Education, Faculty of
Education, School of (Okanagan)
DSpace
UBCO
Shayer, Leslie Pauline
Doctor of Philosophy - PhD
University of British Columbia
Mathematics has long been known as a gatekeeper subject in post-secondary programs, erecting barriers to mathematical success and career. Bleak attitudes surrounding mathematics as well as math anxiety, or negative emotional or physiological responses to mathematics content, compound this subject’s challenges, diminishing life chances for many post-secondary students. The result is not only the spread of negative mathematics attitudes, but also poor performance in mathematics as well as avoidance of mathematics, including STEM, subjects and careers. Using a mixed method design including a two-stage data collection process, this study investigated how contemplative pedagogy could support students in post-secondary mandatory mathematics courses. Two sections of the same course formed the sample, with one being the control and the other, the intervention group. The latter group incorporated evidenced-based practices from contemplative pedagogy after the first course test. Qualitative datasets were formed by researcher field notes, Math Origin Stories, and individual semi-structured interviews, while the quantitative ones comprised of pre- and post-test questionnaires (the Five Facet Mindfulness Questionnaire and the Math Anxiety Rating Scale) as well as course grades. After triangulating the themes emanating from the use of grounded theory to the qualitative data with the conclusions drawn from statistical data analyses applied to the quantitative data, positive results emerged. The use of contemplative pedagogy was a viable intervention to reduce math anxiety, diminish math avoidance, improve math attitudes, and increase math performance. Results further indicated that contemplative pedagogy fostered the building of community in the classroom and led to positive transformations by those in the intervention group. The contemplative pedagogy included in this research shows promising results, suggesting that this intervention may be effective as student supports as they confront some of the negative attitudes and anxiety inherent to mathematics content or STEM-related fields. While supporting overall student well-being, the contemplation developed in such a program can also act as an important component of the classroom environment. This could, in turn, ensure the emergence of a culture of acceptance and belonging for students of all math proficiency levels and further foster equity in the classroom.
MATHEMATICS ANXIETY AND CONTEMPLATIVE PEDAGOGY AT THE POST-SECONDARY LEVEL by Leslie Pauline Shayer M.A., The University of British Columbia, 2020 M.Math., University of Waterloo, 1998 B.Sc., University of Guelph, 1996 A DISSERTATION SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE COLLEGE OF GRADUATE STUDIES (Interdisciplinary Studies) THE UNIVERSITY OF BRITISH COLUMBIA (Okanagan) July 2024 © Leslie Pauline Shayer, 2024 ii The following individuals certify that they have read, and recommend to the College of Graduate Studies for acceptance, a dissertation entitled: MATHEMATICS ANXIETY AND CONTEMPLATIVE PEDAGOGY AT THE POST-SECONDARY LEVEL submitted by Leslie Pauline Shayer in partial fulfillment of the requirements of the degree of Doctor of Philosophy. Dr. Karen Ragoonaden, Okanagan School of Education Supervisor Dr. Sabre Cherkowski, Okanagan School of Education Supervisory Committee Member Dr. David Geary, Irving K. Barber Faculty of Arts and Social Sciences Supervisory Committee Member Dr. Javad Tavakoli, Irving K. Barber Faculty of Sciences Supervisory Committee Member Dr. Cynthia Nicol, The University of British Columbia (Vancouver) University Examiner Dr. Nathalie Sinclair, Simon Fraser University External Examiner iii Abstract Mathematics has long been known as a gatekeeper subject in post-secondary programs, erecting barriers to mathematical success and career. Bleak attitudes surrounding mathematics as well as math anxiety, or negative emotional or physiological responses to mathematics content, compound this subject’s challenges, diminishing life chances for many post-secondary students. The result is not only the spread of negative mathematics attitudes, but also poor performance in mathematics as well as avoidance of mathematics, including STEM, subjects and careers. Using a mixed method design including a two-stage data collection process, this study investigated how contemplative pedagogy could support students in post-secondary mandatory mathematics courses. Two sections of the same course formed the sample, with one being the control and the other, the intervention group. The latter group incorporated evidenced-based practices from contemplative pedagogy after the first course test. Qualitative datasets were formed by researcher field notes, Math Origin Stories, and individual semi-structured interviews, while the quantitative ones comprised of pre- and post-test questionnaires (the Five Facet Mindfulness Questionnaire and the Math Anxiety Rating Scale) as well as course grades. After triangulating the themes emanating from the use of grounded theory to the qualitative data with the conclusions drawn from statistical data analyses applied to the quantitative data, positive results emerged. The use of contemplative pedagogy was a viable intervention to reduce math anxiety, diminish math avoidance, improve math attitudes, and increase math performance. Results further indicated that contemplative pedagogy fostered the building of iv community in the classroom and led to positive transformations by those in the intervention group. The contemplative pedagogy included in this research shows promising results, suggesting that this intervention may be effective as student supports as they confront some of the negative attitudes and anxiety inherent to mathematics content or STEM-related fields. While supporting overall student well-being, the contemplation developed in such a program can also act as an important component of the classroom environment. This could, in turn, ensure the emergence of a culture of acceptance and belonging for students of all math proficiency levels and further foster equity in the classroom. v Lay Summary Bleak attitudes surrounding mathematics as well as math anxiety (negative emotional or physiological responses to mathematics), erect barriers and diminish life chances for many. Using a mixed method design, this study investigated how contemplative pedagogy could support post-secondary mathematics courses. Two sections of the same course formed the sample, where the intervention incorporated contemplative pedagogy. The triangulation of the mixed methods data – qualitative (researcher field notes, math origin stories, and semi-structured interviews) and quantitative (pre- and post-questionnaires and course grades) – concluded that the use of contemplative pedagogy was a viable intervention to reduce math anxiety, diminish math avoidance, improve math attitudes, and increase math performance. Results further indicated that contemplative pedagogy fostered the building of community and prompted positive transformations. vi Preface The supervisor, committee members and the student worked in collaboration on this thesis. Specifically, § The design of this research program was co-devised by the student and the supervisor. § The student performed the associated research and analysed the data. § The manuscript was prepared by the student, edited by the supervisor and the committee members. § This research required a harmonized ethics approval from two institutions: § UBC Okanagan Research Ethics Board, Certificate Number: H22-01138-A001 § Okanagan College Research Ethics Board, Reference Number: 22-006 § No generative AI was used in any way or in any form in this research. vii Table of Contents Abstract ................................................................................................................................. iii Lay Summary ........................................................................................................................... v Preface ................................................................................................................................... vi Table of Contents .................................................................................................................. vii List of Tables ......................................................................................................................... xii List of Figures ....................................................................................................................... xiii Acknowledgements .............................................................................................................. xvi Dedication .......................................................................................................................... xviii Chapter 1: Introduction ........................................................................................................... 1 1.1 Background .................................................................................................................. 2 1.2 Positionality .................................................................................................................. 5 1.3. Significance…………………………………………………………………………………………………………….. 9 1.4 The Research Questions ............................................................................................ 11 1.5 Definition of Key Words ............................................................................................ 13 Chapter 2: Review of the Literature ....................................................................................... 16 2.1 Math Anxiety .............................................................................................................. 16 2.2 Contemplative Pedagogy ........................................................................................... 25 2.2.1 Benefits ................................................................................................................ 26 2.2.2 Secular Mindfulness .............................................................................................. 29 2.2.3 Course Implementation ........................................................................................ 31 2.2.4 Commodification .................................................................................................. 38 viii 2.3 Intersectionality ......................................................................................................... 41 Chapter 3: Methodology ........................................................................................................ 51 3.1 Epistemology .............................................................................................................. 51 3.2 Research Design ......................................................................................................... 53 3.3 Qualitative Method – Narrative Inquiry ..................................................................... 55 3.4 Quantitative Methods ................................................................................................ 57 3.5 Population .................................................................................................................. 59 3.6 Data Collection ........................................................................................................... 61 3.6.1 Quantitative Data Collection ................................................................................ 63 3.6.2 Qualitative Data Collection .................................................................................. 65 3.7 Data Analysis .............................................................................................................. 71 3.7.1 Quantitative Data Analysis ................................................................................... 72 3.7.2 Qualitative Data Analysis – Grounded Theory ..................................................... 76 3.8 Validation ................................................................................................................... 83 3.9 Ethics .......................................................................................................................... 85 Chapter 4: Results ........................................................................................................................ 88 4.1 Math Origin Stories .................................................................................................... 91 4.1.1 Theme Re-Focused MOS1: Struggling with Math Due to… .................................. 96 4.1.2 Theme Re-Focused MOS2: Feeling Alone in Math Class and/or How One Feels About Mathematics ............................................................................................. 98 4.1.3 Theme Re-Focused MOS3: Changing Attitude and/or Confidence in Math Due to One Person or Experience .................................................................................... 99 ix 4.1.4 Theme Re-Focused MOS4: Having a Fear of Making a Mistake or of Asking for Help .................................................................................................................... 102 4.1.5 Theme Re-Focused MOS5: Avoiding Math and Math-Related Fields ……………… 103 4.2 Individual Semi-Structured Interviews ..................................................................... 105 4.2.1 Individual Semi-Structured Interviews Control Group Themes ........................... 108 4.2.2 Individual Semi-Structured Interviews Intervention Group Themes ................... 111 4.3 Researcher Field Notes ............................................................................................. 123 4.4 Triangulation of Qualitative Results……………………………………………………………………. 126 4.4.1. Pre-Intervention Triangulation of Qualitative Data ......................................... 128 4.4.2 Post-Intervention Triangulation of Qualitative Data ........................................ 130 4.5 Quantitative Results ................................................................................................. 135 4.5.1 Results from the Pre- and Post-Tests Questionnaires and Course Grades ......... 136 4.5.2 Analysis of Covariance – Intervention Impact ..................................................... 148 4.5.3 Correlation and Regression ................................................................................. 150 4.6 Triangulation of Quantitative Results ..................................................................... 156 4.7 Triangulation of Mixed Methods Results ................................................................ 163 4.7.1 Pre-Intervention Triangulation of Mixed Methods Results ................................. 164 4.7.2 Post-Intervention Triangulation of Mixed Method Results ................................ 165 4.8 Analysis of the Research Question ........................................................................... 168 4.8.1 Intervention Effect on Math Anxiety .................................................................. 169 4.8.2 Intervention Effect on Math Avoidance .............................................................. 170 4.8.3 Intervention Effect on Math Attitude ................................................................. 170 x 4.8.4 Intervention Effect on Math Performance .......................................................... 171 Chapter 5: Analysis and Discussion ...................................................................................... 174 5.1 Summary of Research Findings ................................................................................ 174 5.2 Interpretation of Results .......................................................................................... 176 5.2.1 What Are You Saying? ........................................................................................ 177 5.2.2. Why Does It Matter? .......................................................................................... 183 5.2.3 Why Are You Presenting It the Way You Are? .................................................... 188 5.3 Connections to Extant Literature ............................................................................. 189 5.4 Limitations ................................................................................................................ 197 5.5 Future Directions ...................................................................................................... 201 Bibliography ........................................................................................................................ 207 Appendices ......................................................................................................................... 223 Appendix A: Math Origin Story Assignment ....................................................................... 223 Appendix B: Questions for Individual Semi-Structured Interviews .................................... 224 Appendix C: The Math Anxiety Rating Scale (MARS) ......................................................... 226 Appendix D: The Five Facet of Mindfulness Questionnaire (FFMQ) .................................. 228 Appendix E: Researcher Field Notes .................................................................................. 231 Appendix F: Calculations .................................................................................................... 243 F.1 Chi-Squared ( ) Test for Independence – Impact of Influence .............................. 243 F.2 Paired and Independent Sample t-Tests .................................................................. 246 F.3 Analysis of Covariance .............................................................................................. 256 F.3.1 Intervention and Mindfulness ............................................................................ 257 xi F.3.2 Intervention and Math Anxiety ......................................................................... 259 F.3.3 Intervention and Course Grades ........................................................................ 260 F.4 Correlation and Linear Regression Analyses ........................................................... 261 F.4.1 Mindfulness and Math Performance .................................................................. 261 F.4.2 Math Anxiety and Math Performance ................................................................ 262 F.4.3 Mindfulness and Math Anxiety ........................................................................... 263 xii List of Tables Table 1: Incorporated Practices from Contemplative Pedagogy into a Mandatory Post-Secondary Mathematics Course ………………………………………………………………………………………….…. 34 Table 2: Participant Demographics for Both Control and Intervention Groups…………………….…. 60 Table 3: Convergent Mixed Method Data Collection, Data Types, and Stages ..……………………… 62 Table 4: Initial Coding Themes from the Math Origin Stories (Initial MOS)...………………………….. 93 Table 5: Re-Focused Coding Themes from the Math Origin Stories (Re-Focused MOS) ………..… 95 Table 6: Control Group Themes for the Individual Semi-Structured Interviews ………………….... 106 Table 7: Intervention Group Themes for the Individual Semi-Structured Interviews ………..... 107 Table F1: Breakdown of Source and Nature of Influence, based on the Math Origin Stories (Control and Intervention Groups Combined) …………………………………………………………………….. 244 Table F2: Nature of Influence for Domestic and International Participants, With Teachers as the Sole Source of Influence ………………………………………………………………………………………………………. 245 Table F3: Descriptive Statistics, Paired t-Test Significance, and Effect Sizes for the FFMQ, MARS, and Course Grades (Control Group) …………………………………………………………………………………….. 249 Table F4: Descriptive Statistics, Paired t-Test Significance, and Effect Sizes for the FFMQ, MARS, and Course Grades (Intervention Group) …………………………………………………………………………….. 250 Table F5: Descriptive Statistics, Independent Sample t-Test Significance, and Effect Sizes for the Difference between the Control and Intervention Groups (Pre-Tests) …………………………………. 252 Table F6: Descriptive Statistics, Independent Sample t-Test Significance, and Effect Sizes for the Difference between the Control and Intervention Groups (Post-Tests)….…………………………….. 254 xiii List of Figures Figure 1: The Tree of Contemplative Practices ………………………………………………………………………. 15 Figure 2: Causes and Consequences of Math Anxiety ……………………………………………………………. 17 Figure 3: Visualization of Data Types and Stages ……………………………………………………………………. 89 Figure 4: Visualization of Math Origin Stories Themes Emanating from the Re-Focused Coding………………………………………………………………………………………………………………………………….. 104 Figure 5: Visualization of Individual Semi-Structured Interview Control Group Themes ………. 111 Figure 6: Visualization of Individual Semi-Structured Interview Intervention Group Themes.. 122 Figure 7: Researcher Field Notes Taken When Analysing the Math Origin Stories (Re-Focused Coding) ………………………………………….……………………………………………………………………………………. 124 Figure 8: Researcher Field Notes Taken When Analysing the Individual Semi-Structured Interviews (Intervention Group) ………………………………………….………………………………………………. 124 Figure 9: Visualization of Researcher Field Notes Themes ……………………………………………………. 127 Figure 10: Visualization of Pre-Intervention Qualitative Data Triangulation …………………………. 128 Figure 11: Results From the Pre-Intervention Triangulation ………………………………………..………. 130 Figure 12: Visualization of Post-Intervention Qualitative Data Triangulation ………………….……. 131 Figure 13: Results From the Post-Intervention Triangulation ………….…………………………………... 135 Figure 14: FFMQ Pre- and Post-Test Average Values (Control Group) …………………………………. 137 Figure 15: FFMQ Pre- and Post-Test Average Values (Intervention Group) ………………..………… 139 Figure 16: MARS Pre- and Post-Test Average Values (Control Group) ………………………..……….. 140 Figure 17: MARS Pre- and Post-Test Average Values (Intervention Group) ……………..…………… 141 Figure 18: FFMQ Pre-Test Average Values (Control and Intervention Together) ………………….. 143 xiv Figure 19: MARS Pre-Test Average Values (Control and Intervention Together) ………………….. 144 Figure 20: FFMQ Post-Test Average Values (Control and Intervention Together) ………………… 145 Figure 21: MARS Post-Test Average Values (Control and Intervention Together) ………….…….. 146 Figure 22: Pre- and Post-Test Course Grades (Control and Intervention Groups) …….………….. 147 Figure 23: Pre-Test Course Grade (Test 1) Versus Pre-Test Overall Math Anxiety (For Control and Intervention Groups Combined) …………………………………………………………………………..…………….. 151 Figure 24: Pre-Test Course Grade (Test 1) Versus Pre-Test Overall Math Anxiety (For Control and Intervention Groups Combined) ………………………………………………………………………………..………… 152 Figure 25: Post-Test Course Grade (Final Exam) Versus Post-Test Overall Math Anxiety (For Control and Intervention Groups Combined) ……………………………………………………………..……….. 153 Figure 26: Pre-Test Course Grade (Test 1) Versus Pre-Test Overall Mindfulness (For Control and Intervention Groups Combined) …………………………………………………………………………………..…..…. 154 Figure 27: Post-Test Course Grade (Final Exam) Versus Post-Test Overall Mindfulness (For Control and Intervention Groups Combined) ……………………………………………………………….……… 155 Figure 28: Post-Test Overall Mindfulness Versus Post-Test Overall Math Anxiety (For Control and Intervention Groups Combined) ……………………………………………………………………………..….... 156 Figure 29: Pre- and Post-Test Values That Are Significantly Different (Control Group) …….…… 157 Figure 30: Pre- and Post-Test Values That Are Significantly Different (Intervention Group) …. 158 Figure 31: Control and Intervention Group Values That Are Significantly Different (Pre-Test Stage) ………………………………………………………………………………………………………………………………….. 159 Figure 32: Control and Intervention Group Values That Are Significantly Different (Post-Test Stage) ………………………………………………………………………………………………………………………………….. 160 xv Figure 33: Impact of Intervention Determined by Analysis of Covariance at Post-Test …….….. 162 Figure 34: Significant Correlations …………………………………………………………………………………….... 163 Figure 35: Results of the Research …………………………………………………………………………………….... 173 Figure 36: Results of the Research Reiterated .…………………………………………………………………….. 175 Figure 37: Causes and Consequences of Math Anxiety Revisited …………………………………………. 179 Figure 38: Abridged Version of Figures 2 and 37: Causes and Consequences of Math Anxiety ... …………………………………………………………………………………………………………………………………….………. 190 xvi Acknowledgements I acknowledge with tremendous gratitude the contributions of my supervisor, Dr. Karen Ragoonaden. Flexible and patient, she offered insights and ideas that, as a mathematician, I had never considered. Dr. Ragoonden made me understand the concept of feeling comfortable with being uncomfortable. Had I not embraced this perspective, then I would not have transformed, nor grown. Instead, I am even more committed to being a lifelong learner, especially with the goal of helping others. Thank you, Karen. I recognize the influence and perspectives of Dr. Bill Cohen and Dr. Jo Towers. Together, they inspired me to encourage students to share their math stories. I further acknowledge and greatly respect all life forces found on the unceded, ancestral, and traditional territory of the Syilx Okanagan People. Through their stories (the captikʷɬ), I have learning that every voice matters, no matter how small. Know that I have found my song and I will use it to help the People-to-Be. With gratitude, I acknowledge Dr. Oren Ergas who, with Dr. Ragoonaden, first opened my eyes to mindfulness and contemplative pedagogy. The dualism between mindfulness and mindlessness captivated my attention, then gathered my intention and attitude. Dear friend, I know that when those three things collide, mindfulness truly exists. You are dearly missed. I thank Okanagan College for their support, including Grants-in-aid funds which allowed me to better perform my research and hire students. My Department offered me different points of view as well as a barometer, often reminding me from whence I came. To both my chairs during xvii this time, Dr. Joseph Hobart and Mr. Bryan Penfound, I give my thanks. They supported me in difficult times and in moments where my work greatly differed from departmental goals and plans. I deeply acknowledge the support of my committee members who offered insight through questions and discussions. From Dr. Sabre Cherkowski, I learned that the role of leader often falls upon those who do not wish to wear the mantle, but that leadership is required for good ideas to extend beyond one’s classroom. From Dr. David Geary, I learned the difference between mindfulness and contemplative practices, in particular, the evolution from the individual to the community. Relationships and connections matter. To Dr. Javad Tavakoli, I offer my gratitude for the grounding perspective of how many need assistance in mathematics and motivating me to be the one to find ways to offer it. I thank the Okanagan School of Education for its continued support and optimism, in particular, Dr. Margaret MacIntyre Latta and Dr. Catherine Broom. Thank you, Dr. Sinclair and Dr. Nicol, for having the patience to read through my dissertation, to find connections, and to deeper my learning. I acknowledge the discussions and pauses instigated by my friends and cohort: Chinook, Jody, Tania, Darlene, Liz, Johannah, and Kyle. Thank you. I lastly acknowledge the contribution of the man getting by an alligator and he screams. Thank you, Private Wilhelm. xviii Dedication To Marc who reminds me that though time passes, love does not fade. To Elizabeth who always makes time to meet me wherever I travel. Our search for the best margaritas continue. Rockin’ Baja Lobster forever!!! To Brian for reminding me of what’s important, no matter what else is going on. And when in doubt… play cards. But, maybe just one-offs… To Nero, for making sure that I take breaks, be they hikes, trips to the beach, or belly rubs. 1 Chapter 1: Introduction I have been teaching mandatory mathematics courses, mostly at the post-secondary, for over 20 years. When a student walks into my mathematics course, I anticipate supporting, nurturing, and fostering an inclusive and welcoming environment that will offer them the best chance at success. Yet, time and time again, I have observed high levels of anxiety, stress, and fear in students when trying to solve a mathematical problem or writing a midterm or final. Sometimes that emotional reaction occurs simply by stepping into the classroom. Though most of my observations are anecdotal, the feelings of tension and anxiety that arise when working with/around mathematics seem debilitating. Known as math anxiety, this state of mind can affect self-esteem and confidence, and become a lifelong issue. I have learnt that an overwhelming fear of mathematics, regardless of aptitude, affects students of all ages, from kindergarten to college to graduate school. According to Beilock (2020), “93% of Americans report experiencing some level of math anxiety” (para. 22); furthermore, “a 2016 study found that 11% of university students exhibited ‘high enough levels of mathematics anxiety to be in need of counseling’” (para. 22). In Overcoming Math Anxiety, Tobias (1993) shared that the lack of mathematical ability prevented individuals from not only “doing” math, but from living their lives. Further research in math anxiety also indicates that this state of mind can affect career choices, negatively affecting a student’s full potential (Ashcraft, 2002; Choe et al., 2016; Hembree, 1990; Mighton, 2020; Perry, 2004; Shishigu, 2018; Tobias, 1993). 2 1.1 Background Over fifty years ago, Dreger and Aiken (1957) introduced the concept of ‘number anxiety’, applicable to individuals who were “emotionally disturbed in the presence of mathematics” (p. 344). Their study included over 700 students and determined that: 1. Number anxiety differed from general anxiety (or General Anxiety Disorder, GAD). 2. Number anxiety and level of intelligence were not correlated. 3. Students with high number anxiety performed worse in math. Understanding the correlation between students with higher levels of anxiety and low performance in math, Dweck’s (2006) concept of “Growth Mindsets” offered some insights into understanding math anxiety. Dweck’s (2006) separated learners into two categories – those who have a “fixed mindset”, believing that their ability is set in stone and unable to be improved upon or changed, while others have a “growth mindset”, convinced that their ability can change and improve over time with effort and practice. Boaler (2016) collaborated with Dweck and extended this concept to mathematics students and whether they considered themselves “math people” or not. Generally, those who did not consider themselves math people thought they could not have success, thereby making trying pointless. Sonnert et al. (2020) elaborated upon “fixed mindsets” in mathematics, stating that many students dwell upon the belief that “mathematics is difficult and has a high potential for making students feel miserable” (p. 106). Given a fixed mindset within that negative belief, Sonnert et al. (2020) further connected attitude with anxiety, reinforcing the effects of negative perspectives along with fixed mindsets. 3 Small (2017), a Canadian math education researcher, also wrote about the commonplace belief that people can or cannot do math1. “Students experiencing difficulty could easily attribute their lack of success to their lack of innate ability” (p. 66). To support these students, Small (2017) encouraged teachers to foster growth mindsets in the classroom, where “errors are not cause for alarm but are seen as a valuable part of the learning by the teacher, and where students believe they can succeed if they try” (p. 67). After all, without trying, one can neither learn nor succeed. Within the context of a growth mindset, Boaler (2016), included a discussion about brain plasticity (or neuroplasticity); that is, the brain’s ability to “grow, adapt, and change” (p. 10) and the impact of a growth mindset on a learner’s ability to solve mathematical problems as well as their overall mathematical attitude. Mighton, a Canadian mathematician working with mathematics literacy (or numeracy), wrote that math “can be reduced to simple, logical steps that are accessible to any brain” (Mighton, 2020, p. 28-29). In view of this research and these perspectives, clearly there is consideration that doing math and learning math is possible for everyone. A reasonable extension of that theory would be that everyone is a math person, as Boaler (2016) surmised. Understanding the possible long-term implications of math anxiety and recognizing that the brain’s ability to develop growth mindsets, introducing breathing, relaxation, and focus 1 Exactly what doing math means is a discussion for another day. Herein, “doing math” simply means using math to solve problems, including ones such as splitting a bill in half in a restaurant (fractions and percentages) or using space wisely on a checkout belt at the grocery store (spatial skills and geometry). 