COGNITIVE AND AFFECTIVE COMPONENTS OF UNDERGRADUATE STUDENTS LEARNING HOW TO PROVE By Visala Rani Satyam A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of Program in Mathematics E ducation Doctor of Philosophy 2018 ABSTRACT COGNITIVE AND AFFECTIVE COMPONENTS OF UNDERGRADUATES LEARNING HOW TO PROVE By Visala Rani Satyam Students struggle with proving, a fundamental activity in upper - level undergraduate mathematics courses. Learning how to prove is a difficult transition for students, as they shift from largely computation - based to argument - based work. In response, mathematics departments h ave instituted courses, introduction or transition to proof, designed to help struggles, and some of their strategies at a given point in time, but we know little abo ut in this area, to follow students through the transition to proof. In addition, little is known about the affective side of proving (e.g., attitudes, beliefs, emoti cognitive processes while problem solving and their motivation to value and want to do mathematics. Understanding affective issues are important, as students consider their fu ture participation in mathematical work and communities. Positive experiences at transitional junctions, such as learning how to prove, are crucial for retention of students through the STEM (Science, Technology, Engineering, and Math ematics ) pipeline. Th e purpose of this work was to explore the cognitive and affective factors involved transition to proof course and what kinds of satisfying moments , i.e. positive emotion al reactions, they experienced. Four semi - structured interviews across a semester were conducted with eleven undergraduate students enrolled in a transition to proof course. The resulting data was analyzed using qualitative methods. Findings indicate that students showed growth in fluency, strategy use, and monitoring and judgement over time. Four developments were frequently observed across techniques, (2) awareness about how a solution attempt was going and managing that for their subsequent strategies, (3) intentional exploring and monitoring when unsure about what direction to pursue, and (4) checking examples in conjunction with other strategies as a way to become unstu ck. The variety of developments and the different ways in which they emerged is significant, because it confirms that multiple developments occur in different ways, strongly suggesting that there is no one path that students take through the transition to proof. without struggle, understanding, external validation, as well as interacting with others. A theory for how satisfying moments are elicited was proposed. Expectations and a sense of understanding and sense - making was also prominent. This work provides guidance for curriculum design of transition to proof courses, in just what makes a satisfying moment satisfying is helpful in thinking about how to construct mathematical tasks with opportunities fo r positive experiences with math. Copyright by VISALA RANI SATYAM 2018 v To the love of my life , no matter what m athematics . F or every person who believed in me and my mathematical endeavors, this is dedicated to you. vi ACKNOWLEDGMENTS This has been a long road. I knew I wanted to study mathematics since I was in four th grade and thought cross - canceling when multiplying fractions was the most amazing thing . T here are many people who have brought me then to where I am today. They say we stand but on the shoulder s of giants; the same can be said for our teachers. I thank the math teachers in my life: my fourth - grade teacher Ms. Drouin, my Intermediate Algebra II teacher Mr. Bradman (under whom I failed a math test for the first time due to the Boston Red Sox winni ng the ALCS), and my c alculus teacher Mr. Pasquale. In college: Ben Hescott for the warmth he brought to all things theoretical , Dr. Kim for her energy and that proof about Galois theory that was a true moment of mathematical beauty to me , and Professor Qu into for tea and cook ies and conversations about doing math long after . My ideals of who all of you. To Judah Schwartz, who introduc ed me to mathematics education as a field - he would set students off arguing agains t each other , like a wonderfully ornery grandpa . And t his list would not be complete without Mrs. Mooney , who while she was not my teac her, although she was a teacher (indeed vice - principal), encouraged me from my very fi rst high honor roll in middle school to her end. I was so fortunate to land at Michigan State University at the time I did and experience all the good things that came of my years there . I thank my affective research group (Eileen, Kevin, and Mi ssy) for helping me think about how to study th is crazy thing called emotions and my writing group for being the support I sorely needed my last semester: Amit, Andy, Amanda, Brittany, Jose, Merve, Valentin, and Younggon. vii In terms o f faculty, only time will tell me just what a bounty I had around me . I thank Russell Schwab for allowing me into the course. I am indebted to my committee - Mariana Levin, Shiv Karunakaran, Teena Gerhardt, and previously on my committ ee, Vinc e Melfi - for bein g so thoughtful and often offering the words to describe what was in my head that I myself struggled to find . T o my advisor, Jack S m ith (our fearless leader) t hank you for believing in my work when I do not. I think you read me accurately from our very first meeting and have support ed me as an individual , like for the both of us. But in all seriousness, a heartfelt thank you to you. As a factor in my coming to Michigan, I thank fi gure skating for providing me escape as needed and for friends from that part of my l ife who were outlets as my precise mathematical career aspirations shifted through college . To Big Charlie and Jacqui White, for picking me up the first time I arrived i n Michigan for a grad school visit and from there on became part of my Michigan support system. Our long conversations about passio ns and success a nd failure and how to find joy in the grind stay with me . But this dissertation especially would not have be en possible without my friends around me . Tha nk you to AJ, Amy, and Farheen for long phone calls and texts that felt so c o mfortable but also giving me solutions . To Lu cy, whose joy for my joy of m ath is i nspiring in and of itself ; your cards mean so much to me! I am so grateful for my fellow PRIME graduate students across all the years and staff ; this is a communit y and s upport that is not to be easily imitated elsewhere . ( I include all the math grad stud ents w ho are basically math ed also in this as well ) To Amy, for being my job search and dissertation buddy, Jill for being my dilemma - mas ter through grad school and essentially teaching me qualitative methods , Joanne for mentoring me through viii each step of the dissertation , Kate for our weekly Thursday lunches and asking about things in my life and hearing me , and Molade for our walks to cardio kickboxing and Whats A pp v oice notes - you always know jus t what to say to comfort me. And now for the 5 th floor of Wells Hall (and co) , who steadily straight up friends over the years : Allison for her I bel her drop - by o ffice visits and walks, Fatemah and Loura, Marowan, and Yammn for being the best neighbo rs one could ask for s more rejuvenating for the soul while dissertating than a hug from a child?), Moll ee for reaching out and being kind beyond words , Tyler for always making sure to ask me how the big things in my life were going, Reshma for late night talks and godcat cuddles and check - , Sarah for walks to IM East and salted caramel mochas when I needed them most, and Hitesh for our morning walks from CVS to campus that I cannot imagine this year without for being there. I thank extended family and family friends who supported me in math over the years : my brother Ravi, Anand, Hima, Rupa and last but not least, my uncle SN Satyam, who gifted me books about space and math that nurtured my fledgling dreams. In the e nd, I say t hank you to my parents, Venkata and Sujatha Satyam, fo r financially supporting me all through school and for ensuring that this was my calling. The truth is even having these dreams would not have been possible without your hard work and struggle to come to this country and pr ovide this level of security in life for me to do what I want . I am grateful. I end then wit h a nod to all my participants , the true sta rs of this work. T hrough our conversations across the interviews, I reflected and learned so much about my own collegiate experience with mathematics. The relationships culti vated in that space are the real products of this dissertation. ix PREFAC E It is a funny thing, where we get ideas from how long they sit with us unbeknownst or when the same kind of question about the nature of things emerges in varying contexts. While working on this dissertation, I realized both of these were true. I watch a lot of figure skat ing. Like many others, I watch for those special moments when the crowd would go crazy and ins tantly rise to their feet . I often wondered What guarantees a standing ovation? Why is it that some performances end with good applause but othe rs pull the crowd out of their chairs , as though electric? Perhaps it had something to do with the rise and fall of the music or the way a skater hit certain movements or maybe s feet. Could one purposely design for this ? Without realizing it at the time, I seek to answer the same question in this dissertation but with an eye toward mathematics: What kinds of mathematical experiences bring about an internal standing ovation for an individual, i.e. feelings of satisfaction and elation? Are there features in common across individuals when these events happen and if so, can we as instructors intentionally create learning opportunities with these features embedded? This may be playing with fire; humans are delicate things and their emotions even more so, nowhere nearly deterministic as to be easily managed. But in a simila r way to how Tolstoy wrote that experiences have threads in common too . The above explains the genesis of the affective side of this work. The cognitive side proving changed as they were learning, not x just at one given snapshot in time. But soon into conducting interviews with students, this question was on my mind: why do some people interpret failure negatively and others positively? Throughout these intervi ews, some students were dejected about their solutions when they felt they were wrong. But other students reacted to getting my problems wrong by asking me how it worked and outright saying that now they knew how to do it in the future. They genuinely saw failure as learning opportunities. I was shocked. Co uld this difference be the key ? honest, peruse) these chapters ahead. While this was not the research que stion I set o ut to answer and there may not be enough evidence to truly draw claims, I think it is the deep question at the heart of a ll this. I do believe that math educators, in all their forms, want the same things for their students, to grow and to feel good about math. I hope this work spurs some thoughts and in keepi ng with the theme, feelings too - on these basic goals. xi TABLE OF CONTENTS LIST OF TABLES ................................ ................................ ................................ ................................ ................................ ........ xv LIST OF FIGURES ................................ ................................ ................................ ................................ ................................ .... xvi KEY TO AB BREVIATIONS ................................ ................................ ................................ ................................ ................. xviii CHAPTER 1: Introduction ................................ ................................ ................................ ................................ ....................... 1 Research Questions ................................ ................................ ................................ ................................ ............................... 3 CHAPTER 2: Literature Review ................................ ................................ ................................ ................................ ............ 4 Proof and Proving ................................ ................................ ................................ ................................ ................................ .. 4 Proof and Mathematics Education ................................ ................................ ................................ ............................ 6 Proving as Problem Solving ................................ ................................ ................................ ................................ .......... 9 Transition to Proof Courses ................................ ................................ ................................ ................................ ...... 10 Multiple transitions taking place. ................................ ................................ ................................ ..................... 11 Affect ................................ ................................ ................................ ................................ ................................ ........................ 11 Major Types of Affect in Relation to Mathematics Education ................................ ................................ .... 12 Attitudes ................................ ................................ ................................ ................................ ................................ ...... 12 Beliefs. ................................ ................................ ................................ ................................ ................................ ........... 12 Emotions. ................................ ................................ ................................ ................................ ................................ ..... 13 Affective Work in Mathematics Education ................................ ................................ ................................ ......... 13 Early work on attitudes ................................ ................................ ................................ ................................ ........ 13 ................................ ................................ ................................ ............................. 14 Studies of emotion as rare ................................ ................................ ................................ ................................ ... 14 Why care about studying emotions? ................................ ................................ ................................ ............... 14 CHAPTER 3: Conceptual Framing ................................ ................................ ................................ ................................ ..... 16 Conceptualizing Proving as Problem Solving ................................ ................................ ................................ ......... 16 Conceptualizing Development ................................ ................................ ................................ ................................ ...... 17 Defining Satisfying Moments ................................ ................................ ................................ ................................ ......... 18 Related Constructs ................................ ................................ ................................ ................................ ........................ 19 Mathematical beauty ................................ ................................ ................................ ................................ .............. 19 Aha moments ................................ ................................ ................................ ................................ ............................. 20 Why create a new construct? ................................ ................................ ................................ ............................. 20 CHAPTER 4: Method ................................ ................................ ................................ ................................ ............................... 22 Study Context: Transition to Proof Course ................................ ................................ ................................ .............. 22 Content ................................ ................................ ................................ ................................ ................................ ............... 22 Course Design ................................ ................................ ................................ ................................ ................................ .. 23 Researcher Positionality: My Dual Role as Researcher and Teaching Assistant ............................... 24 Description of Instructors ................................ ................................ ................................ ................................ .......... 25 Pa rticipants ................................ ................................ ................................ ................................ ................................ ........... 26 Recruitment and Selection of Participants ................................ ................................ ................................ ......... 26 Descriptions of Participants ................................ ................................ ................................ ................................ ...... 28 Data Sources ................................ ................................ ................................ ................................ ................................ .......... 30 xii First Half of In terview: Proving ................................ ................................ ................................ ............................... 31 Proof construction tasks: Selection. ................................ ................................ ................................ ................ 32 Think - aloud. ................................ ................................ ................................ ................................ ............................... 34 Ensuring student ................................ ................................ ........................ 37 ................................ ...................... 37 Emotion words. ................................ ................................ ................................ ................................ ................... 38 Emotion graphs ................................ ................................ ................................ ................................ ................... 39 Second Half of Interview: Satisfying Moments ................................ ................................ ................................ . 40 Self - report of satisfying moments. ................................ ................................ ................................ ................... 40 Emotion graphs as recall. ................................ ................................ ................................ ................................ ..... 41 Interview notes. ................................ ................................ ................................ ................................ ........................ 43 Pilot Study ................................ ................................ ................................ ................................ ................................ .............. 43 First and Second Rounds of Pilot Data Collection ................................ ................................ ........................... 44 Methodological issues. ................................ ................................ ................................ ................................ ........... 45 Logistical issues. ................................ ................................ ................................ ................................ ....................... 45 Third Round of Pilot Data Collection: Re - Design of Proving Section. ................................ ..................... 46 Additional Changes from Data Collection ................................ ................................ ................................ ........... 46 CHAPTER 5: Data Analysis ................................ ................................ ................................ ................................ .................. 48 Re ................................ ................................ ................. 49 Research Question 1a ................................ ................................ ................................ ................................ ................... 50 Research Question 1b ................................ ................................ ................................ ................................ .................. 51 Searchin g for Usable Analytic Frameworks ................................ ................................ ................................ ....... 51 Why existing problem solving frameworks proved problematic. ................................ ...................... 51 ................................ .................... 52 Operationalizing stuckness. ................................ ................................ ................................ ................................ 52 Analysis Process ................................ ................................ ................................ ................................ ............................. 53 Difficulties in Analysis ................................ ................................ ................................ ................................ ................. 53 Issues with tasks. ................................ ................................ ................................ ................................ ..................... 53 Same or different stuck points? ................................ ................................ ................................ ......................... 54 Operationalizing strategy ................................ ................................ ................................ ................................ ..... 54 Research Question 2: Kinds of Satisfying Moments ................................ ................................ ............................ 55 Overview of Constructs and Data Analysis for Satisfying Moments ................................ ........................ 55 Assumptions ................................ ................................ ................................ ................................ ................................ ..... 56 Data Sources ................................ ................................ ................................ ................................ ................................ ..... 57 Data Preparation ................................ ................................ ................................ ................................ ............................ 57 Distilling audio to short descriptions ................................ ................................ ................................ ............. 57 Probing about singular satisfying moments ................................ ................................ ................................ 58 Why not code transcripts directly? ................................ ................................ ................................ .................. 58 Data cleaning: Excluded data and separating out independent instances ................................ ..... 59 Data Analysis: Creation of Coding Scheme ................................ ................................ ................................ ......... 60 Coding. ................................ ................................ ................................ ................................ ................................ .......... 60 Testing codes from the literature ................................ ................................ ................................ ..................... 61 Difficulties in coding scheme creation ................................ ................................ ................................ ............ 62 ................................ ................................ ................................ ........ 63 Indicators of Being Stuck ................................ ................................ ................................ ................................ ................. 64 St ................................ ................................ ................................ ..................... 64 Cross - Individual Developments ................................ ................................ ................................ ................................ ... 68 Development A: Changes in Choosing a Proof Technique ................................ ................................ ........... 68 xiii Example A1: Favoring one technique. ................................ ................................ ................................ ............ 69 Summary. ................................ ................................ ................................ ................................ ............................... 72 Example A2: Recognizing advantages of a technique, independent of statement. ..................... 72 Summary. ................................ ................................ ................................ ................................ ............................... 77 Comparing developments in choice of proof techniques. ................................ ................................ ...... 77 Development B: Assessing How the Solution Attempt Is Going and Harnessing It .......................... 78 Example B1: Intuitive awareness lead to restart. ................................ ................................ ...................... 79 Summary. ................................ ................................ ................................ ................................ ............................... 81 Example B2: Awareness lead to finding new strategy. ................................ ................................ ........... 81 Summary. ................................ ................................ ................................ ................................ ............................... 84 Example B3: Aware but stayed on same solution path. ................................ ................................ .......... 84 Summary. ................................ ................................ ................................ ................................ ............................... 86 Comparing developments in awareness and using it. ................................ ................................ ............. 86 Development C: Exploring and Monitoring ................................ ................................ ................................ ........ 87 Summary. ................................ ................................ ................................ ................................ ................................ .... 90 Development D: Using Examples to Get Unstuck ................................ ................................ ............................. 91 Summary. ................................ ................................ ................................ ................................ ................................ .... 94 Longitudinal Case: Leonhard ................................ ................................ ................................ ................................ ......... 95 ................................ ................................ 96 ................................ ................................ ................................ .................. 101 Developments with Limited Data ................................ ................................ ................................ ............................. 103 Development E: Attending to the Goal ................................ ................................ ................................ .............. 103 Development F: More Systematic Ways of Approaching the Statement ................................ ............ 105 Developments Across All Participants ................................ ................................ ................................ ................... 107 Conclusions ................................ ................................ ................................ ................................ ................................ ......... 109 CHAPTER 7: On the Nature of Satisfying Moments ................................ ................................ ............................... 111 Identification and Description of Codes ................................ ................................ ................................ ................ 111 External Codes ................................ ................................ ................................ ................................ ............................. 114 Completing Task(s). ................................ ................................ ................................ ................................ ............. 114 Overcoming Challenge(s): Present and Comparison to Past. ................................ ............................ 114 Partial Progress. ................................ ................................ ................................ ................................ .................... 115 External V alidation: Grades, Assessments, and Authority Figures ................................ ................. 115 Internal Codes ................................ ................................ ................................ ................................ .............................. 117 Understanding: General, Aha Moment, and Seeing the Solution. ................................ .................... 117 Internal Conviction ................................ ................................ ................................ ................................ .............. 119 On my own. ................................ ................................ ................................ ................................ .............................. 120 Properties of Mathematics Codes ................................ ................................ ................................ ........................ 121 Useful. ................................ ................................ ................................ ................................ ................................ ........ 121 Simple. ................................ ................................ ................................ ................................ ................................ ....... 121 Interactions with People Codes ................................ ................................ ................................ ............................ 122 Social Comparison. ................................ ................................ ................................ ................................ ............... 122 Friendly Interactions ................................ ................................ ................................ ................................ ........... 123 Applying the Coding Scheme to the Data ................................ ................................ ................................ .............. 123 Overcoming Challenge and Completing Task(s) Account for a Large Portion of Data ................. 125 External Validation vs. Understanding: Unexpected Results ................................ ................................ .. 125 Interactions wit h People: Friendly Interactions ................................ ................................ ........................... 127 On My Own & Simple ................................ ................................ ................................ ................................ ................. 128 Data that Did Not Fit into the Coding Scheme ................................ ................................ ................................ 129 Partial Progress as Rare ................................ ................................ ................................ ................................ ........... 130 xiv Combinations of Kinds ................................ ................................ ................................ ................................ .................. 131 Completing Task(s) + Simple ................................ ................................ ................................ ................................ . 131 Overcoming Challenge + Understanding ................................ ................................ ................................ .......... 132 Friendly Interactions + Understanding ................................ ................................ ................................ ............. 133 Clustering of Satisfying Moments by Individuals ................................ ................................ .............................. 135 Case A: Students Who Enjoy Completing Task(s) ................................ ................................ ......................... 136 Case B: Students Who Enjoy Overcoming Challenges(s) ................................ ................................ ........... 136 Case C: Students Who Enjoy Understanding ................................ ................................ ................................ ... 137 Case D: Students Who Enjoy Overcoming Challenges(s) and Understanding ................................ . 137 Conclusions ................................ ................................ ................................ ................................ ................................ ......... 138 CHAPTER 8: Discussion ................................ ................................ ................................ ................................ ..................... 140 ................................ ................................ ......... 140 Findings Related to Satisfying Moments ................................ ................................ ................................ ............... 142 Findings Related to Connections Between Proving and Emotion ................................ .............................. 146 Implications ................................ ................................ ................................ ................................ ................................ ........ 147 Theoretical Is sues ................................ ................................ ................................ ................................ ....................... 147 Formal - rhetorical aspects of proving may actually be problem - centered. ................................ . 147 Noticing as crucial to proving. ................................ ................................ ................................ ......................... 148 Role of confidence in proving and its implications. ................................ ................................ ............... 148 Algorithms as satisfying. ................................ ................................ ................................ ................................ ... 150 Methodological Issues ................................ ................................ ................................ ................................ ............... 150 Studying impasses without intruding? ................................ ................................ ................................ ....... 150 Emotion graphs. ................................ ................................ ................................ ................................ .................... 