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To Prove Or Disprove: the Use of Intuition and Analysis by Undergraduate Students to Decide on the Truth Value of Mathematical Statements and Construct Proofs and Counterexamples

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To Prove Or Disprove: the Use of Intuition and Analysis by Undergraduate Students to Decide on the Truth Value of Mathematical Statements and Construct Proofs and Counterexamples To Prove or Disprove: The Use of Intuition and Analysis by Undergraduate Students to Decide on the Truth Value of Mathematical Statements and Construct Proofs and Counterexamples A dissertation presented to the faculty of The Patton College of Education of Ohio University In partial fulfillment of the requirements for the degree Doctor of Philosophy Kelly M. Bubp December 2014 © 2014 Kelly M. Bubp. All Rights Reserved. 2 This dissertation titled To Prove or Disprove: The Use of Intuition and Analysis by Undergraduate Students to Decide on the Truth Value of Mathematical Statements and Construct Proofs and Counterexamples by KELLY M. BUBP has been approved for the Department of Teacher Education and The Patton College of Education by Robert M. Klein Associate Professor of Mathematics Renée A. Middleton Dean, The Patton College of Education 3 Abstract BUBP, KELLY M., Ph.D., December 2014, Mathematics Education To Prove or Disprove: The Use of Intuition and Analysis by Undergraduate Students to Decide on the Truth Value of Mathematical Statements and Construct Proofs and Counterexamples Director of Dissertation: Robert M. Klein Deciding on the truth value of mathematical statements is an essential aspect of mathematical practice in which students are rarely engaged. This study explored undergraduate students’ approaches to mathematical statements with unknown truth values. The research questions were 1. In what ways and to what extent do students use intuition and analysis to decide on the truth value of mathematical statements? 2. What are the connections between students’ process of deciding on the truth value of mathematical statements and their ability to construct associated proofs and counterexamples? 3. What types of systematic intuitive, mathematical, and logical errors do students make during the proving process, and what is the impact of these errors on the proving process? Clinical task-based interviews utilizing the think-aloud method revealed students’ reasoning processes in depth. Twelve undergraduate students each completed four mathematical tasks requiring them to decide on the truth value of a statement and prove or disprove it accordingly. Through analysis of the data, I developed a framework for 4 distinguishing among types of reasoning based on their cognitive and mathematical properties. The framework identifies four distinct categories of reasoning – intuitive, semantic-empirical, semantic-deductive, and syntactic – each with subcategories. The students in this study used all four types of reasoning for deciding on the truth value of the statements in the tasks. Their use of semantic-deductive and syntactic reasoning mirrored mathematicians’ use of these reasoning types for decision-making. With the exception of one task, the students’ decision-making and construction processes were generally connected. Connections in which the construction process was based on decision-making process mostly facilitated proving. However, simultaneous decision- making and construction processes often led to overturned decisions. Regarding intuitive decision-making, only property-based intuitive decisions were connected to the corresponding construction process. The students in this study made numerous systematic mathematical and logical errors, but systematic intuitive errors were limited and occurred on only one task. The systematic conceptual misunderstandings surrounding the concept of function are troubling due to the centrality of this concept to mathematics. Few errors were overcome, but a certain level of uncertainty may aid students in overcoming logical errors. 5 Dedication To my family Your unconditional love and support makes all things possible. 6 Acknowledgments There are several people who I would like to thank for their support during this journey. I would like to thank my dissertation committee members: Dr. Robert M. Klein, Dr. Allyson Hallman-Thrasher, Dr. Jeffery Connor, and Dr. Eugene Geist for their thoughtful suggestions, kind support, and willingness to be a part of this adventure. Special thanks to Dr. Robert M. Klein who picked me up from near disaster and helped me continue forward with this dissertation. You were there for me when I needed you, but trusted me enough to pursue my own pathway. Special thanks to Dr. Allyson Hallman-Thrasher. I would never have made it through data analysis without you. You supported me when I was completely crazy and remained my constant cheerleader. Thanks to Dr. Gregory D. Foley for your ongoing support, encouragement to explore professional opportunities, and incredible advice on writing. Thank you to my family and friends who have offered their ongoing support despite the fact that I have neglected them terribly while pursuing this dream. I hope you all know that I would not have been able to achieve this without you, and the achievement would be meaningless without all of you in my life. Thank you especially to Sam, the love of my life. This work was supported by The Patton College of Education Graduate Study and Educational Research Fund. 7 Table of Contents Page Abstract ............................................................................................................................... 3 Dedication ........................................................................................................................... 5 Acknowledgments............................................................................................................... 6 List of Tables ...................................................................................................................... 9 List of Figures ................................................................................................................... 10 Chapter 1: Introduction ..................................................................................................... 13 Problem Statement and Research Questions ................................................................ 16 Significance .................................................................................................................. 17 Definition of Terms ...................................................................................................... 19 Organization of the Chapters ........................................................................................ 21 Chapter 2: Literature Review ............................................................................................ 23 Intuition ......................................................................................................................... 23 Analysis ........................................................................................................................ 31 Deciding on the Truth Value of a Mathematical Statement ......................................... 41 Summary ....................................................................................................................... 53 Chapter 3: Research Methods ........................................................................................... 54 Clinical Task-Based Interviews .................................................................................... 55 The Think-Aloud Method ............................................................................................. 56 Limitations of Interview Methods ................................................................................ 58 Research Questions ....................................................................................................... 61 Reflexivity: Personal Perspective ................................................................................. 61 Sampling Procedures .................................................................................................... 67 Data Collection ............................................................................................................. 72 Interview Procedures .................................................................................................... 75 Data Analysis ................................................................................................................ 81 Credibility and Trustworthiness .................................................................................... 90 Chapter 4: Results and Discussion .................................................................................... 92 Monotonicity Task ........................................................................................................ 93 8 Composite Function Task ........................................................................................... 124 Injective Function Task .............................................................................................. 147 Global Maximum Task ............................................................................................... 170 Discussion ................................................................................................................... 201 Chapter 5: Conclusions ................................................................................................... 218 Conclusions With Respect to RQ1 ............................................................................. 221 Conclusions With Respect to RQ2 ............................................................................. 227 Conclusions With Respect to RQ3 ............................................................................
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