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TO INFINITY AND BEYOND: TOWARD A LOCAL INSTRUCTION THEORY FOR COMPLETED INFINITE ITERATION by IULIANA RADU A Dissertation submitted to the Graduate School-New Brunswick Rutgers, The State University of New Jersey in partial fulfillment of the requirements for the degree of Doctor of Philosophy Graduate Program in Mathematics Education Written under the direction of Keith Weber and approved by ________________________ ________________________ ________________________ ________________________ New Brunswick, New Jersey October, 2009 ABSTRACT OF THE DISSERTATION To Infinity And Beyond: Toward A Local Instruction Theory For Completed Infinite Iteration By IULIANA RADU Dissertation Director: Keith Weber There is evidence that students’ and mathematicians’ images of and reasoning about concepts such as infinite unions and intersections of sets, limits, and convergent series often invoke the consideration of infinite iterative processes. Therefore, not reasoning normatively about infinite iteration may hinder students’ understanding of fundamental concepts in mathematics. Research suggests that the vast majority of college students are inclined to use non-normative reasoning when presented with tasks that challenge them to imagine an infinite iterative process as completed and define its outcome. However, there is almost no research on how students’ conceptions of completed infinite iteration can be refined in a normative direction. Adopting a design research approach, the purpose of this thesis is to develop a local instruction theory for completed infinite iteration. This design entails multiple cycles through phases of development, implementation, and analysis. The study consisted of a sequence of two constructivist teaching experiments conducted with pairs of ii mathematics majors, during which the students worked collaboratively through a variety of infinite iteration tasks. Each cycle consisted of 6 or 7 sessions lasting approximately 2 hours, together with Pre-test and Post-Test sessions. The data analysis consisted of multiple phases of iterative analysis of the videotaped sessions and written work and was guided by situated learning transfer theories. The participants in this study employed a variety of types of reasoning when solving completed infinite iteration tasks, some normative and some not. One way that the students refined their reasoning on infinite iteration was by making references to previously solved tasks (in the presence of a concern for consistent reasoning across tasks), although the refinement was not always in normative directions. The challenge that a researcher faces consists in helping students use such references to establish “anchors” in normatively solved tasks to which they can then contrast the solution paths proposed for other tasks. A variety of approaches to responding to this challenge are explored. The study culminates with the formulation of a local instruction theory for completed infinite iteration, containing among other things a sequence of instructional activities and a rationale for their appropriateness. iii Acknowledgements I could not have completed this dissertation without the support of a large number of people. I would like to thank my advisor, Keith Weber, who introduced me to research in undergraduate mathematics education early in my doctoral studies by inviting me to join his research projects even before he was officially my advisor. His guidance during my last four years in the program and especially during all stages of working on the dissertation study was invaluable. I would also like to thank Carolyn Maher, my first year advisor, through whom I was exposed to the IML project and thus was able to experience first hand, for the first time, aspects of what it means to do qualitative research. I will never forget my first meeting with Carolyn during which, as a first year doctoral student who had never done research in mathematics education, I asserted with confidence that I would be using quantitative methods in my research; Carolyn was quick to reply that I could not know that yet. How right she was! Warren Crown is another Rutgers faculty member who mentored me during my first three years in the program. Although he was not directly involved in my work on the dissertation study, his guidance and support during my first tentative steps in the world of teaching mathematics education courses and supervising prospective teachers was much appreciated. I would also like to thank Gerald Goldin, who indulged me for countless hours in discussing the infinity-related mathematics behind my dissertation study, and gave me quite a few ideas for how to create variations from existing infinite iteration tasks. I also received much support from fellow students in the Ph.D. and