151 How to get students to stay with a series of interviews. ................................ ................................ ..... 151 Pedagogical Issues ................................ ................................ ................................ ................................ ...................... 152 Curriculum design of undergraduate transition to proof courses. ................................ ................. 152 Noticing when students are and are not stuck in the classroom. ................................ .................... 153 Interview as a vehicle for reflection and rendering knowledge. ................................ ..................... 153 Alternative Explanation(s) ................................ ................................ ................................ ................................ .......... 155 Limitations and Factors Influencing the Findings ................................ ................................ ............................. 155 Suggestions for Future Research ................................ ................................ ................................ .............................. 158 Empirical Work ................................ ................................ ................................ ................................ ............................ 159 Theoretical Work ................................ ................................ ................................ ................................ ........................ 159 Proving process frameworks. ................................ ................................ ................................ .......................... 159 Challenge of studying phenomena by looking at individual components. ................................ .. 160 Conclusions ................................ ................................ ................................ ................................ ................................ ......... 161 APPENDICES ................................ ................................ ................................ ................................ ................................ ........... 163 APPENDIX A: Proof Tasks ................................ ................................ ................................ ................................ ............ 164 APPENDIX B: Interview Protocols ................................ ................................ ................................ ............................ 167 APPENDIX C: Emotion Graph ................................ ................................ ................................ ................................ ..... 174 REFERENCES ................................ ................................ ................................ ................................ ................................ .......... 175 xv LIST OF TABLES Table 4.1: Schedule of Weekly Content for Transition to Proof Course ................................ ........ 23 Ta ble 4.2: Background of Participants ................................ ................................ ................................ ........ 26 Table 4.3: Proof Construction Tasks by Interview ................................ ................................ ................. 32 Table 5.1: Data Use Matrix ................................ ................................ ................................ ............................... 49 Table 6.1: Proof Construction Tasks By Interview ................................ ................................ ................ 63 Table 6.2: Rubric for Scoring Performance on Proof Tasks ................................ ............................... 66 Table 6.3: Performance across Proof Tasks by Parti cipant ................................ ................................ 67 Table 6.4: Developments in Proving, By Participant ................................ ................................ .......... 108 Table 7.1: Coding Scheme for Kinds of Satisfying Moments ................................ ........................... 112 Table 7.2: Frequency and Percentage of Satisfying Moments by Kind ................................ ....... 124 Table 7.3: Co - occurrence of Codes with Completing Task(s) & Overcoming Challenge(s) 131 Tabl e 7.4: Co - occurrence of Codes with Completing Task(s) and Challenge(s) ..................... 132 Table 7.5: Co - occurrence of Interactions with People w ith a Selection of Codes ................... 134 Table 7.6: Kinds of Satisfying Moments by Participant ................................ ................................ ..... 135 xvi LIST OF FIGURES napshots ................................ ................................ ....................... 18 Figure 3.2. How satisfying moments relate to existing concepts regarding intense positive emotion. ................................ ................................ ................................ ................................ ................................ ... 18 Figure 4.1 . Representation of data sources (by color) within each interview ............................ 31 Figure 4.2. Physical arrangement of the 11 affective words for the Emotion Word Task ..... 38 Figure 4.3. Example of an emotion graph for a proof task ................................ ................................ .. 39 ........ 42 Figure 6.1 Task 1 ................................ ............... 69 Task 1 ................................ ..... 71 - Task 1 ................................ ................... 73 Task 1 ................................ ...... 74 Task 1 ................................ ................................ ............. 75 Figure 6.6. Stages of development in how students choose proof techniques to pursue ...... 78 Task 2 (statement provided) ................ 79 t at Interview 2 Task 1 ................................ ............................... 82 he was unsatisfied with his contrapositive proof. ................................ ................................ .................. 83 Figure 6.10. Statement of Interview 3 Task 1 ................................ ................................ ....................... 84 Task 1. She assumed one variable was even and the other two were odd but it lead to a statement that did not h elp her. ........ 84 Task 1 (statement provided). The last few lines are where she experienced multiple stops and starts. ................................ ................................ ......... 85 Figure 6.13. Task 2 ................................ ........................ 89 xvii Figure 6.14. Task 2, sans the final lines. ........... 90 Task 1, after thinking of example exercises from class. ................................ ................................ ................................ ................................ ........... 92 Task 1. He reasoned out loud about the implications of the last line. ................................ ................................ ................................ ............................. 93 Figure 6.17. Timothy weighing proof techniques on Interview 4 Task 2 ................................ . 94 Task 1 ................................ .... 96 work on Interview 1 Task 2. ................................ ................................ ................................ ........................ 96 Task 1 ................................ ................................ ........ 98 Figure 6.21. Beginning of ork on Interview 4 - Task 2 ................................ ........... 100 Task 1 (top left), Interview 3 Task 2 (top right), Interview 4 Task 1 (bottom left), Interview 4 Task 2 (bottom right). His graphs indicated high positive emotions about his work on Interview 3 and Interview 4 Task 2 but his solutions were incorrect. ................................ ................................ .............................. 102 Figure 6.23. Statement for Interview 2 Task 2 ................................ ................................ .................. 103 Task 2. He reached a true statement and thought he had shown the claim. ................................ ................................ ................................ ............... 104 to start problems, from Interviews 1 through Interview 4. He identified the assumption and conclusion of the statement, using parentheses. ................................ ................................ ................................ ................................ ........................ 106 Figure 6.26. Three Categories of Proving Development ................................ ................................ ... 109 Figure 7.1. Representation of how the Understanding sub - codes are related. Aha Moment and Seeing the Solution are different constructs but can overlap. General accounts for all other kinds of understanding. ................................ ................................ ................................ ..................... 1 18 Figure 8.1. Possible model for how satisfying moments occur as a phenomenon. Situations (independent variables) give rise to the feeling of satisfaction (dependent variable). On my own may act as a moderating variable, in that it strengthens the elicitation of satisfaction. Understanding and expectations may act as mediating variables, explaining why tho se situations elicit satisfaction. Note the variable - paradigm is used for illustrative purposes here. ................................ ................................ ................................ ................................ ................................ ....... 145 xviii KEY TO ABBREVIATIONS MLC Math Learning Center RQ Research Question [word] A dditional word(s) for sake of clarity, not said by interviewee 1 CHAPTER 1: Introduction The transition to proof is difficult for undergraduate students (Moore, 1994; Selden & Selden, 1987) . Stu dents struggle with learning how to prove (Iannone & Inglis, 2010; Selden & Selden , 2013 ) . (Schoenfeld, 199 2) to now writing arguments and justifying said answers. Researchers have identified the types of errors students make ( Selden & Selden, 1987) and t heir struggle s ( Harel & Sowder, 1998; Selden & Selden, 2003 ). C ommon proving error s i examples, notation and symbols, quantifiers , and general logic ( Epp, 2003 ; Selden & Selden, 1987) . S tudents struggle with larger issues as well, such as giving empirical rather than de ductive arguments (Harel & Sowder, 20 07) and having difficulty writing formal arguments ( Alcock & Weber, 2010 ) . Another strand of research has focused on proving process (Karunakaran, 2014; Savic, 2012 ) . We know students struggle s and their st rategies while provi ng at singular points in time, but few have looked at how these strategies change over the course of the learning process. Much existing research is about whether students understand logic and proof techniques , such as contradiction and induction . One way to interpret t his work is that gaining the ability to prove statements is about the accumulation of individual techniques . But development is not necessarily about accumulating competencies ; a s Piaget (1964) said, "For some psychologists, development is red uced to a series of specific learned items, and development is thus the sum, the culmination o f this series of specific items. I think this is an atomistic view which deforms the real state of things" (p. 38). T hinking about proving 2 as the sum of skills an d assessing whether or not students have those skills may not be We may be able to tentatively assess their proof competencies at certain points in time, but we do not yet know how students put all these pieces together while they are learning how to prove nor the order in which these proving for how they learn how to prove, as a mathematical activity. There is a n eed for longitudinal work (Smith, Levi n, Bae, Satyam, & Voogt , 2017 ; Bae, Smith, Levin, Satyam, & Voogt, 2018 ), for having frequent interactions with the same students over a reasonable interval to see how they change I n addition, the affectiv e side of learning has largely been understudied in mathematics education teaching and learning ( McLeod, 1992; Sinclair, 2006) . Affect plays a central role in mat hematics learning but especially in problem solving ( McLeod, 1994; Silver, 1985) . Affect can also influence cognitive processes, such as knowing what to do next while problem solving (McLeod, 1988) . McLeod (1992) claimed that any research can be strengthened by examining both affective and cognitive issues together . Within the context of proof learning, Selden & Selden (2013) , called for more research on how their problem solving and proving work . Positive affective moments may provide the intrinsic motivation then (Middleton & Spanias, 1999) for students to con tinue doing and valuing mathematics. Moments of positive affect are therefore educationally desirable. In summary, the field is currently missing developmental and affective examinations of ho w students learn how to prove . How do students learn how to pr ove? Moreover, is this an activity they wish to do more of? 3 Research Questions In response to this gap , the purpose of this study is to examine both the cognitive and affective components involved in how undergraduates learn how to prove. The research qu estions are: 1. How do es undergraduate students' proving develop over the duration of a transition to proof class ? 2. What kinds of satisfying moments do undergraduate students have during the transition to proof ? This study contributes to research and practice about mathematics education teaching and learning in multiple ways. First, this work attempts to describe how undergradua te students learn how to prove . Second, this work examines the nature of affective expe riences in math ematics, specifically at a transition point education. The se results may be of interest to mathematics education researchers with interests in proof, emotional responses, and cognitive approaches to learning in general. Lastly , findings from this work may benefit course developers , by informing the design of future undergraduate transition to proof courses, from managing expectations about the pace and depth of student understanding to engineering opportunities for positive, satisfying moments for students. 4 CHAPTER 2: Literature Review In this chapter , I review literature in order to unpack the two major phenomena in my study , proving and emotion s in regards to mathematics . First, I provide an overview of proving , the need for understanding development, and then proving from a problem solving perspective. Second, I discuss what we know about affect in mathematics education before focusing on emotion . My conceptual framing of the constructs used in this study will be discussed in Chapter 3. Proof and Proving What does it mean to prove? It is difficult to pin down a defin ition of what it means to prove but many have tr ied. One can think of proving in terms of creating a product, a proof. Stylianides (2007) provides us with one definition of a proof : Proof is a mathematical argument , a connected sequence of assertions for or against a mathematical claim, with the following characteristics: 1. It uses statements accepted by the classroom community ( set of accepted statements ) that are true and available without fu rther justification; 2. It employs forms of reasoning ( modes of argumentation ) that are valid and known to, or within the conceptual reach of, the classroom community; and 3. It is communicated with forms of expression ( modes of argument representation ) th at are appropriate and known to, or within the conceptual reach of, the classroom community. ( p. 291; emphasis in original.) A proof can generally be thought of then as an argument with certain norms of expression. A proof also has generality, distinguis hing it from computations which are tied to specific inst antiations of variables. 5 We can also consider what activities constitute proving. proof, we define proving broadly to denote the activity in search for a proof." ( Stylianides, Stylianides, & Weber, 2016) . Indeed, covering all that constitutes pr oving is difficult . In terms of reasoning, d eductive reasoning is often associated with proofs, but inductive and abductive reasoning are at play as well. S ome specific activ ities that constitute proving include : Constructing a proof o E stimating the truth of a conjecture o Justifying a statement estimated to be true Presenting a proof o Taking audience conviction into account o Explaining to an audience o Demonstrating validity o Demonstrating understanding Reading a proof o Proof Comprehension o Proof Evaluation (Mejia - Ra mos & Inglis, 2009 , p. 90 ) . Why should we care about proof? P roving is often thought of as a foundatio n al activity in mathe matics (Harel & Sowder, 2007) . Some pur poses for why we should produce proofs are listed here : Verification ( demonstrate truth ) Explanation/illumination ( why it is true) 6 Discovery ( discovering new things in process of proving ) Systematization (organizing results into a system ) I ntellectual challenge (affective feeling of self - realization and fulfillment ) Communication (Bell, 1976; d e Villiers, 1990) . A proof serves many function s then, beyond just a way of verifying the correctness of mathematical statements. It is important to note that m athematics did exist prior to proof; proof as a notion is attributed to Elements . However, argumentation has been so useful that it has become a staple of mathematics as a discipline and remains today (Harel & Sowder, 2007). Proof and Mathematics Education Within the field of mathematics education, there was some attention to proving in the early 21 st century (Fawcett, 1938 ), but most of the work has come in recent times. Recent educational standards have st aked the importance of proving at all ages, e.g. Common Core State Standards in Mathematics ( NGA & CCSSO, 2010) in the United States. Indeed, there is the notion that proof mathematical education (e.g., Hanna & Jahnke, 1996; Mariotti, 2006) . Proving is a difficult activity, however, and s tudents have a hard time learning how to prove (Baker & Campbell, 2004; Moore, 1994; Selden & Selden, 2013 ) . This is not surprising; in everyday life, people use examples as verification for truth. We are not accustomed to general, formal arguments. It is sensible that a foreign skill would take time to learn . One reason is that they have little experien ce with proving ( Jones, 2000 ). Students are used to computatio ns or following algorithms, as laid out by the curriculum, and 7 precursors s work is not overly common in the curriculum. For students in the United States, high school geometry is typically the first place they encounter the wo rd proof, by way of two - column proofs. But the highly constrai ned nature of two - column proofs make it not an adequate introduction to proving ( Herbst, 2002) . Students st ruggle with proving at all ages: mathematical justification and proof in middle school ( e.g. Bieda, 2010; Knuth, Choppin, & Bieda, 2009; Staples, Bartlo, & Thanheiser, 2012) and geometry proofs in h igh school (e.g. Senk, 1989) . Introducing young children to the ideas of proof is a develop ing topic of interest ( e.g. Bieda, Drwencke, & Picard, 2014; Stylianides, 2007 ) . The majority of research on students proving has been at the undergraduate level. , and t he difficulties are many. One common issue is in u sing empirical instead of deductive argu ments (Harel & Sowder, 1998; Recio & Godino, 2001 ). Harel & Sowder (1998 ) proposed the idea of a proof scheme to be a person's conception of proof, of what counts as ascertaining (r emove one's own doubts) and persuading (removing others' doubts). Another issue is in translating informal to formal arguments (Alcock & Weber, 2010; Pedemonte, 2007; Pedemonte & Reid, 2011). If arguments are too wide, student s struggle to produce a proof (Pedemonte, 2007). Other student difficulties are around proof - specific writing, such as using quantifiers and notation (Epp, 2003 ; Selden & Selden, 1987 ) , proof methods (Stylianides, Stylianides, & Philippou, 2004, 2007) , using theorems (Selden & Selden, 1987), generalization (Selden & Selden, 1987) and u ndersta nding and working with definitions (Dubinsky, Elterman & Gong, 1988; Moore, 1994). 8 Undergraduate students also struggle to tell whether a p roof verifies a mathematical fact as true, i.e. they are not persuaded by proofs (Alcock & Weber, 2005; Inglis & Alcock, 2012; Ko & Knuth, 2013; Selden & Selden, 2003; Weber, 2010) . This effect has been seen in preservice secondary teachers (Bleiler, Thomp son, & Krajcevski, 2014) and also i nservice secondary teachers (Knuth, 2002 ) . Selden and Selden (2007) distinguished between the problem - centered versus formal - rhetorical parts of proving. The problem - centered aspect of proving involves the decisions and key insights that are made in order to solve the embedded problem in the proof, oftentimes with no set procedure. The formal - rhetorical aspect of proving involves the logical structure of the proof. Students learning how to prove encounter difficulties of both of these types. Both aspects are necessary in order to interpret mathematical statements and try to prove them, although students may favor one appro ach to proving over the other (Weber & Alcock, 2004) . Selden & Selden have worked on helping students with the formal - rhetorical difficulties of proving, through the use of their proof frameworks. While difficulties with formal - rhetorical aspects of proving hinder students especially in the beginning, the problem - centered aspect may pose a longer, more on - going struggle to students. There is still much left to be learned in the pro blem - centered aspect of proving, with its emphasis on strategies and decision - making, especially in terms of how students develop this sense in regards to proof. In summary, research has established many of the ways in which underg raduates struggle . Some work has focused on how to help stu dents (e.g. Blanton, Stylianou, & David, 2003). What we need more work on, however, is in what the learning process of proving looks like. What are students able to do and what does the learning process look like? 9 Learning how to pro ve is more than just accumulating individual skills or techniques , so analyzing just what they struggle to do is not enough to understand their learning. D evelopment is not necessarily just about accumulating competencies . What more, d espite the numerous difficulties, students somehow still learn h ow to p rove through experience and with the help of instructors. Students may not gain full mastery o f proving quickly but they do make progress. How can we understand the developm ents students go through in lea r n ing how to prove ? For this, we go t o the closest cousin of proving for which we have an abundance of research: problem solving. Proving as Problem Solving Research on proving has been cond ucted in a myriad of ways, with new approaches emerging especially over the last couple decades (Stylianides , Stylianides, & Weber, 2016) . One way of looking at proving is as a form of problem solving (Savic, 2012 ) . For these reason s , I draw on the literature of problem solving, as well as that of proof. Problem solving as a research area was a common theme among mathematics education researchers of the 19 80s and early 19 90s (Schoenfeld, 1992; Silver, 1985) . Non - routine m athe m a tical p roblem solving may be thought of as situations "in which possessed knowledge of algorithms, facts, and procedures do not guarantee success" (Malmivuori, 2001, p. 7). I briefly describe the evolution of theory on mathematical problem solving below. Polya (1945) , the forefather of mathematical problem solving, described the problem solving process in a linear fashion : u nderstanding the problem, devising a plan, carrying out the plan, and then reflect ing in order to extend it for future 10 problems . A number of theoretical frameworks for investig ating problem solving have been created since then, building off Garofalo & Lester (1985) brought into focus t he importance of metacognition in problem solving, of s own cognition a nd regulation of it . They identified four categories activities people engage in when working on a task - orientation, organization, execution, and verification and how metacognition is involved in each . Schoenfeld (1985 b ; 1992 ) identified five c omponents of problem solving: cognitive resources , strategies or heuristics , monitoring and control, beliefs and affect, and practices. But p roblem solving need not be sequential ; it can be a cyclical process. Carlson & Bloom (2005) found that subjects often go through cycles of reasoning when problem solving : making a plan, executing the plan, checking if the plan continues to work , and then creating a new plan if issues arose . This framework has been used to analyze proving as well (Savic, 2012 ) , due to the similarities between problem solving and proving processes. In summary, various theories of mathematical problem solving have been developed and have built on each other, leading to the refined work we have today. Transition to Proof Courses Now I turn to a discussi on of transition to proof courses. Considering all the difficulties inherent to proving, it is not surprising that mathematics departments have responded with courses designed to help students learn. The formation of introduction or transition to proof co urses can be seen as a department al response . These courses can take on many names, but I call all courses of this nature t ransition to proof for the sake of simplicity. 11 There is great v ariety in the design of transition to proof course s across the United States . David & Zazkis (2017) conducted a syllabus study to categorize the variety of designs . One common design is to teach proving as a sta nd - alone skill , with instruction on formal logic, quan tifiers, proof methods, and propositions . The content of these courses is often around sets, functions, etc. A variation of this is for the majority of the course to be about logic and grammar, with an introduction to a n advanced mathemat ical topic they will encounter in the future near the end. The other course design is to teach proving through a c ontent area to provide some context , with oftentimes little explicit instruction to formal logic. In these courses, students are often expected to pick u p how to prove along the way. On the other hand, there is an advantage to proof in the context o f a content area , wher e proof as a means of discovery of new results is better motivated. Multiple transitions taking place . Transition to proof courses are transitions in terms of content proof - based work in place of computation. There is transition then in terms of cognitive aspects. But transition can also refer to a transition in terms of experience. Mathematics as many students are used to in K - 12, of computations and algorithms, has now been exchanged for mathematical argumentation and writing. This experience, at a socio - emotional level (Smith, Levi n , Ba e, Satyam, & Voogt, 2016 ). Affect Affect is generally thought as the domain involving emotions (Middleton, Janse n, Goldin, 2017 ). McLeod (1992) defined the affective domain feelings, and moods that are generally regarded as going beyond the domain of the 12 One way to think of affect is as a representational system : Affect includes changing states of emotional feeling during mathematical problem solving (local affect)...and more stable, longer - term constructs (global affect), which establish contexts for local affect and which local affect can influence. Our hypothesis is th at affect is fundamentally representational , rather than a system of mostly involuntary, physiologi cal side - effects of cognition. ( DeBellis & Goldin, 2006, p. 133) For example, frustration while working on a problem serves as an indicator that something i s not workin g ( DeBellis & Goldin, 2006). Thus, frustration serves as an encoding of this cognitive noticing that current strategy is not working and trying a new strategy should be taken. Major Types of Affect in Relation to Mathematics Education Three m ajor types of affect include beliefs, attitudes, and emotions (McLeod, 1992). I provide definitions of each, using McLeod dimensions and Middleton, Jansen, & relate. Attitudes . Attitudes are "orientations or predispositions toward certain sets of emotional feelings (positive or negative) in particular (mathematical) contexts. ( DeBellis & Goldin, 2006, p. 135). Some examples of attitudes in mathematics educati on are being bored by algebra, curious about geometry, and disliking sto ry problems. Attitudes are seen as traits, in that they are long - term and relatively stable to an individual, thus difficult to change. Belief s . Beliefs are the attribution of some sort of external truth or validity to systems of propositions or DeBellis & Goldin, 2006, p . 135). One pervasive example of a belief in mathematics education is believing in one is bad at mathematic s . Other b eliefs include self - efficacy and other motivational variables . B eliefs 13 are often highly stable and perhaps the most difficult to change among attitudes, beliefs, and emo tions. Emotions . Emotions are "rapidly - changing sta tes of feeling experienced con sciously or occurring preconsciously or unconsciously DeBellis & Goldin, 2006, p. 135) . Emotions are generally thought of as responses to events. Emotions tend to be short in duration but can reach high intensity, in contrast to attitudes and beliefs tending to be long in duration but low in intensity. Emotions are local and oftentimes bound up in the context at hand. Emotion is the state (rapidly changing) of affect vs. attitudes/beliefs as traits (stable). Emotion s can also function as r epresentations of the consequence of goals, thereby communicating information about the situation. For example, a person feels happy when they make progress or sadness when noticing a lack of progress ( Middleton, Jansen, & Goldin, 2017). In addition, emot ions do not sit in a vacuum away from attitudes and beliefs - term interests and beliefs about a situation at hand can manifest themselves through t heir emotions (Middleton, Jansen, & Goldin, 2017). Affective Work i n Mathematics Education Early work on attitudes . I provide here a brief overview of the history of studying affect in mathematics education. Early research in mathematics education regarding affect This work in the 1970s was largely quantitative, administering questionnaires to large groups to measuring attitudes pre - and post - some intervention. Well - known attitude scales include the Fennema & Sherman (1976) meant to study gender differences but used by many researchers for general research on 14 The second wave of development in affective work in mathematics education came from a focus on problem solving. A ttention was on beliefs about mathematics and how their beliefs influenced their problem solving. Teacher beliefs was also a large avenue of research, but since the focus of this review is on students, I do not discuss this more. Studies of emotion as rare . Emotions are difficult to study. E motions are much shorter in duration and thus fleeting, and thus hard to capture, compared to attitudes and beliefs. Trait - like variables are more easily measurable, due to their stability; survey work is an appropriate method gold th is. This stability means they are not easily alterable, for good and for bad. Studies focusin g on emotion in mathematics education are far fewer than that of beliefs and attitudes (e.g. Gómez - Chacón, 2000 ; , 2007) . Historically, those that existed were typically around math anxiety (e.g. Buxton, 1981 ) but some recent studies have examined how emotions influence mathematical thinking and learning ( e.g. 2007 ). Careful observ ation of students with detailed interviews can help research ers analyze emotional states of mathematics learners (McLeod, 1988, 1992) . Why care about studying emotions? W hy does studying emotions matter , especially if they are fleeting in n ature? One, emotions are the vehicle for changing attitudes and beliefs. Repeated emotional responses may lead to student having different attitudes, which then may be able to alter beliefs. Two, emotions themselves as in - the - moment and states are the most resp onsive to change . Middleton, Jansen, and Goldin (2017 ) In - the - moment engagement, on the other hand, is more easily susceptible to 15 ( p. 691 ). I assert the same is for emotions, a s the affective construct wit h the shortest dur ation. There is a push for more work on in - the - moment affective constructs ( Evans, 2002; Hannula, 2002; McLeod 1992 ). 16 CHAPTER 3: Conceptual Framing In this chapter, I present some of the concepts used in my study which influenced its design . This is a separate chapter fr om the literature review for the sake of reader clarity. First, I conceptualize proving as problem solving, specifically what a person does when stuck with a focus on strategies and monitoring and judgmen t. I also briefly discuss my conceptualization of development which influenc ed the study design. Lastly, I define a new construct, satisfying moments , and relate it to existing constructs about intense positive emotions. Analytical frameworks will be discu ssed in a separate chapter. Conceptualizing Proving as Problem Solving the work of constructing a proof for a given statement. I chose this particular conceptualization of proving for multiple reasons. First, I purposely wanted to keep the phenomenon of focus broad by using the term proving rather than narrowing my focus to a particular skill, e.g. deductive reasoning. Second, I wanted to focus on proving as a process, r ather than the product (Karunakaran, 2014), to look at what students do and their the tasks whether they produced a successful proof at the end of the allotted t ime was not so important in this research; what they attempt to do was more vital. To consider proving to be a subset of problem solving, we must define what is meant by problem solving, given the rich research tradition about problem solving in mathem atics. To keep things simple, I t ake problem solving to be what a person does when stuck. This is equivalent to what activity a person engages in when reaching an impasse (Savic, 2012). Under this definition, a task may elicit problem solving in one student but not 17 another, depending on wheth er or not they become stuck at any point in the proving process. The o perati onalization of what is meant by stuck will be discussed in Methods. When looking at what a person does when stuck, I focus ed on the components of strategies ( heuristics ) and monit oring and judgement of problem solving (Schoenfeld, 19 85 b ; 1992). techniques for making progress on unfa miliar or b , p. 15). Monitoring and judgment can be thought of as s elf - regulation and fall under the umbrella of metacognition (Schoenfeld, 1992), knowledge of and regulation of . I include these here in the conceptual framing because while I did not strictly adhere to Schoenfeld (1985 b frameworks regarding strategy and moni toring and judgment, it did highly influence my thinking and the design of this study (see discussion in Methods chapter about think - aloud). Conceptualizing Development Development refers to change over time, but even that can be thought of in multiple ways. For example, o ne way of thinking about development is in terms of stages, in which a person presumably passes through each stage on their way to full mastery (Piaget, 1971). One famous example of development is the Van Hiele (1959) levels of geometry thinking. Conceptualizing students learning by way of levels remains to this day (Cobb & Wheatley, 1986; Lo, Grant & Flowers, 2008). I conceptualize development as taking a - a characterization of some construct at a point in time - and looking across these at multiple timestamps for change. Figure 3.1 illustrates this idea. 18 Figure 3 . 