4 techniques used in contemplative pedagogy in mathematics courses may hold the key to improve mandatory post-secondary mathematics courses by lowering math anxiety, along with minimizing its causing and consequencing factors (see Figure 2, in Section 2.1). Moreover, the fact that contemplative pedagogy (including mindfulness) also improves neuroplasticity and brain growth (see Section 2.2) certainly deepened my interest in including them. Finally, my own personal journey with contemplative practices (or practices from contemplative pedagogy) played a role in the decision to incorporate contemplative pedagogy in the post-secondary mathematics classroom. After delving into some of the consequences of math anxiety (Shayer, 2020, cf. Section 2.1), possible supports and/or mitigating factors emanating from the use of contemplative pedagogy were discussed. Contemplative pedagogy “focus[es] on practices of mind and emotion that draw on the human capacity to know, specifically through stillness, awareness, attention, mindfulness, and reflection” (Owen-Smith, 2018, p. 1) and “place[s] the student in the center of his or her learning so that the student can connect his or her inner world to the outer world” (Barbezat & Bush, 2014, p. 5-6). Contemplative pedagogy further reintroduces the gap (Biesta, 2004) and space needed for creativity and curiosity, to allow for pondering and wondering (Coles, 1990). The gap further introduces a third space (Bhabha, 2004) or a safe, middle-ground space for all involved. This doctoral research aimed to support students during the learning process, beginning with the fostering of a safe, inclusive space, by inviting them to share their mathematical journey, and by encouraging the use of contemplative pedagogy in the course to reduce math anxiety, and to improve math attitudes, math self-confidence, and math self-efficacy. In this 5 dissertation, the details of this doctoral research will be divulged, including the presentation of the problem and the overarching research question, a review of the literature as well as the methodology that was used. An exposition of the mixed methods results and a discussion of these results within the context of the extent literature will be provided. Future directions and limitations will complete the dissertation. 1.2 Positionality As I live, study, and work on the ancestral, traditional, and unceded territory of the Syilx Okanagan People, the question, shared by Dr. Cohen in 2020 during the graduate course Coyote Stories: Pedagogy and Praxis,“swit askwist?”, which means “Who is your name?”, must be pondered. Our positionality is our story, and it ripples through every aspect of our being. Officially, I am a white, francophone, straight female from Northern Ontario Ojibway land, with roots in Roman Catholicism. My father is a descendent from the Austrian-Hungry Empire (later called, Czechoslovakia), and my mother, a descendent from Les filles du roi, women who were brought over from France to marry male settlers in Lower Canada. I come from a blue-collar, middle-class family, and have one older brother. My parents were both the first in their respective families to attain higher education and both became teachers – my mother, an elementary teacher for 35 years, and my father, a high school math teacher for 33 years. One might say that teaching is in my blood. I would reply that my route toward teaching was not direct. In fact, I spent many years trying to avoid teaching as a career, yet here I am. I have appreciated the journey and continue to revel in its 6 transformative aspects. My journey continues, never on a straight path. Learning is always possible, as long as I pay attention and reflect. As I wrote in earlier paragraphs, I have been teaching mathematics for over twenty years. Prior to that I was tutor, teaching assistant, piano teacher, gymnastics coach, choir master. Since then, I have also been wife/partner, mother, (critical) friend, research assistant, teaching assistant, and colleague. In consequence, I have been a lifelong learner, believing that no day is complete without learning something new, no matter how small. To you mathematicians out there, even an epsilon’s (or delta’s) worth is enough! I have also been a firm believer that there is learning in everything. While I may sometimes feel uncomfortable with my lack of knowledge or understanding, the innate and natural desire to feel comfortable pushes me to the brink of my discomfort then ignites transformation. I have learned to get “pruney” in my discomfort, knowing that comfort begets complacency, a space where no learning occurs. I have also long learned that the more I know, the more I know nothing. Hence, I am a clean slate consciously trying to acquire knowledge and make connections. With age, I have also gained enough wisdom to be patient. When my digestion of content is slow, I am self-compassionate. My familial history and my lived experiences have moulded me into the person that I am today and for that I am grateful. From this standpoint and perspective, I have witnessed the effects of math anxiety on my students and on classroom dynamics – they have been enormous. As the teaching and learning of mathematics is of great interest to me, finding ways to support students to better learn, with lessened stress and anxiety, and gain confidence in their mathematical selves is my passion. Once, in my younger, naïve days, I had hoped to share 7 the love of mathematics. Though I still love it, my expectations have changed. I hope that my students learn tolerance of the subject and avoid propagating negative attitudes toward others. My journey with contemplative practices began in the early 2000s when I encountered chronic pain as the result of an autoimmune condition. Acquiring skills, such as turning toward challenges or breathing through stressful and anxious moments instilled in me the value of having that personal practice. This practice, which became more meaningful and consistent when I began my second master’s degree, also had a great effect on myself and how I see the world around me. Through the years, I have become less judgemental of others as well as of myself, less reactive, more compassionate and self-compassionate, and more aware. Furthermore, I have greater patience and enjoying pausing to simply live in the moment. I have decreased bombarding others by saying “sorry” too many times and have turned instead toward being grateful. I have realized that learning is a powerful and transformative journey, which involves reflection and contemplation, but also unlearning. For me, the process of unlearning involved turning the scrutinizing lens into myself, where I critically questioned my views, perceptions, and beliefs. Though this was performed in the most self-compassionate and non-judgemental way possible, I still needed to work through my shame and my privilege, before emerging with humility and deeper understanding. Part of my inner explorations led to questioning the possible adoption of contemplative practices in the post-secondary mathematics classroom. After all, if I have been the benefactor of positive and transformative results, could some of my students not be as well? I embodied this question and it took root within me, offering an unexpected purpose and direction – to 8 take a chance, incorporate practices from contemplative pedagogy and see if they could improve the mathematical perspectives of students. Positionality affects every facet of us as academics, researchers, educators, and even just people in everyday life. For example, as a researcher, my perspective, beliefs, and lived experiences influence not only what I study, but how I choose to study it, who I include and how I interpret the results of the study revolve around my positionality. Moreover, as an educator, what I teach, how I teach, how I assess, what I assess, what I bring into the classroom as well as what I see in the classroom are also related to my positionality. My most recent graduate learning experience in education has taught me to be more aware of concepts previous ignored or missed by “Leslie, the mathematician”. Being an uninvited visitor on Syilx Okanagan land, I was encouraged to reflect on this gift and to not only appreciate the land but to live in symbiosis with it. This gratitude and awareness heightened my compassion toward all life forces, both inside and outside of a mandatory post-secondary math course. It has inspired me search deeper for connections and relationships within the content and in that context, emphasizing and expanding my belief that mathematics is primarily about patterns and relationships. As written above, positionality affects how we, as researchers, perform research and interpret data. According to Reid et al. (2017), the dissemination of results is an important consideration of research, especially when it relates to equity deserving populations. Knowing thyself allows the researcher to be authentic in the transmission of results. Furthermore, positionality is “a significant force in shaping epistemological and pedagogical stances. Through cognizance of the ways in which who they are affects how they think and teach, educators are 9 in a stronger position to deconstruct the normative assumptions brought to the classroom” (Greenfield, 2007, p. 231-232). During an interactive evening with Dr. Kovach (Feb 17, 2022 at UBC Okanagan), Kovach reminded researchers that their theory/philosophy should be prevalent and woven throughout all their work. How students behave, cope, succeed in a mandatory post-secondary math course is the result of an intersection of many things – their math attitude and math anxiety (Sonnert et al., 2020), the curriculum, their future plans, the classroom environment (including educators, peers, and even the classroom layout), and external influences such as family and community members. My intention is to empower students – not to teach them skills so that they may continue to be under the boot of mathematics, but so that they may break free from its constricting weight and move toward their own mathematical success and overall fulfillment. This section completes the discussion on my positionality and the reasons for undertaking this research. This discussion, therefore, sets the stage for its significance, which will be addressed in the next section. 1.3 Significance Building upon my positionality (described in the previous section), my multiple perspectives as mathematician, educator, and graduate student in education emphasize the needed connections between mathematicians, math educators, and academics in mathematics education. Though having these multifaceted roles is not a common phenomenon, Sinclair (de Freitas et al., 2017) and Noddings (2005) had beginnings resembling mine, with master’s degrees in mathematics before pursuing graduate degrees in education. To truly improve the 10 state of mathematics education, mathematics educators of all levels (K-12, college, university) need to work together toward the common goal of supporting student success in mathematics. Though there are many known conflicts between mathematics teachers (K-12) and post-secondary mathematics educators as well as mathematicians and academics in mathematics education Bass (2005), I will not take sides in this dissertation, nor assert any opinions on any of the contentious issues. Instead, I would like to forge a path ahead and not incite division, as this research could positively affect each of these groups, no matter their stance. Further points on this matter will be picked up again in Chapter 5. As stated by Beilock (2020) and referenced earlier in the Introduction, math anxiety is widespread and affects some post-secondary students so greatly that counselling is required. Academics in education and psychology have been the primary investigators into math anxiety since it became a topic of study in 1957, initiated by Dreger and Aiken. Much research has been done since that time and two consequences of math anxiety have surfaced, especially in educational contexts: 1. Math anxiety affects math performance. The greater the anxiety, the lower the score in the math-related course (Chang & Beilock, 2016; Choe et al., 2016; Hembree, 1990; Ma, 1999; Namkung et al., 2019; Shishigu, 2018). 2. Math anxiety leads to math avoidance. Hence, students deny themselves a future in Science, Technology, Engineering, and Mathematics (STEM)-related careers, simply by avoiding subjects and jobs which include mathematics (Ashcraft, 2002; Choe et al., 2016; Hembree, 1990; Perry, 2004; Shishigu, 2018; Sonnert et al., 2020; Tobias, 1993), depleting the scientifically-skilled workforce (Choe et al., 2016; Hembree, 1990). 11 The significance of this research is multifaceted. First, this research fills a gap in the literature via the examination of how of contemplative pedagogy can impact mandatory post-secondary mathematics courses with a focus on math anxiety. Though math anxiety has been explored and investigated for over fifty years, minimal supports and solutions have been shared. The problem has been well-defined, many times over. Hence, the foremost important result is to provide students with strategies focused on how to manage math anxiety. Secondly, since little has been written about the incorporation of contemplative pedagogy in mandatory post-secondary mathematics, this research would add to the body of knowledge for other researchers and educators. Most importantly, this study could provide students with the necessary confidence, attitude, mindset, and anxiety-fighting tools to pursue courses in mathematics or STEM-related fields to attain greater satisfaction and success in their lives. After all, as Mighton (2020) wrote, “everyone should have the right to fulfill their intellectual potential” (p. 31). This section considered my positionality and detailed the significance of the research. The following section will present the overarching research question and its related sub-questions that were studied to produce this dissertation. 1.4 The Research Questions Up until now, my background, my perspectives, my lived experiences have been shared. Furthermore, the anticipation of the significance of this research has already been divulged. Before diving into the research question, it is important to bring back the problem itself – mandatory post-secondary mathematics courses and the effect they may have on students. In 12 particular, the math anxiety developed through a student’s mathematical career and reveailing itself during a mandatory post-secondary mathematics course will be closely examined. Given the two most renown consequences of math anxiety – poor mathematics performance and mathematics avoidance – then decreasing math anxiety is important, if not vital. Tobias (1993) even wrote that math anxiety prevents people from living their lives and referenced how the negative self-talk could spread to include non-math areas. Instead of allowing these negative feelings to spread, this research aimed toward a bright and equitable future for all by investigating the overarching research question How can contemplative pedagogy support a mandatory post-secondary mathematics course? As will be described in Methodology, an open overarching research question was chosen to allow for space, so that blinders are not simply worn. Instead, by being more ambiguous, the following sub-questions arose during the data analysis phase2: Does it [contemplative pedagogy] a. Decrease math anxiety? b. Reduce math avoidance? c. Improve math attitudes? d. Increase math performance? This section introduced the overarching research question along with its related sub-questions. The subsequent section will divulge pertinent key words along with their definition. 2 It is important to note that these sub-questions had not been chosen beforehand. 13 1.5 Definition of Key Words In this first chapter, a variety of topics, including the research problem and an overarching research question along with its related sub-questions which have very specific terminology have been considered. Before proceeding any further, it is important to specify the language and vocabulary which will used in this dissertation to better understand and communicate ideas. 1. The terms math and mathematics will be used interchangeably. The latter is used in more formal contexts while the former will be used in conjunction with anxiety or other “negative” contexts as much as possible as it is both shorter and more accessible. 2. The expression math anxiety refers to a “feeling of tension and anxiety that interferes with the manipulation of numbers and the solving of mathematical problems in a wide variety of ordinary life and academic situations” (Richarson & Suinn, 1972, p. 551). 3. The term self-report means “to give details about something yourself, rather than having them reported by someone else” (Cambridge Dictionary, n.d.). The concept of self-report will play a role particularly relating to questionnaires completed during the research. 4. The expression mathematics educator encompasses all those who teach mathematics to other individuals. It includes: a. mathematicians – individuals with graduate degrees in mathematics, many whose primary focus is on mathematics research, 14 b. mathematics teachers – individuals who teach children in kindergarten through grade 12 who have at least a bachelor’s degree in education, and c. academics in mathematics education – individuals with graduate degrees in education generally having a focus on preparing future kindergarten to grade 12 mathematics teachers. d. The term contemplative pedagogy signifies “interior qualities of lifelong impact, such as self-knowing and ethical cultivation” (Grace, 2011, p. 99). Or according to Zajonc (2013) contemplative pedagogy “presumes that the capacities of sustained voluntary attention, emotional balance, insight, and compassion are able to be developed through practice… Full comprehension means that we understand the world from inside as well as the outside” (p. 93). e. The expression contemplative practices represents exercises based on contemplation or introspection, which include a wide variety of techniques. Contemplative practices are incorporated in a curriculum to form a pedagogy. Figure 1 demonstrates the diversity and the richness of practices from which one may choose to include contemplative pedagogy in a course. This section completes the Introduction, which presented the background, my positionality, the research question(s) and key words. The next chapter will provide details relating to the extant literature connected to the overarching research question How can contemplative pedagogy support a mandatory post-secondary mathematics course? 15 Figure 1 The Tree of Contemplative Practices Note: The Tree of Contemplative Practices demonstrates the wide array of formal and informal practices which support contemplation. This figure is used with permission. CMind. (2021). The Tree of Contemplative Practices [Illustration]. The Center for Contemplative Mind in Society. https://www.contemplativemind.org/practices/tree 16 Chapter 2: Review of the Literature The previous chapter introduced the problem, the overarching research question and its related sub-questions, my positionality, as well as included definitions relating to math anxiety and contemplative pedagogy. This chapter will provide a review of the literature pertaining to math anxiety at the post-secondary, contemplative pedagogy (including mindfulness) at the post-secondary, and finally the intersection of the two. 2.1 Math Anxiety Richardson and Suinn (1972) defined math anxiety as a “feeling of tension and anxiety that interferes with the manipulation of numbers and the solving of mathematical problems in a wide variety of ordinary life and academic situations” (p. 551). Academics in education and psychology have been the primary investigators of math anxiety since it became a topic of study in 1957, a field initiated by Dreger and Aiken (1957). Much research has been effectuated since that time and two main consequences of math anxiety have emerged: 1. Math anxiety affects math performance. The greater the anxiety, the lower the performance (or grade) in the mathematics course (Chang & Beilock, 2016; Choe et al., 2016; Hembree, 1990; Ma, 1999; Namkung et al., 2019; Shishigu, 2018; Sonnert et al., 2020). 2. Math anxiety leads to math avoidance. Students avoid courses and careers in STEM-related fields, simply by avoiding subjects and jobs which include mathematics (Ashcraft, 2002; Choe et al., 2016; Hembree, 1990; Perry, 2004; Shishigu, 2018; Tobias, 1993). Furthermore, individuals shun all mathematics in their environment: from feeling comfortable calculating a tip to 17 attempting, to estimating the cost of a grocery trip, to understanding the amount of interest paid on a loan (Tobias, 1993). Math anxiety is rife with its causes and consequences. Figure 2 demonstrates the interconnectedness of math anxiety with both its causes and consequences. Figure 2 Causes and Consequences of Math Anxiety Note: Direct and indirect relationships with math anxiety are displayed, where arrows denote direction. 18 It is important to note that most of the studies connected to Figure 2 consider math anxiety at the post-secondary level. When meta-analyses investigating math anxiety, such as Hembree (1990), Ma (1999), or Namkung et al. (2020), numerical values pertaining to their post-secondary were primarily considered below (and discussed in Chapter 5). Moreover, a few of the references found below do relate to K-12 grades, in particular relating to influence, as children at that level can be particularly malleable to those around them. The literature associated with Figure 2 is detailed below, where the enumeration references the arrows (association or correlation) between any two factors (be they cause or consequence). 1. The influence of adults (including teachers) and peers on math anxiety: Ashcraft (2002), Chang and Beilock (2016), Choe et al. (2016), Jay et al. (2018), Shishigu (2018), Small (2017). Instead of focusing solely on the performance of one individual, Chang and Beilock (2016) examined the influence of other students, parents, and teachers, stating that “teachers’ math anxiety and classroom activities, parental math anxiety, support and expectations, and students’ perceived classroom environment are non-negligible social or contextual factors that may affect the math anxiety-math performance relation” (pp. 35-36). 2. The inverse3 relationship between math attitude and math anxiety: Ashcraft and Kirk (2001), Hembree (1990), Jameson and Fusco (2014), Perry (2004), and Sonnert et al. (2020). Individuals with high math anxiety tend to adopt negative attitudes about math and often have negative perceptions of their own math abilities (Ashcraft, 2002). 3 Inverse means that when one increases, the other decreases, or vice-versa. 19 Sonnert et al. (2020) even suggested combining the two into one variable “because mathematics anxiety has consistently been found to be inversely related to positive mathematics attitudes” and that “mathematics anxiety would be located at the negative pole of the spectrum of mathematics attitudes” (p. 106). 3. The influence of math attitude on math avoidance: Ashcract and Kirk (2001), Hembree (1990), Sonnert et al. (2020). Moreover, Hembree (1990) stated that a positive math attitude led to further math courses. Individuals who have positive attitudes toward math, along with low math anxiety, tend to naturally include math in their lives. In consequence, they improve their competence and activate their skills, leading to better performance results (Ashcraft & Kirk, 2001). “In the reverse situation, with poor attitudes and high math anxiety, individuals exhibit global avoidance and attain lower competence, again with obvious effects on performance” (p. 236). 4. The positive association between math attitude and math performance: Previous math performance, particularly poor performance, influences math attitude (Ma, 1999; Sonnert et al., 2020). Hembree (1990), above, also related how poor attitudes affected performance. 5. The inversely proportional consequence of math anxiety on working memory: Ashcraft and Kirk (2001) also examined, using surveys and math tasks in a laboratory setting, the relationship between math anxiety and working memory capacity, where working memory “governs not only storage and retrieval functions involving long-term memory but also planning and decision-making processes” (Hopko et al., 1998, pp. 344-345). Ashcraft and Kirk (2001) determined that there was a negative association between the 20 two in tasks involving arithmetic or algorithms. That is, the greater the math anxiety, the smaller the working-memory capacity to put toward mathematical tasks, which may partially explain the low performance issues. Small (2017) stated that “their worry about math so occupies their thoughts, it is hard for them to actually think about the math” (p. 66). 6. The positive association between working memory and math performance: Worrying leads to diminished recall and problem-solving ability, in turn resulting in decreased performance (Ashcraft & Kirk, 2001; Namkung et al., 2019; Small, 2017). 7. The inverse relationship between math anxiety and math performance: That is, the greater the anxiety, the lower the mathematics score or grade. This relationship has been well-documented in three meta-analyses – Hembree (1990), Ma (1999), and Namkung et al. (2019). At the post-secondary level (based on the 122 studies), Hembree (1990) determined that the sample correlation coefficient (also significant) was r = -0.27. Based on his calculations, only 7% of math performance is directly explained by math anxiety. The Ma (1999) sample correlation coefficient (of 26 studies at the grade school level), was also determined to be significant, and agreed with Hembree’s (1990) value of r = -0.27. Namkung et al.’s (2019) meta-analysis of 131 studies of school aged children aligned with the two previous correlations with r = -0.34, leading to 11.5% of math performance explained by math anxiety. Using statistical analyses, Shishigu (2018) confirmed many of the previously stated hypotheses emanating from the meta-analyses and examined the effects of prior math performance. He linked prior math performance with both math anxiety and current math performance, stating that “prior math 21 achievement was also the strongest predictor of math anxiety” (p. 7), demonstrating that previous experiences affect current math performance and math anxiety. That is, prior positive math performance led to decreased math anxiety and improved current math performance, while prior negative math performance did the reverse – increase math anxiety and decrease math performance, agreeing with Meece et al. (1990). 8. The connection between math performance and math avoidance: Not performing well is discouraging, leading to negative attitudes and avoidance of math in all its forms, with less practice resulting in poorer performance (Hembree, 1990; Namkung et al., 2019; Perry, 2004; Tobias, 1993). Hembree (1990) added that “high math-anxiety individuals enroll in fewer math courses and earn lower grades in the math classes they do take” (p. 225). 9. The well-documented relationship between math anxiety and math avoidance, depleting the country of scientifically minded individuals as they avoid STEM-fields: Ashcraft (2002), Beilock (2020), Choe et al. (2016), Hembree (1990), Jameson and Fusco (2014), Namkung et al. (2020), Perry (2004), Shishigu (2018), Tobias (1993). According to Ashcraft (2002), “Perhaps the most pervasive—and unfortunate—tendency is avoidance. Highly math anxious individuals avoid math. They take fewer elective math courses, both in high school and in college, than people with low math anxiety. And when they take math, they receive lower grades” (p. 181). 10. Test anxiety affects math performance: Ashcraft (2001), Bellinger et al. (2015), Hembree (1988). Via a meta-analysis of 562 studies, Hembree (1988) determined that “test anxiety and performance are significantly related at grade 3 and above” (p. 72). 22 11. The imperfect correlation between math anxiety and test anxiety – a reminder that math anxiety is not simply test anxiety in mathematics: Brunyé et al. (2013), Dreger and Aiken (1957), Hembree (1990), and Namkung et al. (2019). Out of the 151 studies, Hembree (1990) determined that the sample correlation coefficient between math anxiety and test anxiety was significant with a value of r = 0.52, demonstrating a positive correlation between the two. That is, the greater the math anxiety, the greater the test anxiety, and vice-versa. However, neither fully explains the other as the coefficient of determination (the R2 value) is 0.37, allowing for 0.63 of the variation to be explained by other factors. 12. The worse the skills, the lower the performance: The correlation between math skills and math performance is also well studied: Ashcraft (2002), Mighton (2020), Paulos (1988), and Rylands & Perry (2004). Ashcraft and Kirk (2001) found that “high-anxiety participants are simply less competent in math, unable to perform the necessary calculations at the same level of accuracy as low-anxiety individuals” (p. 224), a statement which aligns with Perry’s (2004) suggestion of simply improving math skills to decrease math anxiety. Linked with math avoidance, students demonstrating poor math performance are less likely to practice math and therefore will have weaker math skills (Namkung et al., 2019; Shishigu, 2018). 13. The influence of instructor behaviours on math anxiety: the classroom environment, including instructor behaviors and teaching methods, influence math anxiety (Ashcraft, 2002; Shishigu, 2018; Small, 2017). Ashcraft (2002) described how some of the students’ math-avoidance was related to instructor behaviors, such as “a high demand for 23 correctness but provide[d] little cognitive or motivational support during lessons” (p. 101), showing “annoyance when students gave wrong answers”, or holding students “responsible for their lack of understanding" (p. 102). Ashcraft (2002) shared that “public embarrassment in math class contributed to their [students’] math anxiety” (p. 185). According to Shishigu (2018), “teaching strategies involving independent and competitive mathematics activities” (p. 7) also contribute to math anxiety. Small (2017) added another instructor behavior, that is, “the emphasis on speed, and the fact that the speed has to be demonstrated in a very public way” (p. 66). 14. Age and/or time since last math course and math anxiety: The younger the student, the less anxious they are. In fact, Namkung et al. (2019) postulated that math anxiety generally begins around grade 4 and increased with the grade level, as more negative experiences and more challenging mathematics are encountered. This theory demonstrated that younger students generally have less math anxiety than older ones. Namkung et al. (2019) posited that different stages of education opened the door to new math anxiety, for example, grade 9 (with the beginning of high school) and the beginning of college/university. Namkung et al. (2019) added that “even simply numerical tasks, such as counting, can also elicit MA [math anxiety] among adults” (p. 463). Finally, Jameson and Fusco (2014) added that the time elapsed since the previous math course (which is also linked to age) increased anxiety and affected the retention of adult learners and their ability to complete a degree. 15. The positive association between poor math skills and increased math anxiety: Ashcraft & Kirk (2001), Hembree (1990), Sonnert et al. (2020), and Tobias (1985). 24 Hembree (1990) determined that students with high levels of math anxiety saw a decrease in anxiety levels when they took more math courses. Hence, taking more math courses could only lead to greater math competency as skills become honed. Ashcraft and Kirk (2001) suspected that math-anxious individuals did less math homework because of a lack of confidence and a history of avoiding math, resulting in insufficient practice. Without the necessary practice to develop mathematical skills, math anxious students were less competent in performing mathematical tasks. 16. The association between math self-efficacy and age and/or the time since the last math course: Adult learners “report lower confidence in their ability to complete ‘academic’ math tasks but no in their ability to complete more routine math tasks” (Jameson and Fusco, 2014, p. 314). 17. The inverse relationship between math self-efficacy and math anxiety: Jameson and Fusco (2014) particularly investigated this relationship in older or mature students. Their lack of confidence increases their anxiety level, especially when completing assignments or writing tests/finals, and their anxiety level decreases their confidence in being able to do math. Upon the examination of Figure 2, it is evident that the connections between math anxiety and other factors are multifaceted. In fact, many loops contained therein lead to amplified consequences due to feedback. For example, #5, 6, and 7 are connected: high math anxiety decreases working memory, which in turn, affects/worsens performance, which leads to greater math anxiety (and so on and so forth). It is important to note that a change at any of 25 these factors could ripple through further down the line, making the study and the minimization of math anxiety (or any of its components) valuable. Though many of these connections have been extensively researched, little has been done to alleviate anxiety levels in mandatory post-secondary math courses. The rise of contemplative pedagogy as an approach to decrease stress and anxiety is well-noted in the literature. Since contemplative pedagogy seems to provide some direction in alleviating anxiety and stress, the following section details the theoretical framework and the research relating to how contemplative pedagogy can be used to support mandatory post-secondary mathematics courses. 2.2 Contemplative Pedagogy According to Zajonc (2009), contemplative pedagogy is a form of experiential learning whose underpinning theory is “one that presumes that the capacities of sustained voluntary attention, emotional balance, insight, and compassion are able to be developed through practice” (p. 88-89). Grace (2011) discussed contemplative pedagogy, that she defined as “cultivat[ing] inner awareness through first-person investigations, often called ‘contemplative practices’” (p. 99), where “contemplative practices can help develop greater empathy and communication skills, improve focus and attention, reduce stress and enhance creativity, supporting a loving and compassionate approach to life” (Center for contemplative mind in society, n.d., para. 2). To discover the diverse categories of contemplative practices, consult Figure 1, the Tree of Contemplative Practices. 