COLLEGE STUDENTS’ INTUITIVE UNDERSTANDING OF THE CONCEPT OF LIMIT AND THEIR LEVEL OF REVERSE THINKING DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in the Graduate School of The Ohio State University By Kyeong Hah Roh, Ph. D. The Ohio State University 2005 Dissertation Committee: Approved by Professor Sigrid Wagner, Advisor _________________________________ Advisor Professor Douglas Owens School of Teaching and Learning Professor Jeffery Connor Professor Hea-Jin Lee Copyright by Kyeong Hah Roh, 2005 ABSTRACT This thesis addresses the relationship of students’ intuitive understanding of the limit of a sequence to their reversibility, which is an ability to reverse the order between ε and N as required in the rigorous definition of limit. The subjects of this research were students who had not had any experience with rigorous proofs using the ε − N definition of a limit. Eleven students completed a series of 1-hour semi-structured, task- based interviews once a week for 5 weeks. Monotone bounded, unbounded, constant, oscillating convergent, or oscillating divergent sequences were tested. Students represented each sequence numerically as well as graphically in determining convergence of the sequence. Students also used tools, called ε − strips, specially developed for this study to explore the ε − N relation in defining limits through hands-on activities with the ε − strips. Finally, students were presented the following ε − strip definitions, and were asked to evaluate the propriety of the definitions as statements of the limit of a sequence. ε − Strip Definition A: A certain value L is a limit of a sequence when infinitely many points on the graph of the sequence are covered by any epsilon strip as long as the epsilon strip covers L. ε − Strip Definition B: A certain value L is a limit of a sequence when only finitely many points on the graph of the sequence are NOT covered by any epsilon strip as long as the epsilon strip covers L. ii It was found that students’ understanding of the definition of the limit of a sequence was associated with not only their conception of limit but also their level of reversibility. In addition, there was improvement in students’ reversibility and/or their conception of limit through the ε − strip activity, even though there was no procedure for indicating students’ errors, correcting students’ misconceptions about limit, or confirming the propriety of the ε − strip definitions to students during the interviews. This study implies that theε − strip activity, combined with the various types of sequences used in this teaching experiment, is one instructional method for helping students develop and accommodate their conception of the limit of a sequence. iii ACKNOWLEDGMENTS I would like to thank a large number of people who provided interests, encouragement, and support to my study thus made this dissertation possible. First of all, I would like to gratefully acknowledge the enthusiastic supervision of my advisor, Dr. Sigrid Wagner. Always, she listened to my ideas attentively and provided sound advice to my research through numerous stimulating discussions. In particular, Sigrid's optimistic attitude and strong faith in my project gave me big courage to endure hard time in my research. My sincere and warm thanks also go to my dissertation committee members, Dr. Douglas Owens, Dr. Jeffery Connor, and Dr. Hea-Jin Lee. I would like to thank Dr. Owens always his time for fruitful discussions, general advice, and his thoughtful concern about my career from the beginning of my study at the Ohio State University. Numerous have been those stimulating discussions and general advice for both my teaching and my research in which I have been helped by Dr. Connor. Dr. Lee is thanked for managing to read my dissertation so thoroughly, and for her helpful advice. I would like to express my sincere appreciation for support and hospitality of School of Teaching and Learning at the Ohio State University. I would also like to thank all faculty and staffs, particularly Dr. Patricia Brosnan who sincerely encouraged me to acclimate in new circumstance during the first year of my study in the Ohio State University. I would like to thank Dr. David Haury, Robert Day, and Claudette Price in iv ERIC Clearinghouse for Science, Mathematics, and Environmental Education for their hospitality to support helpful equipments. I wish to express my deep appreciation to Department of Mathematics at Ohio University, for their interests, support, and help with setting up equipment for my research. Especially, I would like to say a big thank you to all students and instructors who participated in my research for this dissertation. I am grateful to all my friends, Eun Jung Part, Jennifer Kaminski, and other my colleague for their constant interest and helpful discussion.. Finally, I am forever indebted to my parents for their constant love, endless patience and encouragement which enable me to stand all difficulties. v VITA September 1, 1970 . .Born - Kwangju, Korea. 