1 . Conceptualization of d evelopment in students proving by capturing snapshots Defining Satisfying Moments I define a satisfying moment to be an emotion al response to a particular moment in time , characterized by intense positive fe elings. I think of a satisfying moment as being located within an experience, which serves as the context or situation which leads up to the satisfying moment. The use of the word moment is meant to suggest this event holds an instantaneous feeling to the individual, regardle ss of whether it is in reality. However, the distinction between a moment vs. an experience, the latter of which implies a duration, is not important . Figure 3 . 2 . How satisfying moments relate to e xisting concepts regarding intense positive emotion. Figure 3.2 shows how I conceptualize satisfying moments , as acting as a superset for other existing constructs in the literature regarding intense positive emotions . One concept that Proving at Time 1 Proving at Time 2 Proving at Time 3 Proving at Time 4 Satisfying Moments Mathematical Beauty Aha Moments 19 falls under the umbrella of satisfying moments is the idea of mathematical beauty ( Hardy, 1940; Sinclair, 2006) . Another example is the a ha or eureka moment (Barnes, 2000; Liljedahl, 2004) . I discuss these related ideas below. Related Constructs Mathematical beauty . There is a well - documented phenomenon of mathematicians writing and talking about beauty in mathematics (Hadamard, 1945; Hardy, 1940 ; Lockhart, 2002; Poincaré, 1952; Thomsen, 1973) and talk about math as being comparable to art in certain ways. Zeki, Romaya, Benincasa, and Atiyah (2014) showed that when mathematicians experience mathematical beauty, this correlates with activity in the same part of the brain associated with enjoying art . There are philosophical differences o ver whether mathematical beauty is an objective characteristic of the piece of mathematics or a projection from the observer (Sinclair, 2006, 2009) . However, I take the approach of the latter and conceptualize mathematical beauty as an emotional response to mathematics. In the same way it is difficult to def ine beauty, it is difficult to define mathemat ical beauty is one of the texts most associated with the idea of mathematical beauty. Hardy claimed that theorems that are beautiful tend to exhibit a triumvirate of inevitability, economy, and unexpectedness. Some commonly stated features of mathematical beauty include simplicity, brevity, inevitabili ty, economy, enlightenment, un derstanding, and surprise, among others (Blåsjö, 2012; Cellucci, 2015; Hardy, 1940; R ota, 1997; Satyam, 2016; Sinclair, 2006). 20 Mathematical beauty is a driving force for doing mathematics, playing a crucial part of engaging in mathematical inquiry (Hardy, 1940 ; Poincaré, 1952) . Sinclair (2004) identified three roles for beauty in doing mathematics: motivational, generative, and evaluative. M athematical beauty reveals the values of mathematicians and the larger mathematical community. Aha moments . An aha moment is an affective response to an unexp ec ted idea or solution, which are cognitive event s (Liljedahl, 2004). One of the most famous stories examples is of Archimedes sitting in a bath and realizing that displacement equals volume eureka moments as well. Aha mom ents are characterized by a sudden realization or insight. Mathematicians like to think of mathematical beauty as a moment of instantaneous enlightenment, like a lightbulb turning on (Rota, 1997). Hadamard (1945) talked about discovery as a flash of insight as well, also using the metaphor of light illum inating the darkness. There has been some work on aha mome nts (Mason, Burton, & Stacey, 1982), due to the hope that they may change attitudes and beliefs (Liljedahl, 2004). Why create a new construct? Moments of m athematical can be relatively rare and aha moments even more so , which makes these phenomena very difficult to capture and study. In addition, bas ed on my past work, I found that a good number of students did not respond well to to describe math. It came across as an odd word to use, perhaps because of in everyday language. I would have needed a different term to u se with them even if I had gone that route . 21 Satisfying moment s are therefore an expanded version of mathematical beauty . It is also just a shorter way of referring to moments with intense positiv e emotio ns. The scope of this construct is kept broad intentionally , so that students may say they do indeed experience this and thus report more of them. This i nvestigation of kinds of satisfying moment s is therefore about the range of these moments which occur . Some may end up being instances of mathematical beauty or even aha moments . Experiences with intense positive emotions provide motivation for students t o continue doing mathematics and thus can be productive . Research is needed on how these experiences can provide intrinsic motivation for students to continue and value doing mathematics (M cLeod, 1988) . 22 CHAPTER 4: Method In this chapter, I outline and justify the methods used to answ er the research questions. I describe the study context of the transition to proof course, the participants, the sources of data, and the methods of data collection. I also provide a detailed description of the pilot study and how that informed the researc h. Data analysis will be discussed in the next chapter. Study Context : Transition to Proof Course The transition to proof course at this university was designed to ease the transition from cal culus - based courses ( e.g. Multivariable Calculus or Differential Equations, where the work was primarily computation and using formulas) to upper - level math courses that involved writin g proofs. T his course was required for undergraduates majoring and minoring in mathematics, unless they chose to enroll in an advanced linear algebra course, which then functioned as their transition to proof course. This course was a p r er e quisite for Linear Algebra , so a variety of STEM (science, technology, mathematics, and engineering) majors were enrolled in this course as well . Cont ent The first half of the course focused on grammar, and the second half introduced students to basic concepts in real analysis, linear algebra, and number theory (see Table 4.1). 1 The course met for 80 minutes three days a week, for fifteen weeks. 1 This weekly content was true at the time of data collection but has since changed. 23 Table 4 . 1 : Schedule of Weekly Content for Transition to Proof Course Week Topics 1 Sets 2 Functions: injection, surjection, and bijection 3 Mathematical statements (negation, and, or) / Induction 4 Truth tables / Implication/ Contradiction/ Proof by contradiction 5 Converse, contrapositive / Proof by contrapositive 6 Conditional statements and quantifiers 7 Review and exam 1 8 Real analysis; open and closed / sequences and convergence 9 Linear algebra; vector space, linear functions 10 Linear algebra; vector space, linear functions 11 Number theory; division lemma, gcd 12 Number theory; modulus, equivalence relation 13 Review 14 Review and exam 2 15 Review for the final Final exam Note. Description of content for each week of the Transition to Proof course. The first half was about proof grammar and techniques, and the second half of the course presented basic concepts from advanced mathematics students had not taken yet. Adapted with permission. Course Design This specific transition to proof thematics course. The instructor (graduate student or faculty) lectured for roughly 120 minutes each week, with typically 1½ days devoted to lecture. For the rest of the class time, students worked on problems in groups of 3 - 4. A graduate or undergraduate teaching assistant also assisted the instructor with the group work portion of the class two days a week. In the past, the amount of lecture had generally decreased over the course of the semester, depending on the concepts. Students were expected to read selected material from How to Think Like a Mathematician (Houston, 2009) and course - created supplementary d ocuments before 24 coming to class, in order to have a first exposure to the content. Online reading quizzes worth a minimal number of points provided the incentive for students to do this reading. Students also had access to the online forum Piazza where they could ask questions, and instructors, teaching assistants, and fellow students could answer through this system. Homework was a central learning activity of the course. Homework was due every week, to be typed in LaTeX, a typesetting software commonly used in mathematics. Each homework typically had three types of questions: answer only, medium justification, and complete justification. T h e pro portion of the three types of homework problems shifted over the semester, toward s more complete justification full proofs . Students could also seek help on their homework from a math learning center (MLC) on campus. Researcher Positionality: My Du al Role as R esearcher and Teaching A ssistant My relationship with th e transition to p roof course was not that of an outside researcher. Thus, I describe my position relative to the course, because it influenced my access to the participants and the nature of the data collected. I was a teaching assistant for the course in Fall 2016 and Spring 2017, the latter of which was the semester of data collection. I was in the classroom two out of the three 80 - minute periods that the class met in order to help with group work. There were weekly course meetings for instructors and teaching assistants, contributing to my knowledge of the intentions behind course decisions. I was also a t utor at the MLC each week, where students of this course visited (including, someti mes, my own par ticipants in this study) , primarily for help with the homework for the course. Lastly, I had also observed the course periodically in the previous year, as a part of a separate research project. Thus , I had 25 observed the nature of this course and how it had changed over time. The participants in the study were not my own students however. This perspective influenced the study in the following positive ways. One, I had easier access to potential participants due to personally knowing all of the other instructors. Two, I was aware of what students had been taught so far in the course, which affected how I conduct my interviews with students and my interpretation of their work. Three, some students already knew me from the math learning center, so there was an added rapport; I could talk about course milestones and what was currently happening in the course with participants, e.g. commiserate over the last homework or exam. My insider status with the course also had some limitations. There was the potential me first as a teaching assistant as opposed to a researcher. To prevent them from potentially asking me for help and answers, as they would a teaching assis tant, I was upfront about my role in my interviews with them and told them I would have to decline helping them during the interview tasks, when it would interfere with the study. All the participants understood the different role I played when conducting the study, and it was not an issue. Description of Instructors Here I give brief descriptions of the two transition to proof instructors whose students I recruited for this study. Pseudonyms were chosen by the instructors. Mr. X was an assistant professor in the mathematics department. He was the coordinator of the transition to proof course and developed much of its structure. At the time of data collection, he had taught the cour se for multiple semesters. 26 Ms. Frye was a graduate student in the mathematics department. At the time of data collection, this was her second semester teaching the transition to proof course. Participants The participants were N=11 undergraduate students taking a transition to proof mathematics course at a large Midwest ern university (see Table 4.2 ). Their ages were from 18 and up. Twelve students were interviewed initially there was one student who only completed the first of the four required interview s and so is not listed here. Table 4 . 2 : Background of Participants Name Major(s) Minor(s) Year Gender Ethnicity Instructor Amy Actuarial Science Entrepren - eurship 2 F -- Ms. Frye Charlie Computational Math Computer Science 3 M Chinese Mr. X Dustin Statistics 2 M White Ms. Frye Gabriella Actuarial Science 1 F -- Mr. X Granger Physics, Math 1 M Caucasian Mr. X Joel Statistics 2 M White Ms. Frye Jordan Math, Secondary Education Chemistry 2 F -- Ms. Frye Leonhard Math 1 M White Ms. Frye Stephanie Actuarial Science 2 F White Mr. X Shelby Statistics 2 F White Mr. X Timothy Math 3 M White Ms. Frye Note. Pseudonyms are used for participants and instructors. Participants self - identified ethnicity using their own terms. The -- notation denotes a participant opted out of self - identifying their ethnicity. Recruitment and Selection of P articipants I recruited participants using the following process. Recruitment was done in person. I asked the instructors of two sections of the course, Ms. Frye and Mr. X, if I could 27 visit their class in order to talk to students about the research and ask for volunt eers. Ms. Frye and Mr. X had enrollments of 22 and 21, respectively. Across both classes, I selected students to vary along the following parameters: instructor, math major or not, and gender. First, I chose an equal number of participants from e ach section of the course, half the sample to be female, the other half male. The demographics of this specific course tended to be 2/3 male and 1/3 female. I chose to not mimic the gender distribution of the course , due to existing research evidence about interaction between ge nder and affect, especially in regard to negative emotions, so gender was an important variable. When choices still remained, I chose participants bas ed on ethnicity, to align with representation in the course, and finally on their schedule availability. Participants self - identified their ethnicit y, via a blank space on the participant form. 20 students volunteered for the study; of these, I selected an initial 12 participants according to the guidelines above. Some participants did not reply, so they were replaced by additional participants, adhering to the above rules when possible. 2 2 A question that may arise for the reader: Why were there so few non - white participants? Th e fact that most of the p articipants were white stood out. A little background: 7 of the 20 volunteers for the study self - identified as an ethnicity other than white . Out of these , I selected 5 for the study : a black female, black male, Asian female, Middle Eastern male, Hispanic female. When I contacted them to follow up, only 1 replied and she could not continue the study after 1 interview. It seemed odd for a num ber of non - white students to sign up and then very few participate in the study itself . T his sample size is small; no real 28 Descriptions of Participants Here I give brief descriptions of the eleven participants who completed the entire intervi ew series. These are portraits of the students as I came to know them over multiple interactions over time, not first impressions. A s such, I include some interesting details specific to them ; these profiles are not meant to be complete . I include this s ection in order to huma nize these participants , as a reminder that these are all individuals with different backgrounds, personalities, and hopes for their future . These detail s do color the data, especially in examining affective issues . Amy was a white female sophomore majoring in Actuarial Science and minoring in entrepreneurship. Amy said she has a love/hate relationship with math; she does not like math when she first starts a problem but then loves it when s he is done. She especially liked the compet i tive and challenging aspects of math ematics, e.g., doing a hard problem that someone says cannot be done. In terms of career goals, her goal was to be an actuary. Charlie was an Asian male ( an international student from China ) junior majoring in computati onal math and science and considering a min or in computer science. He had a penchant for problems he could do in his head and also talked about his thought process using metaphors throughout the interviews. conclusions can be made from it. However, investigating whether there are structural factors that lead to non - participation by students with non - white backgrounds is worthy of future research . 29 Dustin was a white male sophomore statistics majo r, minoring in actuarial science. He want ed to do actuarial science but was majoring in statistics, to give hims elf more career options. He claimed he did not do well on timed te sts. He expressed that he liked using examples as models for proofs and that h e understood math when some one else e xplained it to him. He found that talking ab out math with other people helped him work. Granger was a white male freshman math and physics major. He wanted to be a professor or an indus trial mathematician. He explained that his class had positioned him as o ne of the smart ones. During interviews , he wrote very quickly and expressed that he felt his brain wa Granger felt he was not emotional in gen eral , let alone when doing mathematics . Gabriella was a white female freshman majori ng in actuarial science. She was good friends with Stephanie; they often work ed on homework together. In the beginning, she said she would oftentimes second guess her answe rs, but she stopped doing this as the semester went on. She said she prefer red calculus - bas ed courses to proving; she just wanted to get through this class , as a requirement for her major. Joel was a white male sophomore majoring in statistics and conside ring a minor in math. He tal ked about ho w math used to come easy to him in high s chool. In the beginning, he said he was terrified about the course , but as time went on, he found the material interesting. He did problems, where one can just do them as opposed to having to figure things out. Jordan was a white female sophomore math major with a minor in chemistry. She wanted to be a secondary math teacher. Jordan started the semester off well but seemed to be de moralized by the class as tim e went on. She felt that she had put in a lot of effort an d 30 time into homework yet still received low homework scores and that she did not understand concepts as time went on. Leonhard was a white male fre shman majoring in math. He wanted to be either a high school teacher or a mathematician for aerospace engineering. He had just transferred to school this semester . Leonhard had lots of thoughts about mathematics and used metap hors to explain his thinking often. Shelby was a white female sophomore majori ng in statistics. She had taken some math classes at nearby community college s, for the smaller class size. She liked having steps in mathematics. She also expressed that talking out loud to people helped her when she was stuc k working on mathematics and that she enjoyed working with people. Stephanie was a white female sophomore majoring in actuarial science. She was good friends with Gabriella, and they would work together on homework. She wanted to work in insurance. She cam e across as practical, not swayed by emotions. She said her biggest struggle was in u nderstanding what the problem was asking for. She put stock in high performance and did get good grades but by end of semester, she was worn down. Timothy was a white mal e junior majoring in math ematics. He wanted to work in informational technology ( IT ) afterwards. He found proving to be fun but felt he needed time to learn things, for concepts and definitions to sink in. He was especially good at talking h is thoughts out loud. Data Sources The data were a series of four semi - structured interviews across the seme ster with each participant . Each interview consisted of two halves : the first half was organized around two proof construction tasks , and the second hal f was about satisfying moments 31 with stimuli tasks . Figure 4.1 sho ws the different data sources by interv iew , as the second half of interviews 2 - 4 was different from that of interview 1. Figure 4 . 1 . Representation of data sources (by color) within each i nterview In capturing development of proving, the half is about general proof structures and the second half focused on content (real analysis, linear algebra, and number theory ). I interviewed participants at these four times: middle of the proof structures section, end of the proof structures section, middle of the content section, and end of the content section. Each of the four rounds o f interviews were done over a two - week s pan. In the following sections, I describe the design of each part of the interview in detail, including instruments and stimuli tasks. First Half of Interview: Proving During the first part of each interview, participants worked for no more than 15 - 20 mi nutes on each of two proof tasks. I chose to give participants two tasks, rather than only one, so that they had more than one opportunity to show their thinking at this current point of the clas s, in case they struggled majorly with the particulars of one task. The idea specifics of one tasks threw them 32 off. I told participants they had 15 minutes but gave them a maximum of 15 - 20 minutes to work on each task, in order to give them enough time to show case their thought process and attempt to overcome stuck points when any occurred. I was interested in their thought process, as opposed to analyzing their final written product. If a student was still working at the 15 minute mark, I oftentimes let them w ork for a minute or two until they finished their current train of thought. Proof construction tasks: Selection. The selection of tasks for the proving section were ho w I chose to measure their proving at a given point in time, the nature of the statements and their possible solutions largely determined what the students did. development , a coherent rationale behind selection of tasks was necessary. Table 4.3 lists the proof tasks (full versions given in Appendix A). Table 4 . 3 : Proof Construction Tasks by Interview Interview 1 Statement Task 1 Suppose x and y are integers. If x 2 y 2 is odd, then x and y do not have the same parity . Task 2 Prove the following statement: If a and b are strictly positive real numbers, then ( a + b ) 3 never equals a 3 + b 3 . Interview 2 Task 1 Prove the following statement: If x and y are consecutive integers, then xy is even. Task 2 Prove the following statement: If a , b , and c are non - zero integers such that a divides b and a divides c , then a divides ( mb + nc ), for any integers m and n . Note. Abbreviated versions of each of the proof construction tasks. Un derlined words were new definitions , which were defined for the participant ; f ull versions of tasks given in Appendix A. 33 Table ) Interview 3 Task 1 Prove the following statement: Suppose x , y , z are positive integers. If x , y , and z are a Pythagorean triple , then one number is even or all three numbers are even. Task 2 Prove the following statement without using induction : If n is an odd natural number, then n 2 - 1 is divisible by 8. Interview 4 Task 1 Prove the following statement: If a and b are odd perfect squares, then their sum a + b is never equal to a perfect square . Task 2 Prove the following statement: If x, y are positive then + > 2. The following criteria were used to select tasks. First, the goal of the task was to function progress was likely heavily influenced by instruction, it made sense to pick tasks that were similar in nature to questions they would encounter in the course, generally around the same time perio d (weeks) but before students actually encountered them . For these reasons, the tasks were taken from past homework assignments from previous semesters of the course, specifically Spring and Fall 2016. Second, task s w ith multiple possi ble solution paths, not just one , were chosen when possible . For example, statements that could only be proven easily using a proof by contradiction were excluded; however, statements that could be proven by either contrapositive or contradiction were stil l viable because of the choice in technique. Third, all tasks were from one content area, basic number theory. The goal was for the tasks to not be heavily dependent on content knowledge nor a singular specific proof technique (e.g. induction). Because I solving abilities, I wanted to minimize the effect of a lack of content understanding. For 34 could be due to struggle s in understanding analysis definitions or concepts and not necessarily in pro ving . There is a danger however in making content - free claims about may be specific to this content area. Even still, I would argue that basic number theory, e.g. properties of even and odd number s, is a more broadly accessible content area tha n analysis, so more students can at least start the task. The first task of each interview was designed to introduce a new definition, a novel situation with new information to deal with it. The second task was designed to elicit stuck point s , where students thought they knew what to do but it would not work. I searched for tasks that looked like they would be routine but were in fact not. Interview 3 Task 2 is an example of a task that was especially succe ssful at what I described above. Interview 1 Task 2 was less so, but because it was the first interview, this did not affect the analysis much . Tasks that are novel and/ or have a stuck point built in are problems , as defined in the literature . To select potential tasks, I looked through all homework sets from the previous two semesters and compiled questions that best satisfied these criteria. I used homework questions that students were likely to have not seen by the time of the interview, i.e. , they wo uld run into a homework question of that type later in the course. When I could not find suitable questions from homework, I found some using other textbooks or online resources or made my own. Think - aloud. In order to capture their strategies and reasons for using certain strategies, I used a think - aloud protocol (Ericsson & Simon, 1980, 1981; Schoenfeld, 35 1985 a ), where participants voice their thoughts aloud about a task, either in real time or shortly after the task is complete. Previous work on how stude nts problem - solve and prove has largely used think - aloud methods as a proxy for accessing cognition (Schoenfeld, 1992; Weber & Alcock, 2004). Over the years, researchers have considered and examined the validity of using verbal data to infer about the tho ught process (Ericsson & Simon, 1980, 1993; Schoenfeld, 1985 a ;). In other words, to what extent does asking a subject to verbalize their thought process affect their thought process? This issue is called reactivity (Leighton, 2009) and affects certain kind s of experimental set - ups and questions (Schoenfeld, 1985 a ). Certain experimental variables can impact the data produced, such as the number of people being interviewed, the degree of interviewer intervention, and the environment under which task is being given (Schoenfeld, 1985 a ). A major issue then in administering a think - aloud is the level of interviewer intervention: more vs. less and the character of it. More intervention can mean more verbalizations and thus evidence , especially of metacognitive behavior. However, asking students to reflect on their problem solving process in the moment can affect their questions during a task s behavior (Schoenfeld, 1985 a ). It safe r then questions during a performance, such as asking them to identify what they just did. Ericsson and Simon (1980) have argued that asking subjects to verbalize their thoughts but not asking for any ex Other non - intervening moves during task performance include "I haven't heard you say 36 much in the past couple minutes. Are you still working on the problem?" and answering specific student q uestions. Based on the affordance and constraints of asking probing questions, I chose to minimize interviewer intervention during task performance. This was because my phenomenon of interest was the proving/problem solving process itself so keeping the pr ocess intact from a validity standpoint as much as possible was of the utmost importance. This was especially the case since my phenomenon of interest was what students do when stuck, and there was a high chance that talking would get them unstuck. In othe I asked students right before they started a task to verbalize their thoughts out loud, as long as they felt it did not interfere with their thought process. If the student had been silent for a few minutes, I would sometimes remind them to say what they were thinking. Otherwise, I remained silent. Over time, I developed a sense for which participants were comfortable talking while working and which participants were less so, an d I held back on nagging the latter group. This is one place where familiarity with the individual was helpful for minimizing the interference on each performance, even if it meant I had to act slightly different across participants. I then de brief ed with the student immediately after they said they were done working. During this debrief, I asked all probing questions: to explain their thought process and any Subjects can talk about their thought pro cesses about a given task if asked immediately after task completion (Ericsson & Simon, 1981). I tried to ask probing questions in a way that did not let on if their solution attempt was correct or not, since there was more data to be collected regarding t heir solution after 37 the debrief. Asking students to think - aloud but not pushing them to do so and then asking probing questions immediately after they had finished t heir work optimized the benefits and pitfalls of think - alouds for studying proving as a phe nomenon. It was a major goal that students felt comfortable throughout the interview. This was especially important during the proof tasks section of the interview, where I video recorded each student while they worked on difficult tasks. It can be difficult to work with someone watching over you , let alone the fact that participants feeling pressure would affect the data. Most of my behavior throughout the interview was centered around making them feel comf ortable, for instance by having students sit in my chair at the center of the desk rather than relegating them to a small chair off to the side. I took the following steps to decrease the likelihood of set the video camera as far away from the participant in the room as I could while still being able to capture their written work. I also stood far away from the student while they were working on one of the proof tasks. In times when students became very frustrated, I left the room momentarily while keeping the camera rolling in the hopes that my temporary absence would decrease the pressure they felt in that moment. Students adapted to the experimental set - up very well none of them glanced back at the camera out of self - aware ness or observable self - consciousness. Collected but not a nalyzed: roving. In this section, I problem solving. This data is not analyzed in detail in this dissertation, except for one instance in the roving re sults chapter. 38 annoyed curious disappointed surprised sad joyful indifferent frustrated satisfying ashamed proud went through whi le working on the task and to draw a graph of their emotions. The purpose of these were to help participants describe their emotional responses to their proof tasks, as a way to capture affective data about cognition (their proof work). Emotion words. Participants were shown cards with an affective word written on each (as shown in Figure 4.2). Five negative - ne neutral emotion word were chosen with the intention of capturing the range of possible emotions while problem solving, based on literature review. Participants were free to choose other or none of these words; these given words were only meant to provid e a base to start with . Figure 4 . 2 . P hysical arrangement of the 11 affective words for the Emotion Word Task The e leven index cards with one emotion word on each were laid out on a table in front of the participant, as shown in Figure 4.2 . Participants were asked to select which emotions on the cards they experienced and to say why. By putting the emotion words on tan gible cards, participants could point to or handle each one physically. They oftentimes put the cards with emotions they felt in temporal order. As the interviewer, I circled the words they chose on a pre - printed piece of paper and then asked if there were any other 39 emotion words the participant would have picked that were not present. Their emotion words expressed the emotions that the participant experienced, and their reasons for picking each word helped me to identify the conditions that led to that emo tion. Emotion g raphs (adapted from McLeod, Craviotto, & Ortega (1990) and Smith, Levi n, Bae, Satyam, & Voogt (2017 ) ) . The participant was given a blank graph and asked to chart their emotions during the entirety of the proof construction task. The graph allowed for a temporal look at the ups and downs in emotion over the course of their solution attempt. Graphing emotions is a technique that can be used to describe variations in Ortega, 1990). Participants were also asked to mark on the X - axis and/or annotate their graph with short captions at the points at which their feelings changed. Figure 4 . 3 . Example of an emotion graph for a proof task 40 Second Half of Interview: Satisfying M oment s The purpose of the second half of the interview was to capture data for my second research question , about the nature of satisfying moments related to the transition to proof. The data in this half of the interview consisted of questions about satisfying moment s, discussion over emotion graph tasks and then the emotion word task (only on interview 1). The inte rview protocol in Appendix B lists the questions that were asked. The goal of the se questions was for students to describe in full detail any satisfying moment s the students had encountered in relation to the course, whether through homework problems or in class. Self - report of satisfying moments. Self - report was an appropriate method for perception of their own satisfying moment , which was most important. For example, if a person truthfully perceived a n experience as satisfying, then I could argue that this experience was satisfying to that person, even if an outside observer watching the entire experience unfold did not see it as response, which is internal and personal. The instructions given to students naturally then influence what they report. In designing the interview questions , I therefore introduced the idea of a satisfyi ng moment with few constrain ts, so that students would report back according to however they defi ne d it for themselves . However , in the first interview, after introducing the idea of a satisfying moment, I asked fol low up questions about situations that could be satisfying, e.g. , proble ms that feel rewarding, flashes of understanding/ insight. These questions were meant to probe , to help students if they could not think of satisfying moments on their own, 41 but also to try to capture some relat ed affective concepts, such as aha moments. I t is possible though that these follow - up questions may have influenced what students reported in the future as satisfying. As we shall see, however, t he fact that students still talk ed about performance and that flashes of insight were stil l relatively rare sugges ts these follow up q uestions did not affect the data unduly. Emotion graphs as recall. It is possible that students may not remember satisfying moment s, without some kind of record. In the first interview, I asked participants in de tail about satisfying moment s in relation to their transition to proof course. For the most salient experience, I asked them to draw an emotion graph. This emotion graph had small variations in wording from the emotion graph for the proof task (see Figure 4.4 ). 42 Figure 4 . 