26 Hart (2004) laid the foundation for the incorporation of contemplative pedagogy in the classroom, by stating the importance of informing students of the scientific evidence supporting contemplative pedagogy, planting the seeds for secular contemplative practices to be incorporated. The next section will describe benefits proven to result from the incorporation of contemplative pedagogy. 2.2.1 Benefits Contemplative pedagogy has been incorporated in post-secondary programs since 1995, after recognizing its positive influences in health care settings (Barbezat & Bush, 2014). According to Barbezat and Bush (2014, p. 11), benefits of contemplative pedagogy include: 1. Improved focus and attention 2. Deeper understanding of the course content 3. Greater connection with peers and compassion 4. Augmented meaning-making ability, insight, and creativity Furthermore, Zajonc (2013) added that contemplative pedagogy encourages critical thinking, inviting educators to not only question what they are teaching but what kind of knowing they want to foster. Hart (2004) shed light on the concept of “how we know is as important as what we know” (p. 28). He inferred that “contemplative knowing” is lacking in education, something that greatly affects understanding and performance. Ergas (2015) elaborated upon how instilling first-person perspectives of attention can promote that common ground. “Meta-pedagogy is interpreted here as the explicit or implicit instruction that tells students where to orient their attention moment after moment” (emphasis in original, p. 211). 27 Grace (2011) expounded upon the importance of contemplative pedagogy by “emphasiz[ing] the power of lived truth rather than theoretical rationale” (p. 100), adding that contemplative pedagogy was linked to lifelong learning. Zajonc (2013) provided an overview of contemplative pedagogy at the post-secondary level, including its benefits, from attention improvement to better cognition, to emotional self-regulation, increased compassion, and greater creativity to deeper understanding of course material. Furthermore, Zajonc recognized mindfulness as a form of contemplative pedagogy, and stated that mindfulness (including Mindfulness-Based Stress Reduction, MBSR, and meditation, see Section 2.2.2) is the most researched component of contemplative pedagogy. Ergas (2019) compared mindfulness in education to mindfulness as education, called the latter the “less common yet more transformative strand and manifests in contemplative pedagogy in higher education” (p. 757). Contemplative pedagogy includes many other practices beyond secular mindfulness such as concentration training/focused attention (to increase focus), open awareness (leading to more creative moments) and the ability to sit with opposites/contradictions without the need to choose one or the other (developing acceptance) (Zajonc, 2013). Contemplative pedagogy, including mindfulness, is used in diverse settings and at a variety of academic levels, as described in detail throughout Shapiro et al.’s (2011a) substantial literature review. Though all the discussed effects are supported by scientific experiments and the results of empirical evidence, the main interest to this doctoral research is their middle hypothesis, that is, the hypothesis relating to mental health and well-being. “The stresses of higher education have been related to numerous mental and physical health problems … and 28 may adversely affect academic and cognitive performance” (p. 506). According to Shapiro et al. (2011a), though certain amounts or types of stress may improve performance, others may be detrimental to one’s overall health and well-being. Similar to the upcoming findings of Ashcraft and Kirk (2001) on the relationship of math anxiety and working memory capacity (cf. Section 2.3), Shapiro et al. (2011a) explored the relationship between stress levels and working memory drops, which in turn affected course performance and learning. Their analysis considered forty years of research and demonstrated that meditation “reduces negative mental health symptoms, including stress and anxiety, and enhances psychological well-being” (p. 507). Bellinger et al. (2015) seconded this premise, which will be detailed in Section 2.3. Shapiro et al. (1998) effectuated research which investigated the benefits of meditation included a nine-week MBSR course involving medical students. “Results indicated decreased levels of anxiety and depression in the MBSR group as compared to the wait-list control group” (Shapiro et al., 2011a, p. 15). Hence, it is necessary to consider “longer-term” interventions relating to mindfulness and contemplative pedagogy, examining the longer-term effects of a consistent practice on self-regulation and performance, all aspects that formed the support offered to students via this research in a mandatory post-secondary math course. Hart (2004) elaborated upon the benefits in higher education, dividing them between state and trait effects. The “main state effects (immediate changes) of meditation are physiological relation and slowed metabolism, a heightened self-awareness, and feelings of calm” (p. 31). Hart (2004) continued and wrote that “the main trait effects (changes that endure over time) are improved concentration, empathy perceptual acuity, a drop in anxiety 29 and stress symptoms, and more effective performance in a broad range of domains” (p. 31). Furthermore, at the post-secondary level, a decrease in stress levels was demonstrated, even with interventions shorter than the regular MBSR course (Brunyé et al., 2013; Mackenzie, 2015), the speculation related to whether those decreases could be sustained. O’Donnell (2015) stated her preference for longer-term programs (eight or more weeks), but also the need for contemplative practices to be included as part of the class’s curriculum to increase attention, awareness, and inquiry. 2.2.2 Secular Mindfulness As discussed in the previous subsection, contemplative pedagogy has many benefits, some researched under the label of “(secular) mindfulness”. In fact, mindfulness has been the most scientifically studied practice under the contemplative pedagogy umbrella (Zajonc, 2013), exploding exponentially in recent years (Ergas, 2019). Consulting Figure 1 in Chapter 1, one may see many contemplative practices which include (secular) mindfulness practices, such as gratitude, loving-kindness & compassion meditation, and walking meditations. Secular mindfulness dates to 1979 with the creation of Mindfulness-Based Stress Reduction (MBSR), a programed designed to support cancer patients whose pain management did not suffice. MBSR and (secular) mindfulness4 evolved from one of the Buddhist lineages, where scientific research is put to the forefront and Buddhist tradition is relegated to a philosophy (Batchelor, 2012; McMahan, 2008; Wilson, 2015). According to Kabat-Zinn (1994), mindfulness signifies “paying attention in a particular way: on purpose, in the present moment, 4 Going forward, when mindfulness is referred to, it means secular mindfulness. 30 and non-judgmentally” (p. 4). Mindfulness began to be incorporated in courses and by educators, starting in medicine and liberal arts (Barbezat & Bush, 2014; Grace, 2011), then spread to a wider audience, to include law, business, and science programs (Magee, 2019; Zajonc, 2013). Shapiro et al. (2011b) cited numerous benefits for medical students, such as improvements in cognition, emotional self-regulation, stress management, and resilience – concepts which align with Hyland (2016), Mackenzie (2015), Ragoonaden (2017a), Schonert-Reichl and Roeser (2016). There were further positive associations in education and learning, such as increased concentration and focus (Zajonc, 2013) and better use of working memory (Bellinger et al., 2015; Shapiro et al., 2011b), both resulting in an amelioration in academic performance. Mackenzie (2015) also touted the outward-extending benefits of mindfulness and contemplative practices, such as increased creativity, critical reflection, and self-awareness (aligning with Shapiro et al., 2011a, and Zajonc, 2013). Mindfulness has had a great influence on the practice of pedagogy, supporting students, faculty, and staff coping and resilience skills, in both academic and real-world settings. Having my own practice and reflecting upon these benefits in an academic setting, I was inspired to investigate whether contemplative pedagogy could have similar benefits in a post-secondary mandatory mathematics course. Due to the fact that anxiety in mathematics far exceeds anxiety in other courses (Hembree, 1990; Ma, 1999; Namkung et al., 2020), I could not assume that the advantages would be the same. Hence, the study which ensued. The dualism of mindlessness and mindfulness, investigated by Ergas (2014), Langer (2014), and McCown (2014), demonstrated the role that mindfulness can play in life and in 31 education. O’Donnell (2015) speculated that attention in the classroom had greatly deteriorated, referring to it as a problem, not only a psychological disorder, but a very real component of modern life. Society has become overstimulated with information, an “implication of the colonization of the ‘attention economy’” (p. 191). Langer (2014) linked mindfulness to increasing attention, something supported by O’Donnell (2015). The latter used mindfulness in the classroom to focus on the present moment, to increase concentration, and to rewire the brain through neuroplasticity. A problem with mindfulness, according to Hyland (2016 & 2017), Van Dam et al. (2017), and Wilson (2015), is that its definition is not consistent. McCown (2015), however, disagreed. He stated the importance of allowing different perspectives, definitions, and discourses to encourage greater understanding and compassion as well as the significance of language to provide a voice to communicate. I believe that adopting McCown’s (2015) perspective is particularly influential to pedagogical praxis. So far in this section, contemplative pedagogy, including mindfulness, has been defined along with its scientifically determined benefits. Before considering the other side of the scale, that of the commodification of mindfulness, the subsequent subsection will divulge the framework used and the practices from contemplative pedagogy incorporated into the mandatory post-secondary mathematics course. 2.2.3 Course Implementation As the benefits of integrating contemplative pedagogy (including both mindfulness and contemplative practices) into a course (and thereby into the classroom environment) are 32 discussed, it is important to consider how to incorporate these practices, especially into a mandatory post-secondary mathematics course. The incorporation of contemplative pedagogy in post-secondary courses has also been well-research, starting with Hart’s (2004) and Barbezat and Bush’s (2014) suggestions. Moreover, Burack (2014) presented a framework, used in Shayer (2020) and which was foundational in this doctoral research as well. It involves four steps: 1. The benefits of practices from contemplative pedagogy and simple ways to introduce them in a course: Introducing practices from contemplative pedagogy in a mathematics course proved to be challenging, especially as they do not seem as “mathematical” or logical by nature for the students. However, Burack did encourage the sharing of the benefits of the practices as something which may interest uncertain participants. 2. The well-being of students: Burack prompted educators to invite students to participate in contemplative practices, not to tell them. Such a simple concept is very powerful and aligns with Berila’s (2014) idea of “teaching the language” to students. By inviting students instead of telling them, they have the freedom of choice – offering empowerment to oppressed or marginalized students. 3. The requirements for the instructor relating to experience, knowledge, and skill: Benefits are particularly noticeable in the classroom when educators themselves have a personal practice (Barbezat & Bush, 2014; Burack, 2014, Hyland, 2015 & 2016). Burack stated that “the best preparation for leading contemplation is having one’s own contemplative practice and being committed to one’s own holistic growth 33 and transformation” (p. 45). This requirement aligns with content knowledge at the post-secondary. If an instructor is always consulting an app or googling a possible concern or response, then the class may not achieve the previously discussed benefits (Ergas, 2015). 4. The separation of church and state: As most post-secondary institutions are public and not affiliated with any particular religion, it was important not to impose religious beliefs on anyone, in particular, vulnerable students. Since contemplative pedagogy has roots in wisdom religions, such as Buddhism, the removal of any or all religious spirituality from the practices and the language used was mandatory (Barbezat & Bush, 2014; Burack, 2014, Hyland, 2016). The research performed by Burack (2014) is fundamental in prioritizing students’ needs when integrating contemplative pedagogy into the classroom. Though some of his statements seem obvious, they are an excellent reminder to meticulously plan so that the result is both ethically sound and useful for the students. Hence, his framework and his perspectives were instrumental to the design and implementation of this doctoral research as were the ideas brought forward by Berila (2014). Berila (2014) investigated ways of incorporating contemplative pedagogy including mindfulness into post-secondary courses, by using her own classroom as the experiment. Her focus was on educating the whole student, including both the inner and the outer life of an individual. Due to her own experiences, Berila warned about being proactive when incorporating contemplative pedagogy into a course. Instead of being surprised by negative 34 emotions or reactions, she encouraged preparations for these possibilities beforehand and provided suggestions. Hart (2004) offered a diversity of practices which support the post-secondary educator, clearly identifying associated intentions and possible benefits. In Shayer (2020) where the sample was gendered (all participants were female), loving-kindness practices were particularly effective in boosting (math) self-confidence and self-compassion, based on qualitative data analysis. However, some prospective participants avoided the study due to their perceived understanding of contemplative practices (but especially mindfulness). Hart’s (2004) variety of practices were more inclusive of participants from a variety of backgrounds and having different levels of comfort with the concept of contemplation. In consequence, this research (see Table 1) incorporated a greater diversity of practices (including prompts for discussion or writing), emanating mainly from Barbezat and Bush (2014), Hart (2004), and the Stress Management and Resilience Training Curriculum (2015). Table 1 Practices Incorporated from Contemplative Pedagogy into a Mandatory Post-Secondary Mathematics Course Practice Source 4-7-8 Breathing Sternberg & Weil (2023, p. 73) & Weil (2000) 5-4-3-2-1 Grounding Exercise Oellerich (2023, p. 42) & Shukla (2019) Freely Writing Hart (2004, p. 42) Gratitude Lai and O’Carroll (2017, p. 13) 35 Practice Source Loving-Kindness Kindness and Compassion* Barbezat & Bush (2014, p. 178-179) SMRTC (2015, p. 86, 91, 139-144) Mindful Eating SMRTC (2015, p. 12-15). Mindful Listening Mindful Listening and Speaking in Dyads Deep Listening Barbezat & Bush (2014, p. 144-145) SMRTC (2015, p. 24-25, 107-108) Hart (2004, p. 36) Mindful Movement SMRTC (2015, p. 34, 117) Not Doing* Doing versus Being* Hart (2004, p. 35) SMRTC (2015, p. 23, 167) The Pause Practice & Triangle of Awareness SMRTC (2015, p. 2-3, 23) Setting Intentions* SMRTC (2015, p. 26-27) Sitting Practice, Awareness of Sensations, Emotions, and Thoughts SMRTC (2015, p. 109) Three Questions Visualization SMRTC (2015, p. 27-28) Where are you now?* Hart (2004, p. 36) Notes: Asterisks (*) above represent prompts that were used in mindful dyads or writing freely; SMRTC is the Stress Management and Resilience Training Curriculum 36 Building upon Hart (2004) perspectives, Burack’s (2014) and Berila’s (2014) suggestions, this research was also informed by Mackenzie (2015). Mackenzie introduced the researched-based overall benefits of mindfulness training, which were numerous and included improvements in, amongst others, cognition, emotional self-regulation, stress management, and resilience. There were further positive associations in education and learning, relating to increased attention and focus, better use of working memory, resulting in an amelioration in academic performance. She further linked mindfulness to neuroplasticity, a concept which has been research by Shapiro et al. (2011a) and links with Dweck’s (2006) concept of growth mindsets. In fact, as mentioned in the Introduction, Dweck worked with Boaler (2016), who used growth mindsets to improve math confidence and proficiency. Brain plasticity is a common theme in both mathematics education (Boaler, 2016) and in contemplative pedagogy including mindfulness (Mackenzie, 2015; Shapiro et al., 2011a). At the post-secondary level, Mackenzie (2015) saw that a decrease in stress levels was demonstrated, even with interventions shorter than the regular MBSR course (which generally include 8-9 weeks of class and between 40-60 minutes of homework per night). This shorter intervention involved only four 75-minute sessions with approximately 10 minutes of homework per night, allowing for easier incorporation into student life. Mackenzie also touted the outward-extending benefits of mindfulness and contemplative pedagogy, such as increased creativity, critical reflection, and self-awareness (aligning with Zajonc, 2013 and Shapiro et al., 2011a). The work included here summarized mindfulness benefits at all levels of education, allowing direction for further study at any or all. Supplementary neurobiology is included for the cognitive scientist, or anyone who would like to 37 delve deeper into the neurological changes which occur with mindfulness. This information was important to convey in a post-secondary course, in part to demonstrate the scientific evidence related to mindfulness as well as to put at ease those feeling threatened by the incorporation of Buddhist traditions into a public education environment. By incorporating Mackenzie’s (2015) perspectives and conclusions into a mandatory post-secondary mathematics course, a greater diversity of students could be accommodated and feel welcome, providing a greater opportunity to affect change. Also, her investigations associated with pre-service and in-service teachers are important as teachers are known for having some of the highest math anxiety levels (Hembree, 1990) amongst college majors. Further discussions on this point will be included in Chapter 5. In the last ten years, mindfulness has been used in post-secondary courses to support challenging discussions relating to privilege and power (Berila, 2014; Magee, 2019, Ragoonaden, 2017b). Berila (2014) pointed to a holistic, more integrative form of education that is the result of contemplative practices. Berila (2014) stated that contemplative techniques, including mindfulness, could support students in courses where oppression was discussed, or which involved marginalized students. For this research, the conditioned responses of students, including fear, anxiety, low self-confidence and/or self-deprecating attitudes form, what I consider, mathematical oppression. This research considered how contemplative pedagogy may support a mandatory post-secondary mathematics course, which includes helping students suffering from mathematical oppression at the post-secondary level. Benefits of integrating contemplative pedagogy including mindfulness into a post-secondary course are not the only possible outcome. From a logical and scientific approach – an 38 important perspective to consider when addressing students in post-secondary mathematics courses – prospective negatives to contemplative pedagogy (including mindfulness) must also be investigated. These prospective negative outcomes are discussed in the subsequent subsection. 2.2.4 Commodification Of course, the benefits in a course are not the only perspective through which mindfulness (or contemplative pedagogy) has been considered. The darker side of secular mindfulness led to the coined term McMindfulness (Ergas, 2015; Hyland, 2016; O’Donnell, 2015; Purser, 2019; Safran, 2014; Wilson, 2014), related to the commodification of mindfulness, its misuse, and its promise to solve all problems. As Wilson (2014) wrote, “It promises everything: it can allegedly improve any conceivable activity and provide unlimited practical benefits. Perhaps it can even save the world” (p. 73). Though the ethical concerns associated with the misuse of mindfulness practices in business and the military are divulged, the systematic oppression of students and teachers are emphasized even more (Hyland, 2016; Purser, 2019). Oppression of school stakeholders include pushing stress onto the individual (student or teacher), rendering students more docile and obedient; obtaining better standardized tests grades, with lessened learning (Ergas, 2015; Hyland, 2016; O’Donnell, 2015; Purser, 2019). “By failing to address collective suffering, and systemic change that might remove it, they rob mindfulness of its real revolutionary potential, reducing it to something banal that keeps people focused on themselves” (Purser, 2019, p. 8-9). Furthermore, Wilson (2015) also wrote about the need to not be subdued by mindfulness. 39 Instead, he shared “the need to combine mindfulness with action. ‘When bombs begin to fall on people, you cannot stay in the meditation hall all of the time,’ [citing Thich Nhat Hanh] Hanh says. ‘You have to learn how to help a wounded child while still practicing mindful breathing.’” (Wilson, 2015, p. 191). A challenge, even with its original incorporation in post-secondary courses in the 1990s, was the concern relating to contemplative pedagogy and spirituality in public institutions. This is an often-discussed issue (Barbezat & Bush, 2014; Burack, 2014; Hyland, 2016; Purser, 2019; Wilson, 2015). Hart (2004) elaborated that “opening the contemplative mind in schools is not a religious issue but a practical epistemic question” (p. 29). In fact, Hart sets the stage for the incorporation of contemplative pedagogy in courses, by reminding educators to inform students of the evidence supporting contemplative pedagogy and lists possible secular practices that may be easily adapted for inclusion. Further suggestions are made in Barbezat and Bush (2014) and Burack (2014). Outside of meditation, where, according to Hart (2004), the majority of research has been performed, he included a diversity of practices such as techniques to improve focus, concentration, creativity, to remind of the nature of impermanence as well as the value of deep listening. Furthermore, many (Barbezat & Bush, 2014; Bai et al., 2009; Fisher, 2017; Grace, 2011) believe that education has become a commodity, where employment is desired over knowledge. They also emphasized the importance of receiving something in return for money paid – grades, degrees, etc. By introducing contemplative pedagogy, a more holistic learning environment may be fostered, making learning easier and longer lasting (Miller, 2019; Zajonc, 2013). Perhaps in the end, since contemplative pedagogy involves making space (Ng and Walsh, 40 2019), and allowing/encouraging silence (Coburn et al., 2011), a gap is created where learning happens (Biesta, 2004). This space informs pedagogical theory and supports learning by student and teacher alike. Fisher (2017) reiterated how contemplative pedagogy is often used as a means to an end, instead of deepening connections to others, acquiring knowledge, and becoming more compassionate. Fisher (2017) examined, via her personal classroom experience, four claims which are often present in contemplative pedagogy, considered prospective issues that may emanate in the classroom, and included questions that educators may ponder before undertaking their incorporation: 1. Students are searching for some kind of spirituality. She asked whether all truly were. 2. Contemplative pedagogy teaches students about how to think and improves critical thinking. Fisher pondered whether first-person learning was always better than third. 3. Students who know themselves have greater empathy than those who do not. Fisher asked how many new-to-college/university students truly knew themselves. 4. The emergence of a new epistemology – the ‘epistemology of love’ (Zajonc, 2013). Fisher wondered whether this new epistemology, founded, according to her, in spirituality, would be accepted in secular institutions. As in previous contexts, a balanced approach to understanding the use of contemplative practices, including mindfulness, in pedagogical contexts is an important consideration. Though Fisher (2017) did use a few absolutes, all or always, contemplative pedagogy’s influence in pedagogical theory prods us to think critically and to ask our own questions, based on our own (subjective) experiences and the facts. 41 As contemplative practices are the superset (extending beyond mindfulness), Shayer (2020) as well as Ng & Walsh (2019) used “contemplative practices” in lieu of “mindfulness”. Moreover, as concerns relating to the commodification of mindfulness have been brought to light, this pivot became more common. Nonetheless, mindfulness has been studied extensively, in a wide variety of setting, including education. Scientifically determined pedagogical benefits from mindfulness relate primarily to the well-being of the student, the educator and the classroom including: stress and anxiety management, emotion regulation, and self-compassionate and inclusive environments (Barbezat & Bush, 2014; Eva & Thayer, 2017; Magee, 2019; Ragoonaden, 2015; Ragoonaden, 2017b; Roeser et al. 2013; Shapiro et al., 2011ab; Zajonc, 2013). The use of contemplative pedagogy, including mindfulness, in post-secondary courses has numerous benefits. Considering all this research as well as the results from this section, a more focused emphasis will now be placed on the contemplative pedagogy in post-secondary mathematics courses. In the literature, there are four studies which consider mindfulness or contemplative pedagogy in mathematics at the post-secondary level, all which will be discussed in the following section. 2.3 Intersectionality So far, math anxiety and contemplative pedagogy have been investigated separately, with a focus on the post-secondary level. Consider now the intersection of the two – math anxiety and contemplative pedagogy – in the extant literature. So far, we have already discussed one of the commonalities, that of neuroplasticity (Boaler, 2016; Dweck, 2006; 42 Mackenzie, 2015; Shapiro et al., 2011a). Knowing that making mistakes in mathematics and learning from them as well as performing practices from contemplative pedagogy can grow the brain, a search for other intersectionalities led to the ambiguous yet overarching research question: How can contemplative pedagogy support a mandatory post-secondary mathematics course? One study did not explicitly consider contemplative pedagogy in a math course; hence, it did not count as one of the four studies discussed in detail. However, Galla and Wood (2012) determined that the effects of the relationship between math anxiety and math performance could be minimized when students felt in control of their emotions. As emotional regulation is often a result of contemplative pedagogy including mindfulness (Mackenzie, 2015; Ragoonaden, 2017b; Shapiro et al., 2011a), the relationship between math anxiety and math performance should be minimized in consequence can be deduced via transitivity. Though this was not directly investigated by Galla and Wood (2012), it certainly may be a hypothesis which may be compared with our results. Now, onto the four studies incorporating contemplative pedagogy (including mindfulness) at the post-secondary. The first, Brunyé et al. (2013), undertook the most involved investigation. The authors considered four interventions that could assist in the regulation of negative emotions accompanying math anxiety. Brunyé et al. confirmed what Ashcraft and Kirk (2001) previously stated: students with high math anxiety waste cognitive resources on worrying, rather than on the math task at hand, affecting overall performance. This connection between working memory deficiency and high anxiety or stress is well discussed in mindfulness circles (Shapiro et al., 2011a) and in contemplative pedagogy research (Zajonc, 2013). Both 43 interventions have demonstrated a reduction in stress and/or anxiety, thereby increasing working memory and overall performance. The primary focus of the Brunyé et al. (2013) study was to examine possible reductions in math anxiety and their consequential effects on cognitive resources deficits during high-pressure situations, such as test or exam-writing. Prior to writing a timed arithmetic test, post-secondary students participated in one of four short-term interventions – three behavioral and one nutritional – to determine whether interventions requiring small amounts of time and are easily implementable could support students in these situations. Brunyé et al. determined that the most effective intervention was focused breathing, though it did not make a big enough difference to balance the scales between a math anxious person and a non-math anxious one. As speculated by the authors, this may be caused using a short-term intervention instead of a longer-term one (as used by Shapiro et al., 2011a). Brunyé et al. (2013) wanted to determine whether short-term practices from contemplative pedagogy could also have positive effects on math anxiety, since short-term interventions are easier to incorporate both in the course and into the lives of busy students. Hence, prior to writing a timed arithmetic test, post-secondary students participated in three behavioral interventions and one nutritional intervention. The goal was to devise supports for students that required little time commitment and which were easy to implement. The interventions were: 1. A focused breathing exercise (a guided exercise, based on a script from Kabat-Zinn, 1995), 2. An unfocused exercise, where students let their minds wander, 44 3. A ‘worry exercise’, where students were asked a list of anxiety-inducing questions, and 4. The consumption of teas containing different levels of L-theanine. This study was a study across six sessions involving 36 participants. The participants needed to complete a number of self-reported questionnaires: one measuring their psychological state at four different moments during each session, two measuring their mindfulness levels – the Five Facets Mindfulness Questionnaire (Baer, 2008) and the Mindfulness Attention Awareness Scale (MAAS, Brown & Ryan, 2003), one measuring their ability to “maintain information in working memory while also solving simple arithmetic tasks” (Brunyé et al., 2013, p. 3), and one measuring their math anxiety level (the Math Anxiety Rating Scale, MARS, Richardson & Suinn, 1972). In the end, the completion of many questionnaires was required. The authors determined that a significant positive difference existed when highly math-anxious students participated in breathing exercises, particularly focused or guided ones. When compared to low math anxiety participants, highly math-anxious participants scored lower on math tests. Participants who performed the focused breathing exercise right before the test increased their accuracy when compared to those performing the worry exercise and to those doing the unfocused breathing, narrowing the gap with those participants with lower math-anxiety. The purpose of the focused breathing exercise was to decrease anxiety and to distract participants from their negative emotions relating to their math test. Though focused breathing did improve anxiety levels, it did not increase math performance enough to erase the difference between the high and the low anxiety groups, though, perhaps “longer-term mindfulness practices might prove effective in narrowing this gap” (p. 5). Following the 45 evidence surrounding MBSR-style courses, a longer-term, say 8-9-week course may be advantageous overall, but may also begin to instill new frameworks of thinking for the students which involve self-regulation and lower anxiety and stress. Brunyé et al.’s (2013) research provided a foundation for Shayer (2020) and was also used when planning the methodology and the data collection process for this research. Furthermore, the investigation of a longer-term intervention (unlike their short-term ones) was encouraged to determine whether their results could not only be replicated but also if they lasted for longer periods of time. Unlike Brunyé et al. (2013) who evaluated the effectiveness of diverse interventions, Bellinger et al. (2015) studied the relationship between mindfulness levels and working memory changes during high-stake tests. This second study’s primary focus was on determining mindfulness levels via questionnaire (MAAS) and then evaluating how students performed in different situations – homework, tests, and in tests in the lab. All their conclusions are based on this data. No interventions or practices from contemplative pedagogy were included to modify math anxiety. Bellinger et al. (2015) undertook two different investigations, evaluating the effects of mindfulness on working memory resources as well as math performance. As mentioned earlier, Shapiro et al. (2011a) and Ashcraft and Kirk (2001) discussed the drop in working memory when anxious or stressed. According to Bellinger et al. (2015), “individuals who received mindfulness training, and extensively practiced, did not show this drop in working memory” (p. 124). This conclusion was based on the fact that students who performed higher on the MAAS had received mindfulness training. They also elaborated that mindfulness’ benefits may also extend 46 to stressful academic environments. No change was seen during low stress (homework) situations. Bellinger et al. (2015) determined that students with higher mindfulness levels had lower math anxiety associated with stressful testing conditions and greater accuracy when solving math problems requiring working memory. The more mindful the student, the more likely the student may succeed in stressful testing situations. This study is one of the few investigations performed at the post-secondary level involving mindfulness. Though it does not include any interventions, the evidence is still of value – that there is a relationship between mindfulness levels and working memory and consequentially, performance. Hence, if mindfulness levels can be improved by including an intervention, then perhaps these positive results can be reproduced. As mindfulness is included in contemplative pedagogy, the research performed by Bellinger et al. (2015) will continue to inform my research when monitoring the changes in mindfulness levels of the students as well as any differences in performance. A third study in mathematics at the post-secondary level was performed by Samuel and Warner (2019) who evaluated the extant literature and summarized findings related to both fixed and growth mindsets. Their analysis involved relationships between low self-esteem and fixed mindsets, along with the fear of making mistakes and of failure. Growth mindsets were associated with academic resilience, the ability to master goals, and to cultivating skills for success. Self-esteem was inversely related to math anxiety and greater self-efficacy was demonstrated by those who had growth mindsets. The authors studied students taking a first-year mandatory statistics course – some taking the course over one semester and others across two, the latter offering supplementary 47 support and remediation. Participants were chosen from both groups then divided into an intervention group and a control group. The intervention group involved both mindfulness techniques (such as guided breathing and positive affirmations) and growth mindset videos. Discussions followed the videos, where fixed mindset comments were reiterated as growth mindset ones supporting the students in their understanding. The instructor also practiced the guided breathing and the positive affirmations before class to prepare. At the beginning of every class, a deep breathing exercise was undertaken and positive affirmations relating to growth mindsets and acceptance were spoken. On exam days, the deep breathing continued, but the positive affirmations changed somewhat to simply boost confidence. The control group received regular instruction only. At the beginning of the first semester, both groups completed two surveys, one measuring math anxiety (the MARS, Alexander & Martray, 1989) and one examining math self-efficacy (the MSES, Betz & Hackett, 1993). Qualitative data was collected via a focus group containing four participants from the intervention group. From the semi-structured interviews, it was ascertained that students appreciated the calm before starting class as well as having a regular routine. Quantitative analysis determined that students involved in the intervention group demonstrated lower test anxiety than those of the control group. Thematic analysis of the qualitative data identified greater self-reported confidence in math by those undergoing the intervention. The work effectuated by Samuel and Warner (2019) provided a foundation for the fostering of growth mindsets in addressing math anxiety at the post-secondary level and for me, to encourage self-reflection on one’s mathematical positionality. After all, an awareness of 48 whether one has a fixed or growth mindset can greatly affect one’s learning (Boaler, 2016; Dweck, 2006), be it in K-12 or post-secondary courses. Moreover, Samuel and Warner (2019) included a longer-term intervention, with a practice, video, or positive affirmation occurring at the beginning of every class. Their methods of data collection and analysis did, however, resemble some of the quantitative data collection and analysis which occurred in this research (cf., Methodology). It is important to point out, as they did, that their research was performed in a statistics course, which is known to have less anxiety than mathematics ones. The final study investigating math anxiety and contemplative practices at the post-secondary is Shayer (2020). There, a case study was used, which incorporated contemplative practices into the lives of the participants, outside of mandatory post-secondary mathematics courses. Practices were chosen and included based on the framework suggested by Burack (2014) and the cautions of Berila (2014). The term-long study was used to investigate the question – Do contemplative practices reduce math anxiety at the post-secondary level? The examination of the research question via data analysis provided the foundation for this doctoral work encouraging deeper investigations for more valuable and longer lasting supports. Based on Shayer (2020), contemplative practices did decrease math anxiety in a very local and specific context, with a very small group (n = 6), and with gendered participants (all were female). Shayer (2020) also brought to light different possibilities which could not only improve participation (e.g., inclusion in the course instead of outside), but also attract a wider diversity of participants (e.g., incorporating a greater assortment of contemplative practices). 49 Based on the affirmative answer to Do contemplative practices reduce math anxiety at the post-secondary level? (Shayer, 2020), this research took wider stance and investigated the deeper overarching question How can contemplative pedagogy support a mandatory post-secondary mathematics course? By asking “how”, I do not simply determine whether contemplative pedagogy can help, but truly examine the differing ways that it may support the learning of mathematics. As was presented in the Introduction and will be reiterated over the subsequent chapters, sub-questions emerged from the data analysis to deepen with the overarching research question: Does it [contemplative pedagogy] a. Decrease math anxiety? b. Reduce math avoidance? c. Improve math attitudes? d. Increase math performance? This chapter has detailed relevant literature associated with math anxiety and the incorporation of contemplative pedagogy, especially in post-secondary mathematics courses, along with the intersection of the two. The collection of evidence examined herein informed this research as contemplative pedagogy was included in a mandatory post-secondary mathematics course to investigate the overarching research question and its related sub-questions. 50 Chapter 3: Methodology The previous chapter examined relevant literature associated with math anxiety, contemplative pedagogy, as well as their intersection at the post-secondary level. As shared therein, a detailed investigation of mandatory post-secondary mathematics courses, with a focus on math anxiety, has not been undertaken. In this chapter, specific details relating to how the methodology will be used to inform the overarching research question How can contemplative pedagogy support a mandatory post-secondary mathematics course? along with its related sub-questions will be provided. As a mathematician, an educator, and more recently, a doctoral candidate in education, I have the benefit of seeing and connecting my mathematical perspectives of teacher and learner. Based on these insights and to eliminate the variability due to different instructors, a term-long study was undertaken in fall 2022 across two sections of a first-year, mandatory mathematics course, where the researcher was instructor to both. One section was the control section, where no contemplative pedagogy was incorporated, and the other was the intervention section, where contemplative pedagogy was included into the course content after the first course test. The control and the intervention groups were formed of interested students from those respective sections. The main purpose of delaying the incorporation of contemplative practices was to offer a baseline for the students in the intervention section so that changes, if found, could be investigated in terms of their significance to the implemented intervention. Furthermore, this intentional delay allowed for a comparison between the control and intervention groups at the first test, which will be elaborated upon in Discussion and Analysis. 51 This doctoral study adopted a convergent mixed methods approach, incorporating both quantitative and qualitative tools. A convergent mixed method is “a form of mixed methods design in which the researcher converges or merges quantitative and qualitative data in order to provide a comprehensive analysis of the research problem” (Creswell & Creswell, 2018, p. 15). The “convergent” aspect relates to the timing of the data collection, which will be discussed further in Section 3.6 and in Table 2. This data collection included a Math Origin Story (MOS, cf., Appendix A), two pre- and post-test questionnaires, one measuring mindfulness (the Five Facet Mindfulness Questionnaire, FFMQ, cf., Appendix D) and one measuring math anxiety levels (the math anxiety rating scale, MARS, cf., Appendix C), individual semi-structured interviews (ISSI, cf, Appendix B) , course grades (test 1 and final exam), and researcher field notes (RFN, cf. Appendix E). The collected data was analysed following both quantitative techniques, by using descriptive statistics (Moore & McCabe, 2017) and statistical inference (Judd et al., 2017), and qualitative techniques, specifically grounded theory (Charmaz, 2006; Glaser & Strauss, 1967), to inform the research and investigate the overarching research question: How can contemplative pedagogy support the learning of mathematics at the post-secondary level? along with its related sub-questions. Details of this analysis are found in Results and discussions in Chapter 5. 3.1 Epistemology Given the nature of this research, with its wealth of qualitative data, a social constructivist epistemology was used. Though a positivist approach was theoretically adhered to while working with the quantitative data, since the quantitative data emanated from self- 52 reporting and course grades (the latter a result of learning, studying, and collaborating), and was used to support qualitative results, social constructivism remained at the forefront. According to Creswell and Creswell (2018), social constructivists believe that “individuals develop subjective meanings of their experiences—meanings directed toward certain objects or things… The goal of the research is to rely as much as possible on the participants’ views of the situation being studied” (p. 8). Researchers understand that, in a social constructivist arena, knowledge is formed via the interaction with others and recognise that their backgrounds, beliefs, and values influence the interpretation of the phenomenon. In this context, it made sense to anticipate that meaning would present itself to the participants via their lived experiences, both individually and collectively in the classroom. Having an epistemological stance is then connected to how phenomena will be interpreted and, especially in (doctoral) research, how it will be conveyed. Van Manen (2006) offered insights regarding writing within the context of qualitative methods – to hone interpretive skills, be open minded, and creative, to consider the researcher’s underlying assumptions as well as those of the method, constantly verifying the latter for its applicability. Being aware of these writing challenges along with the importance of well-chosen, concise, and accessible words prepares the researcher to consider not only words, but also to pay attention to the ‘naked now’ – everything which is presenting itself in the moment, such as tone, body language, hesitations, and silences. Notes were taken throughout the fall 2022 term (also known as the Researcher Field Notes, cf. Appendix E) as well as researcher field notes taken during the data analysis (cf. Figures 7 and 8 in Section 4.3) in an effort to convey the ‘naked now’. 53 3.2 Research Design As a mathematician, I understand the desire to use quantitative methods – for generalization, for extension, to test hypotheses, etc. Unfortunately, given the learning (and the challenges of that learning) of mathematics at the post-secondary, mere numbers were not enough to truly understand the problem. In consequence, qualitative methods, which delve deeper and offer insights into the habits of mind, frameworks of thinking, and foundations for the challenges (and successes) were also considered. The research design for this doctoral work used a mixed methods research design, that is, both quantitative and qualitative methods were included. Several different forms of mixed-methods designs exist, of which Creswell and Creswell (2018) detailed three – convergent mixed methods, explanatory sequential mixed methods, and exploratory sequential mixed methods. The two sequential methods (explanatory and exploratory) compartmentalised and performed one type of research first, then the other: explanatory (first quantitative, then qualitative) and exploratory (first qualitative, then quantitative). As neither of these sequential methods were used, no further details will be provided herein. Instead, convergent mixed methods were used, which incorporated both quantitative and qualitative tools. The “convergent” feature of this method considers that “the investigator typically collects both forms of data at roughly the same time and then integrates the information in the interpretation of the overall results. Contradictions or incongruent findings are explained or further probed in this design” (Creswell & Creswell, 2018, p. 15). This method encourages the triangulation of data, by discovering the common ground found between each 54 qualitative and quantitative data, at different stages, instead of isolating one type from another, at one moment from another. For example, this structure fits well as the Math Origin Story (cf. Appendix A), as part of qualitative data was collected at about the same time as the quantitative self-reported pre-tests – the MARS (cf. Appendix C) and the FFMQ (cf. Appendix D). Furthermore, participants were invited to participate in individual semi-structured interviews (cf. Appendix B) after the completion of the course which occurred shortly after the post-tests of both the MARS and the FFMQ as well as the final exam. Researcher field notes were taken across the duration of the entire study and included details relating to these timing convergences as well as to any challenges or success that presented themselves. Specific timing details of the data collection are found in Table 3 (Convergent Mixed Method Data Collection, Data Types, And Stages). The convergent mixed method design seemed the best fit to not only answer the overarching research question How can contemplative pedagogy support a mandatory post-secondary mathematics course? but also its related sub-questions, including Does it: a. Decrease math anxiety? b. Reduce math avoidance? c. Improve math attitudes? d. Increase math performance? Though changes in math anxiety and math performance can be quantified, the remainder of the questions require qualitative data. Furthermore, even math anxiety which can be measured quantitatively using the Math Anxiety Rating Scale (MARS), also has physiological 55 responses which can be identified via qualitative data. Therefore, these questions necessitated a mixed method design to be fully addressed. This section discussed the convergent mixed method approach used in the study and which empowered both employed qualitative and the quantitative methods in informing the overarching research question as well as its related sub-questions. Those two methods and their application in this doctoral research will be detailed in Sections 3.3 and 3.4, respectively. 3.3 Qualitative Method – Narrative Inquiry Despite the official statement of using convergent mixed methods as a methodology in this research, the qualitative data (and hence its method) spearheaded this study and took on a narrative form – from the researcher field notes to the Math Origin Stories, to the individual semi-structured interviews (cf. Section 3.6 for data collection details). Furthermore, most of the quantitative data (outside of the participants’ course grades) was self-reported data, which may be considered subjective and narrative in some sense (based on how someone sees themselves in each of the given situations). Hence, the main method used was narrative inquiry. The remaining quantitative data – the course grades – was used for comparison, correlation, and validation purposes with extant literature, and played a supporting role in this research. “Narrative inquiry in the social sciences is a form of empirical narrative in which empirical data is central to the work” (Clandinin & Connelly, 1990, p. 5). In fact, Craig et al. (2018) added that it should not only be relegated to the social sciences and “questioned whether it may be the ‘secret sauce’ of which STEM education has not fully taken advantage” (p. 5), adding that there may be “symbiotic relationship” between narrative and scientific inquiry (p. 6). According to Clandinin and Connelly (1990), field notes, autobiographies/stories, 56 and interview transcripts are all forms of narrative inquiry data. In the end, “the sense of the whole is built from a rich data source with a focus on the concrete particularities of life that create powerful narrative tellings” (Clandinin & Connelly, 1990, p. 5). Narrative inquiry is both phenomenon and method (Clandinin & Connelly, 1990; Ragoonaden et al., 2020). Narrative inquiry guided the thought processes while considering the qualitative collection of data emanating from the Math Origin Stories, the researcher field notes, and the individual semi-structured interviews (cf. Section 3.6 for data collection details). According to Clandinin and Connelly (2000), “narrative inquiry is aimed at understanding and making meaning of experience” (p. 80). In short, when reading and rereading the Math Origin Stories and the individual semi-structured interviews then applying Grounded Theory (cf. Section 3.7.2), meaning-making occurred, based on the participants’ lived experiences (Huber et al., 2013). Re-storying involved representing what they had conveyed in as many of their words as possible so as best to have their true voices heard. Stanley and Temple (2008) stated that “lives are always ‘read’ and interpreted through the stories told, and untold, about them” (p. 278). Narratives and stories are very powerful, meaningful pieces which can offer a lot of insight, possibilities, and future directions. I felt so very honoured to be entrusted with the participants’ stories. Hence, throughout this dissertation, I wanted to not only acknowledge that power, meaning, insight, and vulnerability, but promise that their words, voices, and strength were not wasted. Merriam and Tisdell (2016) reminded the researcher to judge “the value of a data source” by asking “whether it contains information or insights relevant to the research question and whether it can be acquired in a reasonably practical yet systematic many. If these two 57 questions can be answered in the affirmative, there is no reason not to use a particular source of data. Documents or artifacts have been underused in qualitative research, however” (p. 180). Of course, the Math Origin Story included options relating to how it was shared or presented. Before this data was collected, preparations were made for all possibilities as all forms of narratives were respected, considered, and then pondered. If the stories were completed in an oral or video fashion, then transcripts would be provided before beginning the data analysis. If the story was shared in a non-typewritten way, further insights could be provided, from vocal inflections to facial expressions, to the fear or joy present. All these options were offered and were considered before moving to the overall data analysis. Given the resulting responsibility emanating from Narrative Inquiry, Section 3.6.2 elaborates upon the careful collection of qualitative data. Section 3.7.2. further explores the qualitative component of the research and details its analysis via grounded theory, in particular, the latter’s exploratory and recursive nature and how it supports narrative inquiry (Burck, 2005). The next section discusses the quantitative methods used in this research and will refer to its own data collection and forms of analysis. 3.4 Quantitative Methods The previous section discussed the use of qualitative methods supported by researcher field notes, the Math Origin Stories, and the individual semi-structured interviews. Since this research employed a (convergent) mixed-methods approach, then quantitative methods also needed to be considered. 58 Quantitative methods use quantities or numbers to represent observations or phenomena. In this case, numbers emanated from two self-reported questionnaires – the FFMQ (see Appendix D) and the MARS (see Appendix C) – as well as the course grades (test 1 and final exam). Both questionnaires were used twice – at the beginning and again near the end of the course; hence, these were labelled “pre-“ and “post-“ tests. The main purpose of collecting these numbers was to determine whether there were noticeable or significant (in statistics language) differences between the pre- and post-test values associated with the inclusion of contemplative pedagogy in the course. The first set of questionnaires, the FFMQ (cf. Appendix D), measured the level of mindfulness in relation to five facets: observing, describing, acting with awareness, nonjudging of inner experience, and nonreactivity to inner experience (Baer et al., 2008) and was rated on a 5-point scale (where 1 = almost never to 5 = almost always). As contemplative practices include mindfulness practices (cf. Figure 1 and the Tree of Contemplative Practices), the use of the scale was deemed both appropriate and applicable. The second set of self-reported questionnaires, the MARS (cf. Appendix C), measured math anxiety levels and used the modified 25-item brief version (Alexander & Martray, 1989). Richardson and Suinn (1972) created a MARS with 98 points, exploring three main components of math anxiety, that is, math course anxiety (MCA), math test anxiety (MTA), and number anxiety (NA). However, the length of this scale was found to be a barrier in terms of time constraints. Alexander and Martray (1989) devised a new and shorter scale with only 25 items, but which still accessed the three components of math anxiety. Before these two scales, the 59 discussion of math anxiety surrounded only two different components of math anxiety – math test anxiety and math found in everyday life anxiety (Cipora et al. 2015). Outside these two sets of self-reported questionnaires, the course grades (test 1 and final exam) were also considered. The course grades also allowed the investigation of relationships between math anxiety level and mathematics performance as well as mindfulness levels and mathematics performance, two findings existing in the extant literature. No contemplative practices were incorporated into the course prior to the first course test, to remove possible confounding of variables and to obtain a baseline for the intervention group. Further details associated with investigations and related hypotheses of the quantitative data analysis are discussed in Section 3.7.1. Before elaborating upon the data collection and its analysis, consider the population from which the purposeful and specific sample was formed. 3.5 Population This study examined a population contained within a college found in the Okanagan Valley, in Western Canada, offering various trades, health, and technology programs as well as Associate diplomas in Arts, Business, and Science which may be converted into the first two years of a bachelor’s degree at an accepting university. Students attending this college have diverse backgrounds, ranging from those entering immediately after high school graduation, to those with full-time jobs and/or families, to those who have been outside of the school system for two or more years (called mature students), from domestic, to international students, to Indigenous students, to middle-upper class families capable of affording post-secondary education, to students on scholarships or government-assisted funding programs. It is 60 important to note that class sizes at this college range from 20-40 students, which are much smaller than most university classrooms. Table 2 demonstrates the diversity of participants that were involved in the research and specifies the gender of the participant and whether they were an international, mature, or racialized student, or whether they had a physical or learning disability. As is demonstrated in Table 2, the proportion of mature and racialized participants as well as their gender identity is quite similar. What is noticeable is the difference between international students (8% between the control and intervention groups) as well as those with physical or learning disabilities (14%). Table 2 Participant Demographics for Both Control and Intervention Groups Section Gender International Mature Student Racialized Physical or Learning Disability Control (n = 36) 20/36 Female (56%) 16/36 Male (44%) 18/36 (50%) 5/36 (14%) 20/36 (56%) 4/36 (11%) Intervention (n = 32) 17/32 Female (53%) 14/32 Male (44%) 1/32 Non-Binary (3%) 15/32 (42%) 4/32 (13%) 18/32 (56%) 8/32 (25%) Now that the population has been detailed, the quantitative and qualitative data collection forming the basis of the convergent mixed method will be described in the subsequent section. 61 3.6 Data Collection As written earlier, a study using convergent mixed methods (Creswell & Creswell, 2018), involving both quantitative and qualitative tools was used across one academic term in a first-year mandatory mathematics course, involving two sections of the same course. One section was the “control”, with no intervention, while the other section was the “intervention” section, where contemplative pedagogy was incorporated into the course after the first test. Before this implementation, the Math Origin Story assignment, the first course test along with the FFMQ and the MARS pre-tests were completed to determine a baseline. Having all this data collected at approximately the same time aligns with the use of the convergent mixed method, as timing matters in such a design (Creswell & Creswell, 2018). The second set of data collection occurred later in the course, after the incorporation of contemplative pedagogy in the intervention section. It included the final exam, the FFMQ and the MARS post-tests, followed shortly by the individual semi-structured interviews. Researcher field notes (cf. Appendix E) were taken throughout the fall 2022 term. Table 3 presents the overall timing of the data collection, with a focus on the convergence of two times or “stages” (pre- and post-tests) to adhere to the convergent mixed method approach. Further details regarding the data collection as well as its analysis are found in the subsequent sections. 62 Table 3 Convergent Mixed Method Data Collection, Data Types, and Stages Stage 1 (Pre-Test) Stage 2 (Post-Test) Quantitative • Test 1 Grade • FFMQ Pre-Test • MARS Pre-Test Quantitative • Final Exam Grade • FFMQ Post-Test • MARS Post-Test Qualitative Math Origin Stories Qualitative Individual Semi-Structured Interviews Researcher Field Notes Note. The Researcher Field Notes formed part of the qualitative dataset but was taken throughout the entire term and data analysis process. As depicted in Table 3, this study included a range of qualitative and quantitative data: 1. The Math Origin Stories (MOS, Appendix A), 2. The pre- and post-Five Facet Mindfulness Questionnaire (FFMQ, Baer et al., 2008, Appendix D), 3. The pre- and post-Math Anxiety Rating Scale (MARS, Alexander & Martray, 1989, Appendix C), 4. The pre- and post-math course grades (test 1 and the final exam), 5. The Individual Semi-Structured Interviews (ISSI, Appendix B), 6. The Researcher Field Notes (RFN, Appendix E and Figures 7 & 8 in Section 4.3). 63 Further details regarding each component of the data collection will be provided in Sections 3.6.1 (Quantitative) and 3.6.2 (Qualitative). Information surrounding the analysis of this data will be presented in Section 3.7. 3.6.1 Quantitative Data Collection As listed above, the quantitative dataset was comprised of two sets of self-reported questionnaires as well as course grades. The latter was used to measure math performance. The first set of questionnaires, the FFMQ (see Appendix D), is a measure of the level of mindfulness in relation to five facets: observing, describing, acting with awareness, nonjudging of inner experience, and nonreactivity to inner experience (Baer et al., 2008). Items are rated on a 5-point scale (where 1 = almost never to 5 = almost always). Though a number of self-reported mindfulness scales exist, the FFMQ is the second most commonly used scale, after the Mindfulness Attention Awareness Scale (MAAS, 5,054 citations to 2,660, based on Google Scholar, October 20, 2016, cited in Van Dam et al., 2017). The FFMQ was chosen as if offers greater depth into the different facets as the MAAS only measures a single factor based on awareness and attention. In Shayer (2020), the inclusion of all five facets proved valuable. In particular, with an awareness of math self-esteem, the facet of “non-judgement” could play an important role. The FFMQ pre-test was completed online via Qualtrics during the first week of the course, while the FFMQ post-test was completed during the final week of classes. Though the questionnaires are identical, the FFMQ pre- and post-tests allowed for comparison between the five facets of mindfulness as well as the “overall” mindfulness level and demonstrated the 64 changes that occurred over the course of the term, for both the control and intervention groups. The second set of self-reported questionnaires (the MARS pre- and post-tests) measured math anxiety levels. The MARS used in this study was the modified 25-item brief version (Alexander & Martray, 1989, cf. Appendix C), instead of the 98-item MARS by Richardson and Suinn (1972). This brief MARS version still tests the three main factors of math anxiety mentioned above, MCA, MTA, and NA. These questionnaires were also offered online via Qualtrics, where the MARS pre-test was available during the first week of classes and its counterpart, the MARS post-test, was given during the last week of classes. Like the FFMQ, these MARS pre- and post-tests allowed for comparison, to investigate change between math anxiety levels from the beginning of the course and those at the end, for both the control and intervention groups. Since mathematics success is intricately linked with improving one’s life chances (Mighton, 2020; Gutiérrez, 2008), a quantitative analysis of math performance was also considered to not simply rely on self-reported tests. Mathematics performance was measured using the test 1 and the final exam grades. To exclude possible lurking variables, no contemplative pedagogy was introduced before the first test to provide a baseline of student math performance without intervention. In totality, the quantitative data provided a numerical foundation with which to correlate, triangulate, and strengthen the qualitative data. 65 3.6.2 Qualitative Data Collection Focusing in on the qualitative component of the mixed methods approach, Merriam and Tisdell (2016) specified that “qualitative researchers are interesting in understanding how people interpret their experiences, how they construct their worlds, and what meaning they attribute to their experiences” (p. 6). Hence, unlike quantitative methods which analyse experiments, compare with pre-determined hypotheses, and deductively generalize to predict future outcomes/results, qualitative methods aim to understand the experiences of others and inductively make meaning of a phenomenon or of phenomena. In this context, understanding the learning of mathematics at the post-secondary level and paying attention to see if contemplative pedagogy can facilitate that learning, by reducing anxiety, improving attitude, and/or producing better results through increased working memory or focus was investigated. As included in Table 3 (Convergent Mixed Method Data Collection, Data Types, And Stages), the qualitative data set was comprised of researcher field notes (cf. Appendix E), the Math Origin Stories (cf. Appendix A), and the individual semi-structured interviews (cf. Appendix B). Qualitative data offered thick descriptions and delved deeper than quantitative data, offering opportunities to hear the stories and lived experiences of the participants. The following subsections detail each of these qualitative components forming the qualitative dataset. 