UNDERGRADUATE STUDENTS’ CONCEPTIONS OF INEQUALITIES by Elena Filofteia Halmaghi Licence in Mathematics, University of Bucharest, Romania, 1985 Thesis Submitted in Partial Fulfilment of the Requirements for the Degree of Doctor of Philosophy In the Faculty of Education © Elena Halmaghi 2011 SIMON FRASER UNIVERSITY Spring 2011 All rights reserved. However, in accordance with the Copyright Act of Canada, this work may be reproduced, without authorization, under the conditions for Fair Dealing. Therefore, limited reproduction of this work for the purposes of private study, research, criticism, review and news reporting is likely to be in accordance with the law, particularly if cited appropriately. APPROVAL Name: Elena Filofteia Halmaghi Degree: Doctor of Philosophy Title ofThesis: Undergraduate Students' Conceptions of Inequalities Examining Committee: Chair: Michelle Nilson, Assistant Professor Peter Liljedahl, Associate Professor Senior Supervisor Rina Zazkis, Professor Committee Member Stephen Campbell, Associate Professor Internal/External Examiner David Pimm, Professor Emeritus, University ofAlberta External Examiner Date Defended/Approved: ii Declaration of Partial Copyright Licence The author, whose copyright is declared on the title page of this work, has granted to Simon Fraser University the right to lend this thesis, project or extended essay to users of the Simon Fraser University Library, and to make partial or single copies only for such users or in response to a request from the library of any other university, or other educational institution, on its own behalf or for one of its users. 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Simon Fraser University Library Simon Fraser University Burnaby, BC, Canada Last update: Spring 2010 Abstract Inequalities are vital in the production of mathematics. They are employed as specialized tools in the study of functions, in proving equalities, and in approximation or optimization studies, to enumerate only a few areas of mathematics where inequalities are put to work. The concept of inequality, however, is problematic for high school and university students alike. Moreover, the school curriculum seems disconnected from the role of inequalities in mathematics and mostly presents inequalities as a subsection of equations. The placement of inequalities in the school curriculum and the disconnect between school mathematics inequalities and a mathematician‟s approach to inequalities take the blame of research in mathematics education reporting on students‟ misconceptions when dealing with this concept. This study moves from the theory of misconceptions to a framework of undergraduate students‟ conceptions of inequalities. In an effort to learn more about what students „see‟ when dealing with inequalities, three research questions are pursued: What are undergraduate students‟ conceptions of inequalities? What influences the construction of the concept of inequalities? How can undergraduate students‟ conceptions of inequalities expand our insight into students‟ understanding of, and meaningful engagement with, inequalities? Data for this study was produced mostly through learner-generated examples of inequalities that satisfy certain conditions. The participants in the research were undergraduate students enrolled in two mathematics courses – a foundations of mathematics course and a precalculus course. The results of this research are five iii conceptions of inequalities. It was also found that the undergraduate students‟ conceptions of inequalities mostly occupy the lower regions of Tall‟s „Three Mental Worlds of Mathematics‟. The speculation is that what Tall calls the „met-befores‟ as well as what I call the „missed-befores‟ influence the construction of the concept of inequalities. Curriculum suggestions for preparing the ground for the work on and with inequalities are presented. This study contributes to ongoing research on mathematics concept formation. Keywords: inequality; conception; understanding; learner-generated examples; met- before; missed-before iv Dedication To the loving memory of my mom, Ana, my first partner in problem solving. To the person who declared me a mathematician in grade 5 and taught me how to prove by ‘reductio ad absurdum’ in grade 6, Professor Lazar Samoila, my middle school mathematics specialist teacher. Domnului Profesor Lazar Samoila, cel care m-a declarat matematician in clasa a 5-a iar in clasa a 6-a m-a invatat sa demonstrez prin reducere la absurd. v Acknowledgements My wholehearted thanks extend to my Senior Supervisor, Dr. Peter Liljedahl, whose passion for teaching, support of learning, and devotion to research in mathematics education were inspirational. Moreover, his guidance, encouragement, feedback, and insight throughout the completion of my program and dissertation were tremendously helpful. A special thanks and my gratitude to Dr. Rina Zazkis, who‟s timely and valuable feedback and suggestions, as well as her continuous support, helped me navigate my way around the program and find the focus of my work. I am so grateful to her for pushing me to take action and to make decisions in crucial moments. I would also like to extend my earnest appreciation to Dr. Ami Mamolo for her collegiality, support, and encouragement. A profound thank you goes to Dr. Tom Archibald for helping me locate the most valuable references for the historical account on inequalities – event that pushed my research and thinking into new and productive directions. The participants in this research also deserve my sincere recognition for their contribution in the two studies that are part of my dissertation. I am truly grateful to Dr. Stephen Campbell and Dr. David Pimm, my external examiners, for their willingness to take me to a defence on a short notice, for their insightful questions during the examination, and for their valuable feedback on my work. vi Table of Contents Approval ............................................................................................................................. ii Abstract .............................................................................................................................. iii Acknowledgements ............................................................................................................ vi Table of Contents .............................................................................................................. vii List of Figures ..................................................................................................................... x List of Tables .................................................................................................................... xii Chapter 1: Beginnings ...................................................................................................... 1 1.1 Introduction ................................................................................................................... 1 1.1.1 The Protagonist .................................................................................................... 1 1.1.2 Background .......................................................................................................... 