4 . One of Stephanie s emotion graphs for a satisfying moment in Interview 2 After the first interview, I gave participants 2 - 4 blank emotion graphs to take home and asked them to fill it o ut (i.e. draw a graph) whenever they had a satisfying moment before the next interview. In interviews 2 - 4, participants came to the interview with already filled out emotion graphs and were ready to talk about satisfying moment s they had ex perienced outsid e the interview. 43 The pu rpose of the emotion graphs was to (a) have a record of a satisfying moment presumably in real time or at least not too long after of a satisfying moment and (b) serve as a stimulus for discussing the satisfying moment during the interview. On the first interview, students were asked to pick emotion words from the index cards for their most salient satisfying moment, which I chose in real time based on the subject the previous questions about satisfying moments. The selection of the experience went as follows: if the subject discussed only one satisfying moment in the interview, I used that experience. If the subject talked about multiple satisfying moment s, I picked the most intense one or the one they talked about the most. If the subject did not talk about any On interviews 2 - 4, I did not ask them to pick out emotion words for their satisfying moment s, in order to (a) avoid task fatigue, as this would be their third time choosing words during the interview, but also (b) students were now comfortable using emotion words when talking about their experience. The e motion word task was no longer needed then, as an artificial stimulus for talking about satisfying moment s. Interview notes. Interview notes were taken on paper during interview. They were then recorded digitally with more observations as soon as possibl e after the interview. I also took notes about things to ask them next time, to keep continuity across interviews. These interview notes became a source of data for some of the analyses. Pilot Study I conducted pilot interviews with three participants in order to test the research design and instruments the semester before the real data collection occurred. In this version of the study, the research focus was on asking students about homework problems 44 as a central place of learning in the class and as a so urce for satisfying moment s. The interview protocol asked students to talk about problems from the homework set they had completed that week, specifically a problem I pre - chose for its non - routine characteristics and a problem they personally found challen ging. The goal was to conduct three interviews with ea ch of the participants, to simulate doing multiple interviews over the semester to study development. All three were ate possible variation due to differences in instruction. Interviews were conducted soon after they passed in their homework, to account for their memory of the proving process receding over time. Homework was due on Wednesdays, so participants were interv iewed anywhere from Wednesday to the following Monday. In the end, two of the participants conducted the set of three interviews and the other participant conducted only one. Interviews occurred in weeks 8, 10, and 14 of the class. Pilot participants were paid $20 per interview as compensation with a $15 bonus for completing the full series of 3 interviews as an incentive. First and Second Rounds of Pilot Data C ollection Based on the first interview, I found that when asked about satisfying moments on the most recent homework, students pointed to a problem almost instantly, i.e. , they could point to a specific experience. They picked a variety of words from my selection to describe the experience, sometimes adding one more of their own, and drew detailed em otion graphs of their experience. There was some confusion over what the x - emotion, represented. At my request, they added annotations for the ups and downs in the 45 emotion graph, sometimes adding more when discussing the graph with me. All in all, the satisfaction portion of the interview went well. A number of issues emerged, however, in the proving section of the interview. I found that it was difficult for students to discuss their thought process about problems after they had been compl eted, even when the interview was done on the same day as the homework had been due. Participants had a tendency to describe their answers quickly, with quick comments at the start about initial strategies that proved fruitless. After two sets of two inter views were completed, I compiled a fu ll list of challenges that appeared. Methodological issues. First , of the process of developing their proofs , likely because I was examining the process after the fact. This lead to discussing proof as a fi nished product, not process. Second , s tudents often did not finish or do some of the homework problems, leading to loss of comparison between participants on pre - chosen problems. Third , students were able to get outsid e help on homework (from professor, teaching assistant, tutoring center, other students, online, etc.), so their answer was not necessarily a representation of their own thinking . Logistical issues. The tight interview window (in order to mitigate memory loss) led to a lower probability of accomplishing the full set of interviews per participant and a These pr oblems led to a redesign of the proving section of the interview, where I asked participants to work on a proof during the interview. Instead of completing the full set of three interviews with a design that had deep flaws, I chose to implement my revamped interview protocol for the third interview. 46 Third Round of P ilot Data Collection: Re - Design o f Proving Section. The purpose of this round of interviews was to test the new proving section of the interview. N=2 participants completed this interview. The priority was testing whether the verbal data produced by participants working on a proof during the interview itself, in real time, would be of better quality in answering my research questions than in the previous hought processes and strategies was much easier using this method, because (a) I could see how they approached the problem on paper, (b) they would talk aloud as they thought, and (c) I could ask clarifying questions in the moment. Another goal of this t hird interview was to test out the proof tasks themselves. In collection) course homework from around this same time in the course schedule. One task turned out to be very similar to what was done in class and thus was done relatively quick ly and without impasses for the students. This task was therefore not very was changed for the actual data collection. The other task was more of a problem , as that term is characterized by Schoenfeld (1992), in that there were times where participants were momentarily stuck. Both pilot participants correctly completed both proofs in the end ho wever. Additional Changes f rom Data Collection Another major change that came out of this pilot data collection was deciding to but I found that participants commonl y pointed to their work and homework problems 47 while talking about them, which was lost in audio. Video of their work and hands provided another source of data validation, especially when any doubt arose from the audio. 48 CHAPTER 5: Data Analysis In this chapter, I d escribe how I analyzed the data. Recall that my research questions were as follows: 1) How does undergraduate students' proving develop over the duration of a transition to proof class? 2) What kinds of satisfying moment s do undergraduate students have during the transition to proof ? The data analysis is described here in a stand - al one chapter because a good deal of work went into deciding how to analyze this data. My two phenomena of interest were quite different, but I faced similar difficulties in addressing them . One phenomenon, satisfying moment s, was a construct I conceptualized myself, so no ready - made analytical frameworks existed. The other phenomenon, proving, was backed by research especially in thinking about proving as problem solving , yet analytical frameworks that served my purpose were difficult to find . In doing this work, I consider the data analysis itself and the challenges I ran into to be a major finding in and of the mselves , which is typical for qualitative work. I detail my journey through these challenges here. I used qualitative methods, because I sought to describe and understand how the phenomena of proving and satisfying moment s occurred. The Data Use Matrix (see Table 5.1) summ arizes how various data from the interview s was used to answer the research questions and the analyses that were done. 49 Table 5 . 1 : Data Use Matrix Research Question Method Data Used Analysis 1. How does undergraduate students' p roving develop over the duration of a transition to proof class? 1a. What problem solving strategies do students use in their attempted solution when stuck? Characterize proving at a snapshot in time Proof Tasks (2) - Writ ten work - Think - aloud - Debrief Triangulation: Interview - About current approach to proofs Look at tasks on which a student becomes stuck. Record their strategies (proof - specific intentions) in response to being stuck use of problem solving strategies when stuck change over time? Compare the snapshots over time Triangulation: Interview - Question about reflecting on change over semester (only interviews 2 - 4) Look for change over a across tasks 2. What kinds of satisfying moments do undergraduate students have during the transition to proof ? Identify moments Describe them Categorize them Interview - Questions about satisfying moments Triangulation: - Emotion Words - Emotions Graphs Bottom - up generation of codes. Add more codes from literature. Apply coding scheme & create new codes as needed (modified open coding). Note: The coding scheme itself answers this RQ. Note. The Data Use Matrix summarizes which pieces of data were used to answer each research question (RQ) and their associated analyses. In this section, I describe the process by which I analyzed my data to answer my first research question: How does undergraduate students' proving develop over the duration 50 of a transition to proof class? In general, the data collected in the proving section of the interview was used to answer my first research questi on. My d . Because there were two stages involved in the way I viewed development, I split this research question into two sub - questions for the sake of describing the analysis: (1a) What problem solving strategies do students use in their attempted solution when stuck? stuck? The results chapter fo r develo addresses the original research question, not split up. Research Q uestion 1a The purpose of research sub - a certain point in time. All of the data generated during the first half of the interview was - aloud, and their responses to questions about their reasoning afterwards. This question was also used when possible: How would you sa y you currently approach proofs right now? a perception. 51 Research Q uest ion 1b The purpose of research sub - question 1b was to compare the snapshots created from research sub - question 1a. This question from the interview was also used: How do you feel your ability to write proofs has changed since the last time we met? The goal of this question was to have students reflect on how they had developed since the last meeting and see how their sense compared to what was revealed by the tasks. Again, the intention their work had changed, as a complement to what was observe d as the researcher. Searching for Usable Analytic Frameworks In studying development, I needed a way to characterize proving at a snapshot in time (RQ 1a) and compare these snapshots across time (RQ 1b). My initial plan for capturing these snapshots of proving was to use an existing problem solving framework, have argued proving to be a subset of problem solving, and probl ems solving as a phenomenon was backed by copious research, it made sense to use existing frameworks for problem solving if they were appropriate for the data. I originally wanted to use Carlson & Bloom for getting these snapshots, because it would provide a thorough way to code all the behaviors that appear in a problem solving attempt. Why existing problem solving frameworks proved problematic. I soon ran into two problems . One, there were frameworks to identify what phase of problem solving a student was in at various times in a task , but characterizing their ove rall problem solving process proving , and general problem solving frameworks wo uld not do that because of their 52 general ity naturally . Any problem solving framework would not pick up proof intricacies, a nd s omething would be lost by staying in the general problem solving analytical frame. What I really needed was a proof - specific framework. This is not a new issue; Savic (2012) has called for the need for a proof framework for conducting research on provi ng. Analytical Framework: Looking a I nstead , I look ed at what students did when stuck and how that changed, with close attention to strategy and monitoring and judgement components of problem solving (Schoenfeld, 1992). S tuckness is an aspect of problem solving, not all, but I argue that without being stuck, a person is not truly in problem solving into authentic proving, when a person is in a state of uncertainty about how to proceed , looki ng at what a person does when stuck is key. I operationalize what it means to be Operationalizing stuckness. Operationalizing what it meant for a person to be stuck on a problem was tricky because it required finding some observable behavi ors to a person (1) realizes there is an issue that needs to be resolved and (2) are not sure what to do. These two criteria had to be present. A key insight into telling when someone was stuck was hesitation over what to do next. Savic (2012) differentiated between an individual se this term despite it not being a word for the sake of simplicity) can manifest itself through silence (via audio) and through body language (via video). In my data analysis, I chose to operationalize being stuck as no written or verbal activity for at l east 15 seconds. Body language instead became more 53 important in telling whether a person was stuck, justifying the need to collect video. Body language behaviors which suggest ed a person was stuck on a proof construction task are reported later. Analysis Process To analyze the data, I watched the video recordings of a select number of written or verbal activity for over 15 seconds). I recorded what behaviors indicated they were stuck, as judging whether someone is stuck can be difficult. When this happened, I recorded (a) my observable evidence that they were stuck, (b) why they were stuck, based off their think - aloud, the later debrief, or my own inferences, (c) actions the y took (as observable on screen, on paper data, or verbally spoken during think - aloud or explained in debrief later), and (d) the intention or strategy I could infer from the action. After this, I looked over the strategies the students enacted when they w ere stuck and looked for patterns of change. I only used tasks where students became stuck, i.e. problems, unless noted otherwise. The notion of actions and intentions while proving came from Karunakaran (2014), as an analytical tool. It was sometimes dif ficult to infer their strategy. In the best case, students stated their strategy out loud during think - aloud or talked about it in debrief. In the worst case, I had to infer their strategy myself from little to no observable data. Difficulties in Analysis Issues with tasks. Some of my interview tasks had unexpected pitfalls. For example, in Interview 3 - 54 students having an incorrect proof, but reasoning from there on could be high - or low - quality problem solving, so it was not particularly relevant to this analysis. If anything, it provided a point of uncertainty to students whic h allowed for more insight into what students do when unsure. In Interview 4 - Task 1, there was some ambiguity over what exactly was odd in the assumption: if a, b were odd or a 2 , b 2 were odd. However, many students did ask for clarification this being the last interview over a semester suggests they have been more comfortable asking me question and it did not affect the final product. Because this or ambiguity m ay in fact work in our benefit. Same or different stuck points ? One issue that had to be resolved was whether to group together multiple stuck points, if and when they really addressed the same challenge. Interview 2 - Task 1 with Timothy is an example of this: He became stuck, took a step, became stuck again, and took another step. His progress throughout this time had a stuttered nature to it, with lots of stops and starts. In cases like these, I considered all of these actions to be in response to one st uck point, as his strategies with each step were all responses to the same stuckness. I counted it therefore only as one stuck point. Operationalizing s trategy to analyze local or more global strategies. For Try to solve the problem Try a different method 55 Try a different proof technique Switch to proof by contrapositive These are all strategie s, from the most concretely actionable (local) to the most overarching (global). Switch to proof by contrapositive is the most concrete strategy, but their goal really is to try to solve the problem . However, try to solve the problem was not helpful for sh edding any light on my research question. Looking at the in - between levels, try a different method is more general than proof technique. My answer to the issue then was to use the most local strategy that was proof - specific but not task - specific . In the ch smallest - sized intention specific to proving but not tied to the specifics of that task and thus diff record. Research Question 2: Kinds of Satisfying M oment s Here I describe how I a nswered my second research question: What kinds of satisfying moment s do undergraduate students have during the transition to proof work? Overview of Constructs and Data Analysis for Satisfying M oment s As a reminder, I operationalized the key terms in this research question as follows. By satisfying moment , I mean an experience characterized by significantly positive emotions, such as an aha moment. By kinds of experiences, I mean experiences that share some set of similar characteristics. Under this conceptualization of kinds and considering the limited existing research about experiences of this nature, grounded theory (Glaser & 56 Strauss, 1967) methods w ere appropriate for identifying key themes in part experiences. I therefore report here my method for identifying kinds, as well as the kinds themselves. In other words, the process by which I identified kinds of satisfying moment s was as much a result as the identification of the satisfying mome nt s themselves. I answered this research question using three steps. First, I identified sections of the audio interview where students discussed satisfying moments: questions 8 - 19. I also looked a t the Emotion Word and Emotion G raph tasks themselves and discussion around them, as needed. Second, I described these moments, according to partic The word selection and emotion graph tasks helped me in describing how the situation unfolded as well, as triangulation for the audio. Lastly, I categorized all these different satisfying moment s. This categorization process was done bottom - up, using techniques from grounded theory. Assumptions sense of their own experiences and what they say is satisfying. There was a choice: Do I report what participants are aware of and claim to be satisfying, or is it better to code what I, as researcher, saw as evidence of a satisfying moment that they were not consciously aware of ? It is tempting to do the latter, to un cover things that participants themselves are reading of their experience (e.g. Satyam et al, 2018). For this reason, I report on what the students identified verbally as satisfying. 57 Data Sources There were two sources of instances of satisfying moment s: in - interview proof - of - interview instances. This analysis focuses on the latter. Across all four interviews with the eleven participants, there were N = 75 instances of satisfying moment s; that is, 75 times participants reported some experience related to their work in the course as satisfying. Of these 75 instances of satisfying moment s, 56 had emotion graphs associated with them. This discrepancy in number comes from two main sources. One, in the first interview, I had participants draw an emotion graph for only on e of the satisfying moments discussed. Two, sometimes students would talk about moments that they said were satisfying but did not draw a graph for it. A minor source is particular to one participant, who drew graphs of how he felt about each question in a n entire homework assignment, not for the specific question that did feel satisfying. These graphs were not usable and therefore not counted. Regardless, this analysis does not rely on the emotion graphs. satisfying moment s, I reduced the data to be analyzed through a careful process to preserve relevant meaning. This is described below. Specifically, I produced 1 - 2 sentence descriptions of what exactly felt satisfying in each experience, which were distill ed representations of their experience. Data Preparation Distilling audio to short descriptions . To prepare the data for analysis, I listened to the audio of each satisfying moment in each of the four interviews for each participant. Af ter listening to th e full retelling of each satisfying moment , I wrote (a) a summary of what 58 had occurred and (b) a short 1 - 2 sentence description of what was satisfying to that student, based on what they said. An example of one of these short descriptions is: Getting a pr oblem you ve been stuck on for a while and getting it yourself (Joel - 1 - 1). The parenthetical identification lists participant, the interview, and a number associated with that satisfying moment. Thus, in the description above, this satisfying moment was th e first one Joel talked about in the first interview. The goal of this two - step process was to carefully identify and keep what felt satisfying to the student. To maintain validity, I later rechecked each of my summaries and 1 - 2 sentence descriptions again st each other, to check that no important relevant information had been lost that would influence coding. Probing about singular satisfying moments . In many cases, I explicitly asked students whether there was a singular moment within this entire experienc e that felt satisfying. I did not always remember to ask this question, as it was a question that arose over the course of data collection. When I did ask it, I included their answer into my sentence description. When I did not ask it, I stuck to the summa ry as close as possible when writing my sentence description, trying to minimize my inferences while also trying to not lose important information about the situation. Given the emergent nature of my analysis, I did not know what information would be signi ficant ahead of time, so minimizing inferences was a non - trivial task. Why not code transcripts directly? In grounded theory methods, it is typical to depart from this tradition in my analysis: My short descri ptions (the data to be coded) w e re by nat ure already interpretive; they were colored by what I noticed while listening and were written by me, not my participant. This was a purposeful decision, however, for 59 ma prolonged and spread out, because I would probe with questions at different points in time. This is not surprising, as their retellings of their experiences were akin to story - telling, and story - telling does not always occur in a straightforward fashion. Secondly, as stated earlier, I sometimes would ask the participant directly what moment exactly felt satisfying within their e xperience, but this (a) presumed that there was a s ingular moment to the participant and (b) may have introduced pressure to the participant to find something to say. Thirdly, incredibly important for analyzing emotions ; tone would have been lost by using transcripts that only included the spoken language. Given these concerns, I thought it better to (a) listen to the entire event and then (b) summarize what seemed to be the satisfying moment to the student, sticking clo se to their interpretations of events. This meant listening to participants intently, paying special attention to aspects such as tone of voice. An outside researcher could verify this by listening to the audio as well. But admittedly with this analysis, I took into account my familiarity with each student what I picked up about their personalities and how they communicate, much of which is not present in a transcript. Considering the nature of qualitative analysis, taking into account familiarity with pa rticipants is appropriate here. Data cleaning: Excluded data and separating out independent instances . After listening to recorded audio, there were 75 satisfying moment s. Of these, one entry was excluded as it concerned why a participant had had no satisfying moment s. In another case, one instance of a satisfying moment was actually two: the participant talked about understanding equivalence classes being satisfying, and also that talking to fellow students 60 about math was satisfying as well. An assumption that underlies this dataset is that each satisfying moment is independent from the next. I therefore separated these instances into two. In contrast, experiences that h ad some element in common were kept together, even if there were multiple different aspects that were satisfying. For example, Stephanie said that doing and understanding homework feels satisfying. She also expressed that re - explaining the homework to fell ow students and getting better grades than others on the homework felt satisfying. Even though there were multiple things Stephanie found satisfying, I did not separate them into different satisfying instances because both concerned the same event, homewor k. If I had done so, these three instances would not have been completely independent of each other. Through this process of excluding one instance and splitting one instance into two, I arrived at N = 75 experiences to be coded. Dat a Analysis: Creation o f Coding S cheme Coding. I coded all N = 75 descriptions of satisfying moment s using grounded theory methods to create a preliminary coding scheme. This bottom - up coding scheme was created in the following fashion: 1. Assigning raw keywords to each instanc e for what participants felt was satisfying 2. Aggregating all the raw keywords together 3. Consolidating keywords similar in meaning 4. Repeatedly grouping similar keywords into larger categories, and 5. Applying coding scheme and looking within each cat egory for variation. Steps 4 and 5 were done cyclically in many rounds, until reaching the coding scheme detailed below. The goals of this cyclical process of defining categories was to minimize 61 overlap between top - level categories while retaining concept ually relevant categories. For example, while the Understanding category had subcategories that could be separated, there was variation left within the Understanding category. This was purposeful in that illuminating. Testing c odes from the l iterature . While a framework for satisfying moment s did not previously exist, there are similar idea s in the literature, including aha moments, mathematical beauty, and self - efficacy. It made sense then to build off existing theory and connect to the literature, in order to grow our knowledge collectively. I derived codes from the following sources: Mathematical beauty: Sinclair (2006), Hardy (1940), Inglis & Aberdein (2014), Blåsjö (2012) Aha moments: Liljedahl (2004) Self - efficacy: Bandura (1977) These sources were chosen based on their thoroughness or uniqueness in examining the topic. subset of the en tire data: N = 30 of the satisfying moment s. I chose instances that exemplified typical instances or were unique. With the combination of these two, the goal was to have relatively high theoretical saturation of all the experiences within my dataset. After this test, I dropped some of the codes because there were no recorded instances in the representative dataset. A few codes from this round were retained in the final set. 62 Difficulties in coding scheme creation . A number of issues arose while in the proce ss of creating the coding scheme using bottom - up methods. First, it was confusing to keep straight coding for what was satisfying versus why something felt satisfying. For example, doing well on homework is what is satisfying but because it is an indicator of my understanding is why that event was satisfying. Second, instances had multipl e codes , which meant any instance could fall into multiple groups, making constant comparison difficult. Nevertheless, I focused on the criteria and what to include and exc lude with each additional instance. Coding along multiple dimensions e.g. code for type of success, difficulty, and people involved did not work because this started to capture contextual elements and not main elements; this was too much information th at it was obscuring the main themes. Lastly, there was a large amount interrelatedness between codes, which made refining the coding scheme difficult. In the following chapters, I discuss the results to both research questions. Chapter 6 concerns developme , and Chapter 7 is about kinds of satisfying moments that students encountered in relation to the transition to proof. 63 CHAPTER 6: roving In this chapter, I answer my first research question, How do es undergraduate students' proving develop over the duration of a transition to proof class? I discuss a selection of some of the major kinds of productive changes that occurred, illust rating each using participant ( s ) as examples . I focus on four important, prevalent developments: (1) sophistication in how students chose proof techniques and their rationales for their choices , (2) awareness about how a solution attempt was going and harnessing that awareness for subsequent strategies , and (3) using examples to notice patterns an d kickstart insight when stuck, and (4) becoming comfortable with exploring and monitoring . Next, I present a longitudinal profile of one individual, to highlight how growt h in reasoning and performance do not necessarily happen together. I then discuss s ome less prevalent developments and end with a cursory analysis for developments across the sample . I interviewed each of the 11 participants four times over the semester and in each interview, they worked on two proof construction tasks. This chapter uses the data from the eight proof tasks for each of the 11 participants ( see Table 6.1, a repeat of T able 4.3 ). Table 6 . 1 : Proof Construction Tasks By Interview Interview 1 Statement Task 1 Suppose x and y are integers. If x 2 y 2 is odd, then x and y do not have the same parity . Task 2 Prove the following statement: If a and b are strictly positive real numbers, then ( a + b ) 3 never equals a 3 + b 3 . Interview 2 Task 1 Prove the following statement: If x and y are consecutive integers, then xy is even. Task 2 Prove the following statement: If a , b , and c are non - zero integers such that a divides b and a divides c , then a divides ( mb + nc ), for any integers m and n . 64 Table 6.1 Interview 3 Task 1 Prove the following statement: Suppose x , y , z are positive integers. If x , y , and z are a Pythagorean triple , then one number is even or all three numbers are even. Task 2 Prove the following statement without using induction: If n is an odd natural number, then n 2 - 1 is divisible by 8. Interview 4 Task 1 Prove the following statement: If a and b are odd perfect squares, then their sum a + b is never equal to a perfect square . Task 2 Prove the following statement: If x, y are positive real numbers and + > 2. Indicators of Being Stuck As I watched video s behaviors contributed to my judgment that students were stuck. A list of these are included Silent No writin g Stares at paper o Holds paper closer o Sits back from paper, to look at it from a distance Taps/plays with pen Touch face with hand or pen These behaviors were not exhaustive and individuals exhibited different behaviors specific to themselves, but I believ e these behaviors cover much of what we see when a person is stuck. provide some attention first to the quality of the written arguments they produced in the 65 four interviews. One common way to think about development in proving is in terms of performance are students more successful at proving as the course goes on? ks for correctness. I assigned their attempts to one of three categories: Correct, Partially Correct, and Incorrect. The idea behind each category was to match standards set in the course, i.e., what students would receive as a score for their written work if they passed it in for Correct proofs were those that would receive full credit on homework, Parti ally Correct proofs would likely get at least half credit on homework, and Incorrect would get less than half credit. In assigning sistant in the course. Table 6.2 below provides more clarity on the criteria for each correctness category, in terms of content of the written work as well. Note this is a rubric for proof as a product, whereas my analysis regarding problem solving is about process. Therefore, conceptual , logical and expression issues were all considered errors. 66 Table 6 . 2 : Rubric for Scoring Performance on Proof Tasks Table 6.3 below quantified here in the following way: 1 denotes Correct, ½ denotes Partially Correct, 0 denotes Incorrect. The last column shows their correctness score across interviews (out of a possible total of 8). Correct Partially Correct Incorrect Criteria in terms of the course standards Would receive full credit in course Would receive at least half credit in course Would receive less than half credit in course Criteria in terms of content Correct proof, with no conceptual or major logical errors. May contain trivial mistakes (e.g. using same variables, minus sign, etc.) that do not affect validity of proof. Overall idea of proof is clear and correct. May contain 1 - 2 conceptual or expression errors, depending on severity . Anything less than Partially Correct or at leas t 2 severe errors Common errors - Proved something more general than given statement - Minor expression issues that do not affect validity or logic of proof - Wrote negation incorrectly - Stated contrapositive incorrectly - Did not justify a step (unless thi s is the point of the proof, in which case Incorrect) - Incomplete Cases: Forgot a case - Minor expression issue that do affect validity or logic of proof - Informally written in words (but idea is correct) - Started with the goal rather than proving it - Pro ved the converse 67 Table 6 . 3 : Performance across Proof Tasks by Particip ant Participant 1 - 1 1 - 2 2 - 1 2 - 2 3 - 1 3 - 2 4 - 1 4 - 2 Total Amy 0 1 1 ½ 1 ½ 1 1 6 Charlie ½ 1 1 1 ½ 0 0 ½ 4 ½ Dustin 0 0 ½ ½ 0 0 0 0 1 Granger 1 1 1 1 1 0 ½ 1 6 ½ Gabriella ½ 1 1 0 ½ 0 0 0 3 Joel 1 ½ 1 ½ 0 0 0 1 4 Jordan 0 ½ 1 0 0 0 0 1 2 ½ Leonhard ½ 0 1 ½ 0 0 0 0 2 Shelby 0 ½ 0 ½ 0 0 0 0 1 Stephanie 1 1 1 ½ 0 0 0 0 3 ½ Timothy 1 1 1 1 ½ 1 ½ 0 6 Note . The headings denote Interview - Task (e.g. 1 - 1 means Interview 1 - Task 1). Cells: 1 denotes Correct, ½ denotes Partially Correct, 0 denotes Incorrect. Looking across the interviews, Granger had the most success across the eight tasks (6 correct, 1 partial ly correct), followed by Amy (5 correct, 2 partially correct) and Timothy (5 correct, 2 partially correct). Dustin and Shelby got the least correct across all interviews (only 2 partially correct each). It is important to note that the difficulty of the it ems was equated across the interviews; in my estimation, difficulty generally increased over time . Later proof tasks were more reliant on increased content knowledge of basic number theory and had more complicated solution paths. If item difficulty had sta yed the same throughout, we would expect that performance would improve as interviews progressed. poor scores on later tasks should not necessarily be taken as an i ndicator that they had not improved. In the following analyses, I only discuss tasks where students were at least partially or completely correct. 68 Cross - Individual Developments I present the most pervasive developments and practices that appeared, i.e. the developments that occurred across the largest sh ares of my eleven participants . The goal is to describe the developments that have the most grounding in data. F or these developments, I take a cross - individual analysis: I discuss one to three student examples to (a) illustrate what that development looked like as it unfolded but also to (b) highlight any variation in how that development occurred. I provide a summary of the changes seen f or each development. I would like to note that my choices of participants are not meant to say that other participants did not show these developments nor even that these are the best examples across participants. The students discussed as examples of each development are merely to be illustrative of the development, serving the reader. Development A : Changes in Choosing a Proof Technique One common development that occurred across participants were changes in how they chose what proof technique to pursue, when approaching constructing a proof. By proof technique , I mean tools such as direct proof, proof by contradiction, proof by contrapos itive, cases, and proof by induction the techniques that were taught in the course. For the sake of redundancy, I will often refer to proof by contradiction as just contradiction and proof by contrapositive as just contrapositive. The first half of the cou rse was about learning proof techniques, so naturally many students generally thought about what proof technique to use as a major way of approaching constructing a proof. In addition, homework tasks were often written in such a way that one of the proof t echniques led to an easier proof, over usi ng other proof techniques. Eight of eleven participants showed signs of this development, based on data from the interview notes and 69 across all tasks (not only the ones where they became stuck). I discuss two parti cipants here, as illustrative examples of this development. Example A1: Favoring one technique. F rom the beginning (I nterviews 1 and 2) , Stephanie favored proof by contradiction over all other technique s when constructing a proof . In Interview 1 Task 1, she immediately jumped to trying proof by contradiction - When I see the if - then statement, I immediately think She explained she felt comfortable usin g this technique. Figure 6 .1 shows her work, where she immediately identified the assumption and conclusion negation . Figure 6 . 1 . Beginning of Stephanie s work on Interview 1 Task 1 Note that Stephanie technically wrote the negation incorrectly; the correct negation 2 y 2 wrote the negation as an implication, a common error. However, this error did not affect the rest of her proof and her reasoning for picking proof by contradiction is unaf fected by her execution. It is interesting that already by the first interview Stephanie felt most comf ortable with proof by contradiction, considering that this was new knowledge they had recently learned in class, not something they came to the course already knowing. 70 In the next in terview, Stephanie go - to method was still proof by contradiction. Upon st arting Interview 2 Task 1, she said "I can see that this is an if - then statement, so automaticall y I'm going to try to use contradiction, but I don't know if it will work or not." hen I read an if - then statement, I 'm most comfortable using negation or a contradiction. So then I just try that, even though I know it doesn't always work, but I just try it." Note how the use contradiction is automatic for her, and she herself said outright she does not always know if pr oof by contradiction will lead to a correct solution. The general structure is enough to determine that she can use her favored technique, but she did not make use the statement in any further way to guide her choice of technique. Stephanie did indeed get stuck on her proof by contradiction, so she switched to proof by contrapositive (see Figure 6.2) . 71 Figure 6 . 2 . Stephanie s move to contrapositive on Interview 2 Task 1 I'll try contrapositive and then I fe lt a little better after I tried contrapositive just because I thought [out of] both of them, probably one of them was gonna be right." Stephanie did not give a rationale for why specifically proof by contrapositive, just that it was another technique. 72 Summary. Interviews 1 and 2 wh enever they can. Stephanie did have a condition for when to use proof by contradiction, when she sees an if - then statement. However , this applies to nearly all statements to be proven in the course that we can safely say this is her general technique. Step hanie becomes less dependent on proof by contradiction and her rationales do become more sophisticated over time, but her work was unfo rtunately incorrect on all four tasks on Interviews 3 and 4 . W e turn then to a differe nt student in order to better see h ow choice of proof technique and rationale changed over time . Example A2 : Recognizing advantages of a technique, independent of statement . I now present the case of Timothy, to show development that extends what we sa w through Stephanie . T imothy was similar to Stephanie in having favored proof techniques in the beginning , but his rationales became more sophisticated and based on the statement itself as his interviews progressed , in addition to producing correct or partially correct proofs. F igure 6.3 - Task 1 to construct a proof for When stuck in the beginning, he re - read the question and wrote what was known. At this point he switche d from his direct proof attempt to proof by contrapositive. 73 Figure 6 . 3 . Beginning of Timothy s work on Interview 1 - Task 1 When asked why he selected contrapositive, he explained it was a method from class but also that it was a logically equivalent tool to direct proof that he could use: Timothy: Interviewer: So actually, so how did you come up with contrapositive? Timothy: Looking at it straight we learned in class that the contrapositive is basically not B implies not A. I knew we said that was logically equivalent, so if I could prove the contrapositive was true, then I could prove the original s tatement was true was kinda my thinking with that. He explained that a direct proof method was not helpful in generating a proof, but he gave no specific ration ale for choosing contrapositive over other proo f techniques. His explanation implied that contr apositive was a legitimate tool from class, so why not use it? While it is possible he may have had some internal reason for using contrapositive, he neither mentioned this on his own nor articulated any f urther reasons when questioned. Later in this int - methods and why: 74 Timothy : I always go about it with either contradiction or induction or straight up so I kinda knew that I might be able to contradict this never equaling that, so I wrote out the contradiction...I guess contradiction is a little easier for me to think about. You just say the first part of the implication is true and the second part is false. So around the implication, negating both parts. Interviewer : Okay Timothy: Timothy exp ressed here that contradiction wa s easier for him than contrapositive, which involves negating the assumption and conclusion. His insight about the work involved in setting up the two different proof techniques contradiction vs. contrapositive was true. It is important to note that he had some rationale for why he might use contradiction, but it was couched in terms of ease of u se , first and foremost. The idea of ease of use as determining choice of proof techniques showed up in latter interviews. In his work for Inte rview 2 - Task 1 (see Figure 6.4 ), Timothy started by defining x and y using the definition of consecutive numbers and in ca lculating xy, became stuck over what to do. Figure 6 . 4 . Timothy s switch to contrapositive on Interview 2 Task 1 75 on some level for him but not for any reason s specific to the statement and did not further articulate why. What exactly made this method easier remained unknown to him or at least was not clear enough to him to easily articulate when asked. (In the end, his contrapositive proof was not to his likin g and also not correct). But by the end of the interviews, Timothy showed sophisticated thinking in considering which proof techniques to use. In Inte rview 4 - Task 1 (see Figure 6.5 ), Timothy became stuck after computing the goal (a+b) directly. Figure 6 . 5 . Timothy s work on Interview 4 Task 1 76 same task. He then gave this further rationale for why contradiction: experiences, so mething. So I tried to use contradiction because I knew I could say then it is a perfect square. His argument was that he wanted to be able to work with an equality, much like Leonhard. Timothy also gave a rationale for not using another method, contrapositive: I thought about contrapositive, too, but then it would say that A and B are not not work when I know like a straight definition of something. So if I could keep this, I knew if I could keep this, like they are perfect squares and say this is a perfect square, th to work with. His explanation was similar to his prior one about equality of objects being easier, i.e. knowing things are not equal is not as helpful. His subgoal then was to find a proof technique that would give him a+b is a perfect square. This task is notable however for draw contradiction: I never really thought about it this way but I realized when you use the contradiction, you don t really have the assumption and conclusion anymore you can actually pick any part of that statement you want and work with it . Rather than with an if/then statement, you start with the assumption and try to work to the conclusion . So y Timothy gave a high - level explanation of the nature of proof by contrad iction. He found proof by contradiction to be freer than other techniques, due to being able to work with all parts of the statement. This stood in contrast to starting with the assumption and trying to prove the conclusion as is done in direct proof but a lso proof by contrapositive. It is of separate note that this revelation came about during this interview context, based on the "I 77 never really thought about it this way but..." clause. The interview served as a vehicle for reflection on proof techniques f or Timothy. Summary. went from picking a proof technique (1) because it existed as a tool to (2) having a fuzzy sense that it would be easier to (3) explaining how the content of the statement drives the problem solving approach to (4) articulat ing understanding at the meta - level of how a technique functions as logical tools. His later interview s revealed insights for when to use contradiction that did not depend on statement content but instead meta - level structure . Comparing d evelopment s in choice of proof t echnique s . Both Stephanie and Timothy showed similar growth in how they chose proof techniques to pursue through most of their interviews. Both discussed liking and being drawn to certain technique s, as - some level of weighing the utility of different tech niques, to think about which would be better , w hether it be a cleaner proof or just easier. He noticed that being able to set things equal provided the prover with more to work with ; contradictio n wa s therefore the most useful technique, based on the content of the statement. The d ifference between the two lies in where they end ed : Timothy came up with a general insight for when contradiction was useful . By looking acros s these two students , we can see this genera l trajectory in how students gre w in how they cho se techniques to use. If we conceptualize this specific development a s a series of stages, Figure 6.6 illustrates the stages students tend ed to step through. 78 Figure 6 . 6 . Stages of development in how students choose proof techniques to pursue To use an analogy, let us think of proof techniques as hammers. In the beginning, students have a certain hammer they like for reasons that tend to be personal and not mathematical, and they use this hammer for all tasks, regardless of the nature of the task at hand. After some time, they start using different hammers other than their favorite but have no clear rationale for why one over another ; they just pick up a different one when the need arises . They then start using specific hammers for specific task s ( attending to content of the statement to be proven) , but without yet explaining why they are do ing so . Finally, some students see when to use certain hammers over others, understand the advantages of each , and can explain why. The same way different hammers work better in different situations, different proof techniques can lead to more straightforw ard proofs. Development B : Asse ssing How the Solution Attempt I s G oing and Harnessing It Another common development among participants was a grow ing metacognitive awareness of how their solution attempt was going , usually when they felt they were on the w rong track. Four of the eleven participants showed development of this kind. Being aware 79 w hat is important is examining what students did in response to their awareness (albeit involuntarily) and how that guide d them to better solutions. For these reasons, I highlight individual tasks where students showed they had this awareness, with the implication that this did not appear in ea rlier interviews. I discuss three studen ts to show variation in h ow students harnessed metacognitive awareness : Granger, Timothy , and Jordan. Example B 1: Intui tive awareness l ead to restart . Granger was another student who was aware when things were going wrong, even if the reason why was not cl ear. In Interview 2 and it was indeed a difficult task for him. He became stuck at some point and took multiple attempts, as can be seen by all the cross - outs in his scratch work in Figure 6.7 . Interview 2 Task 2 Prove the following statement: If a , b , and c are non - zero integers such that a divides b and a divides c , then a divides ( mb + nc ), for any integers m and n . Figure 6 . 7 . Granger s scratch work on Interview 2 Task 2 (statement provided) 80 The following exchange happened during the debrief, in which he showed awareness that things were off: Granger: So I was like, "What am I doing? This isn't right. Something's not right here." Interviewer: But it sounds like you had a sense that... You knew that like, "I am not doing this the right way." Granger: Yeah, I definitely did. I don't know. I just know... I d on't know how to explain that. You just know when something isn't right. Interviewer: Is it like when it's [this attempt is] not helping you get anywhere or it's not clarifying things? Or is it really just like an intuition? Granger: Yeah, just like an intuition, like, "That does not... This statement absolutely doesn't make any sense with this," and I was like, "It can't be r ight." .. .But you know it's just like, "This does not agree with the definition at all, so what am I doing?" And then I just reassess the situation and I'm like, "Okay, let's start fresh." Granger knew something was wrong, intuitively. He could not pinpoi nt what exactly was wrong but had an awareness that this could not be a correct way to go about it . He also explained his strategy o f starting over : Granger : Usually, on homework I would pick a page and start going and, I don't know, it's a weird thing, I 'd be writing or something, and if it's wrong, I'd cross it out and I'd try again if it's wrong... Eventually, if I get to this much space where I've gotten ... I just flip to a whole new page and it's like a refresher like, "Okay, you start a whole new... What's going on." Interviewer : So that kind of helps, it sounds like. Granger : Yeah, yeah, definitely. I don't know . It's intimidating when you see a whole bunch of crossed out marks and it's just like your brain is focusing on what you got wrong and... Ye ah. Interviewer : As opposed to fresh ideas or trying new things. Granger : And we learned in... Ironic, I learned in psychology, when you're trying to figure out a problem, your unconscious mind is also thinking about it but you don't re alize, it's unconscious, but... So as I'm flipping the paper over and just like resetting myse lf, also, my unconscious is thinking about what I did wrong already, so it doesn't matter, I already know what not to do. Interviewer : So you don't need to look at it to... Granger : Yeah, exactly. And looking at it, it messes up consciously what I'm doing unconsciously. 81 His subsequent strategy was to abandon his past attempt and ways of thinking completely and start afresh. It worked on this task: N ear the bottom of his scratch work, he started working in a more helpful direction and was able to get to a c orrect proof. Summary. helpful here, in that it led him to let go of what he had done and star t something afresh, leading to a correct solution. Example B2 : Awareness l ead to finding new strategy. I now highlight one task - Task 1, he became stuck multiple times over the course o , t hen xy is ). 82 Figure 6 . 8 . Timothy s first attempt at Interview 2 Task 1 In response, he started reasoning out loud about the ight now. He was assessing his attempt while working: After getting stuck twice more, he ended up with proof but he was unconvinced about it; he did not feel good about it. 83 contrapositive]. And then I went back because I really wanted to do something with this directly. I liked that better." (The latter proof will be discussed in a lat er section). His use of being off rather than fully about the correctness of this proof. His first proof did not sit well with him then, enough to lead him to look for a different solution. This sense served him well, as indeed his first attempt was not correct but the one he came up with later was. This affective metacognitive sense playing a role in his work was further indicated by his emotion graph and words fo r this task. Figure 6 . 9 . Timothy s emotion graph for Interview 2, Task 1. Note that the dip occurs when he was unsatisfied with his contrapositive proof. when unsatisfied with his proof. In fact, his dip in emotion in the graph came from dissatisfaction about his contrapositive proof specifically. It is possible that he was dissatisfied because he thought his proof was not correct and that manifested itself through his emotions. However, even if this is true, it is interesting (in light of other research questions in this dissertation), that he spoke about the acceptability of his proof affectively. 84 Summary. In summary, Timothy showed a metacognitive, affective awareness and monitoring of his proof attempt. It drove him to keep thinking and look for another way, even though he had reached an end in his work. In a later section, I examine how Timothy was able to act on his awareness and find a better proof, more to this liking. Example B3 : Aware but stayed on same solution p ath . I now present a contrasting example, of a student who was aware when something was wrong but continued her strategies, not changing direction. Jordan beca me stuck and was aware that something was not working, but she would move past it and continue with her current strategy. In I nterview 3 Task 1 (see Figure 6.10), J ordan was stuck in the beginning, stating she felt like she did not understand what she was proving. Interview 3 Task 1 Prove the following statement: Suppose x , y , z are positive integers. If x , y , and z are a Pythagorean triple , then one number is even or all three numbers are even. Figure 6 . 10 . Statement of Interview 3 Task 1 She had an idea about using two cases, where one case would be setting one of x,y, or z to be even and the other case would be setting all three of these variables to be even. She was confused though because she felt she was starting with what she normally would show. Regardless, she forged ahead with using cases on x,y, and z for the equation x 2 + y 2 = z 2 . Figure 6. 11 shows her work on this first case: Figure 6 . 11 . Jordan s beginning work on Interview 3 Task 1. She assumed one variable was even and the other two were odd but it lead to a statemen t that did not help her. 85 She became stuck again after she expanded her terms, because she d id not know what to do with the last line in Figure 6 . 11 . She ignore d it and skipped ahead to the second case, whe re x, y, and z were all even . Afterwards during the debrief, she said I feel like maybe I got the Jordan was unsure about her outcome; she was aware that something was wrong and that there must be a better soluti on but that never changed her strategy of using these particular cases on x, y, and z throughout her attempt. incorrect in the end. On Interview 4 - Task 1, she began by checking some examples and did some algebra but then became stuck with how to show that the expression was not a perfect square. From then on, her work was a stuttered series of stops and starts (see Figure 6.12) : factoring out a 2, getting stuck, taking the square root, getting stuck, factoring out , and getting stuck. s solution here was also incorrect in the end . Figure 6 . 12 . Jordan s work on Interview 4 Task 1 (statement provided). The last few lines are where she experienced multiple stops and starts. Interview 4 Task 1 Prove the following statement: If a and b are odd perfect squares, then their sum a + b is never equal to a perfect square . 86 During the debrief, she was honest: How are you feeling about it overall? I just don't think I'm allowed to do that. I don't think I did it right. Jordan was aware that her solution attempt was off (and in fact believed she had taken invalid mathematical moves) . When she was stuck during the proving process, she ignored that something was off and kept pushing forward via algebraic manipulation . Summary. These tasks from the latter two interviews showed that by the end of the course, Jo rdan knew when her attempt was off and had some idea of why (e.g. not knowing how to formally show something), but she would ignore it and move past it and/or not alter her current path. Comparing developments in awareness and using it. All three of these students show ed awareness when things were not going well . working, feeling intuitively somethin g was wrong in this process, while Jordan and focused more on not liking t he outcome. However, these three react ed differently to feeling something was w rong: Jord an would continue on with her current plan of attack , Timothy would re - assess what he was doing by reasoning out lo u d about the relationships , and Granger would start on a fresh page in order to not be in fluenced by this past thinking. Anoth er way to examine this i s to look at the level at which they worked : Jordan stay ed grounded at t h e level of the algebra to try to make her way of thinking work , whereas both Granger and Timothy went back to the top level of the problem . Timoth y in fact ed of the problem (he physically would lean awa y fr om the paper) and muse about the ta sk as though with a . B oth Granger and Timothy stumbled upon correct proofs for their tasks here proofs discussed here were incorrect . In summary, a ll three 87 were a ware of how their attempt was going, but Granger and Timothy used that to guide them selves to better solutions successfully. These three students serve as vari atio aware ness of how their proof attempt is going looks like and the ir subsequent metacognitive strategies: (1) continuing with the plan (Dustin falls into this category too), (2) abandoning the current path completely , and (3) playing aro und with what one is drawn to in order to find a new path . This last variation i n particular may be a n example of the inquiry - driving role of mathematical aesthetics in leading the mathematician to investig ate certain avenues of solution attempts over others (Sinclair, 2004 ) . This discussion may make it seem like Jordan did not experience development . I t is important to note that having awareness that one may have used invalid mathematical moves (as she worried about on Interview 4 Task 2) is far better than assuming solution is always correct. Jordan may have continued on her current approaches when stuck because she thought them the most likely path to success or did not know what else to do, in the same way that Timothy and Granger thought changing their approach would lead to a correct proof and/or did not what else to do . The key difference was in how Jordan did no t know what to do but stay ed in that confused state, whereas Timothy in particular engage d in practices that helped him go from not knowing what to do (same state as Jordan) to figuring out what to do. well is t he first step ; using that effectively to get one self unstuck is the next. Development C : Exploring and Monitoring Working without already knowing how a solution would go was another development seen ac ross participants. Students were used to tasks in their past 88 mathematical courses, from K - 12 through calculus in college, that lend themselves to clear methods and procedures upon reading the task. As will be discussed in a latter chapter, students also fo und it satisfying being able to see the entire solution path ahead of time. But during the transition to proof course, students became stronger at careful, intentional working and exploring without knowing what will happen in advance and no ticing when a key piece of information for constructing the proof arose . be better to start working and see what comes up. This practice is about effectively manag ing oneself when there no clear strategy is apparent . Four of the eleven students showed growth along these lines. I only discuss one student Amy for this development because she served as a representative for the changes seen in the participants analy zed. But more so, Amy is a case of a student who was high achieving from the start and did not change much throughout the interviews. R e call from Table 6.2 that she got 5 tasks correct and 2 partially correct, out of 8. Her performance therefore already ha d little room to grow, but moreover, her approach when stuck did not undergo serious changes except for one singular change described below. Amy considered herself as a planner, always thinking ahead. Over time , she became more comfortable with working without a specific strategy in mind. She was a strong performer and confident in the class from the start, often finishing tasks quickly. Amy was outwardly confident in her mathematical ability and oftentimes saw how to do tasks right away. For example, sh e wrote her proof for Interview 2 Task 1 in under four minutes. While discussing other things at the end of the second interview, Amy said this about herself: 89 Amy: I just plan, I don't know, I plan everything super far in advance. Interviewer: Oh, okay. So when you go in... Amy: I just feel like for everything, I just look ahead. Even when I'm doing math problems. I just like, in my brain, I think about what I'm gonna do before I start doing it. Amy specifically noted that this was how she did mathemati cs, always planning out her mathematical actions in advance and thinking ahead in the problem. But even with her disposition to wards plan ning , Amy became comfortable with workin g on her feet as interviews progressed . In Interview 4 Task 2, she decided to use became stuck briefly after that because she did not know what to do now. She said out loud that she did not have a plan while working but that she would fi gur e something out (see Figure 6.13 ) . Figure 6 . 13 . First half of Amy s proof for Interview 4 Task 2 Amy intentionally chose to explore the mathematical situation, manipulating the equations algebraical ly, with no clear purpose. This proved fruitful , as she noticed the contradictory 90 nature of 0 < ( - x+y)(x - y). After this, t he rest of her work argued why this was an impossible situation for ( - x+y) and (x - y). Thus, she had found a contra diction, as shown in Figure 6.14 . Figure 6 . 14 . Second half of Amy s proof on Interview 4 Task 2, sans the final lines. During the debrief, she talked about what was going on when she was stuck early on: Interviewer: Okay, ar e there any points in this problem where you feel like you got stuck? That you'd call stuck? Amy: I feel like this whole portion, I was kind of stuck, but I was just like, "Just check through the algebra until you can get to something." I was like, "I don't see this going anywhere, but I'm sure it will. Just keep going." Her proving process on this task showed how she did not know at the beginning what she was going to do but was able to roll with the punches. She worked without a specific goal in mind and when a potential avenue appeared, she pursued it and found the contradiction. Summary. With Amy , she moved f rom planning out steps ahead (based on her own words) to being comfortable exploring and monitoring her work when unsure what to do. The important thing was her noticing an insight when it arrived. Some of this may have been due to t he difficu lty of proof tasks; these tasks were no longer so easy that their solutions could be seen right from th e beginning (compared to traditional K - 12 math), so 91 some level of working without knowing what will happen is part and parcel o f a true problem in provin g. Development D : Using Examples to Get Unstuck As this analysis has focused on what students do when stuck, it is sensible to pose the que stion: Do students develop effective ways of becoming unstuck? One could say this is a hidden goal of a transition t o proof or any problem solving course. A productive practice specific to mathematics emerged during some of the latter interviews, where student would check examples as strategy located within a temporal string of strategies. Three participants - Charlie, Granger, and Timothy - showed this behav ior during interviews, through a cursory analysis. I present Timothy here as a representative , to showcase how example checking was used to become unstuck . Timothy developed a robust practice of using examples when stuck, over the course of interviews. In an earlier section, I indicated Timothy was not happy with his first solution to Interview 2 Task 1. As a result, he was silent for some period of time, which I interpreted as some version of being stuck. He looke d back at his work, reasoned out loud what would happen. While imagining, he thought of "plenty of examples like this [from class] where you give a generic odd and even v alue in this case and then solve it out." He insight about taking even and odd cases on k (see Figure 6.15) . 92 Figure 6 . 15 . Timothy s new solution on Interview 2 Task 1, after thinking of example exercises from class. H is pro of was indeed correct. In this task, from example exercises from class; he drew on past mathemat ical situations he had seen. The last interview provided a view of Timothy using example checking, however, to get out of tough situations . The last line of his work in Figure 6.16 . for Interview 4 Task 1 s hows he got to a formula f or m but then became stuck again. This was in fact his fourth stuck point on this ta sk . 93 Figure 6 . 16 . Timothy s work on Interview 4 Task 1. He reasoned out loud about the implications of the last line. He then started reasoning out loud: If m has to be an integer, what does n have to 2 is an integer. out from his work, thinking about what he needed conceptually. He checked an example ou t loud and noticed n 2 /4 would have to have a ½ in it, to cancel out the - 1/2. He had some realization about why the claim was true, based on his examples, but did not know how to officially show m was not an integer within the allotted time . In the end he gave up, but his work was partially correct and his strategy got him quite close to noticing what the contradiction was in a way that other students did not , that m could never be an integer in this situation. 94 On the final task of the fourth interview (Task 2), his example checking came t o full fruition (see Figure 6.17 ). Figure 6 . 17 . Timothy weighing proof techniques on Interview 4 Task 2 He became stuck on the first item beca use he was not sure how to negate the statement. He then reasoned out loud what his issue was and possible decisions he could take ( see the contrapositive and contradiction set - ups in his work. ) This is akin to parallel processing in assessing which of man y solution paths is a good idea. He then checked some examples: What he h ad done in this instance was reason out loud about the issue - > i magine multiple paths - > check examples - > try a different proof method. incorrect due to multiple algebraic errors, but his approach of using examples in conju nction with other strategies is unaffected by this. Summary. Timothy developed a practice of what to do when stuck, whether knowingly or not, as interviews progressed. He did some combination of these strategies in this relative order: Look back over work - > reason out loud - > imagine what would happen if certain things were true - > check examples - > have an insight that establishes a direction. It is important to note that Timothy put reasoning ou t loud to good use here, based on how 95 frequently he became u nstuck after doing so. His use of talking out loud stood out across the sample , even when considering that participants were explicitly asked to think out loud. Timothy used examples as a way of instigating insights, whether intentional or not. He reasoned out loud about what was known, musing about the content at hand, in a sense looking for something to work with. The important thi ng, however, was that he was attentive enough to notice something when it came up. An example is not a proof, but it can provi de an idea for a proof, and he used examples in this nuanced way. It is of note that this practice may have originated from his instructors instructor, Ms. Frye , reported 3 that she spoke to her class about using examples to get an intuition about why a statement was true but that examples did not count as a proof. Another participant (Granger) also said that Mr. X suggested checking examples as well. Longitudinal Case: Leonhard In contrast to the cross - individual discussion of developments, I now present a profile of development by following one individual across the interviews. The purpose of development. Here I follow the c hanges seen in Leonhard because over the in terview series he showed growth in certain areas his affect and his proving process but his performance declined. 3 (M s. Frye [pseudonym], personal communication, May 19, 2018) 96 s Process f or Choosing Proof Techniques ine was to choose proof techniques based on what he knew and was familiar with. In Interview 1 - Task 1, Leonhard chose to use proof by contradiction to approach this problem, despite being a little stuck because he was not being sure how to negate the con clusion (see Figure 6.1 8 ). Figure 6 . 18 . The beginning of Leonhard s work on Interview 1 Task 1 His rationale for that contradiction because that is what they used in class and he wa s used to it. In approaching Task 2 of that sam e interview (see Figure 6.19 ), he used proof by Figure 6 . 19 . Writing out contradiction and not finding it helpful; beginning of Leonhard s work on Interview 1 Task 2. 97 contr apositive this time when stuck. At first he wrote the contradiction statement but h is how nice looking it would be to use a contradiction 4 He had a sense avoid it and use the contrapositive. This could mean that proof by contradiction would not be so clean or would require more work. In fact, Leonhard wanted t o use contradiction, as estab lished on the l - nly because he was worried about it did he switch to contrapositive. His move to co ntrapositive specifically was motivated then but only because it was another technique; his rationale used general terms and he did not articulate it in more detail. In Interview 2, Task 1, he wanted to do direct proof but became stuck because he was unsure whether what he wanted to do would work. He applied the definitions to x and y and then was stuck again over what method t o use, direct proof vs. proof by contradiction. He became stuck again in choosing whether to do direct or contradiction. contradiction becau nhard chose what met hod to use based off what he felt he could do at that point in time, his own sense of fluency with methods and . 