1993. B.S. Mathematics Education, Ewha Womans University, Seoul, Korea. 1995 . M.S. Mathematics, Seoul National University, Seoul, Korea. 2000 . .Ph.D. Mathematics, Seoul National University, Seoul, Korea. 2000 – 2001 . .Senior Researcher Ewha Educational Research Institute, Ewha Womans University, Seoul, Korea. 2002 – 2003. Graduate Teaching and Research Associate, ERIC Clearinghouse for Science, Mathematics, and Education, The Ohio State University. 2003 – 2005. Visiting Assistant Professor, Department of Mathematics, Ohio University. PUBLICATIONS 1. Roh, K. (2004). The role of reverse thinking in the contexts of limit. In D. E. McDoull & J. A. Ross(Eds.), Proceedings of the twenty-sixth annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 191-192). Ontario, ON; University of Toronto. 2. Roh, K. et al. (2003). Curriculum development for mathematics and information science in a knowledge based society, Seoul, Korea; Ewha Womans University Publisher. vi 3. Roh, K. (2003). Problem based learning in mathematics education. ERIC Digest. EDO-SE-03-07. Columbus, OH; ERIC Clearinghouse for Science, Mathematics, and Environmental Education. 4. Roh, K. (2002). Inquiry by mathematics. In L. A. Milbourne (Ed.), The connected teacher’s companion to inquiry in science and mathematics (pp. 25-30). SE067123. Columbus, OH; ERIC Clearinghouse for Science, Mathematics, and Environmental Education. 5. Haury, D. L., Lee, H., Lorson, M., Roh, K., & Sonmez, D. (2002). The connected teacher’s companion to problem solving in science and mathematics. SE067122. Columbus, OH; ERIC Clearinghouse for Science, Mathematics, and Environmental Education. 6. Choi, H. Han, C. Moon, H., Roh, K., & Wee. N. (1999). Medial axis transform and offset curves by Minkowski Pythagorean hodograph curves, Computer-Aided Design, 31, 59-72. 7. Choi, H. Han, C. Moon, H., Roh, K., & Wee. N. (1998). A new algorithm for offset of plain domain via MAT, Differential/topological techniques in geometric modeling and processing ’98, 161-185. FIELDS OF STUDY Major Field: Education vii TABLE OF CONTENTS Abstract…………………………………………………………………………… ii Acknowledgments………………………………………………………………… iv Vita……………………………………………………………………………….. vi List of Figure…………………………………………………………………….... x List of Table……………………………………………………………………….. xiii Chapter 1 STATEMENT OF THE PROBLEM……………………………… 1 The Problem………………………………………………………. 1 Research Questions……………………………………………….. 11 Theoretical Framework…………………………………………… 12 Chapter 2 LITERATURE REVIEW…………………………………………. 24 Readiness for Instruction on Limits………………………………. 24 Misconceptions About Limits…………………………………….. 29 Cognitive Obstacles Influencing Misconceptions About Limits…. 35 Instructional Approaches…………………………………………. 42 Conclusion………………………………………………………... 47 Chapter 3 RESEARCH METHODOLOGY…………………………………. 49 Preparation of the Research………………………………………. 49 Data Collection Methods…………………………………………. 54 Data Analysis……………………………………………………... 72 viii Chapter 4 RESULTS…………………………………………………………. 79 Summary of Results by Interview Task........................................... 79 Students’ Conception of the Convergence of a Sequence………… 91 Students’ Conception of the Limit Value of a Sequence………….. 120 Relationship Between Concept of Limit and Definition of Convergence………………………………………………………. 134 Students’ Reversibility in the Context of the Limit of a Sequence.. 136 Chapter 5 CONCLUDING REMARKS AND DISCUSSION...…………….. 183 Results in Terms of Research Questions………………………….. 185 Implications of the Research for Teaching and Learning ………... 201 Future Research……………………………………………………206 Conclusion………………………………………………………... 210 References …………………………………………………………………….. 211 Appendices …………………………………………………………………….. 218 Appendix A. Description of Research Project….…………….…... 219 Appendix B. Student Consent Form…………………………….... 221 Appendix C. Survey ……….……………………………………... 222 Appendix D. Pretest ………..…………………………………….. 228 Appendix E. A List of Task Activities & Interview Protocols……. 230 Appendix F. Posttest ……………………………………………… 254 ix LIST OF FIGURES Figure Page 1.1 Formal cognition versus intuitive cognition…………………………...…... 15 1.2 Formal cognition versus two kinds of intuition: Primary intuition and 19 secondary intuition………………………………………………………… 3.1 Procedure of the teaching experiment and data analysis for they study….... 53 3.2 Mirror activity…………………………………………………………….. 59 3.3 An example of a symbolic representation of a sequence: