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Tracing Middle School Students' Understanding Of 2008 KATHLEEN B. SHAY ALL RIGHTS RESERVED TRACING MIDDLE SCHOOL STUDENTS‟ UNDERSTANDING OF PROBABILITY: A LONGITUDINAL STUDY by KATHLEEN B. SHAY 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 Education written under the direction of Dr. Carolyn A. Maher and approved by _____________________________________ _____________________________________ _____________________________________ _____________________________________ _____________________________________ New Brunswick, New Jersey October, 2008 ABSTRACT OF THE DISSERTATION TRACING MIDDLE SCHOOL STUDENTS‟ UNDERSTANDING OF PROBABILITY: A LONGITUDINAL STUDY By Kathleen B. Shay Dissertation Director: Dr. Carolyn A. Maher This study traces the probabilistic reasoning of five students from an urban middle school who attended an after-school mathematics enrichment program through grades 6, 7, and 8. Case study methodology is used to describe the ways of thinking and development of ideas of these students as they were presented with open-ended tasks intended to engage them in building ideas about chance. The tasks called for the students to investigate dice games to determine whether or not they were fair, and to devise strategies to make the games fair. Students were encouraged to discuss their ideas and justify their conjectures in small groups and with the whole class. The data for this study come from videotape records of seven after-school sessions and interviews in the Rutgers Informal Mathematics Learning project (IML) during the spring of 2004 and 2005, when the students were in grade 6 and 7. The video data were transcribed and analyzed along with student work according to the model for studying the development of mathematical thinking proposed by Powell, Francisco, and Maher (2003). ii Analysis of the data revealed that students exhibited the use of common judgmental heuristics such as representativeness, availability, and the equiprobability bias. At least three of the students combined the representativeness heuristic with the outcome approach to create what I call the hybrid heuristic for chance events. The application of this heuristic to assessing the fairness of games is the belief that if either player is able to win a game, then the game must be fair. All of the students studied came to reject the idea that dice sums are equally likely. They reached conclusions based on both classical and experimental approaches. Each student produced a sample space or worked with a partner who did. Though small samples were used, all of the students used experimental data to inform or provide support for their conjectures about fairness. In grade 7, the question of whether permutations of dice outcomes should be counted as different events was raised repeatedly, and, despite persistent challenges and questions by graduate interns, the students did not change their beliefs about this issue. iii ACKNOWLEDGEMENTS I am deeply grateful to Dr. Carolyn A. Maher, my dissertation director and lifelong mentor, for her constant support and encouragement over these many years. My life has been enriched in so many ways by my association with her. To my committee, Dr. Joseph I. Naus, Dr. Arthur B. Powell, Dr. Harold B. Sackrowitz,, and Dr. Keith H. Weber, I greatly appreciate your participation in this endeavor. Having such esteemed scholars reading my work inspired me to strive for excellence. I am thankful to the students who participated in this project: Adanna, Alia, Brionna, Chanel, Chris, Danielle, Dante, David, Ian, Jerel, Justina, Kianja, Kori, Nia, and Terrill. Many thanks to Marjory Palius, Robert Sigley, and the staff at the Robert B. Davis Institute for Learning for your assistance. I am also indebted to Anoop Ahluwalia, who helped me to develop analytical codes, and Barbara Tozzi and Jim Neuberger, who reviewed my results. I hope to return the favor one day. Thank you to Christopher Beattys, Judith Leonard, and Jeremy Milonas for your help in verifying transcripts and reporting on the debriefing videos. Middlesex County College provided tuition support and granted me a sabbatical leave to work on my research, for which I am profoundly grateful. This work would not have been accomplished without the love and support of my husband, Jim Miller. Thank you, darling, for everything. iv DEDICATION for Fannie, Dottie, Dana, and Amy v TABLE OF CONTENTS ABSTRACT OF THE DISSERTATION ....................................................................................................... ii ACKNOWLEDGEMENTS ...........................................................................................................................iv DEDICATION ................................................................................................................................................ v LIST OF TABLES ...................................................................................................................................... viii LIST OF ILLUSTRATIONS .........................................................................................................................ix CHAPTER 1- INTRODUCTION ................................................................................................................... 1 1.1 THE IMPORTANCE OF LEARNING TO REASON PROBABILISTICALLY ....................................................... 1 1.2 CONCEPTIONS OF PROBABILITY ........................................................................................................... 2 1.3 THE PROBLEM ...................................................................................................................................... 4 1.4 PURPOSE OF THE STUDY AND RESEARCH QUESTIONS .......................................................................... 6 1.5 SIGNIFICANCE AND LIMITATIONS ......................................................................................................... 7 CHAPTER 2 - THEORETICAL FRAMEWORK AND LITERATURE REVIEW ....................................... 8 2.1 THEORETICAL FRAMEWORK .................................................................................................................. 8 2.1.1 Rutgers Longitudinal Study ......................................................................................................... 8 2.1.2 The Growth of Mathematical Understanding ............................................................................ 9 2.2 LITERATURE REVIEW .......................................................................................................................... 11 2.2.1 The Development of Probabilistic Reasoning ............................................................................ 11 2.2.2 Misconceptions ......................................................................................................................... 16 2.2.3 Effects of Instruction .................................................................................................................. 25 2.2.4 Assessment ................................................................................................................................. 49 2.2.5 Directions for Future Research ................................................................................................. 51 CHAPTER 3 – METHODOLOGY ............................................................................................................... 53 3.1 SETTING ............................................................................................................................................. 53 3.2 SAMPLE .............................................................................................................................................. 55 3.3 DATA COLLECTION ............................................................................................................................ 56 3.3.1 Observations .............................................................................................................................. 56 3.3.2 Documents ................................................................................................................................ 56 3.3.3 Interviews ................................................................................................................................... 56 3.4 DATA ANALYSIS ................................................................................................................................ 57 3.4.1 Video analysis ........................................................................................................................... 57 3.4.2 Coding ...................................................................................................................................... 59 3.4.3 Reporting Results ...................................................................................................................... 61 3.5 VALIDITY ........................................................................................................................................... 61 CHAPTER 4 - RESULTS ............................................................................................................................. 62 4.1 PROBABILITY SESSIONS AND INTERVIEWS IN GRADE 6........................................................................ 63 4.1.1 Activity 1- A Game With One Die .............................................................................................
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