4 It should be noted that Leonhard made some errors here: a and b strictly positive means they cannot be 0, so his written work should state that a > 0, b > 0, not greater than or equal to. always equals, which the equal sign implies . 98 As time progressed, there was clear growth in his reasoning for his choices even though his solutions were overall incorrect . Interview 3 Task 1 is an example where Leonhard cycled through a few options for proof techniques, as seen in his wr itten work (see Figure 6.20 ). Figure 6 . 20 . Leonhard s work on Interview 3 Task 1 99 He used proof by contradiction but then became stuck in writing the negation, because his realized he had the sa me issue with how to negate the conclusion, as before. So, he switched again to direct proof. His rationale for why contradiction in the first place was as follows: ther e [to contradiction]. I like it the most because...at some point you usually run into something that just comes out sounding weird. So then you have to be right I guess. Leonhard admitted that contradict ion wa s his favorite, so he tended to use it wheneve r he could. He liked it, because of its unique nature in producing something nonsensical. He ran into the same issue. Leonhard knew he liked certain methods over other s and had some rationale - in how proof by contradiction results in a nonsensical claim and that he should have kn own to use contrapositive. His rationale was still general, however, in that contradiction was a technique he liked and that his fondness for it drove his usage of it. Interestingly, he mused out loud about how his underlying idea may have been to check which proof techniques did not work well here and see what is leftover: o so you can find the things that suggests we should not put too much stock into t h is claim about his thinking . 100 By the fourth interview, Leonhard showed growth in the precis ion and detail given in his rationales for his choice of p roof technique. In Interview 4 - Task 2, he was stuck in the beginning and his subsequent actions were to identify the assumption and conclusion, test a couple examples for x and y, and then try pro of b y contrapositive (see Figure 6.21 ). Figure 6 . 21 . Beginning of Leonhard s work on Interview 4 - Task 2 H is rationale for contrapositive was, useful than not equal to in pro ving, and neither direct proof nor proof by contradiction provided an equality. He decided what proof technique to use based on specifics of the statement to be proven. In addition, his rationale also explicitly explained why another proof technique (contr adiction) would be less useful here. In the end, Leonhard had a rationale for why his chosen proof technique was a helpful approach and why other techniques would be less helpful. In the end, his proof was incorrect, as reaching a true not the same as showing the conclusion, but his rationale for why use contrapositive was coherent . 101 Making Sense of Over the course of these interviews, the rationales Leonhard gave for why he chose the proof techniques that he did became more sophisticated. He moved from choosing certain methods (1) for little to no reason to (2) having some rationale, with a general s ense of one technique being better than others to (3) based on the statement itself. - that Leonhard (see the excerpt from Table 6.1 below). Participant 1 - 1 1 - 2 2 - 1 2 - 2 3 - 1 3 - 2 4 - 1 4 - 2 Total Leonhard ½ 0 1 ½ 0 0 0 0 2 Across the interviews, he got 1 task correct and 2 partially correct. Moreover, h is work for the last two interviews (four tasks) was all incorre ct according to the scoring rubric, due to making substantial errors and/o r missing crucial pieces of the proof. Interestingly, Leonhard his work was correct on three of these four tasks ; h e showed great confidence , as can be seen in his emotion graphs for these tasks in F igure 6.22 . Over the interviews, even though his success on tasks stagnated, Leonhard showed progres s in terms of affect , of having confidence in his work . There are some good things to this, in how he had a positive orientation towards his work, but it is also worrying when a student does not notice major flaws in their work. Leonhard is an example then of where a e is high and their reasoning and rationale for their decisions is high, but these do not necessarily lead to correct work. 102 Figure 6 . 22 . Leonhard s emotion graphs for Interview 3 Task 1 (top left), Interview 3 Task 2 (top right), Interview 4 Task 1 (bottom left), Interview 4 Task 2 (bottom right). His graphs indicated high positive emotions about his work on Interview 3 and Intervi ew 4 Task 2 but his solutions were incorrect. 103 There is a difference then between reasoning and execution: Leonhard reasoned well but his execution was flawed. Can we say Leonhard understands contrapositive? Another interpretation of this profile is that progress in terms of process does not always manifest itself in terms of performance, as measured by objective correctness . Judging a student based on solely their w ritten work does not necessarily capture the thinking and reaso ning behind their choices th at was valid, which alone is valuable growth in proving . Developments with Limited Data Here, I talk briefly about some other developments that occurred but were less pervasive across participants. These developments are not particular responses to bein g stuck but are approaches to proving in general. Develo pment E : Attending to the Goal In contrast to working without a plan, some students were more attentive to the goal while proving, as opposed to just working . Charlie only showed signs of being stuc k on two tasks of the eight, but one of the tasks serves as a great example of getting stuck and then unstuck. On Interview 2 - Task 2 (see Figure 6.23 ) , Charlie experienced multiple wrong directions and multiple stuck points, due to interpreting the defin ition of divides incorrectly. Interview 2 Task 2 Prove the following statement: If a , b , and c are non - zero integers such that a divides b and a divides c , then a divides ( mb + nc ), for any integers m and n . Figure 6 . 23 . Statement for Interview 2 Task 2 a equal to each other ( b ut realized what he was doing was not helpful for the goal: 104 with his goal and work backwards and this lead him to a correct proof . Charlie attend e d to the goal even when not stuck. During Interview 3 Task 2, he thought about whether what he wa s doing was helpful for what he had to show this is not a good idea for prov[ing] this [statement] something his instructor , Ms. Frye, recommended in class at some point, likely explaining this development. In contrast, Leonhard had a habit of working on a proof and reaching a true statement, thinking that meant his work was correct. This is an example of how not attending to the goal can lead to incorrect proofs. An example of this occurred on Interview 4 Task 2 (s ee Figure 6. 24 ). Figure 6 . 24 . Leonhard s work on Interview 4 Task 2. He reached a true s tatement and thought he had shown the claim. 105 believed confidently that he had written a correct proof. Reaching a true statement does not mean one has proven the cl aim, however. Paying close attention to what needs to be shown i s important. Develo pment F : More Systematic Way s o f Approaching the Statement The last observed development I discuss was in how students systematically broke down problems. Here, I pro vide t wo cases: Leonard who did this fro m the start vs. Timothy who did th is over time and reported it as an area in which he felt he had grown. From the beginning until the end, Leonhard had his own proc ess whenever he read a task: I dentify the assumption and conclusion, oftentimes assigning them P and Q (as is sta ndard nomenclature). Figure 6. 25 shows this consistent practice across interviews . 106 Figure 6 . 25 . Example of Leonhard s systematic approach to start problems, from Interviews 1 through Interview 4. He identified the assumption and conclusion of the statement, using parentheses. In contrast, Timothy became more systematic in his approach over the course of the class, specif ically in how he bro ke problems down. In Interview 2 , he revealed how he was just now vergence, open/closed) : Timothy: I felt like a lot of kinda new definitions were kinda thrown at us quickly Interviewer: Yeah Timoth y: So it was kinda like sorting through and learning each definition kinda one at a time. He expressed that he was taking his time, because it was a lot of new definitions and information to sort through at once. 107 However by Interview 3 , Timothy said he had noticed changes in himself. He explained that he now knew how to break definitions down, think ing about each part separately. He now looked at definitions and proved them according to the order of the quantifiers that appeared: Timothy : I look at the d efinition now and actually try to go quantifier by quantifier so it helps a lot because now I rules or what d out, just so I can understand... like understanding why He said that quantifiers made more sense now, whereas befo re he would follow steps in his notes from class and not really be sure why he was doing what. Interviewer: Yeah, so before, it sounds like maybe in class, you guys would do like a convergence proof Timothy: U h huh I nterviewer: And like you could do it but Timothy: Yeah, problem the same I nterviewe r: Yeah Timothy : . Systematically breaking down definitions and approaching proving that way helped him understand what he was proving better . Based on how often Timothy appears as a case of development in this chapter , it is clear that he reaped the benefits from this . Developments Across All Participants I focused on the four c ommon developments , a longitudinal profile, an d two less pervasive developments across my sample . Table 6 .4 shows the developments that occurred across all eleven participants, based on interview not es. It is imp ortant to note that of all the participants, Timothy seemed to grow the most over the inte rviews . Some 108 Table 6 . 4 : Developments in Proving, By Participant Changes in how one chooses a proof technique Harness awareness of how solution attempt is going Check examples in conjunction with other strategies Work without a plan Imagine multiple paths Approach becomes more systematic Check work using other methods Draw on familiar examples Try multiple methods Attend to goal Amy X Charlie X X X X Dustin X Granger X X X Gabriella X X Joel X X Jordan X Leonhard X X X X Stephanie X X Shelby X X Timothy X X X X X X X 109 participants e.g. Amy and Granger started the interviews high performing and so did not show much development. Others, e.g. Dustin and Jordan, became more dejected as the course went on and showed the same one development, in how they chose proof techniques. Future analyses will delve into the other d evelopments listed here a nd other participants. Conclusions The developments shown here can be group ed into three broad categories (see Figure 6. 26 ) . Fluency and/or wielding them quickly, without struggle. Strategy refers to students inte ntions in trying to solve a problem. M onitoring and Judgment to pay attention and collect information about how the solution attempt is going (monitoring) and making decisions on what to change (judgment). It is theoreticall y possible to have monitoring and not judgment or vice versa. M onitoring without judgment would be being aware that your work is not going well but not knowing why or what to do next. This in fact describes Jordan , who monitored her work but did not use that information to change course when stuck . Judgment without mo nitoring would be making arbitrary decisions, not based on any of the informati on from Fluency Monitoring & Judgement Strategy Figure 6 . 26 . Three Categories of Proving Development 110 the attempt. This latter idea is difficult to imagine and was not seen in this sample but may be possible. The fact that these major categories mapped back onto Strateg y and Monitoring & Judgment of b ) components of problem solving was a validity check of my conceptual framing. In addition, even though fluency with proof techniques and logic was not a develop ment I had set out to look for, it is sensibl e that it showed up here. It is difficult to imagine students showing strategy and monitoring & judgment wi thout some proficienc y in proof techniques and logic. 111 CHAPTER 7: On the Nature o f Satisfying Moments In this chapter , I answer my second research question, W hat kinds of satisfying moments do undergraduate students have during the transition to proof ? Informally, this question led to identifying what events felt satisfying to participants , (i.e. led to significant positive emotions), and t hen categorizing them. First, I discuss the kinds of satisfying moment s and how often each kind occurred. Then, I present more foc used analyses: combinations of codes that co - occur red together and student profile s of satisfa c tion . Identification and Descr iption of Codes The coding scheme is described in Table 7.1 . Each code is a kind of satisfying moment , as emerged from the data or derived from literature. I discuss characteristics of each of the kinds , providing prototypical exampl e(s) from the dataset as needed for the purpose of illustrating what each kind is conceptually. Results of applying the coding scheme will be discussed after. 112 Table 7 . 1 : Coding Scheme for Kinds of Satisfying M oments Code Sub - code Keywords in Data Criteria for Code Example External Satisfaction is about some external element to the individual, such as a task or situation. Completing Task(s) Figure it out No stuck Know how to do Figuring out how to do something, typically a mathematical task. Emphasis is on the accomplishment of solving a task. Excludes struggle. Overcoming Challenge(s) Present Struggle Stuck Hard Struggling on a task and overcoming it. This includes problems that are perceived as hard to the participant. A nything where you struggl e first and then figure it out (Jordan - 3 - 1) Comparison to Past Something I struggle with Not good at X Present day struggle and overcoming it is set against the backdrop of a previous struggle on a similar kind of task or situation . Participant compares two time points: the present to existing history. G etting one side of induction to look like another , som ething she struggles with (Stephanie - 1 - 2) Partial Progress Better Improvement Best I can Incremental or partial mastery . Includes improvement and doing better than I did before or to the best my present capability. U n derstanding a problem better (Gabriella - 3 - 1) External Validation Grades Self and Authority Points / Full credit Receiving good scores, grades, or other outcomes as the source of satisfaction. G etting good grades (Jordan - 1 - 1) Assessments Self and Authority Exam/Mini - exam Doing well on a significant assessment, specifically an exam. Excludes homework. D idn't get stuck on mini - exam (Jordan - 1 - 2) Authority Figures Self and Authority Praise Authority figure (often instructor) giving praise to the person specif ically, e.g. saying work looks good. TA saying her wor k was on a hard pr oblem she worked on by herself (Amy - 3 - 1) Internal Satisfaction is about some internal state. Understanding General Making sense Understanding (I) get this Understanding how or why something works, usually a concept, task, or method; a sense of things falling into place or order. U nderstanding real analysis becaus e it's understanding a concept (Jordan - 4 - 1) 113 Table Aha Moment Realize Turning point Enlightenment / Revelation A singular moment of mathematical understanding . Often characterized more intensely as realization or insight. I nstructor's other explanatio n clicked for him, a revelation (Granger - 1 - 1) Seeing the Solution See hor for knowing the solution path or what to do. F elt good about this proof, could see it (Dustin - 2 - 1) Internal conviction Expressing personal conviction in the veracity of right answer or what they found was true Know ing he'd gotten it right. . . before getting the grade (Timothy - 4 - 1) On my own By myself On my own No help Doing something in present time on their own, without any help (people, notes, etc. ). This idea has to be explicitly expressed by participant. G etting a homework problem right all by myself (Jordan - 1 - 3) Properties of Math Satisfaction is located within the mathematics itself by the participant. Useful Applies Universal This technique or way of thinking is useful for other problems, e.g. applies to another problem. Learning the method & applying it to another problem (Granger - 3 - 1) Simple Simple Easy Familiar Task marked by a sense of ease and effortlessness. This can b e throughout the entire time or a task becoming easy after an event. Que stions that are easy, simple (Charlie - 4 - 1) Interactions with People Satisfaction comes from an interaction with other people specifically. Social Comparison Self VS Others Only one/me Doing better at X than others Proving people wrong Compared An interaction of an adversarial nature among peers, e.g. involving competition. This code includes situations such as: being the first or only one to do/know X being/doing better at X than others proving other people (classmates, authority figures, etc.) wrong TA said no one would get the only one in class to get it (Amy - 4 - 2) Friendly Interactions Self AND Others Helping Explaining to others Contributing An interaction of a non - adversarial nature with peers, often helping or working together. Examples include: helping, teaching, or explaining to others contributing or debating ideas vicarious experiences B eing able to explain a problem to so meone else s uch that it makes sense to them (Jordan - 1 - 4) 114 External C odes External codes were situations in which satisfaction was about some external element to the individual, such as a task or situation. Completing Task(s). An instance was coded as Completing Task(s) if it referred to figuring out how to do something, typically a mathematical task. The emphasis in these cases was on the accomplishment of solving a task. This category included situations like erence to struggle, so the codes Completing Task(s) and Overcoming Challenge(s) were mutually exclusive. Overcoming Challenge(s): Present and Comparison to Past. In contrast to Completing Task(s) , this category contained all instances that described a cha llenge, in that there was direct reference to an obstacle or struggle. This category is essentially a more problematic version of Completing Task(s) under this category. It is important to note that the sense o f challenge was specific to participant and their relationship to the ta sk at hand; the same task was a challenge to one student and not implied that perhaps the ta sk was not personally challenging to them but appeared challenging . In these cases, I looked at the associated emotion graphs, and the graphs started with negative levels of emotion. Hence, data triangulation with the emotion graphs in these instances show perceived as challenging when students experienced negative emotions at the start. Within this category of Overcoming C hallenge ( s ) , two clear subcategories emerged. Some instances referred to occasions where parti cipants discussed facing a challenge in the 115 present time ( Present challenge), whereas other instances referred to a history of challenge or struggle on a similar type of problem ( Comparison to Past ). A common example of the latter subcategory was working o n a kind of problem that has been a struggle in the past. If the instance referred to a past struggle, I coded it as Comparison to Past ; if not, I coded it as Present . Therefore, Present and Comparison to Past are mutually exclusive within the Overcoming C hallenge (s) category. Partial Progress. The idea of incremental growth or experiencing progress as good is a common idea ( Dweck, 2006) . The criteria for Partial Progress is incremental or partial External Validation: Grades, Assessments, and Authority Figures . In contrast to satisfaction coming from an internal sense of accomplishment of a task, External Validation of extrinsic motivation, where the motivation to do s omething comes from outside rewards (Middleton & Spanias, 1999). I conceptualize External Validation as taking place between the participant and some authority, whe ther that authority be a person or an assessment . Because external validation as a concept can be quite broad, I separated three types of external validation into individual sub - code s: Grades, Assessments, and Authority Figures. These codes were not mutually exclusive, in that an instance could fall under multiple of these sub - code s. An instance was coded as Grades if satisfaction came from external performance, a common one being receiving good grades. An instance was coded as Assessment if the instance referenced a significant assessment, constrained here to an exam or m ini - exam. Even though ho mework wa s also an important assessment in the 116 course , I exclude d homework from this category because homework was frequent and therefore a normal occurrence to the student, whereas the mini - exam and exam in this course happened less frequently. In additio n, I wanted to separate out significant and rare events, such as exams. While seem ing quite similar, Assessment was different from Grades in that Assessment concerned the background in which a satisfying moment took place whereas Grades focuses on the out come as satisfying. Another difference is that knowing how to do something on an assessment served as an indicator of mastery to participants, irrelevant of the actual grade assigned. This did mark a slight departure from my principle to code only what the students themselves verbally mark as contributing to satisfaction. Instead, if the instance took place during an assessment, I coded the instance as Assessment, regardless of whether the participant explicitly referred to the exam context being a factor. The reason for my departure from sticking close to the part thought this wa A third sub - code was Authority Figu res, was assigned w hen a person of authority offered praise or other forms of validation to the participant. People of authority tended to be instructors or teaching assistants for the course. An example of this wa instructor's question in the way he was looking for because your correct response means you understand th e topic and validation from him (Stephanie - 1 - 2 ) . Instances that included talking with an authority figure but where that person was left in the background and not the foreground of the experience were excluded from this category. 117 Internal C odes Internal codes were situations which did not refer to some objective external element, like a problem, but instead were more internally located to the participant . Understanding: General , Aha Moment, and Seeing the Solution. Understanding as a whole can be a source for mathematical be auty (Sinclair, 2006). One can think of understanding resulting from However, the difficulty here is that the term understanding on its own can take on multiple meanings. Just the words asking, knowing how to do the question, or graspin g how the concepts in the question relate to each other. What students mean when they use the term understanding did not necessarily match what mathematics education researchers mean by the term . I t wa s difficult to tease apart these different meaning, e specially knowing vs. understanding. For g what he did wrong ( Granger - 2 - 3 ) , did understanding actually mean knowing? Because of these difficulties, I decided to not tease apart these different meanings into sepa rate sub - code s. I f elt v ery little would be gained conceptually by separating Understanding into multiple sub - categories. T wo ideas were however worthy of being separated into sub - codes: Aha Moment and Seeing the Solution . Aha Moment captures those experiences of understan ding with a short temporal duration, often characterized as a realization, a revelation, or something I decided that this idea wa s worthy of its own sub - code because (a) temporal duration can be inferred from how participants discuss ed their ex perience and (b) sudden rushes o f understanding as satisfying was an idea present in mathematical beauty literature ( Sinclair, 2006 ). Seeing the Solution was generated by noticing that several 118 instances referred to situations where participants were working on a task and then could Seeing the Solution is a metaphor for things coming into focus, borrowing from visu al language. (1) Felt good about this proof, could see it (Dustin - 2 - 1 ) (2) Saw how to do induction problem on exam, which he initially thought he could do it but had go tten stuck using his usual ways (Joel - 4 - 2) (3) When someone says the problem in a way that makes it click for her, making it easier to visualize the situation and see if it's true (Shelby - 1 - 3 ) I did not code instances as Seeing the Solution and (b) there were no other second backwards, so the word seemed central to his experience. Figure 7.1 show s the relationship am ong the subcategories of Understanding . Figure 7 . 1 . Representation of how the Understanding sub - code s are related. Aha Moment and Seeing the Solution are different constructs but can overlap. General accounts for all other kinds of understanding. Aha Moment Seeing the solution General 119 Aha Moment and Seeing the Solution are not mutually exclusive, as there can be overlap. Understanding: General was used if Understanding: Aha Moment and Unde rstanding: Seeing the Solution did not apply ( Understanding: General is mutually exclusive with Aha Moment and Seeing the Solution ) except in rare circumstances when the satisfying moment involves multiple types of und erstanding. One such instance was uring out specific moves on a proof he couldn't see how to do at first: cases and factoring. Made sense (Dustin - 4 - 2) . the satisfaction came also that they make sense in a more general w ay, so it fell under Understanding: General and Understanding: Seeing the Solut ion . Another instance of this was convergence question (which he's generally not comfortable with, unsure what to do), looking back on the proof as a whole, seeing it made sense. Had a realization that was a turning point in the problem (Timothy - 2 - 1) . This instance fe ll under Understanding: General and Understanding: Aha Moment because there was a realization while proving but also general sense - making from looking over his proof as a whole after finishing it. Internal C onviction . Internal C onviction refers to the par what they have done is correct or true. The instance that best illustrates this concept is he exam before grade came back (Timothy - 4 - 1) . The satisfaction occurred during the situation itself, immediately knowing that he had found the answer; he did not need a grade in order to know it. Another example of this didn't get stuck anywhere on - 3 - 1) . to a certain immediate conviction 120 This code originated from the data itself, but the notion of conviction h as ties to mathematical beauty . Conviction in finding the correct answer may be related to a a more related to mathema tical beauty. On m y own. This code refers to a participant doing something on their own or without help. Common keywords include: by myself, himself/herself/ourselves, on my own, no help. Instances had to refer to events that had already happened in orde r to be included in this code; future exp ectations were excluded ( e.g. believing they can do this problem on their own next time). This code was difficu lt to apply because students were very often working alone i n many of the experiences they discuss ed if the participant explicitly used those words or some of the other keywords, rather than code all instances where the participant happened to work alone. This decision aligns with my principle o tisfying, not mine. My ass umption was that if accomplishment happening on their own was vital to what felt satisfying, the participant would use those words. For example, - exa m (Jordan - 1 - 2) . the participant doing some thing on their own, because it wa s an exam and exams are solitary effor ts in this course. However, Jordan emphasized that the heart of the satisfaction was not getting stuck, not that she w as able to do this on her own., so this was not coded as On my own. 121 Properties of Mathematics Codes This cluster of codes refers to times when participants spoke to techniques, methods, or the mathematics itself as being satisf ying. This is in contrast to prior categories which involved relationships between the mathematics and the participant. Code s about what properties of mathematics are commonly seen as beautiful were added from the existing literature at one point, but after a test coding, most of them were removed because they had no instances. The only two concepts that are properties of math ematics originated from the data itself and are described below. Useful. This code refers to a mathematical technique or way of thinking as being us eful for doing other tasks in the future. An example of an instance that spoke to the idea applying it to another problem (Granger - 3 - 1) This category had a large overlap with the notion of mathe matical utility or u niversality , which is a common characteristic of mathematical beauty (Sinclair, 2006). Common keywords for this code were applies and universal. This category could have been called Applicability or Universality, but the term Useful captures the utilit arian and practical nuances as well. Simple. Simple captured instances when a participant emphasi zed satisfaction coming from the simplicity of a task or when a task becomes simple. Common keywords include simple, easy, and familiar. is characterized by a s ense of ease in doing the task such that it i s effortless. This is a code that comes from mathematical beauty, the long - held idea that simple mathematics is beautiful (Hardy, 1940 ; Wells, 1990 ). It could be argued that talking abou t simple tasks actually refer to the relationship between the solver and the task, as what one person finds simple another person may not. 122 However, many of the participants talk about the mathematics itself being simple, so I am sticking close to their int erpretation in considering Simple a property of the mathematics. At first, one may think that this category should fall under Completing Task(s) , because simple tasks tend ed to mean the participant did not get stuck. While this wa s true, this category also includes experiences where a task became simple, i.e. the task was challenging and then something happened where it became simple and was from there on easy or clear to do. Because of this variation, Simple exists as a category that can factor across Comp leting Task(s) and Overcoming Challenge(s). Interactions with People Codes These codes refer red to experiences where interactions with other people were described as satisfying. In order to fall into this general cluster of codes, the participant had to re fer to the interaction itself as being satisfying. For example, a clause like the rest of the r eferences people but was used just to set up that the mathematical task was a challenge. An instance with this clause would therefore not n ecessarily count as an interaction then. Social Comparison. This category refers to satisfaction that comes from comparing oneself to others and coming out ahead. Here, the comp arison is between the self v s . peers. Social comparison also has links to se lf - efficacy, as a type of vicarious experience (Bandura, 1977). Typically, social comparison has an adversarial or competitive nature and involves at least one other person. The three main situations that occur are: being the only or first one (compared t o others) or to do/know X being or doing better at X than others proving other people ( especially classmate s or authority figures ) wrong 123 Whil e many of these situations had challenge s inherent in them, instances were c oded Social Comparison when it wa s the interaction or the comparison with others that wa s satisfying. If the people in the experience had disappeared, would the experience still be satisfyi ng? If no, then the experience was not coded as Social Co mparison . Friendly Interactions . This category concerned non - adversarial interactions, as different from Social Comparison . Typical situations that count ed as friendly interactions were working with, talking to, or helping fellow students in the course. The core idea of this category wa s of a person working with their peers , rather than working versus peers . Instan ces where the satisfaction came from interac tions with instructors or TAs did not fall in this category because they were authority figures, so they would be categorized as Ext ernal Validation: Authority Figures. Applying the Coding Scheme to the Data After finalizing the coding scheme, I then went back and uniformly applied the coding scheme to the dataset. The results of the cod ing process are given in Table 7.2 , from most to least frequently occurring kinds of satisfying moment s. Instances contained multiple kind s, so each instance could and frequently did take on multiple codes. The average number of codes per instance was 2.3 codes, so each instance on average had 2 - 3 codes assigned . 124 Table 7 . 