3.6.2.1 Researcher Field Notes. Since contemplative pedagogy was incorporated into the course after the first test, researcher observations played an important role within the context of the qualitative data collection. In essence, while introducing practices into the classroom, I found myself in that ‘naked now’, where what happens can be just as important as what does not. Moreover, field notes can be considered phenomenological observations, where 66 phenomena may be difficult to translate into words. “The problem of writing is that one must bring into presence this phenomenon that can be represented only in words—and yet escapes all representation (Van Manen, 2006, p. 718). Clandinin and Connelly (2000) also wrote about researcher field notes, not only how to write them, but also how to translate them into research writing. Clandinin and Connelly (2000) considered how researcher field notes can “assist memory to fill in the richness, nuance, and intricacy of the lived stories and the landscape” (p. 80), adding extra weight and meaning to taking good field notes. The researcher’s positionality also plays a role when taking field notes, as we, as human beings, rotate to things with which we are familiar or for which we are looking. Having seen students struggle in the past, a balanced approach focused on noting overall attitudes in both sections as well as the incorporation dates and practices was taken. As will be elaborated upon in Section 3.9 (Ethics), since I did not know who the participants were during the intervention, careful thought was put into not identifying individual reactions, comments, or statements. As observers, researchers cannot be fully removed from the experience; in fact, they are part of the experience. They may detail another’s experience; however, they also write what they are thinking and feeling while doing so, offering a personal connection to the experience. Furthermore, field notes can help us retell the story. Perhaps not all details are written down; hence, the writings can support the reliving of the observational experience, to reconstruct the past as best we can (Clandinin & Connelly, 2000). Clandinin and Connelly (2000) further elaborated upon the dualism of turning inward versus observing outward when performing research and writing field texts, the prospective challenges that this dichotomy may bring, and 67 also the richness of notes and of the experience. Taking good field notes requires paying attention to what is going on around you, inside you, but to also recognize what is not happening, the silences or spaces which may present themselves. Clandinin and Connelly (2000) wrote that field notes are “the ongoing, daily notes, full of the details and moment of our inquiry lives in the field, are the text out of which we can tell stories of our story of experience (p. 104). Given the options of interpreting the field experience, my preference was to write the field notes, formed of notes for each week, with more information once the incorporation of contemplative pedagogy had begun. These notes were based on both sections, the control and the intervention, and pertained to general classroom behaviours, what was taught as well as my perception of events. These notes were added to outside of class, sometimes dependent upon on student questions, reactions, or emails, when they related to the lived experience. Due to ethics, I was not aware of whom was participating; hence, these observations were my own of the overall course and the classroom dynamic. Finally, Clandinin and Connelly (2000) reminded researchers not to forget the purpose of the field notes nor of the research at every step of the process. The next section will examine the first qualitative data collection from participants, which steered the research toward which practices would be included and at which time. 3.6.2.2 Math Origin Stories. As written at the beginning of this chapter, part of the qualitative data collection involved an assignment where students were invited to share their Math Origin Stories (cf., Appendix A). Though the title was meant to be particularly welcoming to students, 68 it was, in essence, a narrative about one’s mathematical journey, or mathematical autobiography. The purpose of this assignment was to provide information to both the student and myself regarding how they perceive mathematics, what types of challenges/successes (if any) they have had, who had an influence (positive or negative), when these challenges or influences occurred, and why students feel the way they do about mathematics. Such an assignment provided a wealth of data toward understanding the students’ personal journey in learning mathematics. This sharing of their journey/experience was a first step toward meaning-making and deepening understanding. This assignment also provided an opportunity for students to not only share what they thought and felt, but also to have their voices heard. The assessment of this assignment loosely followed the guidelines set in Owen-Smith (2018), that is, not judging the quality of the responses but rather determining whether the prompts were addressed. Students were also warned beforehand that spelling and grammar would not be evaluated. I felt that this last statement was required to satisfy my desire for students to feel comfortable with making mistakes and focusing on the ideas, instead of the format and structure. Furthermore, in a mathematics course, it was clear that expecting high calibre writing skills could further exacerbate any anxiety. An initial thematic analysis of the Math Origin Stories was performed to determine rudimentary themes associated with mathematical challenges encountered by students throughout their mathematical journey. The thematic analysis entailed reading through the Math Origin Stories several times and searching for common statements or experiences related to challenges faced by the students. This initial analysis was the beginning of the iterative process that is Grounded Theory (see Section 3.7.2 for further details) and rooted the 69 researcher in the context of that particular course. Furthermore, it offered the opportunity to reflect on the overarching research question How can contemplative pedagogy support a mandatory post-secondary mathematics course?, and to contemplate related sub-questions. As it was important to be purposeful with this specific group and to be as supportive as possible with their particular challenges, various evidence-based practices from contemplative pedagogy were chosen for incorporation. As discussed in Section 2.2.3 (Course Implementation), the Stress Management and Resiliency Curriculum (Stress Management and Resiliency Curriculum Facilitator Manual, 2015) as well as Barbezat and Bush (2014) were the main sources for the practices. After the end of the course and after the last day for grade appeal, a more in-depth analysis was performed with the Math Origin Stories, to hear the full voice of the participants and the sentiments attached to them. Further details surrounding this secondary iteration of analysis are provided in Section 3.7.2.1. 3.6.2.3 Individual Semi-Structured Interviews. The qualitative dataset of this doctoral research was rich as it not only included researcher field notes (cf. Appendix E) and Math Origin Stories (cf. Appendix A), but it also involved individual semi-structured interviews (ISSI, cf. Appendix B) that were performed early the winter 2023 term, after the last date of grade appeal. Zoom interviews were undertaken with interested participants from each group (control and intervention) to offer flexibility, but also to simplify the transcript-creation process. Due to the online environment, interviewing properly and efficiently was of even greater importance than 70 interviews done in person to obtain valuable and valid data as well as to make the participants feel as comfortable as possible. The individual semi-structured interviews produced thick qualitative data, as the interviews were well done, in part due to following both Stake’s (2010) and Patton’s (2002) hints toward efficient and effective interviewing. Stake (2010) stated that “if there is expectation that one or several interviewees will produce quotable materials, then the interview should be tailored to what is special about that person” (p. 95), encouraging the researcher to ask open-ended questions to allow interviewees to relate their own stories or perspectives. Stake (2010) further encouraged the interviewer to practice beforehand, and especially be prepared if using recording devices or forms of technology, for they do not always work as well as planned. Since not everything may be observed externally by another, Patton (2002) wrote that “we cannot observe feelings, thoughts, and intentions. We cannot observe behaviors that took place at some previous point in time. We cannot observe situations that preclude the presence of an observer. We cannot observe how people have organized the world and the meanings they attach to what goes on in the world. We have to ask people questions about those things” (p. 341). Here, many things related to this research that may not be seen fully by the outsider, for example, how mindful someone is or what influenced their positive/negative/neutral attitude about mathematics. In consequence, interviews were used to truly know AND understand. Patton (2002) also encouraged the use of an interview guide, where certain questions are predetermined and have some structure, instead of being completely open. This way, some 71 common ground may be found, making coding and triangulation more effective. Saldaña and Omasta (2017) encouraged the use of semi-structured interviews as they allow “significant latitude to adjust course as needed” (p. 82). However, before jumping into more open-ended questions, Patton (2002) suggested beginning with descriptive questions before moving onto ones pertaining to feelings or opinions, to allow time to the participant to settle in and to feel more comfortable with the interviewer. Asking clear and concise questions, avoiding jargon, and using common language to the subject area all lead to better responses (Patton, 2002; Stake, 2010). Saldaña and Omasta (2017) also encouraged interviewers to pay attention not only to the interviewee’s responses, but also to their body language and to the tone and timber of their voice. In the end, we interview to hear someone’s story, to enter their world, to see life through their lens. This opportunity is a gift that each researcher/interviewer should cherish, and of which should be respectful. By waiting until after grade appeal, I was able to perform the interviews myself. By this point in time, I knew the students well enough, from performance to comfort, to future plans. Knowing this information helped me to recognize more of the nonverbal cues and allowed me to give them space to reflect on questions. 3.7 Data Analysis The culmination of all the data described in Sections 3.6.1 and 3.6.2 resulted in rich and thick descriptions, with a variety of numerical data offering great opportunities to address the overarching research question: How can contemplative pedagogy support a mandatory post-secondary mathematics course? and its related sub-questions. Empowered by a convergent 72 mixed method design, Sections 3.7.1 and 3.7.2 will discuss how both the quantitative and qualitative datasets were analysed, so that their conclusions were interwoven. 3.7.1 Quantitative Data Analysis As a mathematician who has taught a variety of statistics courses for over twenty years, I know that quantitative data provides a certain level of comfort to researchers as they may enter numbers into software and obtain binary (accept or reject) results. This subsection considers the range of data analysis performed on the quantitative dataset. Both descriptive and inferential statistics were used in this research. First, descriptive statistics, such as the average and the standard deviation, of the pre- and post-tests (FFMQ, MARS) were computed. The purpose of computing such statistics was to develop intuition for the data, but to also examine the facets of the FFMQ and the components of the MARS individually. Inferential statistics were also undertaken. Inferential statistics empower the researcher to make predictions (or inferences) about the population based on the sample or to test hypotheses by comparing the sample and the population. This form of statistics offers the opportunity to generalize and to compare with extant results. Testing of relevant hypotheses was of primary interest during the data analysis. The relevant hypotheses are called the null hypothesis, denoted H0, (read “H-naught”) – what we see currently/the baseline – and the alternative (or research) hypothesis, denoted HA, (read “H-A") – what the researcher believes exists or is occurring. In theory, the researcher hopes to “reject” the null hypothesis in favour of the alternative/research hypothesis. The determination of accepting or rejecting a null hypothesis is based in the concept of statistical inference and p- 73 values (and significance). Effect sizes are also be presented along with p-values, as both play an important role in “understand(ing) the full impact of (the) work” (Sullivan & Feinn, 2012, p. 282). Where p-values are connected to chance and can inform the researcher whether an effect size exists, the effect size is calculated to determine the magnitude of the effect (Sullivan & Feinn, 2012). Sullivan and Feinn (2012) state that reporting p-values and effect sizes together is essential. “Unlike significances tests, effect size is independent of sample size” (p. 280). As written earlier, the overarching research question How can contemplative pedagogy support a mandatory post-secondary mathematics course? has related sub-questions: Does it [contemplative pedagogy]: a. Decrease math anxiety? b. Reduce math avoidance? c. Improve math attitudes? d. Increase math performance? Though the overarching research question cannot be expressed in terms of quantitative hypotheses, as it does not have a binary (or yes/no) response, the sub-questions can. Consider the quantitative rewriting of the sub-questions as follows: H0 (null hypothesis): Contemplative pedagogy has either NO EFFECT or DOES NOT decrease math anxiety, reduce math avoidance, improve math attitudes, or increase math performance. 74 HA (alternative hypothesis): Contemplative pedagogy DOES decrease math anxiety, reduce math avoidance, improve math attitudes, or increase math performance. To investigate the null hypothesis, the quantitative data (the FFMQ & MARS pre- and post-tests as well as the pre- and post-math course grades) underwent the following inferential statistics tests (Judd et al., 2017; Moore et al., 2016): 1. Independent two sample t-tests5 to compare the FFMQ, MARS, and the course grade averages for two distinct moments in time – Stage 1 (pre-test) and Stage 2 (post-test) – for the control and intervention groups, 2. Paired t-tests to determine the significance of the difference between the pre-test (Stage 1) and post-test (Stage 2), for the FFMQ and the MARS questionnaires as well as the course grades, for the control and intervention groups separately, 3. Multiple analyses of covariance (ANCOVA) to investigate of the effect of the intervention on the two post-tests (FFMQ and MARS) as well as the final exam, while holding the pre-test values constant. (The latter values were held constant since no contemplative pedagogy had been incorporated at Stage 1 (pre-test), therefore, no differentiation was made between the groups.) 4. Correlation and linear regression analysis to explore whether values are correlated and/or have linear relationships with one another. 5 A separate independent two sample t-test was also performed (see Appendix G) to compare the pre-test number anxiety level for regular students and mature students to compare with extant results (Hembree, 1990; Namkung et al. 2019). 75 In this research, whether the average difference between the control and intervention groups (for both the FFMQ and the MARS) was significant is not the sole focus. Instead, tests to determine whether the math anxiety level decreased for the intervention group, when compared to the control were used. Moreover, whether the mindfulness levels increased for the intervention group versus the control group was investigated. To support this line of investigation analyses of covariance (ANCOVA) were undertaken. Hypotheses relating to ANCOVA are: H0 (null hypothesis): There is no post-test difference in the mindfulness/math anxiety/test grades between the control and the intervention groups, while controlling for the pre-test values HA (alternative hypothesis): There IS a post-test difference in the mindfulness/math anxiety/test grades between the control and the intervention groups, while controlling for the pre-test values If the null hypothesis is rejected, then post-hoc tests are used to determine the direction of the difference (i.e., increase or decrease). Chapter 4 shares the results of these tests, with extra details found in Appendix G for the more complicated calculations. Further quantitative analyses were performed to explore correlations between variables. These analyses are also connected to hypothesis tests, with hypotheses as follows: 76 H0 (null hypothesis): There is not a significant correlation between the variables. HA (alternative hypothesis): There IS a significant correction between the variables. In this context, statistical analyses examined the correlation between the following variables: 1. Mindfulness and math performance (Bellinger et al., 2015; Galla & Wood, 2012), 2. Math anxiety and math performance (Hembree, 1990; Ma, 1999; Namkung et al., 2019), and 3. Mindfulness levels and math anxiety (relating to working memory, Ashcraft, 2002, Bellinger et al., 2015). As written previously, the quantitative dataset was but a mere snapshot into our true investigations. The depth of understanding for both the type of practice to be included as well as the real effect on mandatory post-secondary mathematics courses, including math anxiety, will only arise via the qualitative data analysis which will be detailed in the next section. 3.7.2 Qualitative Data Analysis – Grounded Theory The previous section described the numerous statistical tests which were undertaken to examine the quantitative data. However, as previously mentioned, the anticipated role of that data was to support (or to question) the results of the qualitative data, which was collected in a number of different forms – the researcher field notes, the Math Origin Story, and the individual semi-structured interviews. Deeper discussions surrounding the roles of the results from both data types are found at the end of Chapter 4 when the overall research question is addressed. 77 As math anxiety is defined as an emotional or physiological response to mathematical stimuli (Dreger & Aitken, 1957; Richardson & Suinn, 1972), qualitative data analysis provided the most meaningful results since emotions and physiology are not mere binary variables (not simply on or off) nor are they easily quantifiable. It is also difficult (impossible?) for an outside observer to accurately measure both mindfulness and math anxiety levels. Since there are differing levels of difficulty when learning mathematic, using tools that can better measure these (subjective) graduated differences and investigate how individuals responded in consequence was of utmost importance. This research is based in the belief that the best tools to truly understand the lived experiences of the participants is through their narratives. To analyse these narratives as well as the researcher’s narrative (researcher field notes), grounded theory was used. When considering how to analyse the collected qualitative data, Glaser and Strauss’s (1967) seminal text, The Discovery of Grounded Theory, presented a solid option, especially given the iterative approach of the data collection. Glaser and Strauss (1967) introduced researchers to the “discovery of theory from the data” (p. 1) and hence, Grounded Theory. The multifaceted realizations of grounded theory were debated in Walsh et al. (2015) – is it a paradigm, a methodology, a coding procedure? The result, according to the authors, was that grounded theory is a bit of everything. In fact, the authors in Walsh et al. (2015), included Glaser, also discussed the (disappointingly) narrow use of grounded theory, applied primarily to qualitative data since the 1980s (Bryant & Charmaz, 2007). Walsh et al. (2015) added that grounded theory applies to all data types – quantitative, qualitative, or mixed – where the main goal continues to be theory-building. As a mathematician who sees mathematics broken down 78 primarily into three components – language, pattern recognition, and identifying relationships – Glaser’s declaration that “Grounded Theory is simply the discovery of emerging patterns in data” (Walsh et al., 2015, p. 593) certainly hits home. As a nod to Glaser, his concept of “emergence” was adhered to throughout the data analysis and discussion. After all, the overarching research question, How can contemplative pedagogy support a mandatory post-secondary mathematics course?, allowed room for flexibility, growth, and emergence. “Grounded Theory is just a set of steps that take you from walking in the data knowing nothing to emerging with a conceptual theory of knowing how the core variable is constantly resolved” (Walsh et al., 2015, p. 594). Creswell and Creswell (2018) defined grounded theory as “a design of inquiry from sociology in which the researcher derives a general, abstract theory of a process, action, or interaction grounded in the views of participants. This process involves using multiple stages of data collection and the refinement and interrelationship of categories of information” (p. 13). Glaser and Strauss (1967) argued to use data as a starting point for theory generation instead of the previously used methods of logical deduction done beforehand, in particular, using deductive reasoning with quantitative data. Glaser and Strauss (1967) further considered the verification versus the generation of theories and discussed the conflict between these two theory-generating techniques. Glaser (1992) laid a firm foundation by letting the research question emerge from the data. He suggested to researchers to choose a more general area instead of a specific problem, to then allow the question to materialize from the collected data. Glaser supported a positivistic stance and encouraged researchers to start analyzing data from 79 day one. With the initial analysis of the Math Origin Stories, this form of grounded theory, labelled "Classic GT", certainly did have an influence. Corbin and Strauss (1990) introduced and extensively detailed their version of grounded theory, called “Straussian GT”, relating that change should be built into the method and that neither a deterministic nor non-deterministic stance should be taken (but something in between). As this form was not used, further details are not provided herein. A third type of grounded theory and the most pertinent to this research is “Constructivist GT”, created by Charmaz, a graduate student of Strauss. Charmaz’ (2006) approach differs from both Glaser’s Classical GT and the Straussian GT in part due to the chosen epistemology. The previous two approaches used positivism, while Charmaz (2006) incorporated constructivism, offering some flexibility in the coding. Charmaz’ approach incorporates ambiguity, where the researcher “becomes receptive to creating emergent categories and strategies” (p. 168). The incorporation of ambiguity in research resembles introducing ambiguity in the classroom as a form of mindfulness and hence contemplative pedagogy in Ritchart and Perkins (2000) “looking closely, exploring possibilities and perspectives, and introducing ambiguity” (p. 31), further compounding that this epistemology and approach aligned best with the research problem and question at hand. Finally, as narrative inquiry is rooted in social constructivism, this version of GT aligned particularly well with the qualitative method. The framework of Constructivist GT applied to this mixed method design was as follows: 1. At the onset of research, only decisions relating to initial data collection were made (based on Glaser & Strauss, 1967), the problem was left somewhat open, and the 80 question was not too specific. In this research, as will be presented in Results, an initial analysis of Math Origin Stories was performed to determine appropriate practices to include. Furthermore, the somewhat vague research question How can contemplative pedagogy support a mandatory post-secondary mathematics course? offered numerous possibilities in terms of direction and interpretation. 2. The initial dataset was coded to determine further lines of investigation and evident gaps, which altered the original direction of the project. Charmaz’ (2006) coding procedure has two steps, like Classic GT, but is more flexible and subjective than the two other forms of Grounded Theory (Kenny & Fourie, 2015). Charmaz’ coding process involves two-stages: a. Initial/Open Coding: Charmaz utilizes the questions from Glaser and Holton (2007) to begin “What is the main concern being faced by the participants? and “What accounts for the continual resolving of this concern?” (para. 48). When coding the responses, Charmaz (2006) considers actions and looks for hints of theory, instead of searching for themes. She encourages the use of “gerunds, that is, noun forms of verbs such as revealing, defining, feeling, or wanting, helps to define what is happening in a fragment” (Charmaz, 2006, p. 164), and the participants’ own language for codes. b. Re-Focused coding: Notable or repetitive codes were identified and transformed into theoretical categories. The latter form the foundation of theoretical sampling, saturation (see step 3), and memo writing (see step 5). 81 3. Further data was then collected at this point, in the prospective new direction (called, “theoretical sampling”), continuing until data saturation occurs, that is receiving enough data with no new directions. 4. The constant comparison method was used, including three parts: comparing codes with codes; comparing codes with new or emerging categories; and comparing categories with categories, where codes were data labels which were then classified into categories. Constructivist GT also compared with the extant literature at every step, including a complete literature review, after the data analysis was complete. This delay of the literature review allowed me, the researcher, to fully immerse myself in the extant literature based on where the research ended up instead of where it may have been, before the initial data collection, offering a stronger view of gaps and possible found answers. 5. Concurrent with the constant comparison method was the technique of memo writing, where the researcher writes reflections on the data and the decision making/classification/comparison process. This technique is analogous to field notes which, in this case, involves reactions and responses to the data, things that are noticed and are notable. According to Charmaz (2006), memo writing can help to identify “gaps in the data”, assist to gain insights and see connections as well as constructing the grounded theory. As written at the beginning of this section, an important theme in grounded theory is to begin coding early to help determine directions. Almost as consequential as early coding is the loosening of parameters, 82 instead of having very strict boundaries of where your research may go. In other words, let the data drive the direction and evolve into the theory. Charmaz further emphasized being true to the participants’ experiences, something that Glaser (2007) posited removed the ability to conceptualize an accurate grounded theory. Glaser further disagreed with Charmaz’ use of constructivism, stating that it “promotes narration to the extent that is ‘neglecting the fundamental properties of abstraction analysis’” (Glaser, 2002, para. 19, cited in Kenny & Fourie, 2015, p. 1280). Furthermore, the more interpretivist and subjective interview techniques required by Charmaz is an “unwarranted intrusion of the researcher” (Glaser, 2002, para. 8), making the “researcher’s interactive impact on data more important than the participants’” (Glaser, 2002, para. 20). I disagree with Glaser’s previous claim, as I feel that the relationality during the interview process supported greater communication (and connection) between researcher and participant as well as the divulging of deeper truths. The researcher may have an impact, but if the interviewing process is done well, it should only empower the participant’s voice, not overshadow it. Furthermore, this is expected when working with a social constructivist epistemology as learning does not happen in a void. All in all, the data obtained from the Math Origin Stories supported by early researcher field notes laid the foundation for the framework and the timing of the practices that were incorporated in the intervention section of the course. Moreover, the recursive and iterative nature of grounded theory used feedback from early data analysis (Burck, 2005) to deepen directions for the post-test narrative inquiry (i.e., the individual semi-structured interviews). Hence, at the end of the data collection, the individual semi-structured interviews were poised 83 as an opportunity for ‘meaning-making’ between myself (the researcher) and the participants. Along with the final exam grades and the two post-tests, the individual semi-structured interviews by both the control and intervention group supported the evaluation of the effects contemplative pedagogy had on the learning of mathematics. If math anxiety prevents people from living their lives as Tobias (1993) stated, then, as described in this section, qualitative data may have been the best option to provide insights on the effects of contemplative pedagogy on the math anxious. In consequence, grounded theory was the most effective tool for evaluating those effects. The next section explores the validation of those tools and methods. 3.8 Validation The previous section detailed the diverse data analysis techniques that were used on both data types, quantitative and qualitative. However, an important aspect of data collection, before beginning any analysis, is determining the validity of that data. The latter, called validation, is a multifaceted process. First, after collecting qualitative data, such as individual semi-structured interviews, performing member checking is encouraged. This action allows the interviewees to check the facts/story and determine whether their words and perspectives were considered appropriately (Stake, 2010). Sometimes member checking is not possible; however, it is suggested as much as is possible. In this situation, member-checking for the semi-structured interviews did occur and formed an important way to validate the data. Despite the numerous options provided, all Math Origin Stories were submitted in a written form; hence, member checking was not necessary. 84 Member checking is also form of triangulation (Stake, 2010), where the latter is also related to validation. Data triangulation measures the quality of the analysis. Triangulation is the examination and analysis of different types of data emanating from different sources and which may align with diverse theories or perspectives (Yazan, 2015; Stake, 2010). Flexibility in the design allowed for modifications and adjustments that arose as participants insights and their reactions via observations were acquired (Stake, 2010). Triangulation also allows the researcher to investigate the phenomenon from a multitude of perspectives. Stake (2010) further encouraged the incorporation of four different types of triangulation in any study: 1. Triangulation arising from diverse data sources. 2. Triangulation emanating from different investigators. 3. Triangulation of various theories. 4. Triangulation of diverse methodologies. The incorporation of three of these four types of triangulation occurred in this research. As there were several data sources (as depicted in Table 3), various theories emanating from education, mathematics and contemplative studies are incorporated, and diverse methodologies (narrative inquiry, grounded theory, deductive reasoning), the listed triangulation was achieved. As triangulation and validation are intricately linked, using quantitative data results to support qualitative theories also formed part of the triangulation and validation process, a component which is further discussed in Results. 85 3.9 Ethics A harmonized ethics approval including both the Western Canadian University home to my doctoral program and the College at which I teach was required since the research involved human subjects (participants) at the college under the purview of the university. To satisfy ethics, to respect the confidentiality of students, and to protect their privacy so that they felt safe in their sharing, all collected data was coded. A teaching assistant (TA) was hired to minimize undue influence as I was the course instructor as well as the one leading the practices from contemplative pedagogy. The TA discussed consent with students and provided their contact information to the interested participants. The participants were then coded by the TA using a combination of letters and numbers. CPMM was used as the root of the code, where CPMM stands for Contemplative Pedagogy in Mandatory Math, and participant 01 is labelled as CPMM01. Codes CPMM01 through CPMM68 were used, where CPMM01-CPMM36 represent participants in the control group and intervention group members are labelled CPMM37-CPMM68. Consent forms and all records associated participants’ names with coded numbers have been and will continue to be kept separate from the anonymized data. Participants were regularly reminded that they could recuse themselves from the research at any point, without any penalty or consequences from either institution. To follow grounded theory procedure, initial coding of the data began immediately after the Math Origin Stories submission to determine which practices would be used, whether different investigations needed to be undertaken, to make sure that nothing was omitted, and to saturate the data. Individual semi-structured interviews were held after the last date of 86 grade appeal so that interested participants were not in the middle of final exams and had time to digest their thoughts on the course. Finally, due to this timing, I was able to conduct these interviews. All paper and electronic documents (consent forms, stories, and interview transcripts) are and will continue to be stored in a secure location in my supervisor’s office. All paper documents will be archived for five years following any publication and will then be destroyed via shredding. All digital data are and will continue to be stored on a password-protected, encrypted, digital storage device, also in a secure location for five years following any publication until all files are deleted and the device is securely formatted. Though no one removed themselves, participants were informed that they could withdraw from the study in progress and have their data destroyed. Not all members participated in all components of the study. In consequence, the participant rate is provided for each collected dataset in Chapter 4. This was a minimal risk research study because the probability of possible harms implied by participation was no greater than those encountered in everyday aspects of regular academic life. For example, there were no further physical risks, psychological or emotional risks, and social risks associated with this study that do not already exist in regular life or classes. The consent letter included my contact information for anyone who desired to speak with me directly or voice their concerns in a private setting, after the last day of grade appeal. During the research, they also had the contact numbers of research ethics boards of both 87 institutions, of my supervisor and of the TA should they have any issues with me either personally or professionally. Finally, for the well-being of participants, only evidence-based innocuous practices from contemplative pedagogy that promote well-being and/or improve learning were included in the classroom, as was detailed in Section 2.2.3 (Course Implementation). Practices that could trigger trauma in individuals who have experienced difficult life experiences, such as the body scan (Berila, 2014), were avoided. Nonetheless, campus counselling and other health and wellness contact information were shared with the students on multiple occasions, in person as well as through the learning management system. This chapter discussed the convergent mixed method approach used in the study and which empowered both the quantitative and the qualitative methods detailed in Sections 3.3 and 3.4, respectively. Mixed methods data was collected from the sample (depicted in Table 3) following the methods described in Section 3.5 and analysed according to the processes presented in Section 3.6. In the next chapter, the results emanating from this convergent mixed methods research based on the overarching research question How can contemplative pedagogy support a mandatory post-secondary mathematics course? and its related sub-questions will be shared. Using grounded theory, themes relating to the qualitative data – the researcher field notes, the Math Origin Stories, and the individual semi-structured interviews – will be divulged. Also, the quantitative data will be investigated and explored using descriptive and inferential statistics. Discussions related to these results, followed by a comparison with extant hypotheses and literature will be presented in Chapter 5. 88 Chapter 4: Results The previous chapter laid the foundation for the convergent mixed-method study undertaken at a Western Canadian College in fall 2022 with the ensuing individual semi-structured interviews occurring in winter 2023. As described Chapter 3, two sections of the same first-year mandatory mathematics course formed the study: one section became the control section and the other, the intervention section. The latter section incorporated practices from contemplative pedagogy after the first course test. From each of the mentioned sections, students were invited to participate in the study. A total of 68 students participated in the study, with 36 forming the control group and 32 the intervention group. The main question guiding this research was How can contemplative pedagogy support a mandatory post-secondary mathematics course?, supported by its related sub-questions Does it [contemplative pedagogy]: a. Decrease math anxiety? b. Reduce math avoidance? c. Improve math attitudes? d. Increase math performance? Based on the above questions, mixed methods data was collected. This chapter presents the results from that data collection. As presented in Table 3 (Convergent Mixed Method Data Collection, Data Types, and Stages) and displayed in Figure 3 (below), the collected data emanated from the following sources: 89 1. The Math Origin Stories (MOS, Appendix A), 2. The pre- and post-Five Facet Mindfulness Questionnaire (FFMQ, Baer et al., 2008, Appendix D), 3. The pre- and post-Math Anxiety Rating Scale (MARS, Alexander & Martray, 1989, Appendix C), 4. The pre- and post-course grades (test 1 and final exam), 5. The Individual Semi-Structured Interviews (ISSI, Appendix B), and 6. The Researcher Field Notes (RFN, Appendix E) and Figures 7 and 8 in Section 4.3. Figure 3 Visualization of Data Types and Stages Qualitative Researcher Field Notes Stage 1: Pre-Test Stage 2: Post-Test Quantitative Pre-Test FFMQ Pre-Test MARS Pre-Test Course Grade (Test 1) Qualitative Math Origin Stories Quantitative Post-Test FFMQ Post-Test MARS Post-Test Course Grade (Final Exam) Qualitative Individual Semi-Structured Interviews 90 The three qualitative datasets (the Math Origin Stories, the individual semi-structured interviews, and the researcher field notes) will be examined separately in Sections 4.1 through 4.3. Then the quantitative data will be explored (all pre- and post-test data – FFMQ, MARS, and course grades) in conjunction to test relevant statistical hypotheses in Section 4.5, with supporting calculations provided in Appendix F. As the mixed method data is analysed, the use of “we” will replace “I” in the exposition. The purpose of this replacement is to emphasize the weight of the analysis; that is, to respect the vulnerability and trust gifted by the participants, but also to recognize that I, myself, am being transformed by what I observed and what I experienced during this study. Furthermore, there is an awareness going forward that I am no longer just myself on this journey – we are us – and together we are meaning-making of the experiences arising. Triangulation of the data and results are presented in three sections: Section 4.4 for the qualitative data, Section 4.6 for the quantitative data and Section 4.7 for both data types, that is, for all mixed methods data. Triangulation is offered as a way to both summarize and validate the mixed methods data. Connections between the themes and hypotheses will be made in Section 4.8 to address the overarching research question How can contemplative pedagogy support a mandatory post-secondary mathematics course? and its related sub-questions. Discussions relating to prominent hypotheses and themes from this research as well as gaps in the extant literature will occur in the Chapter 5. 91 4.1 Math Origin Stories The Math Origin Stories (MOS) offered students the opportunity to share their mathematical journey, in what was devised to be a safe and judgement-free environment. Adhering to the insights gained from the night with Kovach (on Feb 17, 2022 at UBC Okanagan) mentioned in Section 1.2 (Positionality), I, the researcher, prepared myself to embody the shared stories. To do so, I reflected on my positionality and grounded myself in my purpose – to improve the experience of mandatory post-secondary mathematics courses. Due to this pause, Kovah’s three questions came to the forefront, guiding the research and my mindset: “What are you saying? Why does it matter? Why are you presenting it the way you are?” (Kovach, 2018, p. 228). Before being entrusted with the stories, I used these three questions as a mantra to keep me focused on my purpose and to represent the participants’ voices as respectfully and authentically as possible. Kovach (2018) further added that “researchers must care for the stories and those who offer them. In asking for individuals’ stories, it matters to respect their dignity, their voice, and their experience on their terms” (p. 227). Experience, according to Clandinin and Connelly (1994) is “the stories people live. People live stories and in the telling of them reaffirm them, modify them, and create new ones” (p. 415). Upon hearing the voices of the participants, both gratitude and honour were felt with the demonstrated vulnerability and trust. In consequence, reverence was offered while being a guardian and a caretaker of their stories. Narrative inquiry, the study of the experience of the story, then, is first and foremost a way of thinking about experience” (p. 375). Hyde-Clarke added that “the narrative of the self is thus largely used as a tool to make sense of personal experience or expectations” (p. 80). 92 Bruner, cited in Hyde-Clarke, posited that the narrative goes beyond merely choosing events in a specific way (as stated above). “The particulars selected need to be constituted as relevant to the story, becoming its parts and functions. In this way, the construction of reality and experience has two clear phases: how it is created as a purpose of explanation; and then how it is interpreted” (p. 80). This chapter focuses primarily on the interpretation of these shared stories and experiences. As mentioned in Methodology, a Constructivist Grounded Theory approach (Charmaz, 2006) was used when analysing the qualitative data. The Initial/Open Coding began with the Math Origin Stories, where initial questions such as “What is the main concern being faced by the participants?” and “What accounts for the continual resolving of this concern?” were suggested by Glaser and Holton (2004, para. 48). These questions played an important role when exploring the shared stories of the students. When coding the responses, gerunds (verbs ending in “-ing”, and demonstrating action) were used to better represent the students’ lived experience along with the students’ own language (Charmaz, 2006). Following Charmaz’s (2006) approach, the subsequent initial themes arose when searching for common concerns in the Math Origin Stories (see Table 4). For the purposes of keeping all the qualitative themes separated according to their respective datasets, acronyms are used in the labels. In Table 4, the word “Initial” and the acronym MOS (Math Origin Stories) are used as these themes emerged from the initial coding of the Math Origin Stories: 93 Table 4 Initial Coding Themes from the Math Origin Stories (Initial MOS) Initial Themes from the Math Origin Stories (Initial MOS) Theme Initial MOS1 Struggling with the effects of math anxiety Theme Initial MOS2 Living with the inner conflict of negative self-talk and/or negative math self-esteem Theme Initial MOS3 Grappling with/feeling stuck in a negative math attitude Theme Initial MOS4 Perceiving being alone in negative feelings and sensations surrounding mathematics Theme Initial MOS5 Blanking out (forgetting everything) in test/exam situations Considerations regarding the Math Origin Story and its initial analysis were of paramount importance, especially as it was the first notable qualitative data collected. The participation rate by the control group was 90.00% (or 36 out of 40). For the intervention group, 84.21% (or 32 out of 38) students participated, as two students dropped the course before the drop date – one never attended and the other dropped the course for health reasons. As this data collection occurred before the incorporation of contemplative pedagogy, the overall participation rate across both sections was computed to be 87.18% (or 68 out of 78). As described in Section 3.7.2 (Qualitative Data Analysis – Grounded Theory), grounded theory is an iterative process emerging from a somewhat general problem and reasonably open research question. In this case, the main research question being investigated was How can contemplative pedagogy support a mandatory post-secondary mathematics course? After 94 reflecting upon the first iteration of grounded theory which produced the initial MOS themes, a few sub-questions arose: Does it: Decrease math anxiety? Reduce math avoidance? Improve math attitudes? Increase math performance? As discussed in Section 2.2.3 (Course Implementation) and Chapter 3, evidence-based practices for the intervention section of the course were taken from the Stress Management and Resiliency Techniques Curriculum (i.e., the Stress Management and Resiliency Techniques Facilitator Manual, 2015) and Barbezat and Bush (2014). The implementation of practices from contemplative pedagogy began after the completion of all the pre-tests (or Stage 1 from Table 3 (Convergent Mixed Method Data Collection, Data Types, And Stages). Diverse data (post-tests or Stage 2 of Table 3) was collected to determine the value of the integration of contemplative pedagogy to a mandatory post-secondary math course. After the end of the course (the 13 weeks of instruction followed by the two-week exam period), a secondary analysis (or iteration) of the Math Origin Stories occurred, this time a Re-Focused coding (Charmaz, 2006). Not only is such a coding encouraged by Charmaz (2006), but the purpose was different. This Re-Focused coding went beyond the original focus to dig deeper and assigned all statements greater weight, not only those related to challenges (as was desired by the Initial MOS coding). This secondary coding offered the opportunity to revisit themes and extended them with the purpose of saturating the data and confirming that no further direction needed to be investigated before or during the individual semi-structured interviews. Using this lens, the themes found initially (Initial MOS Themes) were elaborated upon and other themes were added (see Table 5). The emergent themes from this Re-Focused coding are discussed one 95 by one over the next subsections. The labels for each of the themes are connected to the secondary analysis of the Math Origin Stories; hence, they are labelled Re-Focused MOS. Table 5 Re-Focused Coding Themes from the Math Origin Stories (Re-Focused MOS) Re-Focused Coding Themes from the Math Origin Stories (Re-Focused MOS) Theme Re-Focused MOS1 Struggling with math due to: a) math anxiety b) negative self-talk and/or negative math self-esteem c) having a negative math attitude and/or feeling stuck in math d) blanking out when writing a test or exam Theme Re-Focused MOS2 Feeling alone in math class and/or how one feels about mathematics Theme Re-Focused MOS3 Changing attitude and/or confidence in math due to one person or experience (positive or negative) Theme Re-Focused MOS4 Having a fear of making a mistake or of asking for help Theme Re-Focused MOS5 Avoiding math and math-related fields Charmaz (2006) wrote of the importance of keeping notes while performing qualitative analysis. Hence, while performing this Re-Focused coding, notes were taken and added to the Researcher Field Notes. They are presented in Figure 7 (Researcher Field Notes Taken When Analysing the Math Origin Stories (Re-Focused Coding)) in Section 4.3 (Researcher Field Notes). 96 In the following subsections, each theme emanating from the Re-Focused coding of the Math Origin Stories will be more closely inspected. Participant quotations from both the control and intervention groups will be provided to support these themes. As no practices were introduced when the Math Origin Stories were collected, no distinctions are made between the data collected from the control and intervention groups. To ensure participant confidentiality and anonymity, the participants’ names were coded with CPMM (Contemplative Pedagogy and Mandatory Mathematics) and participant 01 was labelled as CPMM01. Moreover, codes CPMM01 through CPMM36 represent the control group, while CPMM37 through CPMM68 constitute the intervention group. 4.1.1 Theme Re-Focused MOS1: Struggling with Math Due to… As written above, during a revisiting of the Math Origin Stories, a Re-Focused approached was used to consider all the ideas that were shared by the participants and to best represent their voices. This Re-Focused coding delved deeper into the stories of the participants, which differed from the previous, more superficial initial coding, as the initial coding had a time-sensitive purpose. The thick and rick data presented during this second qualitative analysis offered meaningful themes, where the first one includes a combination and slight modification of initial themes listed above. 97 All the components of Theme MOS1 consider challenges related the learning of mathematics, specifically, Theme Re-Focused MOS1: Struggling with math due to… a) math anxiety b) negative self-talk and negative math self-esteem c) having a negative math attitude and/or feeling stuck d) blanking out when writing a test or exam Participant quotes related to Theme Re-Focused MOS1 are as listed below, the letter appearing in the parenthesis references which of the sub-themes is being touched upon: “Math often gets a reputation of being only for “smart people” and just being overly difficult. This does not help inspire confidence as I internalize these ideas and question if I am “smart enough” to be successful at math. Most of the time, I know I am not.” [bc, CPMM09, Control] “I feel a lot of anxiety when it comes to math. My feelings about math changed in middle school, when the idea that “girls can’t do math” became internalized. I grew to believe it, so why try?” [ac, CPMM17, Control] “I study really hard and my brain is empty when I write exams. I get really nervous and forget everything. I hate math. It makes me feel really bad.” [abd, CPMM21, Control] “I am bad at math and will never succeed at math. I think this was caused by a mixture of anxiety, fear of failure and a bad mindset. I’ve given up trying.” [abc, CPMM27, Control] “Math classes and tests stress me out so much. I forget formulas all the time and get confused which to use when. This makes me feel dumb.” [abd, CPMM38, Intervention] 98 “I am a believer that you are either a math person or you are not. Or maybe I’m just lazy and stubborn.” [c, CPMM42, Intervention] “I felt anxious about math due to feeling rushed and even humiliated by it starting in middle school.” [a, CPMM51, Intervention] “When I think about math my thought… is that it is going to be very difficult and probably too difficult for someone like me to do.” [bc, CPMM62, Intervention] These quotes are a representative sample that clearly reflects the theme of struggle from varied perspectives. Early in the course, it is clear that many participants had challenges surrounding mathematics, from physiological concerns to more cerebral ones. The next section will offer details regarding feeling alone in a math class or feeling alone due to how one feels about mathematics. 4.1.2 Theme Re-Focused MOS2: Feeling Alone in Math Class and/or How One Feels About Mathematics This section explores quotes related to the loneliness theme, that is Theme Re-Focused MOS2: Feeling alone in math class and/or how one feels about mathematics. While performing this Re-Focused coding, the depths of emotions, particularly related to feeling alone, were hard to read. A lot of negative feelings were brought up, with many respondents blaming themselves for shortcomings (and hence having some overlap with the negative self-talk/self-esteem in the previous theme) or presenting strong sentiments of isolation. It was clear that a number of these thoughts and feelings had been entrenched for some time. A few quotation exemplars are: 99 “I did not feel like my teachers supported me in my precalculus classes. I felt alone and would get stuck on questions. I was unable to make sense of the solutions when it felt like everyone else did.” [CPMM14, Control] “I asked myself why I had to be the dumb one. The rest of the class got it. Why couldn’t I?” [CPMM20, Control] “I hated going to math class. I didn’t understand anything, and the teacher picked on me so I felt worse. I felt different and alone.” [CPMM31, Control] “Everyone seemed to understand, adding to my feelings of isolation.” [CPMM38, Intervention] “I felt alone and would get stuck on questions, unable to make sense of the solutions. I was also too scared to ask for help, adding to my feelings of isolation and helplessness” [CPMM46, Intervention] The anonymous quotes above described how some participants felt alone, either in the math class or how they felt about math. The next section will provide supporting quotes where participants shared either a positive or negative math experience due to one person. 4.1.3 Theme Re-Focused MOS3: Changing Attitude and/or Confidence in Math Due to One Person or Experience The previous section considered feeling alone in a mathematics setting or in the way that a participant felt about mathematics. This section considers the influences people or experiences can have on a participant’s attitude surrounding mathematics. In particular, the Theme Re-Focused MOS3: Changing attitude and/or confidence in math due to one person or 100 experience (positive or negative) arose among all the submissions. A mix of both positive and negative influences and experiences emerged. Hearing these experiences, I felt as though I was “bearing witness”, making the stories and the sharing more meaningful and more uplifting or difficult (depending on the context). Bringing these influences to light, in their particular contexts, may affect a change in perception for those influencing. A variety of influences are presented below. “My grade 10 precalculus teacher had a large influence on my attitude toward math. She took the time to help me when I needed it. She would teach me other ways or methods to solve problems if the first way didn’t make sense to me.” [positive, teacher, CPMM03, Control] “I didn't take math in grade 12 and everyone kept asking me why, but I just could deal with the pressure and that feeling I got when I didn't understand something. In conclusion, the pressure teachers put on me was good motivation when I was younger but ended up leading me to not like math.” [negative, teacher, CPMM17, Control] “Because I was ahead of everyone else, my teacher decided to take me out of math class for a month or two until I was at the level the rest of the class was at. It felt like a punishment to sit by myself in an empty classroom doing nothing for a few months. When I returned to math class I made sure to never raise my hand or finish the problems too fast. I was scared to be removed from the class again.” [negative, teacher, CPMM23, Control] “One of my friends encouraged me to do maths for my further study. She said that she was also very week (sic) in maths during her primary school but with great efforts she 101 scored high percentages in secondary school. After hearing my friend’s talk, I got confidence that I can also do maths.” [positive, friend, CPMM29, Control] “My mother helped me a lot. She would spend time at night after work learning stuff so she could teach me. I don’t know how I would have passed without my mom.” [positive, family, CPMM38, Intervention] “After a math assessment to see where our math skills were I was put in a higher level math class. It was the first couple of classes and I was struggling… Instead of moving me down a level or two, the teacher put me in the bottom level, which ended up being incredibly too easy for me. When I got to the next grade, everything was new and hardly recognizable. I fell behind and essentially failed the course.” [negative, teacher, CPMM44, Intervention] “My math teacher really did a number on me. … He would humiliate us if we got a question wrong… I felt physically ill before his class.” [negative, teacher, CPMM53, Intervention] “I associate math with so much stress because it was really pushed into my head that a student’s worth is based on grades. When I stopped getting good grades in math it made me feel like a bad student.” [negative, other, CPMM62, Intervention] “My relationship with math deteriorated because of my precalculus teacher. It wasn’t that she was mean or rude to me, it was just that she did not know how to teach math. Myself as well as many of my classmates took summer school to improve our grades or pass the class. It was a rough year for all of us. I dreaded it afterward.” [negative, teacher, CPMM66, Intervention] 102 The quotes above relate to positive or negative influences or experiences during the participants’ previous mathematical journey. The following section considers fear surrounding making mistakes or of asking for help and offers supporting quotations. 4.1.4 Theme Re-Focused MOS4: Having a Fear of Making a Mistake or of Asking for Help The previous section considered positive and negative influences or experiences had by participants. This section is closely linked as some quotations concern experiences as well, and some could easily be connected to a person or experience depending on the situation. The Theme Re-Focused MOS4: Having a fear of making a mistake or of asking for help presented itself in different contexts, with a few participant quotations found below. “With a fear of getting questions wrong/not understanding them, I would push my math practices to the side because I didn’t want to feel upset with myself” [CPMM20, Control] “A main emotion is my lack of comfort when being called upon out of fear of not knowing and admitting I dont (sic) understand basic concepts that many people expect me to know.” [CPMM27, Control] “The main part that makes me nervous about math is unexpected quizzes/questions, as well as tests in general. I am scared of ‘failing’. I felt judged based on my test grades and didn’t feel comfortable asking for help.” [CPMM37, Intervention] “I was too scared to ask for help, making me feel even more helpless” [CPMM46, Intervention] “Most of the time I don’t know where to start. It’s not worth trying because I always get everything wrong.” [CPMM57, Intervention] 103 The participant statements above related to fears surrounding attempting problems alone or in class, along with the challenges associated with asking for help. The next section considers the last theme, that of math avoidance. 4.1.5 Theme Re-Focused MOS5: Avoiding Math and Math-Related Fields The last four sections have considered themes arising during the Re-Focused coding of the Math Origin Stories. This section provides details pertaining to the last emerging theme, that is, Theme Re-Focused MOS5: Avoiding math and math-related fields. Math avoidance, discussed in the Literature Review, has many causes, such as math anxiety and negative math attitudes. Unfortunately, math avoidance can be detrimental to one’s self-esteem and can also have consequences which reach outward, as shortages in STEM fields are often discussed or the need to bring in individuals from elsewhere to work in STEM careers. A few respondent quotations are found below: “I was relived (sic) when I saw that the math class I was in at university had the same content as I went through in high school. I had failed the class, which was such an embarrassment, that I felt compelled to not only quit the class, but to drop out of sciences entirely. I switched to arts.” [CPMM13, Control] “I do not feel super confident about my math abilities which I believe has led to my aim of avoidance.” [CPMM23, Control] “When I learned that I needed at least calculus courses to major in biology, I changed my major to history. I hate math and math hates me.” [CPMM44, Intervention] 104 “When I think about mathematics, I am filled with anxiety. I chose to do a degree in arts so that I would be able to do minimal math and sciences classes.” [CPMM62, Intervention] “I dreaded math afterward. And that ultimately caused me to not take any more math.” [CPMM67, Intervention] The avoidance of math and math-related fields was investigated in this section, completing the inspection of each emerging theme from the Re-Focused coding of the Math Origin Stories (see Figure 4 for a depiction of the themes). Further comments and discussions relating to determined practices and the emergent themes will be elaborated upon in the Chapter 5. Figure 4 Visualization of Math Origin Stories Themes Emanating From the Re-Focused Coding Themes Re-Focused MOS1. Struggling with math due to...2. Feeling alone in math class and/or how one feels about mathematics3. Changing attitude and/or confidence in math due to one person or experience (positive or negative)4. Having a fear of making a mistake or of asking for help5. Avoiding math and math-related fields 105 4.2 Individual Semi-Structured Interviews The previous section (and included subsections) closely examined the Math Origin Stories, a source of rich and thick qualitative data. This section now considers the second source of qualitative data, collected after the completion of the course. More specifically, Individual Semi-Structured Interviews (ISSI) occurred in late winter 2023, following the last date for grade appeal of the fall term. Interviews were performed via Zoom and were transcribed based on the transcripts offered by Zoom. The transcripts were shared with the participants for the sake of member-checking and no changes were requested by any of the participants. Participants from both the control group (10 out of the original 36 control group participants, 27.78% participation rate) and the intervention group (26 out of the original 32 intervention group participants, 81.25% participation rate) engaged in the individual semi-structured interviews. As noted in Appendix B, only the intervention group was asked questions related to the practices. The constant comparative method described in Section 3.7.2. (Qualitative Data Analysis – Grounded Theory) was used with the transcripts of the individual semi-structured interviews. That is, the constant comparing of codes with codes, codes with new categories, and categories with categories until themes emerged and were formulated. As was done in Section 4.1 (with Theme MOS1), sub-themes have been grouped together under an overarching theme when applicable, to be more concise and focused. Themes attributed to the control group are presented first in Table 6, followed by ones emerging from the intervention group (in Table 7). Moreover, themes emerging from the Individual Semi-Structured Interviews have ISSI labels, along with either a “C” for a Control theme or a “I” for an Intervention theme. For example, 106 Theme ISSI1C is an Individual Semi-Structured Interview theme emanating from the control group. Table 6 Control Group Themes for the Individual Semi-Structured Interviews Control Group Themes for the Individual Semi-Structured Interviews Theme ISSI1C Discovering Negatives Theme ISSI1Ca Noticing Knowledge Gaps and Having Struggles Theme ISSI1Cb Feeling Stuck Theme ISSI1Cc Feeling Disappointed with Self Theme ISSI2C Not Giving Up The themes presented in Tables 6 and 7 were the result of applying the constant comparative method to the transcripts of the individual semi-structured interviews, as described above. The following subsections will examine each of these themes in order and will share supported quotations from the respondents. Section 4.2.1 will present themes attributed to the control group, while Section 4.2.2 will provide the three overarching themes, along with their respective sub-themes, that emanated from the intervention group. 107 Table 7 Intervention Group Themes for the Individual Semi-Structured Interviews Intervention Group Themes for the Individual Semi-Structured Interviews Theme ISSI1I Changing Frameworks of Thinking (Beginning of Internal Changes) Theme ISSI1Ia Becoming More Open-Minded Theme ISSI1Ib Unlearning Theme ISSI1Ic Improving Math Attitude Theme ISSI1Id Increasing Math Self-Confidence Theme ISSI1Ie Demonstrating Acceptance and Self-Compassion Theme ISSI2I Recognizing External Changes Theme ISSI2Ia Decreasing Fear of Trying, of Making Mistakes, of Asking for Help Theme ISSI2Ib Decreasing Math Avoidance Theme ISSI2Ic Positive Physiological Effects of Practices Theme ISSI3I Becoming Aware of Internal and External Changes Theme ISSI3Ia Decreasing Math Anxiety Theme ISSI3Ib Making Connections, Feeling Included, and Building Community Theme ISSI3Ic Continuing Practices and Gratitude for Flexibility 108 4.2.1 Individual Semi-Structured Interviews Control Group Themes As written above, the following subsections will consider each of the themes individually, beginning with those emanating from the control group. The first theme is an overarching one called Theme ISSI1C: Discovering negatives. This theme label was chosen as all the sub-themes are negative. In fact, outside of the first theme Theme ISSI1Ca: Noticing knowledge gaps and having struggles, the remaining sub-themes resembled some of the original (negative) themes from the Math Origin Stories, thereby displaying minimal change or growth throughout the term. This first sub-theme clearly arose after self-reflection as control group participants who responded to the semi-structured interview questions referenced gaps in knowledge due in part due to the pandemic and shared what they felt were ensuing struggles, which continued even at the end of the course. A few of their quotations are as follows: “I was really successful in school but when the pandemic had caused us to shift to virtual learning, I struggled a lot and fell behind because of that.” [CPMM02, Control] “I feel like I missed a lot because of the pandemic and that made me more anxious when I started this class, and it still did at the end.” [CPMM24, Control] “Learning got harder with the pandemic. Math in particular. I feel far behind and that I will never make up on lost time.” [CPMM33, Control] The above quotes are representative of several participants from the control group, airing their frustration with the consequences of pandemic learning. The next sub-theme, still under the Discovering negatives umbrella, Theme ISSI1Cb: Feeling stuck dives deeper into the 109 negative thoughts some participants had surrounding mathematics and not foreseeing possible change. A few quotations are as follows: “For a long time, I would associate it with negative thoughts of self-doubt and always just looked at it as something I wasn’t confident that I could overcome. I was ashamed.” [CPMM05, Control] “I started the course with a little bit of dread honestly and then I barely l passed the year.” [CPMM11, Control] “I didn’t feel any more confident about retaking this course, even though I had done it already.” [CPMM27, Control] “Seeing concepts that I knew before was okay but learning new concepts was the most difficult portion of the class for me. I felt stuck.” [CPMM31, Control] Finally, the last sub-theme of the Discovering negatives, Theme ISSI1Cc: Feeling disappointed with self, builds upon the previous sub-theme and compounds well-known concerns surrounding negative self-talk (Tobias, 1993). This sub-theme emerged from participants feeling disappointed and discouraged, both during and after the course. A few exemplar quotations are: "I thought I could do better on my own with YouTube and Khan [Academy], so I skipped a bunch of classes. I should’ve just attended. It was hard to go back to class.” [CPMM05, Control] “In the beginning I felt nervous and unsure of myself in the class. It can be discouraging looking at content you could understand at a younger age and no longer understanding and lacking competence in the course material at a later age.” [CPMM18, Control] 110 “I remember hoping the notes I was taking would be family (sic) and that something would click and I would be able to follow at least. Still, I wasn’t able to make connections and I was really disappointed with myself” [CPMM22, Control] “After finishing the class, I recall being unhappy. I felt like I had let myself down and was probably going to have to switch career paths because I really thought I couldn’t understand math anymore.” [CPMM27, Control] “By the end of the class, I was even more disappointed in myself because I feel I gave up just because of my anxiety with math.” [CPMM34, Control] The three previous sub-themes were grouped together and related to the overarching Discovering negatives theme for the control group. Another theme emerged from the control group, but as a solitary theme, that is, Theme ISSI2C: Not giving up. It was clear from the discussions with participants that some wanted to “try something different next time to maybe do better or feel better about myself” [CPMM34, Control]. Another wished that they had had the option when registering. “If I knew when signing up for the course, I would’ve probably tried the section with stuff added that had a chance to help me and my feelings about math. If I see it, I might try it next time.” [CPMM24, Control] As displayed in Figure 5, we see that the main overarching theme for the control group could be linked to ‘disconnection’ – disconnection from self, others, and mathematics. The second theme provides seeds of hope for something better in the future. 111 Figure 5 Visualization of Individual Semi-Structured Interview Control Group Themes This subsection focused on the control group themes which emanated from the individual semi-structured interviews. The next subsection will begin to elaborate upon themes arising from conversations with participants of the intervention group. 4.2.2 Individual Semi-Structured Interviews Intervention Group Themes Until this point in Section 4.2, we have presented all themes emerging from the individual semi-structured interviews as well as have supported the control themes with quotations from the participants. Now, we turn toward themes emanating from the interviews of participants in the intervention group. Notes were taken while analysing the individual semi-structured interview data. As the intervention group interviews provided information noticeably different than the Math Origin Stories, only notes pertaining to the intervention section were retained. Again, as these relate to researcher field notes, they are presented in 112 Figure 8 (Researcher Field Notes Taken When Analysing the Individual Semi-Structured Interviews (Intervention Group)) in Section 4.3 (Researcher Field Notes). 4.2.2.1 Theme ISSI1I: Changing Frameworks of Thinking (Beginning of Internal Changes). The first main theme, Theme ISSI1I: Changing frameworks of thinking, will be explored via its linked sub-themes. In particular, the focus of this overarching theme relates to processes occurring within the respondent, related to mathematics. The first sub-theme Theme ISSI1Ia recognizes the internal changes by Being more open-minded. Quotations supporting this theme include: “My overall feeling towards mathematics was frustration and fear… Now, I can see that math is not something that needs to be scary and frustrating, but it can be fun.” [CPMM38, Intervention] “Instead of feeling like there is only one right way to do a problem, I can now see that there are many right ways.” [CPMM45, Intervention] “Going into future math classes I have now learned to be open-minded in learning new things” [CPMM49, Intervention] “Having a flexible attitude and open-minded perspective allowed me to apply this lesson (of accepting change and adapting to your environment) to my mathematical journey this year. I had a great term.” [CPMM52, Intervention] “When first starting this course, I held a very negative view of math and believed it was not for me and that I was just going to work to pass the class. I have a completely different view of mathematics after completing the course. I now view it in a positive light and am confident in my skills. It was very hard to change my mindset but with each class, I walked in a little more positively and used an ‘I can mindset’ as opposed to an ‘I 113 cannot’. This mindset changed because I started to use practices and found the best one that worked for me. Hint, it uses numbers!” [CPMM56, Intervention] “I did not have a good relationship with math before this course. It’s fair to say if math had a face, at the time, I would’ve like to punch it. I 100% react differently now. Having math meet me where I’m at and not force me along at a pace I can’t follow has changed my perspective on math” [CPMM59, Intervention] The next sub-theme connected to Changing frameworks of thinking is Theme ISSI1Ib: Unlearning. Some participants in the intervention group found that stepping back and looking from a new perspective offered insights to previously held beliefs, for example: “I thought that getting something wrong was super embarrassing and made me dumb and that the only right way to solve a problem was the one the teacher showed on the board. Learning that this is not the case was very significant to me and will help me in my future.” [CPMM41, Intervention] “I unlearned that math is equally accessible to people. Not only is it inaccessible geographically (attending a school without proper funding I.e. reserves schools and schools in impoverished areas) but that even in the same classroom students can be given different access to success in mathematics. I hadn’t thought about how colonial math excludes a lot of people most heavily Indigenous students. Having time for self-reflection helped me understand that there is more than one way of knowing and that they may be valued differently by society, but both have an important place in making the world (or on a small scale the classroom) a more equitable place.” [CPMM58, Intervention] 114 Another sub-theme linked to changing one’s framework of thinking is Theme ISSI1Ic: Improving math attitude. Several participants perceived a change in their attitude toward mathematics. A more positive perspective of mathematics was acquired for some members of the intervention group. During the interviews, many smiles, celebratory statements, such as ‘woohoo’, and gestures, such as arms extended in the air above the head, were displayed. A few relevant quotes supporting this sub-theme are: “I have become a lot more inspired and encouraged to be happy about math and not feel as though it is a nuisance.” [CPMM41, Intervention] “My understanding of the importance of math and my attitude has changed in a good way.” [CPMM47, Intervention] “With less fear, real learning is able to happen! This is the first time in I felt comfortable expressing my thoughts and misunderstandings aloud in a math classroom” [CPMM54, Intervention] “With math you really do have to apply yourself and have a good attitude in order to succeed. A couple of times, I did feel confused and reacted to myself poorly as when you don’t understand something it can be frustrating. After using practices such as the Pause, 5-4-3-2-1, or loving kindness, I got over the frustration with myself and turned it into learning. I felt better about math then too.” [CPMM58, Intervention] “This semester has changed my view of mathematics. I walked into the classroom very scared on the first day but after changing my view on math I am now so much more confident in my skills and ability to pass along a positive view of math in the classroom” [CPMM63, Intervention] 115 “My attitude towards, feelings about and confidence in my math ability are leaps and bounds above what they were!” [CPMM66, Intervention] Some of the sub-themes are more interconnected than others. For example, the previous sub-theme (Improving math attitude) is intimately connected to Theme ISSI1Id: Increasing math self-confidence. However, since math attitude and math self-confidence are often considered separately when exploring the effects of math anxiety, for example (Hembree, 1990; Ma, 1999, Tobias, 1993), we have kept them separate here as well. Supporting quotations for increasing math self-confidence are provided below and further discussions will be included in the next chapter: “Going forward, I have newfound confidence in math.” [CPMM45, Intervention] “The practices in this coursed (sic) helped me feel confident in my math abilities.” [CPMM56, Intervention] “I have never had more confidence speaking in math class and my hands don’t (sic) sweat when I have to give customers change at a table anymore!” [CPMM67, Intervention] Finally, the last sub-theme of this internal transformation theme is Theme ISSI1Ie: Demonstrating acceptance and self-compassion. Both acceptance (Chin et al., 2019) and self-compassion (Neff, 2009) have been linked with benefits extending beyond regular mindfulness practices, as will be further discussed in Discussion. An example emanating from a participant: “my understanding of math has changed, and I do not feel as intimidated by it. I do not understand all of it, and don’t think I ever will, but I am okay with that” [CPMM63, Intervention] 116 This subsection has focused on internal changes demonstrated, linked with changing old frameworks of thinking (and unlearning). The next subsection relating themes emanating from the intervention group will concentrate on the recognition of external changes. 4.2.2.2 Theme ISSI2I: Recognizing External Changes. Section 4.2.2.1 focused on the first set of sub-themes for the Intervention group, those connected with internal changes. We will now consider external changes, whether they be visible to others or simply outside of the body. The first sub-theme for external changes is Theme ISSI2Ia: Decreasing fear of trying, of making mistakes, of asking for help. Participants in the semi-structured interviews vocalized a change related to their self-consciousness and their ability to try or make mistakes (on their own or in front of others), or to ask for help (inside or outside of class). Some quotations shared by participants are as follows: “By focusing less on my anxiety and fear of math, I learned it is okay to make mistakes when doing math, it doesn’t make you dumb.” [CPMM37, Intervention] “Mindful listening taught me that I didn’t have to prepare when waiting. I could just listen and trust that I wouldn’t say something dumb when it was my turn to talk or ask questions.” [CPMM40, Intervention] “I ask questions in class now and don’t worry about people thinking that I’m stupid.” [CPMM41, Intervention] “Asking for help seems safer now. I don’t feel judged by my peers or my teacher. I don’t feel like they are against me. I think that some might even want me to do well.” [CPMM48, Intervention] 117 “Talking in pairs [mindful dyads] helped me to listen to you and those in class. I gained courage to ask my own questions that way.” [CPMM51, Intervention] “I felt supported by others in the class by students and [the teacher]. I knew that they wouldn’t make fun of me if I screwed up. I put my hand up more after realizing that. I’m not afraid to try.” [CPMM53, Intervention] The next external sub-theme Theme ISSI2Ib: Decreasing avoidance discusses a decrease in math avoidance, a common phenomenon when someone does not like math, has a bad math attitude, or high math anxiety. A few shared quotations supporting this sub-theme are: “I feel more confident, so I need to rethink my major. I’d really like to become a doctor. I think I can do it now.” [CPMM49, Intervention] “Math really is everywhere. I feel I can walk towards it instead of running away.” [CPMM53, Intervention] “After this class I will not shy away from math, but instead look for opportunities to include it into my life” [CPMM61, Intervention] Finally, the last sub-theme representing external changes is Theme ISSI2Ic: Physiological effects of practices. Some participants in the intervention group recognized some changes in their physiological response to mathematics due to the practices. A few supporting quotations are found below: “I felt more relaxed going into a test. I was still nervous, but my heart wasn’t racing.” [CPMM41, Intervention] 118 “Thanks for posting the practices online. I tried more of them that way and didn’t worry about what others thought. Loving kindness has really helped. I feel lighter.” [CPMM49, Intervention] “Come the final exam, I remembered to breathe. I was more calm than ever before. I believed that I could do it.” [CPMM53, Intervention] “After using 54321 and holding onto my formula sheet, I sat up straight. I didn’t hunch or cringe. Bring it on! I knew that I had this.” [CPMM60, Intervention] “I didn’t blank after breathing. Yes! [big smile]” [CPMM68, Intervention] The third sub-theme above completes the trifecta associated with external changes, whether they are visible to others or seen outside of one’s mind. The next subsection will present themes examining both internal and external changes. 4.2.2.3 Theme ISSI3I: Becoming Aware of Internal and External Changes. The previous subsection considered external changes, while the one before that, contemplated internal changes. This subsection will delve into the overlap of the two, that of both internal and external changes. The first sub-theme considered here is Theme ISSI3Ia: Decreasing math anxiety. Given the causes and consequences of math anxiety, detailed in Chapter 2, we note that the effects of decreasing math anxiety can be both internal – changes in attitude and (math) self-confidence – and external – lessened physiological responses to the anxiety. The following are a few quotations from the intervention group: “During the last class, my anxiety calmed down. I am getting better at it.” [CPMM42, Intervention] 119 “At the end of the course, my heart and respiration rate returned to normal. I feel less stressed in math class now. Good timing. Ha ha.” [CPMM53, Intervention] “I liked practising mindfulness and Loving Kindness practice to divert my anxiety. I feel like I can control my anxiety instead of it controlling me.” [CPMM57, Intervention] “The Pause Practice and the 54321 Practice take away the anxiety and frustration towards math that I once felt.” [CPMM62, Intervention] “I liked including the topic of mindfulness in my class. I think it is very important for students to incorporate mindfulness in all subject areas, but I believe it could be particularly valuable for reducing math anxiety and facilitating a positive learning environment. It did for me.” [CPMM68, Intervention] The next sub-theme linked to internal and external changes is Theme ISSI3Ib: Making connections, feeling included, and building community. This sub-theme contemplates the transformation occurring when making connections, feeling like one is included, or when building community. This sub-theme is a counterpoint to Theme MOS2 which presented felt isolation and loneliness, something that will be discussed further in Chapter 5. A few quotations supporting connection, inclusion, and community include: “Throughout the course, I felt comfortable and safe reaching out to my peers when I needed help, as well as my teacher.” [CPMM37, Intervention] “The practices in this class helped me to realize that a big part of how I feel about math is the environment. The environment is what sets the students up for failure or success and this year was a success” [CPMM47, Intervention] 120 “Hearing others struggle with the same concept or helping me learn the concept I was struggling with was helpful and I didn’t feel ‘stupid’ or ‘incompetent’ but rather grateful and less alone.” [CPMM54, Intervention] “When I began this course, my reaction to mathematics was one of dread and annoyance. Now that the end of the semester has arrived, I think that I now look at problems with more creativity and wonder. I also think more positively about group work in math and see more value in collaborative learning opportunities.” [CPMM61, Intervention] The final sub-theme representing internal and external changes is Theme ISSI3Ic: Continuing practices and gratitude for flexibility. Brought upon by over eight weeks of practice, some participants foresaw continuing these practices going forward, building upon their perceived strength or changes across the term. Their commitment is evidenced by the following quotations: “Loving kindness was a bit slow but more helpful in the long term.” [CPMM38, Intervention] “I really appreciated the time to talk and listen with someone. Not interrupting was really hard at first, but I got more out of it after. I’m going to try to continue this technique going forward.” [CPMM42, Intervention] “The practices available on Moodle [the College’s Learning Management System] was nice. I started to work full-time, so I couldn’t always be there. I even put the audio on my phone and still use it now.” [CPMM45, Intervention] 121 “Breathing and 54321 were practices that worked for me. I’ll keep them up.” [CPMM46, Intervention] “If someone was struggling with learning math, I would tell the person to start with breathing, the Pause Practice, and then use 54321 and then loving kindness. I do and will continuing doing so.” [CPMM49, Intervention] “Flexibility and accepting change are crucial to a successful math journey and I will include practices in my future academic journey.” [CPMM53, Intervention] “I have a better understanding of how mathematics is connected to so much of a person’s life. Pausing and breathing helped with that. It’s grounding. I can use these connections and expectations to pause and focus on the here and now.” [CPMM57, Intervention] “Thanks for posting the audio of the practices. I could go back as often as I wanted and when I wanted. I think that helped me continue after the course ended.” [CPMM59, Intervention] “The ‘yes and…’ exercise helped me take risks when solving problems. Math can be confusing, but as long as you understand the process for a problem, you can break it down step by step in order to answer the question.” [CPMM62, Intervention] “I will continue to use practices in the future. They created space for supportive conversations and problem solving.” [CPMM65, Intervention] “Posting things on Moodle [the College’s Learning Management System] really helped me, especially at the beginning. I wasn’t sure what others would think of me if I tried stuff in class. I still use it. Thanks!” [CPMM67, Intervention] 122 Figure 6 presents a visualization of the Intervention Group Themes for the Individual Semi-Structured Interviews. Starting with Changing frameworks of thinking (Beginning of internal changes), we can envision the evolution to Recognizing external changes, then Becoming aware of internal and external changes. Figure 6 Visualization of Individual Semi-Structured Interview Intervention Group Themes This section closely examined themes that emanated from the individual semi-structured interviews which were held over Zoom during winter 2023, for both the control and intervention groups. While performing the interviews and well as reading the Math Origin Stories, it was important to consider the responses or submissions in the contexts of the participants sharing their story. Moreover, as stories were told and retold, they evolved and were modified, often with less negative representations of participants themselves or with a decreased amount of uninteresting content (Hyde-Clarke, 2016). Storytellers adhere to what is most meaningful to them and not to the original story. In this case, whether it be in the Math 123 Origin Stories or in the individual semi-structured interviews, what is most meaningful to the participant is the most important part to be shared. What is most meaningful or prominent on their minds are feelings/opinions that are strong – positive or negative – and those deserve to see the light of day. The themes which emerged in the two previous sections attempted to honour that perspective, the voice of the participants. The following section will share the last of the qualitative data, that of collected field notes throughout the fall 2022 and early winter 2023 terms. 4.3 Researcher Field Notes The last two sections have explored and analysed the qualitative data submitted by the participants, through Math Origin Stories or via the individual semi-structured interviews. This section considers the last of the qualitative data that was acquired throughout the fall 2022 term and early 2023 winter term, that is, the Researcher Field Notes (FRN). Researcher Field Notes were taken by me, the researcher. Some of the notes were based on general observations made surrounding the two sections of the mandatory mathematics course. They were written immediately after class so that nothing would be forgotten. An abridged form of these notes is found in Appendix E (Researcher Field Notes). Figures 7 and 8 (below) include researcher field notes taken during the data analysis of the Math Origin Stories and the individual semi-structured interviews, respectively. 124 Figure 7 Researcher Field Notes Taken When Analysing the Math Origin Stories (Re-Focused Coding) Figure 8 Researcher Field Notes Taken When Analysing the Individual Semi-Structured Interviews (Intervention Group) 125 The researcher field notes included observations occurring in both sections of the course, control or intervention. When and which practices were introduced in the intervention section were also documented for posterity and self-reflection (cf. Appendix E). Furthermore, through the act of taking researcher field notes, thoughts on social barriers emerged, while trying to support prospective participants to feel comfortable and to make the incorporation of contemplative pedagogy as seamless as possible. Since this was happening in the ‘naked now’, I became very aware of some of these barriers and offered greater flexibility, such as posting flash cards and audio of practices for students who could not attend consistently or could be uncomfortable practising in front of others. A few overarching themes that emerged from the researcher field notes are as listed below. Again, labels were added to these themes to separate them from those previously determined. In this case, the label RFN is used to refer to the Researcher Field Notes. • Theme RFN1: Challenges relating to attendance, especially with international students, which occurred in both sections. For the intervention section, this led to a pivot which saw the increase of informal practices incorporated in class and more formal ones posted on the Learning Management System. (Prospective) participants could then access practices whenever they desired and had the opportunity to try formal practices in their chosen environment. • Theme RFN2: Blame took different forms – self and others. Negative self-perceptions surrounding mathematics was particularly noticeable for international students. 126 • Theme RFN3: Community building in the intervention section. Even the simple incorporation of informal practices in the classroom, including mindful dyads and self-reflection, changed the atmosphere in the course. Students were more supportive of each other in the intervention section. Students asked more questions or offered suggestions more often. Participants valued the community and its positive effects. • Theme RFN4: Vulnerability and voice were offered and respected. The sharing of participants was visceral, and they appreciated having their voices heard. All students were acknowledged and heartfully thanked by me regularly. • Theme RFN5: Transformation occurred and was felt by the intervention participants, physically and mentally. Figure 9 (below) displays a visualization of the themes emanating from the Researcher Field Notes. In particular, the themes are classified as “positives” or “negatives”. As detailed in Sections 4.1-4.3, thick and rich data emerged from the qualitative dataset, presenting a variety of diverse and related themes. Themes arising from the qualitative data analysis will be triangulated at both stages (pre- and post-test) in the next section, before advancing to the quantitative data and its analysis. 127 Figure 9 Visualization of Researcher Field Notes Themes 4.4 Triangulation of Qualitative Results The past three sections have examined the qualitative data emanating from the Math Origin Stories (MOS), the Individual Semi-Structured Interviews (ISSI), and the Researcher Field Notes (RFN) and presented emergent themes. The themes connected to each of these datasets include the acronyms provided above (MOS, ISSI, RFN) for differentiation purposes. According to Table 3 (Convergent Mixed Method Data Collection, Data Types, And Stages), the data was collected at two specific moments in time, as part of the convergent mixed methods design. Section 4.4.1 is the first of two subsections triangulating the qualitative data at different stages and its focus is on Stage 1 or at pre-test, that is, before the incorporation of contemplative pedagogy into the course. The purpose of triangulation is to consider the results from different perspectives, thereby validating the data as well as the results. Researcher Field Notes(Themes)Negatives1. Challenges with attendance2. Blaming self and othersPositives3. Community building4. Respectful sharing5. Positive transformation 128 4.4.1 Pre-Intervention Triangulation of Qualitative Data This subsection focuses on the triangulation of data gathered before the integration of practices from contemplative pedagogy in the course, that is, at Stage 1 or at pre-test. By examining Figure 10, we point out that pre-intervention triangulation occurs in the overlapping areas between the Re-Focused coding of the Math Origin Stories and the early researcher field notes. Figure 10 Visualization of Pre-Intervention Qualitative Data Triangulation Based on the themes from the Re-Focused coding of the Math Origin Stories and early researcher field notes, we posit that: 1. Participants from both the control and intervention groups feel the effects of math anxiety: Themes Re-Focused MOS1 (Struggling with math due to…) and Re-Focused MOS5 (Avoiding math and math-related fields) demonstrate common consequences of math anxiety (cf. Figure 2: Causes and Consequences of Math Anxiety), from having a Re-Focused Math Origin StoriesEarly Researcher Field NotesTriangulation 129 negative attitude towards mathematics or avoiding mathematics or mathematics related fields. 2. Participants crave a supportive environment in which to learn mathematics: Themes Re-Focused MOS2 (Feeling alone in math class and/or how one feels about mathematics) and Re-Focused MOS4 (Having a fear of making a mistake or of asking for help) illustrate the need for participants to not feel alone in their struggles with mathematics and that an inclusive and compassionate environment is made available to them. Furthermore, Theme RFN3 (Vulnerability and voice were offered and respected) recognized the need to respect each voice as well as the vulnerability offered by participants, something which requires a safe, compassionate, and inclusive environment. Figure 11 displays the results of the triangulation of the collected qualitative data (and their results) occurring before the incorporation of the intervention (at Stage 1 or pre-test). The next subsection will triangulate the qualitative data arising after the intervention was included in the course, that is, Stage 2 or post-test. Triangulating at these two moments in time emphasizes the “convergent” nature of the mixed methods design. 130 Figure 11 Results From the Pre-Intervention Triangulation 4.4.2 Post-Intervention Triangulation of Qualitative Data Analogous to Section 4.4.1, the triangulation of qualitative data occurs herein, this time post-intervention (at Stage 2 or post-test). Figure 12 demonstrates the area where triangulation occurred post-intervention (that is, in the shaded area, common to both datasets). Pre-Intervention Triangulation1. Participants feel the effeccts of math anxiety.2. Participants crave a supportive environment in which to learn math. 131 Figure 12 Visualization of Post-Intervention Qualitative Data Triangulation Based on the themes from the individual semi-structured interviews and the researcher field notes, we postulate that: 1. There was no noticeable improvement for the control group: Overarching Theme ISSI1C (Discovering negatives) shared the frustration and self-disappointment felt by the participants. In particular, feeling stuck in their predicament and the absence of change was at the root of those emotions. Furthermore, based on comments made by participants (Theme ISSI2C), some members of the control group wanted to try to improve their situation the next time around or try something different, compounding their awareness that they had not improved across the term. Finally, Theme RFN2 (Blame took different forms – self and others), though initially grounded in the Math Origin Stories, connected with the themes of self-blame or frustration with self. Individual Semi-Structured InterviewsResearcher Field NotesTriangulation 132 2. The intervention group saw a decrease in math anxiety and its effects (consequences): Theme ISSI3Ia (Decreasing math anxiety) presented supporting quotations relating to a decrease in perceived math anxiety. Moreover, as presented in Figure 2 (Causes and Consequences of Math Anxiety), math anxiety has many consequences, such as negative math attitudes, negative math self-confidence, and math avoidance. Hence, Themes ISSI1Ic and ISSI1Id (Improving math attitude & Increasing math self-confidence) as well as ISSI2Ib (Decreasing math avoidance) demonstrated a decreased effect of math anxiety. For example, a more positive math attitude and lessened math avoidance (or a desire to take further mathematics courses) are signs that math anxiety is not affecting participants in anticipated ways. Though not explicitly discussed in Section 2.1 (Math Anxiety Causes and Consequences, Theme ISSI2Ic (Positive physiological effects of practices) described decreased physiological effects of math anxiety due to practices adopted by the intervention group. Finally, Theme RFN5 (Transformation occurred and was felt by the intervention participants, physically and mentally) offered a supporting perspective of decreased math anxiety and its effects. 3. The intervention group saw the building of community: Themes RFN2 (Community building in the intervention section) and RFN3 (Vulnerability and voice were offered and respected) saw the building of community building in the intervention section and where contemplative pedagogy fostered a safe and inclusive environment. This idea is further supported by Theme ISSI3Ib (Making connections, feeling included, and building community), where conversations in dyads (or in the overall group) resulted in greater 133 awareness of shared humanity (or at least, shared feelings of mathematics) and encouraged connection and belonging. 4. The intervention group experienced positive effects from its community: Beyond the last point, the intervention group not only built community but encountered positive effects in consequence. For example, Theme ISSI2Ia (Decreased fear of trying, of making mistakes, of asking for help) became a natural consequence of feeling safe and comfortable in the course. Theme RFN3 (Vulnerability and voice were offered and respected) described an inclusive and safe environment where participants allowed themselves to be vulnerable and share their true voice. 5. The intervention group underwent positive changes/transformation due to practices from contemplative pedagogy. Beyond the positive effects from the practices listed in points 2 through 4, Themes ISSI1Ia (Becoming more open-minded) and ISSI1Ib (Unlearning) further exemplified the effect of (contemplative and reflective) practices emanating from contemplative pedagogy. Theme ISSI1Ie (Demonstrating acceptance and self-compassion) related to self-improvement (or internal changes), but acceptance and self-compassion both are intricately linked to mindfulness (and therefore contemplative pedagogy). Theme ISSI2Ic (Positive physiological effects of practices) demonstrated a variety of changes in the positive aspect for those using practices, especially in a mathematics course. Theme ISSI3c (Continuing practices and gratitude for flexibility) resulted from the desire of participants to continue with their practices and the flexibility of delivery of those practices supported their on-going use. Theme RFN5 (Transformation occurred and was felt by the intervention participants, physically and 134 mentally) offered a different yet supporting perspective on the positive changes and transformation occurring with(in) the participants. 6. The intervention group demonstrated appreciation for the flexibility of delivery. Presenting content, including practices emanating from contemplative pedagogy proved to be challenging at times. To better support the students, a stance of flexibility was undertaken. That is, practices with audio recordings were posted to the Learning Management System. This allowed participants to use the practices from contemplative pedagogy at times and places that were more convenient to them. Theme ISSI3Ic (Continuing practices and gratitude for flexibility) supported this modification and its ensuing gratitude. The six points above, which are the results of the post-intervention triangulation, are summarized and presented in Figure 13. These past two subsections considered the triangulation of the qualitative data at two specific moments in time – pre-test (Stage 1) and post-test (Stage 2), or before and after the incorporation of practices from contemplative pedagogy. Before sharing overarching emergent theories, the quantitative results will be presented in the following section to provide a numerical perspective on the research performed, guided by the overarching research question How can contemplative pedagogy support a mandatory post-secondary mathematics course? and its related sub-questions. 