Reversible Reasoning in Multiplicative Situations: Conceptual

REVERSIBLE REASONING IN MULTIPLICATIVE SITUATIONS: CONCEPTUAL ANALYSIS, AFFORDANCES AND CONSTRA1NTS by AJAY RAMFUL (Under the Direction of John Olive) ABSTRACT While previous research studies have focused primarily on the additive structure , relatively fewer attempts have been made to explore the multiplicative structure from a reversibility perspective. The aim of this exploratory study was to investigate, at a fine-grained level of detail, the strategies and constraints that middle-school students encounter in reasoning reversibly in the multiplicative domains of fraction and ratio. In the first phase of the study a mathematical analysis of reversibility situations was conducted. In the second empirical phase, three pairs of above-average students at the grades 6, 7 and 8 were interviewed in a rural middle-school in the United States. Three selected data sets were analyzed using Vergnaud’s theory of the multiplicative conceptual field, the concept of units, and notion of quantitative reasoning. The findings put into perspective the importance of the theorem-in-action as a key element for reasoning reversibly in multiplicative situations. Further, the results show that reversibility is context-sensitive, with the numeric feature of problem parameters being a major factor. Relatively prime numbers and fractional quantities acted as inhibitors preventing the cueing of the invariant ‘division as the inverse of multiplication’, thereby constraining students from reasoning reversibly. Moreover, the form of reversible reasoning was found to be dependent on the type of multiplicative structures. Among others, two key resources were identified as being essential for reasoning reversibly in fractional contexts: interpreting fractions in terms of units, which enabled the students to access their whole number knowledge and secondly, the coordination of 3 levels of units. Similarly, interpreting a ratio as a quantitative structure together with the coordination of 3 levels of units was found to be essential for reasoning reversibly in ratio situations. This study also shows that students can articulate mathematical relationships operationally but may not necessarily be able to represent them algebraically. The constraints that the students encountered in the multiplicative comparison of two quantities substantiate previous research that multiplicative reasoning is not naturally occurring. Failure to conceptualize multiplicative relations in reverse constrained the students to use more primitive fallback mechanisms, like the building-up strategy and guess-and-check strategy, leading them to solve problems non- deterministically and at higher computational cost. INDEX WORDS: Division, Fraction, Multiplication, Multiplicative Comparison, Quantitative Reasoning, Ratio, Reversibility, Theorem-in-action, Units, Vergnaud’s Theory REVERSIBLE REASONING IN MULTIPLICATIVE SITUATIONS: CONCEPTUAL ANALYSIS, AFFORDANCES AND CONSTRAINTS by AJAY RAMFUL B.S (Hons) Mathematics, University of Mauritius, Mauritius, 1997 M.Phil (Mathematics), University of Mauritius, Mauritius, 2000 MA, University of Brighton, England, 2004 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 2009 © 2009 AJAY RAMFUL All Rights Reserved REVERSIBLE REASONING IN MULTIPLICATIVE SITUATIONS: CONCEPTUAL ANALYSIS, AFFORDANCES AND CONSTRAINTS by AJAY RAMFUL Major Professor: John Olive Committee: Denise S. Mewborn Andrew G. Izsák Electronic Version Approved: Maureen Grasso Dean of the Graduate School The University of Georgia December 2009 iv DEDICATION I dedicate this work to my daughter, Jaylina Aislee who is now supposedly at the concrete operational stage and who, at every given opportunity, teaches me children’s mathematics. v ACKNOWLEDGEMENTS I express my sincere gratitude and thanks to the members of my committee. I am deeply indebted towards my major professor, Dr Olive who has diligently and dedicatedly supervised this project from start to finish. I cannot forget his constant support and guidance starting from the independent study that I did in Spring 2007 when the current research was just in its embryonic stage. I express my sincere appreciation for his willingness to help and for always giving value to my ideas. I thank him for devoting so much time and effort in painstakingly reading my analysis over and over again, making insightful comments that prompted me to rethink about various parts of the dissertation. The three years that I have spent working with him shaped my thinking about children’s mathematics. Thank you Dr Olive. I am equally indebted to Dr Mewborn whose decision helped to shape critical aspects of this study. Her comments and critiques led me to reorient my thinking at various points. I acknowledge her prompt and quick support as a pragmatic head of department. I thank her for the various assistantships (sometimes at difficult moments) and for providing me teaching opportunities. My interest in student’s thinking was very much stimulated by Dr Izsák’s video- recording class (in Spring 2007) that served as the springboard for the current study. I am very thankful to Dr Izsák who has a major contribution in the realization of this dissertation. His sharp and pointed questions served as triggers in designing this piece of research. His writings influenced me in a number of ways, from methodology to data analysis. vi I chose to come at The University of Georgia in 2006 to work with people that I had been reading about from a distant island. I have a special thank to Dr Les Steffe who allowed me to collect data for this dissertation under his project. He taught me a lot about children’s mathematics in the two EMAT classes. Dr Jeremy Kilpatrick has also been an inspiring educator. I would also like to thank the significant others – Dr Jim Wilson, Dr Patricia Wilson, Dr Larry Hatfield, Dr Nicholas Oppong, Dr Dorothy White, and other faculties from other departments who allowed me to audit their classes. I am also thankful to Helen, Rita, Valerie, and Ms Boswell who have always been on the forefront to help with administrative matters. This dissertation would not have been possible without the six participants of the study, Ted, Cole, Brian, Aileen, Jeff and Eric. I would like to thank these six thinkers for allowing me to ‘read’ their minds while they were zealously involved in thinking about the problems I posed to them. I would also like to thank the school principal and one teacher who went out of her way to facilitate the conduct of this study. Unfortunately, anonymity should be maintained. I thank Katy for negotiating access to the school. My thanks also go to Fulbright and the Mauritius Institute of Education for financially supporting me for the first two years of my study. I express my sincere appreciation to Pastor Jeff Thompson, the Clarke’s family and friends who continuously supported me and my family here in Athens. I thank my father and mother who have supported me in various ways throughout the study. Finally, I express my sincere gratitude to my wife, Lovelina who has been by my side during those difficult moments and who provided me with the best conditions to complete my studies. I am particularly thankful to her for her patience. Lovelina and Aislee gave me the strength and motivation to go to the end. vii TABLE OF CONTENTS Page ACKNOWLEDGEMENTS.............................................................................................................v LIST OF TABLES......................................................................................................................... xi LIST OF FIGURES ...................................................................................................................... xii CHAPTER 1 INTRODUCTION .........................................................................................................1 Conception 1: Piaget’s Notion of Negation and Compensation................................3 Conception 2: Reversibility in Mathematics Education Research ............................9 Steffe’s and Olive’s Conception of Reversibility: A Fine-grained Definition Based on Schemes........................................................................................................12 Other Definitions of Reversibility...........................................................................15 Differentiating Between ‘Reversibility Situations’ and ‘Operational Reversibility’….................................................................................................18 A Working Definition for Reversibility ..................................................................20 Problem Statement: The Research Gap...................................................................25 Rationale: The Importance of Studying Reversibility.............................................30 2 THEORETICAL FRAMEWORK...............................................................................35 Vergnaud’s Approach as a Port of Entry.................................................................35 A Second Path: Unit Structures...............................................................................43 viii Mental Operations ...................................................................................................49 A Third Path to Study Reversibility: The Notion of Quantities..............................57 3 METHODOLOGY ......................................................................................................62

Load more