universities in the U.S. using self-report questionnaires and semi-structured individual task-based interviews. Data was triangulated using multiple measures analyzed by three independent experts using self-designed coding sheets to assess students’ externalization of the duality conception of infinity. Results of this study reveal that college students’ experiences in traditional Calculus sequence courses are not supportive of the development of duality conception. On the contrary, it strengthens the singularity perspective on fundamental ideas of mathematics such as infinity. The study also found that coding and assessing college students’ conception of duality is a challenging and complex process due to the dynamic nature of the conception that is task-dependent and context-dependent. Practical significance of the study is that it helps to recognize misconceptions and starts addressing them so students will have a more comprehensive view of fundamental mathematical ideas as they progress through the Calculus coursework sequence. The developed duality concept development framework called Action-Process-Object-Duality (APOD) adapted from the APOS theory could guide educators and researchers as they engage in assessing students’ conception of duality. The results of this study could serve as a facilitating instrument to further analyze cognitive obstacles in college students’ understanding of the infinity concept. vii Table of Contents Acknowledgements .................................................................................................. v Abstract ................................................................................................................. vii Table of Contents ................................................................................................. viii List of Tables ........................................................................................................... x List of Figures ......................................................................................................... xi List of Illustrations ................................................................................................ xii Chapter 1: Introduction ............................................................................................ 1 1.1 Purpose of the Study .............................................................................. 1 1.2 Research Questions ................................................................................ 2 1.3 Significance of the Study ....................................................................... 3 1.4 Theoretical Framework .......................................................................... 4 1.5 Definition of Terms ............................................................................... 6 1.6 Chapter Overview .................................................................................. 7 Chapter 2: Review of Literature .............................................................................. 9 2.1 Cultural-Historical Epistemology of Duality in Calculus ..................... 9 2.2 Infinity Concept ................................................................................... 13 2.3 Duality Concept ..................................................................................
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