4 1.1.3 Finding a Niche .................................................................................................... 8 1.2 Narrowing Down the Topic ........................................................................................ 10 1.3 Outline of the Dissertation .......................................................................................... 14 Chapter 2: Inequalities in Mathematics, History of Mathematics, and Mathematics Education Research.............................................................. 16 2.1 What Are Inequalities? ............................................................................................... 16 2.1.1 Definitions .......................................................................................................... 16 2.1.2 Inequality Manipulation ..................................................................................... 18 2.1.3 School Methods of Solving Inequalities ............................................................ 21 2.1.4 Famous Inequalities ........................................................................................... 26 2.1.5 A Mathematician‟s Approach to Inequalities: A „Loved‟ Inequality ................ 27 2.2 How Did Inequalities Come to Be? ............................................................................ 32 2.2.1 Why Educators and Researchers in Education Are Concerned with the History of Mathematics ...................................................................................... 34 2.2.2 The Evolution of the Inequality Concept ........................................................... 37 2.3 Where Are Inequalities Located in the K-12 Curriculum? ......................................... 57 2.3.1 Inequality in British Columbia Curriculum from K to University .................... 57 2.3.2 Inequalities in Romanian Curriculum ................................................................ 60 2.3.3 School Curriculum and Inequalities ................................................................... 65 Chapter 3: Inequalities in Mathematics Education Research .................................... 67 3.1 Inequalities in Mathematics Education ....................................................................... 68 3.1.1 A Call for Research on Inequalities Was Issued ................................................ 69 3.1.2 Results Are Presented ........................................................................................ 70 3.2 How to Teach Inequalities .......................................................................................... 71 3.3 How Students Perform on Inequalities ....................................................................... 78 3.4 Research Questions ..................................................................................................... 81 vii Chapter 4: In Pursuit of a Theoretical Framework .................................................... 83 4.1 Understanding Mathematics ....................................................................................... 84 4.2 Concept Image – Concept Definition ......................................................................... 87 4.3 APOS Theory .............................................................................................................. 89 4.4 Three Worlds of Mathematics .................................................................................... 90 Chapter 5: Methodology ................................................................................................. 94 5.1 Situating the Preliminary Work within the Discipline of Noticing ............................ 94 5.2 Examples and Learner-generated Examples in Learning Mathematics and Learning about Learning Mathematics ....................................................................... 98 5.3 Conception ................................................................................................................ 102 5.4 General Settings ........................................................................................................ 104 5.4.1 FAN X99 .......................................................................................................... 105 5.4.2 MATH 100 ....................................................................................................... 106 5.5 The Studies ................................................................................................................ 107 5.5.1 Study 1: Creating an Instrument to Analyse Undergraduate Students‟ Work on Inequalities ........................................................................................ 108 5.5.2 Study 2: Using the COIN Instrument to Evaluate Undergraduate Students‟ Conceptions of Inequalities ............................................................. 109 5.6 Summary ................................................................................................................... 109 Chapter 6: Study 1: The Emergence of Conceptions of Inequalities ........................ 110 6.1 Introduction ............................................................................................................... 110 6.2 Setting ....................................................................................................................... 112 6.3 The Task (T) ............................................................................................................. 114 6.4 Initial Results ............................................................................................................ 115 6.4.1 Looking at the Data – Level 1 Analysis ........................................................... 115 6.4.2 Looking at the Data – Level 2 Analysis ........................................................... 119 6.5 Sanding the Lens - The Second Iteration of the Task ............................................... 122 6.5.1 The Refined Task (RT), Results and Analysis ................................................. 124 6.6 Results: The COIN .................................................................................................... 131 6.6.1 Describing the Conceptions of Inequalities ..................................................... 133 6.7 Backwards Forwards ................................................................................................. 141 Chapter 7: Study 2: Undergraduate Students’ Conceptions of Inequalities ............ 143 7.1 Methodology: Portraying the Participants and Presenting the Data ......................... 144 7.1.1 Generate Examples Task (GET) ...................................................................... 146 7.1.2 FAN X99 Inequalities Survey (IS) .................................................................. 147 7.2 Data Analysis ............................................................................................................ 148 7.2.1 Initial Sorting of the Data from MATH 100 .................................................... 148 7.2.2 The Data from FAN X99 Seen Through the COIN ......................................... 158 7.2.3 The Data from MATH 100 Seen Through the COIN ...................................... 164 7.3 Summary ................................................................................................................... 173 viii