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Students' Logical Reasoning and Mathematical Proving

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Students' Logical Reasoning and Mathematical Proving STUDENTS’ LOGICAL REASONING AND MATHEMATICAL PROVING OF IMPLICATIONS By KoSze Lee A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Educational Psychology and Educational Technology 2011 ABSTRACT STUDENTS’ LOGICAL REASONING AND MATHEMATICAL PROVING OF IMPLICATIONS By KoSze Lee Students’ difficulties in reasoning with logical implication and mathematical proving have been documented widely (Healy & Hoyles, 2000; Knuth, Choppin, & Bieda, 2009). Review of the educational and cognitive science studies of students’ reasoning with logical implications and mathematical proving have revealed that their lack of cognizance of counterexamples might be a crucial factor. This study examined the role of logic training and counterexample in enhancing students’ logical reasoning and various aspects of mathematical proving, namely, Proof Construction, Proof Validation and Knowledge of Proof Method. In particular, the study hypothesized that logic training emphasizing counterexamples was better able to improve students’ reasoning of logical implications as well as mathematical proving, in comparison to the other two approaches emphasizing rule violations and truth tables. Using a pretest-intervention-posttest experimental design (3 conditions by 2 test trials), students' written and interview data (N = 60) were collected from three Singapore school sites, each over a four-day contact period (including the pretest and posttest administration days). Experimental results showed that logic training emphasizing counterexamples was significantly more effective in improving students’ logical reasoning of implication than the other two approaches (p = .0007, large effect size). However, logic training was only similarly effective or ineffective in improving some aspects of students’ mathematical proving across conditions. Interview findings from 12 selected students’ works on a new proving task conjectured that students improved their use of deductive inferences in all aspects of mathematical proving after logic training. Moreover, their successes in constructing mathematical proofs were also subjected to two conjectured factors, students’ interpretation of implication and mathematical knowledge. These findings suggested the importance of logic training and counterexamples in mathematics education and pointed to further inquiry about the role of students’ interpretation of implications and mathematical knowledge in mathematical reasoning and proving. ACKNOWLEDGEMENTS From the bottom of my heart, I want to thank the follow people. For without them, completion of this dissertation and the doctoral program is not possible. I owe my accomplishments to my wife, PohChing Chia, and my two wonderful kids, Johan Lee and Joanne Lee. PohChing is a constant source of love and support to me. She has never doubted my abilities and what I can accomplish. Johan and Joanne have inspired me with their joy and contentment in life. They are the wind beneath my wings. I am greatly indebted to my advisor and mentor, Dr John (Jack) P. Smith III, for his insights and wisdom in guiding my scholarly journey. In many ways, he is a good model of a university faculty for me. Lunch conversations with him were always enriching and enjoyable. My special thanks also go to committee members, Dr Kelly Mix, Dr Sharon Senk and Dr Raven McCrory. They were committed and supportive in my dissertation, and provided valuable inputs. I also acknowledge the teachers and students who participated in this study. Their willingness to commit their time and efforts to the training sessions made this study possible. I also want to thank Leo Chang, Lorraine Males, Aaron Mosier, Sasha Wang and Jungeun Park for their timely suggestions and support for my dissertation preparation. I want to express my gratitude to the colleagues of the Mathematics Learning Research Group (MLRG), uniquely here in Michigan State University. My heartfelt gratitude to Dan Ouyang and Sumei Wei, and John and Bonnie Bankson, for their generous hospitality and great friendship support while I was back to complete the study. Most importantly, glory be to God, who has blessed me and my family with His loving kindness during these years in U.S. iv CONTENTS LIST OF TABLES ......................................................................................................................... ix LIST OF FIGURES ....................................................................................................................... xi LIST OF EXCERPTS .................................................................................................................. xiii CHAPTER 1 INTRODUCTION .................................................................................................. 1 Background ..................................................................................................................................... 3 Mathematical Implications as Generalized Conditional .......................................................... 4 Students’ Difficulty with Logical Implication – a crucial barrier ............................................ 6 Effects of Instructions on Students’ Reasoning of Logical Implications ................................ 10 Effects of Logic Training on Students’ Mathematical Proving .............................................. 12 Students’ Difficulty with Mathematical Proving ..................................................................... 13 Aim of this study: In search for an effective logic training..................................................... 15 CHAPTER 2 THEORETICAL FRAMEWORK ........................................................................ 17 Mathematical Implication, its Logical Variants and Counterexamples ........................................ 17 Converse, Contrapositive and Negation of Implication ......................................................... 18 Conceptualization of Mathematical Proving Ability .................................................................... 19 Generalized Conditional and Mathematical Proof and Proving ................................................... 20 Definition of Mathematical Proofs ......................................................................................... 21 Proof Construction.................................................................................................................. 22 Proof Validation...................................................................................................................... 22 Knowledge of Proof Methods.................................................................................................. 23 Leveraging the use of Counterexamples in Reasoning and Proving of Mathematical Implications ....................................................................................................................................................... 24 Constrained Example Generation........................................................................................... 25 Enhancing CEG through task formulations ............................................................................ 26 The Research Questions ................................................................................................................ 27 CHAPTER 3 METHOD ............................................................................................................. 30 Subjects ......................................................................................................................................... 30 Design ........................................................................................................................................... 31 Materials ....................................................................................................................................... 34 Pre-test and Post-test instruments .......................................................................................... 34 Training materials .................................................................................................................. 38 Training Materials used in the Control condition .................................................................. 39 Training Materials used in the PO condition ......................................................................... 41 Training Materials used in the W condition ........................................................................... 44 Students’ reflection of learning ............................................................................................... 47 Materials for Post Study Interview ......................................................................................... 47 Procedure ...................................................................................................................................... 48 v Data Scoring and Coding .............................................................................................................. 50 Scoring of selection task items ................................................................................................ 51 Coding schemes for Students’ responses to proof items ......................................................... 51 Coding for Deductive-proof Construction .............................................................................. 52 Coding Proof-by-counterexamples Construction for mathematically
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