2 : Frequenc y and Percentage of Satisfying M oments by Kind Kind # % (of N = 75) Overcoming Challenge(s) 37 49% Present 26 35% Comparison to Past 11 15% Understanding 34 45% General 23 31% Aha Moment 8 11% See the solution 7 10% Completing Task(s) 21 28% External Validation 18 24% Grades 8 11% Assessments 10 13% Authority Figures 3 4% Interactions 16 21% Friendly Interactions 12 16% Social Comparison 6 8% On my own 13 17% Simple 11 15% Internal Conviction 5 7% Partial Progress 4 4% Useful 2 3% Total Codes Applied 169 - Note. This table l ists the percentage of each code out of N=75 satisfying moment s . The codes are listed from most to least frequently occurring. Data were frequently assigned multiple codes, hence percentages do not add up to 100%. The total number of codes applied is included in the last row. Based on percentages across the full dataset, the mo st common types of satisfying moment s were the following: Overcoming Challenge(s), Understanding, Completing Task(s) , and External Validation. Each of these accounted for m ore than a 20% share of the data. A second tier of codes captured at least 10% of th e dataset: On my own, Friendly Interactions , and Simple . Social Comparison, Internal Conviction, Useful, and Partial Progress each applied to less than 10% of the data. 125 Overcoming Challenge and Completing Task(s) Account for a Large Portion of Data Not surprisingly, many satisfying moment s (49%) involved overcoming challenges. This finding is sensible, because the course was designed to be challenging for students and thus accomplishing a challenge can make an experience out of the norm, s expectations of what would happen. A little over a third of the data 35% - concerned present challenges, making this the larger of the two sub - code s. Nevertheless, 15% of the data involved the comparison of the present to past challenges. This result s uggests the experience, especially overcoming long - standing struggles. The fact that experien ces with a lack of challenge were satisfying too ( Completing Task (s) ) is also not surprising and confirms informal observations that as educators we grapple with: Students often enjoy tasks on which they do not have to struggle. In other words, students enjoy exercises . Accomplishments absent of struggle also confirms the importance of mastery experiences, from self - efficacy (Bandura, 1977). Experiences that involved both overcoming challenges and completing task(s) accounted for 77% o f the entire dataset. This high per centage suggests that satisfying moment s tended to be about mastery, regardless of whether or not there was struggle. External Validation vs. Understanding: Unexpected Results External Validation was a code I expected to account for a large part of the da taset because of the emphasis on grades and performance across society. Indeed, 18 instances - about a quarter of the dataset - fell under External V alidation . Of those, eight instances were about grades, nine took place on assessments, and two were about authority figures. There were only two instances that were both about grades and assessments: 126 convergence proof she'd ever d one, and got full credit for it (Amy - 4 - 1) Scorin g well on the mini - exam (Leonhard - 1 - 1) The overlap between Grades and Assessments within External Validation was therefore minimal , considering how tightly grades and assessments are intertwined. W hile External Validation did account for a little over a fifth of the data, it was not the most common kind of satisfying moment . Understanding took a larger share of the data, a little less than double that of External Validation . There was indeed a large amount of within - code variation for this category, from basic sense - making to knowing how to do something to understanding concepts. History and expectations play a role when it comes to what level of understanding is satisfying, explaining some of the variation seen within the Understanding code . Basic sense - making can be satisfying when even that is difficult to come by. For example , Jordan discussed in the four th interview how her instructor's explanation made sense, to the point that she felt she could do similar problems on her own next time. At this point in time, Jordan was worn down by the course and felt like she was not understanding very much. It is natural then that even just following alon g with what an instructor said c ould be satisf ying, although this example did have a link to then being able to successfully complete a task. What are the implications then of such a large portion of the dataset falling under Understanding ? First, students do find understanding satisfying they want to understand and when they do, it feels good. This is important for two reasons. One, this corroborates mathematicians writing about understanding as a quality of mathematical beauty and that 127 students experience and appreciate at least some version of this. Even if aha moments occur less frequently (10% of instances here), it is heartening that students find regular understanding to be satisfying. Two, reform efforts in U.S. mathematics education have o ften pushed for more understanding in mathematics classrooms (NCTM Standards, Common Core State Standards), while counter - efforts have often called for a return to basic skills and facts with the belief that understanding will come later . That students rep ort understanding as satisfying and are aware themselves that understanding feels good provides support at the student - level for efforts pushing for more understanding in the mathematics classroom. Interactions with People: Friendly Interactions I ch ose to group both Interactions with People codes together in the table, to show that they acco unted for 21% of the data. F riendly Interactions accounted for 16% of the data, more than Social Compa rison . The most common Friendly Interaction reported was in explaining or helping others with a question (6 of the 10 instances). There were some interesting other cases that fell in this c ategory. For example, Shelby voiced two instances of talking and working with others in the MLC: Working with others (not just for getting answers); when student says something that makes problem click, to the point that you can tackle those problems on your own later (Shelby - 1 - 1) Talking about math in the MLC with p eople and writing on the board if they're people wh o she can boun ce ideas off of. (Shelby - 4 - 2) Other type s of friendly interaction came from Gabri ella, about debating with others and also this experience from the beginning of the course : 128 She struggled the first week, feeling like doesn't understand, can't contribute, a nd contribute in groups and get correct answers. This came from someone else telling her she can get through this cours e and others are struggling too . (Gabriella - 1 - 1) A student with the same major as Gabriella who had already taken the course in a previous semester assured her that it was a difficult course but that she could get through it. This is an example of a vicario us experience (Bandura, 1977). Gabriella noticed that someone else similar to her ( with the same major) could also get through this helped her. It is important to note that while these interactions concerned helping one another, personal mastery was still present . Whe n a student was able to explain a homework question to another student and can see evidence t hat they understand, that served as - adversarial peer - to - peer interactions that appear in the data suggest t hat mathematics classrooms, even upper - level undergraduate mathematics classrooms, can benefit from facilitating more interactions between students. This course encouraged the use of st ructure s outside the classroom, where students could meet in a common s pace to work and talk together about homework. Ev en if take the mo st cynical interpretation, that students do this for their own sens e of mastery and understanding, the conclusion from the data is straightforward : Some s tu dents enjoy ed helping each other . On My O wn & Simple On my own and Simple were the most common codes among the second tier , capturing 17% and 15% of the data respectively. There was variation within On my own in regard to what no help rk problem right all by myself (Jordan - 1 - 3) 129 by himself before others because he wants to know why it works and it's more satisfying than havi ng someone tell him how it works (Leonhard - 1 - 3). There was one ins tance that referred to not using because they hadn't done one [a questi on] like this in a few weeks (Stepanie - 4 - 2). There was variation wi thin what was meant by Simple as well. Two instances thought about it differently and it clicked in a way that problem became simple. Now he knows what to do with these problems (Joel - 2 - 1). It was the task becoming simple when the previous state was one of confusion, that is important to noti ce here. In addition, two instanc es referred to thinking : Gets excited about it [induction] now because easier and doesn't have to think (Timothy - 1 - 1) Questions that are easy, simple, and/or that he can do just by thinking about them (Charlie - 4 - 1) The first instance referred induction is th e content mentioned here, as induction has an algorithmic and procedural nature. At face value, this is similar to students finding exercis es pleasing. At a base level, the sentiment makes sense; we want to conserve the amount of resources needed to do a t ask, so when we are able to do a task with little effort, that manifests itself in the form of an aesthetic feeling. Data that Did Not Fit into the Coding Scheme One instance did not fall into any category. It concerned writing homework in the typesetting language LaTeX; the participant said it felt satisfying because it was like 130 programming. This suggests something about the writing of a proof, specifically the product , was satisfying. There was satisfying momen ts: ngthy and hard - looking problem ( 1 - 1) . Timothy also talked about looking back at his proof as a whole and feeling good about it making sense. There was not enough evidence in the text i tself to warrant a code, but there may be a possible code for creating a product or proof as a creation . This may appeal to a more aesthetic take, of evaluating a product for its beauty. This would need to be explored with more data. Partial Progress as Rare There were only two instances that spoke to t he idea of partial mastery as satisfying. Both of t hese came from Gabriella: D oing the best she could on the test, knowing how to do most of the hard ones, after sh, making her emotions drop ( 2 - 1 ) . U that she had no ( 3 - 1 ) . Typically, when refi ning a coding scheme, a code containing only two instances in a datas et of this size wou ld be a likely cand idate for elimination. I did consider removing the Partial Progres s c ode , but this idea of progress could not be subsumed by my other categories easily. Additionally, I think it is telling that incremental growth, which is supposed to be a good thing, is generally not reported as satisfying . Th e two instances reported he re ca me from the same student, which raises questions about its generality . Full mastery and accomplishment, with or without struggle, make up a large share of satisfying moments. This suggests the question is there something about mathematics as a domai n that makes 131 partial mastery not as sati sfying as in other domains, such as learning how to play an instrument or running ? Even taking into account that satisfying momen ts are highly personal, perhaps mathematics educators need to underscore to students th at partial progress in mathematics is something to be proud of, in and of itself. Combinations of Kinds As mentioned previously, getting the codes to a point where the interrelations could be minimized was difficult. It took many rounds of refining the co ding scheme to do this. One way is to look at what codes co - occur with each other. In other words, when I label an instance code A, am I likely to also label it code B? This would suggest that perhaps code A and B should be collapsed. However just becaus e two codes co - occur do not mean code A and B are the same con struct (Bakeman & Gottman, 1997 ). In this section, I discuss codes that seemed to co - occur together but seem to be separate constructs. My judgments are given further backing from a co - occurren ce matrix I constructed. Each cell of this co - occurrence matrix corresponds to a row X and column Y, and each cell represents the co - occurrence of Y with X. Co - occurrence was calculated as the % = all instances labeled code X and code Y / all instances lab eled code X. In other words, each cell is a conditional p robability: Of all instances of code X, what percent were also labeled code Y? Completing Task(s) + Simple One frequent co - occurring pair was Completing Task(s) and Simple . A third (33%) of Completing Task(s) instances were also coded as Simple (see Table 7 .3 ) and in turn, 64% of Simple instances were also coded Completing Task(s) (see Table 7 . 4 ) . Table 7 . 3 : Co - occurrence of Codes with Completing Task ( s ) & Overcoming Challenge ( s ) 132 Code # External Validation Internal conviction Progress On my own Understanding Useful Simple Completing Task(s) 21 29% 5% 0% 24% 29% 0% 33% * Challenge: Aggregate 37 24% 8% 5% 22% 43% * 0% 11% Note. Only selected codes are shown here i n columns . External Validation and Understanding are aggregates across sub - code s. Table 7 . 4 : Co - occurrence of Codes with Completing Task(s) and Challenge(s) Code # Completing T ask (s) Challenge : Aggregate External Validation : Aggregate 18 33% 50% * Internal Conviction 5 20% 60% * Progress 3 0% 67% * On my own 13 38% 62% * Understanding : Aggregate 34 18% 47% * Simple 11 64% * 36% Note. Only selected codes are shown here in columns. This combination makes sense, in that experiences that lack struggle are likely to also be simple or feel effortless to the person. There could be an argument that these constructs are so inte r - related that they are the same, but Simple also includes instances where challenging problems became simple. So Simple t a sks as I have defined in this study are not exactly the same as Completing Task(s) , but they do tend to occur together. Overcoming Chall enge + Understanding Another set of co - occu r ring codes were Overcoming Challenge(s) and Understanding . Table 7.3 shows that 43% of Overcoming Challenge instances involved Understanding and Table 7.4 shows that 47% of Understanding instance s involved Overcoming C hallenges . 133 B oth of these codes have a large number of instances themselves (n=37 and n=34 respectively) . There were also a numbe r of codes that were unidirectional in relation to challenge, a instances were a lso coded as Overcoming C hallenge ( s ) but not vice versa . Table 7.4 shows the codes for which many of their instances were also coded as challenges : E xterna l V alidation (n=18), Internal C onviction (n=5), P artial P rogress (n=3), O n my own (n=13), and Social Comparisons (n=6). The freque ncies of some these codes are quite small, which may explain why Overcoming Challenge(s) accounts for a large share of each . External Valida tion and On my Own have double - digit f requencies , so I will discuss them, as an example of this unidirectional relation. Table 7.4 shows that 62% of O n my own instances were also coded as Overcoming C hallenge ( s ) . It may seem at first th own has an equal effect on accomplishment with or without c hallenge, looking at the similar conditional probabilities (24%, 22% respectively) in Table 7.3 . But when looking at the whole of On my Own instances (see Table 7.4 ) , 62% were also challenges, whereas only 38% were accomplishments without challenge. On my own i s therefore not necessary to feel good about doing a challenge, but when someone is proud of doing something on their own, it tended to be something challenging. Friendly Interactions + Understanding Friendly Interactions tend ed to involve Understanding and to a lesser extent occur red on non - challenging tasks. In T able 7.5 , a third (33%) of Friendly Interactions involved tasks without challenge s. 134 Table 7 . 5 : Co - occurrence of Interact ions with People with a Selection of Codes Code # of items Completing a task Challenge: Aggregate External Validation: Aggregate Internal Conviction On my own Understanding: Aggregate Social Comparison 6 17% 50% 17% 17% 17% 50% Friendly Interactions 12 33% 8% 8% 8% 17% 67% M ore interestingly, 67% of Friendly Interactions also fell under one of the Understanding codes . In turn, 24% of Understanding instances were about Friendly Interactions . This suggests that satisfying Friendly Interactions tend ed to have an u nderstanding component, but satisfying Understanding instances did not require peer interaction. This result makes sense given that t he most common Friendly In teraction s were variations of teaching fellow stud en ts and watch ing them understand. In addition, helping others can produce confirmation o r king and talking with others can sonal understanding. This suggests that one important kind of peer - peer interactions which are sat isfying are the ones that provide deeper understanding of the content. Half (50%) of Social Comparison s were also coded as Overcoming Challenge ( s ) , but there were only n=6 instances of Social Comparison in the entire dataset . This makes it difficult to make any strong inferences about commonly co - occu rring codes for Social Comparison . However, if we consider this co - occurrence to be a claim with limited data, it makes sense because social comparison is often adversarial or competitive . B est ing others at something difficult can be more indicative of mastery than besting others at an easy task. 135 Clustering of Satisfying M oment s by Individuals These results also raise the question from a n individual - centered standpoint: Do individuals tend towards certain kinds of satisfying moment s? In other wor ds, what is the variation of kinds satisfying moments? In this cursory analysis, I present satisfying moment s by participant. Table 7.6 shows the results for each participant, i.e. which code s each of their satisfying moment s fell under and how many. Table 7 . 6 : Kinds of Satisfying Moments by Participant Participant/ Cod e Completing Task(s) Overcoming Challenge(s) External Validation Internal Conviction Progress On my own Understanding Useful Simple Social Comparison Friendly Interactions Total Codes Total Moments Amy 1 5 * 2 2 1 3 14 6 Charlie 7 * 2 3 1 1 5 * 1 20 8 Dustin 7 * 2 2 4 2 17 9 Gabriella 5 * 1 1 2 1 4 * 2 16 6 Granger 1 4* 1 1 7 5 Joel 4 1 1 3 * 1 7 4 Jordan 4 * 1 2 1 4 * 2 14 8 Leonhard 2 4 3 2 2 2 1 2 2 20 9 Shelby 3 1 6 * 1 1 3 15 8 Stephanie 6 * 2 4 * 1 1 1 15 7 Timothy 5 * 1 1 1 4 * 1 13 5 Note . Each cell shows the frequency of satisfying moments per participant by code. The * denotes c odes that made total satisfying moments. Total Moments is t he number of satisfying moments, where Total Codes is the sum of codes applied across all their instances. Empty cells are zeros , which have been omitted for clarity. Overcoming Challenge(s), External Validation, and Understanding are aggregates . By looking at code s that accounted for at a m satisfying moment s, four major profiles are revealed, which I discuss below. 136 Case A: S tudent s W ho E njoy Completing Task(s) One profile is that of the student whose satisfying moment s come mainly from accomplis hing tasks that are not challenges, i.e. exercises. Three of the eleven participants fit this profile: Charlie, Dus tin, and Jordan. Charlie extended t his profile in that simplicity wa s embedded in many of his satisfying moment Truth table elicited posit ive emotions b ecause it was simple and easy (Charlie - 1 - 1) streng (Charlie - 3 - 2) Charlie also talked can do just by thinking about them (Charlie - 4 - 1) Being a ble to solve head implied not needing to expend effort and also spoke to clarity: A n answer that came naturally just from thinking is satisfying. There is an effortless, almost comforting, feeling to the satisfying moment s described by not just Charlie, but all three of these participants. Case B: S tudent s Who E njoy Overcoming Challenges(s) The next profile is that of the student who really enjoys accomplishing challenging tasks. Three of the eleven participants fit this profile: Amy, Stephanie, and Leonhard. In satisfying moment s were about challenge ; w hile that is technically less than half of his instances, he is honorarily included in this category because the greates t sh are of his instances we re this kind. There are other code s that can go with Overcoming Challenges too. Amy focused on Social Compa rison . During the interviews , it was clear Amy was competitive and especially ld get (Amy - 4 - 2). It makes sense that a person who likes challen ges would also be motivated by social 137 challenge and social comparison occurred in separate instances, but there were instances where they occurred in tandem, so the combination of these two kinds makes se nse. Another kind that co - occurred with enjoying challenges was doing them own . Stephanie liked doing que stions on her own, which again wa s sensible because doing something without the help of others can be thought of as a more general form of ch allenge. When a student solves difficult math problem by themselves, t his can be interpreted as being competitive with oneself , in that they have exceeded their own expectations of themselves. Case C: S tudent s Who E njoy Understanding A surprising profile may be that of students for whom understanding is everything. Two of the eleven participants fit this profile: Granger and Shelby. Both of these students talked quite a bit about understanding the mathematics in their satisfying moment s. Th is is corrobora ted by how they we re also the only participants to talk about usefulness and a pplying it to another problem (Granger - 3 - 1) t on 3 problems this past week (Shelby - 3 - While there is an element of being happy about having a procedure instances are fundamentally about liking a certain piece of mathematics for its power to do more. C ase D: S tudent s Who E njoy Overcoming Challenges(s) and Understanding As a combination of the previous two profiles, there we r e also students for whom it both challenges and understanding constitute d most of their satisfying moment s. Three of the eleven participants fit here: Gabriella, Joel, and Timothy. 138 For Gabriella, making sense of the question was a big struggle she talked about throughout the interviews. O nce she understood what the question was asking , she generally knew what to do. Of her four instances of U nderstanding , one con cerned basic sense - making, but the other three were indeed about a deeper level of understanding. For Joel and Timothy, all but one of their instances concerned understanding, but they also reported a relatively low total number of satisfying moment s across the four interviews the solution in problems where he was stuck and one was about knowing wha t to do in the future. While again there is a procedural flavor he sense that the mathematics fell into place for them was apparent. Timothy in fact reported a ha moments (not shown in Table 7 .6) in three of his five sati sfying moment s. Understanding can therefore come in different ways: basic sense making , to knowing what to do , to instantaneous realizations that illuminate the path forward. Taking satisfaction from both challenging problems and understanding, especially when the understanding comes from working on challenging problems, may serve students well for their mathematical future. Conclusion s In this chapter, I answer ed the research question, What kinds of satisfying moments do undergraduate students have during the transition to proof ? Through grounded theory techniques, I developed a system of kinds of satisfying moments. T he most commonly occurring ones in this data were Completing Task(s) , Overcoming Challenges , U nderstanding , and External Validation . The aggregate of interactions with people, both Social Comparison and Friendly Interactions , also applied to a large share of the dataset. 139 Additional an alyses showed how codes related to each oth er and which ones st ood out . Common combinations of co - occu r ring codes revealed the following pairings: Completing Task(s) & Simple , Overcoming Challenge(s) & Understanding , and Friendly Interactions & Understanding . Four s tudent profiles of what students most often found satisfying were revealed: (a) Completing Task(s) , (b) Overcoming Challenge(s) , (c) Understanding , and (d) a combination of Overcoming Challenge(s) and Understanding . Based on all these analyses, accomplishment both with and without challenge, understanding, and working with and/or helping fellow students seem to be major kinds of satisfying moments. 140 CHAPTER 8: Discussion In this chapter, I consider the methods and results of this study in relation to research on proof and problem sol ving and design of introduction to proof courses. First, I summarize the findings to both of my research questions. Then I discuss the implications of this work, to contextualize my results. I also identify the limitations and future research and development suggested by this study, with respect to studying proving and satisfying moments . I end with some concluding remarks with respect to task design and affect in mathematics . As a reminder, the research questions were: (1) How does undergraduate stud ents' proving develop over the duration of a transition to proof course? (2) What kinds of satisfying experiences do undergraduate students have during the transition to proof? Findings Related to the Development Proving My first research question focused proving work ove r the course of the study. Four developments were observed over multiple students in the sample: (1) increased sophistication in how they chose proof techniques to use and their rati onales for why, (2) awareness about how a solution attempt was going and harnessing that to change their strategies, (3) becoming comfortable exploring and monitoring when which strategy to pursue is unclear, and (4) checking examples in conjunction with o ther strategies as a way to trigger new insights when stuck . Some of these developments were specific t o the context of proof, such as choice of proof technique s ; other s were more general problem solving an d thus less proof - specific. 141 Results indicate d that students showed growth in fluency, strategy, and monitoring and judgement in how they reacted when they were stuck. This was evidenced in the approaches they chose to try next, the rationale for their ch oices, and how they monitored their progress. For example, early on, students tended to use a favorite method (proof by contradiction, contrapositive, etc.) for all problems, indiscriminately. But it was not always the case that these developments led to i interviews, yet his solutions were incor rect for the last two interviews. What does it mean then to have positive growth but stagnated performance? Some may see this as evidence that a student did not in fact improve, but I claim that a de - coupling of performance and growth is appropriate here. This is an age - old case question in educational research and remains for the future. Although I only discussed a few of the developments in detail, multiple types of dev elopments could be seen in the students . Beyond types or categories of development , the re were also multiple ways a development could emerge . This is sensible, that different students would grow in different ways and that that growth would look a little different. The phenomenon of multiple and relatively simultaneous developments can be con ceptualized metaphorically as many ropes, each made up of many strands, representing a different way of getting to the development. This is important to acknowledge because oftentimes there is an unspoken assumption that there is one path for learning mathematics and the goal of instruction is to move students along that path. Instead, there are many productive paths of proving development. 142 Findings Related to Satisf ying Moments The second research question was about identifying the kinds of satisfying moments students experienced in relation to the course. These kinds were identified using grounded theory techniques and group ed broadly into external and internal situ ations, properties of the mathematics, and interactions with others. The most common satisfying moments among participants concerned completing task(s), overcoming challenges, understanding (as an aggregate of its various forms), external validation, and i nteractions with people. Some codes direc tly indicated the nature of a satisfying experience, whereas others, like On my Own , appeared to function as a sort of modifier, where its presence seemed to Certain kinds of satisfying moments stood out in the analysis , and certain aspects of experiences tended to co - occur: (a) Completing Task(s) with S imple tasks and (b) Overcoming Challenge(s) with Understanding . It is important to note that Understanding was not necessary for feeling satisfaction at Overcoming a Challenge , but when a person did feel good about Understanding , the situation was typicall y challenging. This nuance in how challenges and understanding give rise to satisfaction was shown by the four profiles that cover this sample of students: those who enjoy (a) Completing Task(s) , (b) Overcoming Challenge(s) , (c) Understanding , and (d) a co mbination of the two, Overcoming Challenge(s) and Understanding . Last, interactions with people were frequently connected other aspects, based on other co - occurring codes: Friendly Interactions with Understanding . These interactions point to overarching ch aracteristics behind the kinds of satisfying moments. 143 In thinking about what lies at the heart of satisfying moments as a phenomenon, two ideas emerged. First, mastery seemed to be an overarching characteristic . Partial progress, confirmations or reassuran ces of present mastery, and expectations of future mastery may explain why the situations discussed above were satisfying. However, there were a few instances where mastery did not fully explain the satisfaction; understanding and interactions with others did. U nderstanding can serve as confirmation of present mastery, but there was something about sense - making that intrinsically seemed to feel satisfying to many of the participants. U nderstan ding involves things falling into place (Sinclair, 2006) which does not fall squarely under the umbrella of mastery. M astery and understanding overlap then, but there is an aesthetic component to understanding that mastery on its own does not seem to capture . The same applies to interactions wit h people, especially Friendly Interactions ; working together with people and helping others to understand has elements that are satisfying which fall outside the purview of pure mastery. Second, expectations likely play ed a large role in what was reported as satisfying moments. The results share much in common with the idea of self - efficacy (Bandura, 1977), the expectation of success. S ed to mediate w hether an experience wa s perceived as satisfying. When a student was successful i n a situation that was expected to be unsuccessful, this positive discrepancy between expected and actual outcom e may have been linked to satisfaction. There is an element of surprise in the expected outcome , which corroborates past work on the importance of surprise in aesthetic responses to mathematics (Satyam, 2016). This difference between expected and actual outcomes may explain then why students remembered these events , elevating events up and out from the milie u of 144 everyday proving that constituted their normal experi ence. An experience that followed may not stand out in memory, including situations w here failure is expected and then indeed felt. In other words, memorable events are more likely later be reported as satisfying. The observation does beg the question: Are there satisfying moments that are not memorable events? What happens be a satisfying m oment. However, I speculate that we have experiences that we do not place importance on in memory but are still felt at a subconscious level, affecting us later. This is likely beyond the test of empirical data with our current methods , so it remains a phi losophical musing. Based on these results, I propose d a theory of the phenomenon of satisfying moments. This theory came out of my observations that some of my codes were of different Completing Task(s), Interactions wit h People ), but a code like On my own acted more like a moderating variable, appeared only in conjunction with other codes and so likely moderated the strength of the main relationship . In addition, codes like Understanding and Partial Progress were more ab stract than other situations. Figure 8.1 illustrates how the different codes may relate to each other, to explain how certain situations give rise to the feeling of satisfaction (satisfying moment). 145 Figure 8 . 1 . Possible model for how satisfying moments occur as a phenomenon. Situations (independent variables) give rise to the feeling of satisfaction (dependent variable). On my own may act as a moderating variable, in that it strengt hens the elicitation of satisfaction. Understanding and expectations may act as mediating variables, explaining why those situations elicit satisfaction. Note the variable - paradigm is used for illustrative purposes here. Accomplishments with and without c hallenge, understanding, external validation, and social interactions with people covered the range of the majority of satisfying moments in this dataset. One can think of them as situations that elicit the emotional response of satisfaction. Working on ch allenging problems by yourself and/or without needing help ( On my own ) may strengthen the feeling of satisfaction , thereby acting as a moderating variable, which moderates the relationship between the independent and dependent variables . But above all, und erstanding and expectations of mastery may be what mediate (explain) how certain situations give rise to satisfaction as an emotional response. This 146 means, without understanding or an expectation of mastery, the situations on the left in Figure 8.1 do not give rise to satisfaction. The depiction of this model was influenced by the independent - dependent variable paradigm, with mediating and moderating variables. This is my speculation; my research methods do not support making any causal arguments. In fact, the situations that give rise to satisfaction as a feeling are not manipula table; they only provide opportunities for situations to happen. Grounded theory is useful for revealing the categories, but not necessarily for unpacking how the categories relate to each other. However, I offer this up as a speculative theory, based on the varying conceptual types of my codes. Findings R elat ed to Connections Between Proving and Emotion Throughout the analysis of proving, connections between affective and cognitiv e presented themselves (though I did not pose a research question to address them) , particularly in how their emotions interacted with their awareness of their soluti on attempts. This corroborates the idea that what students value mathematically (e.g., efficiency, straightforwardness, cleanliness, etc.) may draw and guide them to what (Sinclair 2004). These values manifest themselves though emotion. For example, Granger and Timothy showed strong negative emotions towards solution s they thought were wrong and demonstrated how their emo tions influenced their future attempts . The important result here is in how th ey harnessed strong negative emotions to search for alternate solutions . A cursory examination at the emo tion graphs students drew and emotion words they picked while proving also provided preliminary findings about the relationship between cognition and affect . Analyzing this in full is beyond t he scope of this study, 147 entailing different research questions. More robust frameworks for looking at this need to be developed. Implications Now I discuss some implications of this work, relating my findings to those of existing studies when possible. I separate these issues into theoretical, methodological, and peda gogical foci . Theoretical Issues This work contributes to existing literature on proving, specifically in its focus on (a semester) than most studies examining stude - interventions and with re peated interactions with multiple students, not just one or two. This work was longitudinal in the short - term sense , examining students proving work across one semester. Formal - rhetorical aspects of provin g may actually be problem - centered. Al though this study focused on the problem solving aspects of proving, the development s discussed earlier revealed the amount of decision making that goes into even writing the first line of a proof. Students took the content of the statement to be proven into account when deciding how to begin a proof. This suggests a revisiting of the distinction between formal - rhetorical and problem - centered aspects of proving (Se lden & Selden, 2007). In the formal - rhetorical phases of proof construction, the first and last statements are seen as following logically from the statement to be proven and can be stated without a great deal of thought. Acts that we expect to be formal - r hetorical, such as writing the first line of a proof, may actually be more complicated and dependent on content. Selden & Selden 148 (2007) do not treat these aspects as dichotomous but teaching formal - rhetorical and problem - centered aspects of proving separat ely may not give us the desired results in students if the interplay of these two aspects is lost. Noticing as crucial to proving. The act of "noticing" appeared multiple times in pment, in exploring and monitoring until students noticed so mething helpful and in using examples before noticing patterns. mathematical thinking has been a topic of research (Sherin, Jacobs, & Philipp, 2011; Jacobs, ally key here. How does a student know where to pay attention and to notice certain things? In solving mathematical problems , there can be many mathematical objects and relationships to attend to; noticing as a phenomenon can be quite complicated (Lobato, Hohensee, & Rhodehamel, 2013). One could speculate that successful provers notice important relationships when they appear, since the solution to problems that are truly problems is not clear from the beginning. Many of us have had the experience of notici ng some important piece that makes everything fall into place . In these moments, the solution can circles. How do we teach students to notice when something important arises in their work? As a focus for future research, how can we study the development of student noticing? Role of confidence in proving and its implications. To segue from the discussion of noticing, what role does confidence play in noticing and prov ing in general? Amy was my example student for exploring and monitoring ; she was comfortable just working without a strategy and noticed when something useful appeared. I noted separately that 149 Amy had high confidence in her mathematical work, from the begi nning of the interviews. Did the fact that she had confidence in her work play a contributing factor in her productive noticing , in trusting herself that she would notice important insights when they came up and that she would act on it when she saw it? Ot hers with high levels of confidence levels of confidence in their work, e.g. Dustin and Jordan, struggled. This connection between confidence may have implications th en for the importance of confidence in proving, a process so ridden with failure when held in comparison to most of computation and exercises. Learning mathematics is difficult; students experience repeated failures and th at failure is often taken as an indicator of a lack of (a fixed) ability. Perhaps confidence acts as an insulator of self against these failures? A sane person experiencing the regular failures in learning how to prove would likely quit, finding it not ple asing. This would be a natural reaction, all things considered. Similarly, if we think then about which demographic groups in the United States are culturally associated with confidence, could confidence partly explain why we see mostly white males and a dearth of women in higher levels of mathematics? Sociocultural factors may be at especially in mathematics harder for some than others. We should not be so quick , however, to claim that indiv iduals from underrepresented backgrounds should just be more confident. How mathematical classrooms treat students showing confidence as indicators of correctness and/or intelligence (when not always warranted) should be examined. 