Ed.D. programs in Mathematics Education at Rutgers, many of whom offered themselves as iv “lab rats” for my task-piloting experiments. Special thanks go out to Cathleen Rossman, Kathy Lin, and Cecilia Arias, my fellow “Dissertation Support Group” members, through whose eyes I could look at my study from an outsider’s perspective. During my dissertation work, I found it immensely helpful to set daily goals and report to the group with respect to the weekly progress – brilliant idea, gals! Another thank you goes to the four mathematics majors who participated in this study. I appreciate the time and effort you dedicated to this project, and the perseverance with which you strove to find your way through the maze of infinite iteration tasks I threw at you. I am grateful for the support offered by Harriet Schweitzer and Judy Bornstein, two wonderful educators from the Center of Math, Science, and Computer Education (Rutgers). Through the center, they offered me a graduate assistantship when I was in dire straits regarding financial support for my studies; without them I might not have finished my doctorate! Working at CMSCE also allowed me to try my hand at offering professional development courses to in-service mathematics teachers, which greatly helped my understanding of the public school system and the needs of K-12 teachers. Last but not least I would like to thank my family. My husband Viorel supported me unconditionally through all the ups and downs inherent to a lengthy and challenging process such as a doctoral program, and on top of that, took on a great share of the house keeping and cooking responsibilities. I truly couldn’t have asked for more. Much gratitude to my parents, two dedicated mathematics teachers who instilled in me a love of mathematics from an early age and supported me in everything I set out to do. Finally, a big thank you to my two sisters, Simona and Speranta, who were always there for emotional support and the occasional math talk. I love you all. v Notes On The Text This dissertation contains excerpts from student interviews and classroom dialogue; these excerpts were typed verbatim. All students are referred to in the text with pseudonyms. Chapters 3 through 6 contain stand-alone papers, which is why the heading format in these chapters is different from that used in Chapters 1 and 2. Chapters 3 and 4 are written using the pronoun “we” instead of “I”. This decision reflects the fact that a person other than the author made significant contributions toward the two papers contained in these chapters. vi Table of Contents Abstract………………………………………………………………………………. ii Acknowledgements…………………………………………………………………... iv Notes on the Text…………………………………………………………………….. vi Table of Contents…………………………………………………………………….. vii List of Tables………………………………………………………………………… xi List of Figures………………………………………………………………………... xii Chapter 1- Introduction……………………………………………………………… 1 Background and Research Questions………………………………………... 2 Dissertation Structure………………………………………………………... 4 Chapter 2 – The Historical Development of Mathematical Infinity…………………. 9 Introduction………………………………………………………………….. 9 The Greeks and the Infinite………………………………………………….. 10 The Potential Infinity Era……………………………………………………. 11 Bolzano’s Actual Infinity and Comparison Criteria…………………………. 11 Cantor’s Contribution to the Study of Infinity………………………………. 13 Cantor’s Approach to Size Comparison……………………………... 13 Cantor’s Approach to the Cardinalities of Z, Q, and R……………… 14 The Cardinality of Power Sets………………………………………. 17 Ordinal Numbers and the Alephs……………………………………. 18 Reactions to Cantor’s Theory………………………………………………... 21 Chapter 3 – Paper 1: Formalizing the State at Infinity of Completed Infinite Iteration………………………………………………………. 23 Introduction…………………………………………………………………... 23 Theories on Completed Infinite Iteration…………………………………….. 24 The APOS-Based Theory……………………………………………. 25 The Basic Metaphor of Infinity ………………………………………. 26 What Is Lacking? ……………………………………………………. 28 Defining the State at Infinity………………………………………………… 30 The R1-R3 Approach ……………………………………………….. 31 The Pointwise Convergence Approach ……………………………… 34 Applications …………………………………………………………………. 35 vii Where Did All the Marbles Go? …………………………………….. 35 Handling Oscillating Objects ………………………………………... 37 The Endless Spiral …………………………………………………... 38 Acting on a Number …………………………………………………. 40 Extracting the Relevant Information ………………………………… 41 Discussion……………………………………………………………………. 43 Other Possible Definitions …………………………………………... 43 Cognitive Considerations ……………………………………………. 45 Chapter 4 – Paper 2: Conceptual Changes in Mathematics Majors' Understanding of Completed Infinite Iterative Processes..………………………….. 51 Introduction ………………………………………………………………….. 51 Related Literature …………………………………………………………….

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