150 Algorithms as satisfying. S tudents often spoke about mathematical tasks that had a clear set of specific steps as satisfying. For example, a number of participants spoke about enjoying induction i n their discussion of satisfying moments. Why are st eps and algorithms pleasing? This counters the idea that understanding and other forms of explanation are intrinsically pleasing (Sinclair, 2006). Is it that the reduced cognitive load translates to the affective domain as feelings of contentedness? O ne answer to this may be in what makes a mathematical situation feel like a p uzzle and other situations not. Methodological Issues This work involved a relatively large number of novel constructs and data analysis techniques, adapted from existing literature. I descri be some of the ways in which what I did may help others. Studying impasses without intruding? People tend to get quiet when they are stuck. Going into this study , I had expected that some participants would have a difficult time talking when they were stuc k. I did not anticipate that this would be true across all the participants , but they all fell silent when stuck. This was true even for my one participant, Shelby, who said she found it useful to talk when stuck and would in fact turn to me to say her tho ughts out loud when stuck on the proof tasks, after I offered to be her listener. But e ven she would fall silent at times when she was stuck. Why is this important? For any research that focuses on what students do when stuck, it is important to not force them to talk out loud when stuck . Talking out loud could easily change the nature of the . Students are deep in thought, additional load on their cogni tive focus and capacity (Eric s son & Simon, 1981) . They may 151 lose their train of thought, as happens in everyday life when interrupted. If I had prodded my students to talk when they were stuck , there is a chance that my leading questions could have helped t hem become unstuck. In fact, Timothy developed a habit of talking to himself after he became stuck, where he stated the fact that he was stuck, explained why he was stuck, and discussed some of his preliminary ideas that related to the mathematical situat ion at hand. This often proved successful in leading him out of being stuck. His speech was explanatory, so it was more communicative to others than typical self - - guided talk appeared to help him o ut of tough situations. In conclusion, prodding students to talk may influence the phenomenon of becoming stuck as an object of study, but inviting students to think out loud may be useful pedagogically. Emotion graphs. Asking students to draw a graph of their emotions and pick out words that described their emotions while working on a task proved to be insightful for looking into their experience. With careful choices about data collection and care about the types of claims that can be gleaned, these tool experience. Satyam et al., (2018) examined the affordances of different variations of graphing as a research tool, and these findings corroborate the results reported in that work: Graphing is useful as a stimulus for helping students make sense of and discuss affective phenomena. The analysis of the graphs themselves, taken as a self - report of some phenomenon, must be done carefully. How to get students to stay with a series of interviews. I originally chose 12 partic ipants, with the hopes that 8 participants would complete all four interviews, i.e. , that I would have no more than a 1/3 drop - out rate. I expected that participants would drop 152 out, especially near the e nd of the semester when their work load increased, as they prepared for final exams . Instead, I was surprised when 11 of them continued with this work until the end (and the 12 th participant had medical issues preventing her from continuing the study). In a world where it is difficult to keep participants c o ming back, the question is : Why did they stay? Yes, they were paid for being in the study, but I do not think that modest reward explains their continuing participation . I believe my participants stayed in for the simple reason that they got something out of the interviews. A number of them said outright that they saw these interviews with the proof tasks as providing extra practice for their class. I also think the interviews were useful to them as a space to talk about their thoughts regardi ng the class and math in general. I believe the lesson here for research practice is to think about whether the data collection proce ss is of current value to students, whether that be mathematically and/or emotionally valuable. Pedagogical Issues These r esults also have implications for the design and teaching of introduction to proof courses. Curriculum design of undergraduate transition to proof courses. I argue that knowing the ways in which students develop and what they find satisfying in challenging mathematical work is useful for designing transition to proof courses at the undergraduate level. One way to go about that design would be to think about the types of developments one wishes to happen and design tasks that aid in student problem solving d evelopment. For example, if a goal is for students to come away with knowing when each proof technique makes sense, then one can design a task that asks students to prove a single 153 statement using different proof techniques (e.g. using direct proof, then co ntradiction, and contrapositive) and then reflect on the advantages and disadvantages of each. A less time - consuming variation of this would be to ask students to consider two or three techniques and write down some of the advantages and disadvantages to u sing each , prior to implementing any of them . This same tactic could be used for desi gning assessment items. W e can see here the need for a proving process framework, which could drive the curriculum development of courses like these , meant to help students. Noticing when students are and are not stuck in the classroom. Distinguishing between observable behaviors that indicate a student was stuck vs. thinking silently but not stuck was difficult for me to operationalize. I found in th is study that I could only make that distinction by interpreting body language and having a familiarity with the individual. This distinction is especially important for the classroom how can we tell when a student is unproductively stuck vs. engaged in productive struggle? As math educators, the first we would like to intervene , but productive struggle should be encouraged , not only among college students (Middleto n, Jansen, & Goldin, 2017). I n fact , as instructors , we often may not want to st ep in and interrupt productive struggle . Our task may center more in helping students accept productive struggle as a mathematical virtue and learn to make the responses to struggle more productive. Interview as a vehicle for reflection and rendering knowl edge. Last, I realized at some point through these interviews that there were interesting things happening in this space , beyond the foci of my dissertation . These students were being honest about how they felt they did on exams, their in - class experience, and how they felt the course as well 154 as other courses - were going for them . They were musing out loud about their thoughts on what they were currently learning. My participants expressed their thoughts about mathematics generally and may not have had other people to talk about it with. Leonhard for example would routinely talk to me during the interview, sometimes going over 2 hours, and tell me his thoughts about mathematics as a whole. Now Leonhard was not the norm , ho wever, the fact remains that thi s was a large public research university with only a couple mathematics advisors for the entire student body . S o advice and mentorship is and was likely hard to come by. This speaks to the importance of truly listening to students and taking their experien ces seriously. This is especially important considering that this is a transition for students . The course i nstructors were upfront that the math and their work would be different, but it is not clear if this is generally the case across the country. It is very, very easy for students to make ill - formed inferences about their lack of ability in mathematics and leave the STEM pipeline when there is no intervention by a mentor. How can large institutions institut e opportunities for interaction of this kind fo r their undergraduates? A space like the Math Learning Center, where students can gather to work together and talk about mathematics, would be useful, given the current national focus on STEM education. The interview also acted as a vehicle for reflection for the students. Some students, like Timothy, realized meta - level aspects about proving during the interview. Needing to discuss the mathematics and share their thought processes likely helped students reflect on and render their knowledge. More opportuni ties for this kind of reflection in mathematics education seem useful. 155 Alternative Explanation(s) Here I consider one alternative explanation of this data and discuss why I believe it can be ruled out. An alternate interpretation of the development results is that students naturally bec a me better at proving over time due to the sheer amount of relevant experience alone the material and not due to changes in internal cognitive, affective, or reasoning processes . Becoming better at using tools is not necessarily reflective of deeper mathematical understanding, as Guin & Trouche (1999) noted about students using calculators as tools. I argue that can be ruled out because taking experience as the primary factor does not account for the variation and individual differences seen across students in this sample by the end of the semester. Some students grew in th e problem solving domain while others still struggled by the end. Especially since the course design required students to work on proving tasks in class, it can be argued that all students who attended class had some base amount of experience with proving, at least more so than if regulated. The developments seen in some students and not others and also the variations xperience with the course material is relatively uniform, suggests that repeated practice with tasks is not sufficient to explain growth in proving competence in this context . Limitations and Factors Influenc ing the Findings This analysis was qualitative in nature. My goal was to generate theory and the small sample size was an indicator of that. Generalizations such as how most student learn 156 how to prove are not possible with this data. Future quantitative work assessing the theory generated here will be required to answer questions of that nature. course. For some of the developments, there was evidence of the instructors explicitly encouraging students to engage in certa in helpful practices, e.g., Ms. Frye recommending using examples to gain an intuition for a statement but not to prove it. There is a question then regarding the specificity of these results : How much do they reflect the specific features of this course? T o what extent would we see these same developments in any other transition to proof course? Transition to proof has been organized in a myriad of ways across the United States (David & Zazkis, 2017), so a prototypical transition course does not in fact exi st . Students using strategies like checking examples likely would have The d evelopment s documented in this study was also shaped by the specific proving tasks that students wor ked on in the interviews . This raises the question of how much the developments observed in students were shaped by the nature of those tasks . When a student did something different on a task, was it due to particulars of that task or was it indicative of some internal development? This question holds for much of scientific research (Popper, 1963) and so remains unanswerable here. However, the tasks used in this study were drawn from a reasonable population of tasks similar to those seen in class and on hom ework, so one can argue that development measured via tasks from the course i tself would not have looked substantially different. In addition, my first research question focused on identifying what changes that occur and not necessarily why . I leave future researchers to grapple with that question. 157 Development may well also depend to some extent on the interview context. Would the same developments have been seen if students had work ed on the tasks by themselves without my presence in the room? My presence could have added pressure and thereby impede d problem solving performance, but it also meant they could ask me factual questions easily. In addition, they knew that they would be expl aining their work to me afterwards, so they have tended to write more informal written arguments which could be explained verbally when stuck. As a counteracting force though, my stature and also openness in demeanor may have contributed to making these in terviews a place where students felt comfortable sharing their thoughts when problem solving. If one wishes to minimize interview presence, less intrusive data collection methods are an alternative. audio and their written work may be useful. Timing of tasks was very important and so there are alternative choices that could be made. If I were to repeat the same study, I would ask students to draw emotion graphs immediately after they completed a task , before the debrief, in order to shorten the already short window of time between the proving and affective record of it. Another choice was in asking students to draw a fact. However, I contend that it may not matter much what students actually felt in the moment (that is, while they were working on their proofs). Instead, what matters more was their percept ion of it and remembrance of it afterwards, because those remembered emotions were more likely to stick with them and affect their subsequent work. If emotions in the moment are of interest, one could measure emotion using more physiological 158 methods, such emotions but not the character of the emotion, in the way that direct observation or a person reporting their own can. I acknowledge that these above factors the course, the tasks, the interview setting, and timing of tasks influenced the data. However, all these factors are not so straightforward in their effects as to determine the data one way or another. Lastly, t e study emotions in mathematics education because emotions are the type of affect most responsive to change and can alter attitudes and beliefs, this implies we wish to alter feels to t his author as intrusive and manipulative. Emotional responses are highly personal. Do we wish to be sign a lesson. The take - away from this study is not to expect that if we do X, all students will feel Y. Rather, I argue for creating opportunities for satisfying moments, to at least set the conditions for them to perhaps occur, regardless of whether they do. The contrapositive holds here if we do not provide conditions for satisfying moments to occur, satisfying moments may rarely happen and perhaps only for a few students. Suggestions f or Future Research The empirical results reported here and the spec ulative propositions and frames that arose from those results suggest issues that could be explored in subsequent studies . 159 Empirical Work Future work could examine how satisfying moments change over time for a student. This study produced some , but not sufficient data to make claims about change over time. It would be fascinating to see instances of students learning to enjoy challenges over time and if so, inquire about the factors that orient such change. Is enjoying challenge more of a trait - like aspect to an individual that is resistant to change? Another promising direction for future research is to examin e satisfying moments of groups of students (for example, those working together in a small group), not just individuals. Liljedahl (2004) called for investigation of group aha moments. This is relevant for the classroom, in thinking about how to design instruction around eliciting intense positive emotions for multiple people at once. Considering how frequently satisfying moments involved fe llow students, there may be potential for students to experience this together. Lastly, this work originally sought to seek out the conditions that elicit satisfying moments, i.e. what actually triggers the feeling of satisfaction. Identifying kinds of satisfying moments, in order to get a lay of the land (so to speak) is the necessar y start but getting at what truly causes satisfying moments is the next step. Theoretical Work Proving process frameworks. There is a strong need for a proving - problem solving framework that would support the characterization and assess ment of student s proving process over time . This call is not a new one: mathematics education community a proving - process framework, complete with 160 additional problem - solving attrib utes that a prover experiences. (Savic, 2012, p. 121) Such a framework would be useful for diagnostic purposes in the classroom as well as research : It would be an accomplishment to generate a proving - process framework, similar - solving framework , that would accommodate beginning provers (if not more advanced accomplished students of mathematics) . Such a framework - solving attributes that need improvement. It mi ght identify the phases (Orienting, Planning, Executing, and Checking) or problem - solving attributes (Resources, Heuristics, Affect, and Monitoring) that need work, and focus instruction on that phase/attribute (Savic, 2012, p. 122) . Reliable assessments f recently developed (Mejía - Ramos, Lew, de la Torre, Weber, 2017). A framework that covers all that the proving process entails, much like what has been developed over the years for problem solving, would be useful. Challenge of studying phenomena by looking at individual components. Fundamentally, analysis is the process of breaking complex phenomena down into more basic and separable components, studying each separately, and then putting them all togethe r again. The underlying assumption is that the process of decomposition and recomposition supports insights into what is going on that would not be possible if the researcher simply looked carefully at the whole phenomenon of interest . However, I sensed in my analyses of both proving and satisfying moments that something that was being lost in this process, some gestalt sense of what was going on. I believe this happened for both the analysis of proving and satisfying moments because both are fuzz y constructs they 161 are hard to define, especially in a way that is measurable. Problem solving and satisfying moments each involve numerous highly interrelated processes. Considering the two phenomena, I undertook this study thinking I could effectively s tudy cognition as one component and affect as another . But as the work unfolded, I found that the relations between the two could not be ignored. This study uncovers some of the connections, but future work will likely uncover more. Going forward, theory t hat attends to interactions may be most illuminating. Even in rs . Theory that targets interactions among highly interactive rather than separable components will work, from descriptive to more explanatory understandings. Conclusions As was just stated, it is diff icult to ignore the connections between affect and cognition. My concluding remarks relate these two phenomena and look across the results of each analysis to review what we have learned. The importance of task design is clear. Careful task design, whethe r it be in - class instruction or out - of - development in productive ways and potentially have students feel good about doing mathematics. Similarly, poor task design can fail to support those goals, if not worse. This extends beyond the context of proving and even undergraduate education, into K - 12 schooling as well. Constructing and selecting tasks that support developments we wish to see are important. In the case of satisfaction, sequencing seems c rucial because of the 162 temporal order of emotions. Curriculum is therefore an important force. Careful attention to the storyline of the mathematics (e.g. Dietiker, 2016) and what reactions certain curriculum choices elicit (e.g. Dietiker, Richman, Brackoni ecki, Miller, 2016) may be the key to providing more frequent opportunities for students to feel good about math ematics and themselves as learners of that content . Are there any affective qualities in common across successful provers? While I did not addr ess this directly in the analysis, some observations about my participants began to coalesce. Students whose satisfying moments involved challenges were successful either to start or became successful. The students in my sample whose satisfying moments con cerned accomplishments without struggle and easy tasks tended to struggle with proving and the course, as time went on. Finding joy in struggle may be important for students, as they take on more difficult tasks in life, let alone mathematics. In addition , successful problem solvers in my sample seemed to take failure as opportunities for learning. When they got a problem wrong, they would ask to see how it worked and expressed that now they would know what to do in the future. This speaks to a larger issu e of how we treat mastery , challenge, and non - success in mathematics . While this issue extends far beyond mathematics, how math is often portrayed in this world makes failure more salient and catastrophic than in other domains of human endeavor . A retoolin g of how we teach mathematics and how failures can be progress may help us resolve this. 163 APPENDICES 164 APPENDIX A : Proof Tasks Interview 1 Task 1 We say that two integers, x and y , have the same parity if both x and y are odd or both x and y are even. Prove the following statement: Suppose x and y are integers. If x 2 y 2 is odd, then x and y do not have the same parity. Interview 1 Task 2 Prove the following statement: If a and b are strictly positive real numbers, then ( a + b ) 3 never equals a 3 + b 3 . Reminder: Binomial expansion of (x+y) 3 = x 3 + 3x 2 y + 3xy 2 + y 3 Interview 2 Task 1 Two numbers are consecutive means one number comes after the other. Prove the following statement: If x and y are consecutive integers, then xy is even. Interview 2 Task 2 We say x divides y if kx = y for some integer k. Prove the following statement: 165 If a , b , and c are non - zero integers such that a divides b and a divides c , then a divides ( mb + nc ), for any integers m and n . Interview 3 Task 1 Three positive integers a , b , and c are called a Pythagorean triple if they satisfy a 2 + b 2 = c 2 . Prove the following statement: Suppose x , y , z are positive integers. If x , y , and z are a Pythagorean triple, then one number is even or all three numbers are even. Interview 3 Task 2 Prove the following statement without using induction : If n is an odd natural number, then n 2 - 1 is divisible by 8. Interview 4 Task 1 A perfect square is any number that can be written as n 2 , for some integer n. Prove the following statement: 166 If a and b are odd perfect squares, then their sum a + b is never equal to a perfect square. Interview 4 Task 2 Prove the following statement: If x, y + > 2. 167 APPENDIX B : Interview Protocol s Interview #1 Protocol Logistics: Interview should take place in a quiet room Max time: 90 min o 20 minutes for Proof Task #1 o 20 minutes for Proof Task #2 o 30 - 45 minutes for Satisfying Moments questions, word selection, and graph Ask students ahead of time to bring any scratchwork and a copy of their homework Materials for students: The 2 proof tasks on separate sheets of paper Resources for proof task for s tudent: sheet with definitions, laptop, scrap paper Note: Provide the hw, example sheets, and solutions in paper form. Cards with emotion words on them Emotion graph worksheet Materials for interviewer: Paper for writing down word selection Paper for notes Key for this document: Black plain text is the script, instructions and questions to be spoken to participant Black italicized text is instructions for interviewer, not to be spoken Red italicized text denotes purpose of question, linking everything back to research questions. Warm - up & Basics (1st interview only) with math courses in your past and MTH 299. 1. What mathematics courses have you taken he re before 299? 2. Are you taking other math courses this semester, along with 299? 3. How is MTH 299 going for you, so far? Proof Tasks [RQ1] about the final answer. So I problem. Prete 168 You may know me as a TA for this course, but in this interview, because I am interested in how you are thinking, so I will not be a ble to help you out during the task. This means I er any questions about the math or what to do next if you get stuck. e Proof Task #1 [where the task is written]. You may look at definitions in this supplementary document from class, your notes from class, or online using minutes to work on it and it ave you stop and we can talk about what you did. You can start. Interviewer sits near enough to see their work but a little farther away than when typically asking interview questions, in order to give student space to minimize pressure from being watche d, as much as possible. Give the student max 20 minutes. After student is done, ask the following questions: 4. Can you mark for me any places you would count as scratchwork, as in, you 5. Were there any places where you got stuck? Probe about their rationale at points of interest, where students paused or went in a new direction: Ask student to do the Emotion Word and Emotion Graph (scroll down to that section) for this proof task. Proof Task #2. Repeat above steps. After both tasks are complete: 6. How would you say you currently approach proofs right now? [RQ 1a] Satisfying Moments [RQ 2] 7. Have you had any satisfying moments related to your work in MTH 299 since the last time we m et? Identifying satisfying moments without any influence in certain directions by interviewer 8. Were there moments that felt satisfying while you were working on problems on the last homework set? Clarify and Probe as needed Identifying satisfying moments re cently in time 169 9. How about the rest of this homework set? Identifying satisfying moments a little farther out 10. most rewarding? A different approach to trying to access satisf ying moments 11. How about in class (since the last interview)? Identifying satisfying moments longer ago 12. Can you think of a time when you had a flash of understanding or insight? If they had any A - HA moments , a hypothesized type of satisfying moment . 13. Do yo u have moments of negative emotion, such as frustration? What moments stand out? Moments of intense negative emotion (for sake of completeness) 14. Do these moments (positive or negative) affect your motivation to continue to do math? If so, how? Link between moments of intense emotion and motivation. The Word Selection & Emotion Graph Tasks should be about the same experience. It is up to the interv iewer which experience to choose. Word Selection Task [RQ2] moment> 15. What words did you choose and why? Identifying emotions and conditions 16. included here? answers 15. What words did you choose? Please circle them on this sheet. Identifying emotions behind the problem 16. What made you pick the words you did? Identifying conditions 17. included here? Covering any other emot answers Emotion Graph Task [RQ 2] Description of Task: 11 Words total 5 negative 5 positive 1 neutral annoyed curious disappointed surprised sad joy ful indifferent frustrated satisfying a shame d proud Words in black have no corresponding word pair. Each word is written on a separate small noteca rd, in black. I spread the notecards out in the exact arrangement above in front of the student. 170 The x - axis is time, from when you started to when you stopped working on this proble m. The y - axis is emotions, where positive and negative emotions. Think of the highest mark as indicating strong positive emotions like satisfaction or excitement. The middle mark would be neutral, as in your normal resting state. The lower mark would be st rong negative feelings like frustration or panic. Please also mark what triggered any ups and downs, as in turning points, in your graph, like which strategies you tried and how that corresponds with your emotional reactions. 18. Talk me through your graph he re. Have participant talk through the experience, marking turning points (conditions for changed emotion). Clarify and Probe as needed 171 Interviews #2 - 4 Protocol Logistics: Interview should take place in a quiet room Max time: 90 min o 20 minutes for Proof Task #1 o 20 minutes for Proof Task #2 o 20 minutes for Satisfying Moments questions, word selection, and graph Materials for students: The 2 proof tasks on separate sheets of paper Resources for proof task for student: laptop, scrap paper Note: Provide the hw, example sheets, and solutions in paper form. Cards with emotion words on them Emotion graph worksheet Materials for interviewer: Paper for writing down word selection Paper for notes Key for this document: Black plain text is the script, instructions and questions to be spoken to participant Black italicized text is instruction s for interviewer, not to be spoken Red italicized text denotes purpose of question , linking everything back to research questions . Warm - up 1. How is MTH 299 going for you since our last interview? Proof Tasks [RQ1] w t talking to yourself out loud. You may know me as a TA for this course, but in this interview, because I am interested in how you are thinking, so I will not be able to help you out during the task. This means I or what to do next if you get stuck. mean n e been quiet for some time. 172 After you are done, I will ask to talk me through what you did and why. I may ask you some questions like, Proof Task #1 can start. Interviewer sits near enough to see their work but a little farther away than when typically asking interview questions, in order to give student space to minimize pressure from being watched, as much as possible. If it looks like interviewer presence is c ausing pressure, interviewer will leave and tell participant to talk into the microphone. Give the student max 20 minutes. If present, interviewer should try to take notes about what participant is doing their process, including place they got stuck. A fter student is done, ask the following questions: 2. include if you were typing it up in LaTeX? 3. Were there any places where you got stuck? Probe about their rationale at points of interest, where students paused or went in a new direction: Word Selection Task Emotion Graph Task (leave the room for emotion graph) Proof T ask #2. Repeat above steps , including Word and Graph Tasks . After both tasks are complete: 4. How would you say you currently approach proofs right now? [RQ 1a] 5. Do you think your ability to write proofs has changed since the last time we met? In what ways? Probe as needed. [RQ 1b] Satisfying Moments [RQ 2] 6. Have you had any satisfying moments related to your work i n MTH 299 since the last time we met? 173 Ask them to pull out graphs they did at home and talk me through it . Identifying satisfy ing moments without any influence in certain directions by interviewer 7. Talk me through this experience/your graph. Ask as needed a. Can you find me the exact problem? b. What was happening before this? c. Were you working alone or with others (if not clear) d. If they forgot to return graphs, give them a blank sheet to draw it . If they forgot to do it period, ask the following: 8. Have you had any moments related to MTH 299 that felt satisfying? By satisfying, I mean a super positive feeling, like reward ing or a feeling of joy, etc. Clarify and Probe as needed Identifying satisfying moments recently in time I f they report no satisfying moments: 9. is this true? 10. Is math ever satisfying for you? Prompt for examples and try to suss out situations/properties. 11. If so, what do you think it takes for math to be satisfying for you? Back to MTH 299 12. Have you had any moments of negative emotion, such as frustration, since our last interview? Any moments that stand out? M oments of intense negative emotion (for sake o f completeness) 13. Do/how do these moments (positive or negative) affect your motivation to continue to do math? If so, how? Link between moments of intense emotion and motivation. 174 APPENDIX C : Emotion Graph Please draw a graph of your emotions over the course of this homework problem. The X - axis represents time , from when you started working on the problem to when you finished. Please mark different strategies you used on the x - axis. The Y - axis represents your positive and negative feelings while working on this homework problem. Think of the highest mark as indicating emotions like satisfaction or excitement; the middle mark would b ; and the lower mark would be feelings like frustration or panic. If there were points during the problem when your feelings changed, be sure to mark those points on the X - axis. TIME EMOTION When you started working on the problem When you finished working on it Positive Negative 175 REFERENCES 176 REFERENCES Alcock, L., & Weber, K. (2005). Proof validation in real analysis: Inferring and checking warrants. The Journal of Mathematical Behavior , 24 (2), 125 - 134. 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