135 Figure 13 Results From the Post-Intervention Triangulation 4.5 Quantitative Results Until now, the focus has been on the analysis of the qualitative datasets, that is, the data emanating from the Math Origin Stories, the individual semi-structured interviews, and the researcher field notes. Since this research was performed using a mixed methods design, we now begin to explore the collected quantitative data and perform the appropriate analyses. As discussed in Methodology, several statistical approaches were used with the quantitative data, emanating from both pre- and post-test questionnaires (Math Anxiety Rating Scale, Post-Intervention Triangulation1. There was no noticeable improvement for the control group.2. The intervention group felt a decrease in math anxiety and its effects.3. The intervention group experienced the building of community.4. The intervention group experienced positive effects from its community.5. The intervention group underwent positive changes/transformation due to contemplative pedagogy.6. The intervention group demonstrated appreciation for the flexibility of course and content delivery. 136 MARS, Appendix C & Five Facet Mindfulness Questionnaire, FFMQ, Appendix D) as well as the course grades (test 1 and final exam). The first subsection considers comparisons of the pre- and post-tests for both the MARS and the FFMQ as well as the course grades, for the control and intervention groups separately. It then examines the difference between the control and intervention groups for both stages specified in Figure 3 (Visualization of Data Types and Stages), that is, Stage 1 (the pre-test) and Stage 2 (the post-test). The second subsection evaluates the overall quantitative impact of the intervention. Finally, the last subsection investigates correlation and relationships between different variables examined in this research. 4.5.1 Results from the Pre- and Post-Tests Questionnaires and Course Grades This section begins the analysis of the quantitative datasets through the use of graphs (see Figures 14 – 22). This graphical exploration examines the changes in averages values for each of the FFMQ, MARS, and course grades. Different perspectives will be undertaken, such as the comparison of stages (i.e., pre-test versus post-test) for these datasets as well as the contrast between groups (i.e., control versus intervention). The following subsections focus on each perspective and investigation separately and refer to the results of statistical data analysis presented in Appendix F for support. 4.5.1.1 Pre-Test and Post-Test FFMQ Values (Control and Intervention Separately). This subsection begins the graphical investigation of the pre- and post-test values, starting with the FFMQ, for the control and intervention groups separately. Figure 14 presents the change in FFMQ average values between Stage 1 (pre-test) and Stage 2 (post-test) for the control group. 137 Upon inspection, we see that the non-reactivity facet increased between these two stages. The observe facet stayed fairly constant, while the remaining facets (describe, non-judgement, and awareness) as well as the overall mindfulness level decreased. The averages and standard deviations for all facets of the FFMQ are presented in Table F3 (in Appendix F, Calculations). Furthermore, appropriate statistical tests (i.e., paired t-tests) were performed to determine which of the changes were significant with numerical results also displayed in Table F3. In this instance, none of the FFMQ facet averages were significantly different between the pre-test and the post-test stages for the control group. Figure 14 FFMQ Pre- and Post-Test Average Values (Control Group) 2.52.62.72.82.933.13.23.3Pre-Test Average Post-Test AverageFFMQ Pre- and Post-Test Averages (Control)Observe Describe Non-JudgementNon-Reactivity Awareness Overall Mindfulness 138 Figure 15 graphically compares the average values at the pre-test and the post-test stages for the intervention group. From the graph, we see that the describe facet decreased between these two stages. However, all remaining facets (observe, non-judgement, non-reactivity, and awareness) as well as the overall mindfulness level all increased at the post-test, when compared to the pre-test. Table F4 (in Appendix F, Calculations) displays the averages and standard deviations for all FFMQ facets as well as the results from the statistical test used to compare pre- and post-values. In the case of the intervention group, the non-judgement, non-reactivity, and awareness facets as well as the overall mindfulness levels all increased significantly post-test for the intervention group. Despite what appears in Figure 15, the describe facet is not significantly lower at post-test when compared to pre-test. In statistical terms, they are considered the same. Further discussions are undertaken in Chapter 5 to interpret these results more deeply. 139 Figure 15 FFMQ Pre- and Post-Test Average Values (Intervention Group) The subsequent subsection presents similar graphs and analogous results to those determined herein but will use the MARS dataset. 4.5.1.2 Pre-Test and Post-Test MARS Values (Control and Intervention Separately). Continuing our graphical analysis of the pre- and post-test data, Figure 16 displays the average values at the pre-test and the post-test stages for the control group. From the graph, we observe that all components of the MARS including the overall math anxiety level increased at post-test, when compared with the pre-test averages. Table F3 in Appendix F displays the averages and standard deviations for all MARS components and presents the results of the statistical tests used to determine the significance between these differences. Based on those calculations, we 2.82.933.13.23.33.43.5Pre-Test Average Post-Test AverageFFMQ Pre- and Post-Test Averages (Intervention)Observe Describe Non-JudgementNon-Reactivity Awareness Overall Mindfulness 140 conclude that the math test anxiety and number anxiety components are considered the same, while the math course anxiety and the overall math anxiety values are significantly different. In fact, the latter two values significantly increased at the post-test stage for the control. Elaborations on the consequences of these results are made in Chapter 5. Figure 16 MARS Pre- and Post-Test Average Values (Control Group) Similarly to what performed for the control group, Figure 17 shows the average pre- and post-test MARS values for the intervention group. Figure 17 displays a noticeable decline for all MARS components as well as the overall math anxiety level. In fact, based on the numerical calculations and statistical test results summarized in Table F4 (in Appendix F), ALL MARS components and overall math anxiety levels decreased significantly for the intervention group. 1.51.71.92.12.32.52.72.93.13.3Pre-Test Average Post-Test AverageMARS Pre- and Post-Test Averages (Control)Math Test Anxiety Number AnxietyMath Course Anxiety Overall 141 Figure 17 MARS Pre- and Post-Test Average Values (Intervention Group) Since there are only two course grades (test 1 and final exam) for the two groups (control and intervention), a graphical and statistical investigation will be presented considering all these aspects together in Section 4.5.1.7. Tables presenting the results of the full comparison between the pre- and post-test values (for the FFMQ, MARS, and course grades) are found in Appendix F. Specifically, Table F3 examines the difference between the pre- and post-test values for the control group, while Table F4 repeats these calculations for the intervention group. Both of these investigations are called within group analyses as we are examining one group at two instances in time. 1.51.71.92.12.32.52.72.93.13.3Pre-Test Average Post-Test AverageMARS Pre- and Post-Test Averages (Intervention)Math Test Anxiety Number AnxietyMath Course Anxiety Overall Math Anxiety 142 In the next subsection, we continue our graphical exploration by considering Stage 1 (pre-test) and Stage 2 (post-test) separately, as we compare the averages of the control and intervention groups. Those investigations are called between, as we are comparing two different groups at one specific moment in time. 4.5.1.3 Pre-Test FFMQ Values (Control and Intervention Together). Figure 18 displays the averages for each of the FFMQ facets for both the control and intervention groups at Stage 1 (or pre-test). Based on this figure, we see that the intervention averages are larger at pre-test for the observe, describe, and awareness facets as well as the overall mindfulness levels. The control group demonstrates higher averages for the non-judgement and non-reactivity facets. Despite what seems to be the case graphically, none of these differences are significant (see Table F5 for the results of the independent sample statistical tests). We note that the pre-test stage occurred before the incorporation of the intervention. Based on the results above, there are no noticeable differences between the FFMQ averages for the control and intervention groups at pre-test. 143 Figure 18 FFMQ Pre-Test Average Values (Control and Intervention Together) 4.5.1.4 Pre-Test MARS Values (Control and Intervention Together). Figure 19 exhibits the MARS pre-test averages for both the control and intervention groups. Based on this figure, we see that the intervention group demonstrated higher pre-test average values for all MARS components as well as the overall math anxiety level. When considering the numerical values and performing the associated statistical tests (results displayed in Table F5), we discovered that there was no significant different for the pre-test number anxiety component. However, the intervention group exhibited higher averages for math test anxiety, math course anxiety, and overall math anxiety levels at pre-test when compared to the control group. 2.52.72.93.13.33.5Control InterventionFFMQ Pre-Test Averages (Control and Intervention)Observe Describe Non-JudgementNon-Reactivity Awareness Overall Mindfulness 144 Figure 19 MARS Pre-Test Average Values (Control and Intervention Together) 4.5.1.5 Post-Test FFMQ Values (Control and Intervention Together). Figures 18 and 19 depicted the graphical comparisons at pre-test for both the FFMQ and the MARS datasets. Figure 20 is the first of two figures which explores the post-test values, by examining the difference for the FFMQ dataset. By considering the post-test values, we begin to contemplate the effect of the intervention. Hence, conclusions drawn in the remainder of this section will be further discussed when performing data triangulation in Sections 4.6 and 4.7 as well as when interpreting the results in Chapter 5. Figure 20 presents the graphical differences between the FFMQ averages at post-test (Stage 2) for both the control and the intervention groups. Though we observe larger values for 1.522.533.5Control InterventionMARS Pre-Test Averages (Control and Intervention)Math Test Anxiety Number Anxiety Math Course Anxiety Overall 145 the intervention group for all facets of the FFMQ, only the awareness facet and the overall mindfulness level are significantly larger (see Table F6 in Appendix F for further details). Figure 20 FFMQ Post-Test Average Values (Control and Intervention Together) 4.5.1.6 Post-Test MARS Values (Control and Intervention Together). Analogously to the previous subsection, here we explore the average post-test MARS values for both the control and intervention groups. Figure 21 reveals that the control group demonstrated higher MARS component values as well as overall math anxiety levels. When examining the data and performing statistical tests, we discover that the intervention group exhibited significantly lower MARS components (all three) as well as overall math anxiety levels. Implications of these statements are discussed in Chapter 5. 2.52.72.93.13.33.5Control InterventionFFMQ Post-Test Averages (Control and Intervention)Observe Describe Non-JudgementNon-Reactivity Awareness Overall Mindfulness 146 Figure 21 MARS Post-Test Average Values (Control and Intervention Together) 4.5.1.7 Pre-Test and Post-Test Course Grades (Control and Intervention Together). So far, we have considered the results from the FFMQ and the MARS, both at pre- and post-test, for both the control and the intervention groups. Now, we consider the pre- and post-test course grades for both the control and intervention groups. Since there was not a lot of data to consider, Figure 22 displays the results for all this information at once. Based on Tables F3-F6 (see Appendix F), we notice an increase in the intervention group values, while the control group displays a decrease. Based on calculations from Tables F3-F6, we observe that the control group experienced a significant decrease between the pre- and post-test, while the intervention group underwent a significant increase. When comparing the control and intervention groups at pre-test, the intervention group experienced lower grades, while no difference was exhibited 1.51.71.92.12.32.52.72.93.13.33.5Control InterventionMARS Post-Test Averages (Control and Intervention)Math Test Anxiety Number Anxiety Math Course Anxiety Overall 147 at post-test. Given these changes between stages, further investigations are required (and will be explored in the next section). Figure 22 Pre- and Post-Test Course Grades (Control and Intervention Groups) This subsection considered the FFMQ, MARS, and course grades for the control and intervention groups separately at pre-test and post-test, then the control and intervention groups together at Stage 1 (pre-test) and Stage 2 (post-test) individually. The purpose for these disjoint inspections was to closely examine all the facets and components, before weaving them all together. The next section undertakes that examination to fully report the impact of the intervention in this study. 7273747576777879Pre-Test Average Post-Test AverageCourse Grade Averages (Control and Intervention)Course Grade (Control) Course Grade (Intervention) 148 4.5.2 Analysis of Covariance – Intervention Impact The previous section only allowed for the comparison of two concepts at once – either pre- versus post-test OR control versus intervention. Through this lens, we were able to gather data to direct where deeper investigations were required. As referenced alongside numerous graphs, Tables F3 through F6 in Appendix F (Calculations) displayed descriptive statistics (averages and standard deviations) along with significance and effect sizes linked to appropriate statistical tests. The presented significance levels measured the difference between the various averages – either pre- and post-test averages, or control and intervention group averages. What these tables could not investigate was the overall contribution of the intervention to the FFMQ facets and the MARS components. This section explores the effects of the intervention by using a one-way analysis of covariance (ANCOVA), a statistical tool which allows us to explore the intersection of group and stage comparisons. Further details of the calculations are provided in Appendix F. While controlling for the pre-test overall mindfulness level, an ANCOVA was performed on the post-test mindfulness level, and it was determined that there was a significant increase in the post-test overall mindfulness level due to the intervention. Revisiting the graphs in the previous section (Figures 18 & 20) and comparing Tables F5 and F6 (in Appendix F), we emphasize the noticeable changes in the FFMQ facets of awareness, non-judgement, and non-reactivity. In particular, the latter two facets experienced a change in the direction of significance from positive in the pre-test (meaning that the control average was higher) to a 149 negative in the post-test (meaning the intervention average was higher). Both situations led to further investigations using an ANCOVA (see Appendix F for details) and resulted in: • A significant increase in post-test awareness attributed to the intervention. • A significant increase in post-test non-judgement due to the intervention. • No significant difference in post-test non-reactivity due to the intervention. Re-examining math anxiety and its three components in Figures 19 and 21 (above) and Tables F5 and F6 (Appendix F: Calculations), changes between the pre- and post-tests when comparing the control and intervention groups were demonstrated. Specifically, the direction of significance transformed from negative values (intervention having higher anxiety) pre-test to positive values (control having higher anxiety) post-test. Again, these changes required deeper investigations in the form of ANCOVAs (see Appendix F for significance levels and effect sizes) and resulted in the following: • A significant decrease for post-test overall math anxiety attributed to the intervention. • A significant decrease in post-test math test anxiety due to the intervention. • A significant decrease in post-test number anxiety attributed to the intervention. • A significant decrease for math course anxiety due to the intervention. Finally, though we saw a significant difference between the test 1 grades when comparing the control and intervention averages, we did not see a significant difference between the final exam grades (see Tables 10 and 11). Recognizing this change, from significant 150 to insignificant, another ANCOVA was implemented to see if the intervention could explain why the control group scored higher in test 1, but both groups scored about the same in the final exam. This test resulted in a significant increase in the final exam grade due to the intervention. Further calculations are presented in Appendix F. From all the ANCOVA tests computed above, it is evident that the intervention had an impact on certain mindfulness facets, math anxiety components, and in overall math performance. Further discussions relating these statements in conjunction with the qualitative results as well as with the research question will be elaborated upon in Sections 4.7 and 4.8. The next subsection will investigate correlations between course grades and both mindfulness and math anxiety levels. 4.5.3 Correlation and Regression The previous two subsections analysed, in depth, the differences between the control and intervention groups as well as the differences within each group, for the two self-reported questionnaires (FFMQ, Appendix C, & MARS, Appendix D) as well as the course grades. This subsection will complete the quantitative data analysis by focusing on correlation and regression analysis. That is, an investigation of associations and relationships was undertaken, with results as follows (and details provided in Appendix F): 1. The linear correlation between overall pre-test math anxiety and pre-test course grade (test 1) for all 68 participants, that is, the control and intervention groups combined resulted in a significant correlation. Figure 23 displays the scatterplot of this relationship, with the estimate 151 of the linear regression line and the coefficient of determination (R2) found in the bottom right-hand corner of the graph. Figure 23 Pre-Test Course Grade (Test 1) Versus Pre-Test Overall Math Anxiety (For Control and Intervention Groups Combined) 2. The linear correlation between pre-test math test anxiety (MTA) and pre-test course grade (test 1) for all 68 participants produced a significant correlation (see Appendix F for further details). Figure 24 demonstrates the scatterplot and includes the estimated equation of the linear regression line and the coefficient of determination. We note that the R2 value is smaller in this instance than the previous one, revealing that math anxiety beyond simply math test anxiety contributes to the test 1 grade. y = -8.3549x + 99.368R² = 0.42660.0010.0020.0030.0040.0050.0060.0070.0080.0090.00100.001.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00Test 1 versus Pre-Test Overall Math Anxiety 152 Figure 24 Pre-Test Course Grade (Test 1) Versus Pre-Test Overall Math Anxiety (For Control and Intervention Groups Combined) 3. The linear correlation between post-test overall math anxiety and post-test course grade (final exam) for all 68 participants resulted in a significant correlation (see Appendix F for detailed results). Figure 25 displays the scatterplot of this relationship, along with an estimate of the linear regression line and the R2 value. y = -7.4943x + 99.171R² = 0.37060.0010.0020.0030.0040.0050.0060.0070.0080.0090.00100.001.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00Test 1 versus Pre-Test Math Test Anxiety 153 Figure 25 Post-Test Course Grade (Final Exam) Versus Post-Test Overall Math Anxiety (For Control and Intervention Groups Combined) We note that separate investigations for the control and intervention groups were undertaken with respect to the (linear) correlation between the post-test overall math aniety and the post-test course grade (final exam). The correlation for both the intervention and control groups proved to be significant. The intervention group demonstrated an R2 = 0.3160, while the control group exhibited an R2 = 0.1590, implying that the post-test overall math anxiety level affects the final exam of the intervention group more than that of the control group. 4. The linear correlation between pre-test overall mindfulness and pre-test course grade (test 1) did not produce significant results. With specific results presented in Appendix F, no linear correlation was determined in this context for both the control and intervention groups y = -6.8561x + 90.569R² = 0.2210.0010.0020.0030.0040.0050.0060.0070.0080.0090.00100.001.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00Final Exam versus Post-Test Overall Math Anxiety 154 combined. As the intervention had not begun at this stage, combining the two groups was understandable. The scatterplot of this insignificant relationship is provided in Figure 26. Figure 26 Pre-Test Course Grade (Test 1) Versus Pre-Test Overall Mindfulness (For Control and Intervention Groups Combined) 5. The linear correlation between post-test overall mindfulness and post-test course grade (final exam) also did not produce significant results. Figure 27 presents the scatterplot of this insignificant relationship, with calculation details provided in Appendix F. y = 5.3449x + 59.13R² = 0.04080.0010.0020.0030.0040.0050.0060.0070.0080.0090.00100.001.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50Test 1 versus Pre-Test Overall Mindfulness 155 Figure 27 Post-Test Course Grade (Final Exam) Versus Post-Test Overall Mindfulness (For Control and Intervention Groups Combined) 6. The linear correlation between post-test overall mindfulness and post-test overall math anxiety resulted in a significant correlation. Calculations and significance are provided in Appendix F, while the scatterplot, along with an estimate of the linear regression line and the R2 value, are displayed in Figure 28. y = 5.5226x + 56.293R² = 0.05160.0010.0020.0030.0040.0050.0060.0070.0080.0090.00100.001.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00Final Exam versus Post-Test Overall Mindfulness 156 Figure 28 Post-Test Overall Mindfulness Versus Post-Test Overall Math Anxiety (For Control and Intervention Groups Combined) The above six points presented graphs and shared various correlations between variables to concludes the quantitative data analysis. The next section will triangulate the quantitative data before proceeding to an overall triangulation of all the mixed methods data. 4.6 Triangulation of Quantitative Results Analogous to Section 4.4 where all the qualitative data and results were triangulated, we now undertake the triangulation of all the quantitative data. Unlike Section 4.4, we will not only triangulate at Stage 1 (pre-test) and Stage 2 (post-test) separately, but we will also consider the difference between these moments in time (or Stages, as included in Table 3: Convergent Mixed Method Data Collection, Data Types, and Stages and Figure 3). y = -0.2772x + 3.8119R² = 0.20671.001.502.002.503.003.504.004.505.001.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00Post-Test Overall Math Anxiety versus Post-Test Overall Mindfulness 157 First, we begin by presenting visual representations for the results and conclusions for the quantitative data as concisely as possible. Figure 29 summarizes the significant results from Figures 14 and 16 as well as Table F3 (Appendix F) occurring within the control group. We note that when there was a significant change for the control group, it was in the opposite direction than desired; hence the use of words such as ‘worsened’. Figure 29 Pre- and Post-Test Values That Are Significantly Different (Control Group) Note: All comparisons other than those listed above are considered the same for the control group. Figure 30 summarizes the results emanating from Table F4 (Appendix F) and Figures 15 and 17, that is, the product of the investigation of the difference within the intervention group Significantly Different (Control Group)Math Course Anxiety Worsened (MARS)Overall Math Anxiety Increased (MARS)Final Exam Grade Lower Than Test 1 (Course Grades) 158 at post-test and pre-test. All values which are not included are considered the same between the pre- and post-test stages. We note that there are more circles in Figure 30 than in Figure 29, all representing positive changes for the intervention group. Figure 30 Pre- and Post-Test Values That Are Significantly Different (Intervention Group) Note: All comparisons other than those listed above are considered the same for the intervention group. Significantly Different (Intervention Group)Non-Judgement Increased (FFMQ)Non-Reactivity Improved (FFMQ)Awareness Ameliorated (FFMQ)Overall Mindfulness Increased (FFMQ)Math Test Anxiety Decreased (MARS)Number Anxiety Diminished (MARS)Math Course Anxiety Reduced (MARS)Overall Math Anxiety Decreased (MARS)Final Exam Values Increased (Course Grades) 159 Unlike Figures 29 and 30, which summarize results from within the control and intervention groups, respectively, Figures 31 and 32 outlines the results from between those groups at different moments in time. In particular, Figure 31 presents the significantly different values between the control and intervention groups at pre-test (see Figures 18 and 19 and its associated Table F5). As previously mentioned, the pre-test stage occurred before the implementation of the intervention, while the post-tests transpired after the incorporation of the intervention. Figure 31 Control and Intervention Group Values That Are Significantly Different (Pre-Test Stage) Note: All comparisons other than those listed above are considered the same for the control and intervention groups at pre-test. Significantly Different At Pre-TestThe Intevention Group Manifested Greater Math Test Anxiety (MARS)The Intervention Group Displayed Larger Math Course Anxiety (MARS)The Intervention Group Demonstrated Greater Overall Math Anxiety (MARS)The Intervention Group Revealed Lower Test 1 Values (Course Grades) 160 Figure 32 displays the values which are significantly different between the control and intervention groups at the post-test stage (based on Table F6 and Figures 20 and 21). In essence, Figure 32 provides the most meaningful information so far as it explores the differences between the control and intervention groups at the the post-test stage, that is, after the incorporation of contemplative pedagogy. Figure 32 Control and Intervention Group Values That Are Significantly Different (Post-Test Stage) Note: All comparisons other than those listed above are considered the same for the control and intervention groups at post-test. Significantly Different at Post-TestThe Intervention Group Demonstrated Greater Awareness (FFMQ)The Intervention Group Resulted in Higher Overall Mindfulness (FFMQ)The Intervention Group Exhibited Lower Math Test Anxiety (MARS)The Intervention Group Displayed Smaller Number Anxiety (MARS)The Intervention Group Demonstrated Reduced Math Course Anxiety (MARS)The Intervention Group Exhibited Lower Overall Math Anxiety (MARS) 161 By triangulating the within results with the between ones, certain discrepancies arose and led to investigations using the analysis of covariance (ANCOVA). For example, when exploring the difference between the control and intervention groups at pre-test (Table F5 and Figures 18 & 19), we saw that there were significant differences between some of the components of math anxiety (all but number anxiety) and the test 1 grade. In particular, the intervention group demonstrated significantly higher anxiety (for all but number anxiety) and performed significantly lower on test 1 at pre-test (or Stage 1). When considering the post-tests (or Stage 2), the intervention group resulted significantly higher awareness and overall mindfulness as well as significantly lower math anxiety (for all components) than the control group (see Table F6 in Appendix F and Figures 20 & 21). Investigating more deeply the effect of the intervention using an Analysis of Covariance (ANCOVA), which does not consider only ONE group or ONE stage but everything together, we concluded that there was a significant increase in post-test overall mindfulness, two of the mindfulness facets – awareness and non-judgement – and the final exam for the intervention group. Furthermore, a significant decrease for all components of math anxiety (including math test anxiety, number anxiety, and math course anxiety) for the intervention group were also observed. Figure 33 displays the impact of the intervention through the use of the analysis of covariance instigated by the triangulation of the within and between results emerging from the FFMQ, MARS, and Course Grade data. 162 Figure 33 Impact of Intervention Determined by Analysis of Covariance at Post-Test The last of the statistical tests examined correlations between variables. Significant negative (when one variable increases, the other variable decreases, and vice-versa) correlations were found between course grades and math anxiety levels as well as mindfulness levels and math anxiety. Figure 34 presents the significant correlations between variables determined previously. Elaborations on this front will occur in interpretation along with comparisons with extant literature in Chapter 5. Overall Impact of InterventionGreater Awareness (FFMQ)Larger Non-Judgement Values (FFMQ)Higher Overall Mindfulness (FFMQ)Lower Math Test Anxiety (MARS)Smaller Number Anxiety (MARS)Reduced Math Course Anxiety (MARS)Lower Overall Math Anxiety (MARS)Larger Final Exam Scores (Course Grades) 163 Figure 34 Significant Correlations This section has presented concise quantitative results along with their triangulation. The next section will connect Sections 4.4 and 4.5 for an overall triangulation of the data. 4.7 Triangulation of Mixed Methods Results Based on the triangulations presented in Sections 4.4 (Qualitative) and 4.6 (Quantitative), we now collate the information to perform an overall triangulation of all the mixed methods data and results. The triangulation occurring herein will focus on the two stages, that of Stage 1 (pre-test) and Stage 2 (post-test), further emphasizing the convergent mixed methods design. Closer inspection of the difference between those two stages will be Significant CorrelationsPre-Test Overall Math Anxiety (MARS) and Test 1Pre-Test Math Test Anxiety (MARS) and Test 1Post-Test Overall Math Anxiety (MARS) and Final ExamOverall Mindfulness (FFMQ) and Overall Math Anxiety (MARS) Post-Test 164 considered in Section 4.8 when addressing the main research question and its related sub-questions. 4.7.1 Pre-Intervention Triangulation of Mixed Methods Results As we began in Section 4.4.1, this first section focuses on the triangulation of data (and its results) before the incorporation of practices from contemplative pedagogy, that is, at Stage 1 or pre-test. Beyond the Math Origin Stories and early researcher field notes, we now also include the two pre-test questionnaires (MARS and FFMQ) and the test 1 grades. Based on this information, we pick up the threads from the qualitative data triangulation and add quantitative supports when applicable: 1. Participants feel the effects of math anxiety: From the qualitative triangulation in 4.4.1, we saw that Themes MOS1 (Struggling with math due to…) and MOS5 (Avoiding math and math-related fields) supported this statement as they represented common consequences of math anxiety. Moreover, the MARS pre-test results are correlated with the Test 1 grades with a negative linear correlation coefficient (r = - 0.6531). This means that the greater the anxiety, the lower the score on test 1. Finally, Table 6 demonstrated that the difference between the control and intervention groups for math anxiety levels at pre-test was significant for all math anxiety components, but number anxiety. That is, the intervention group perceived greater anxiety than the control group. 2. Participants crave a supportive environment in which to learn mathematics: In Section 4.4.1, Themes MOS2 (Feeling alone in math class and/or how one feels about mathematics) and MOS4 (Having a fear of making a mistake or of asking for help) supported this statement. Quantitative results connected to this concept could be Math 165 Course Anxiety (or MCA), where both groups had averages in the low-to-mid 2’s (out of a 5-point Likert Scale, where 1 = Not at all anxious and 5 = Very much anxious), meaning that they were not too anxious with the idea of a math course. However, at pre-test, intervention did have a significantly higher MCA value than the control. The two points above described the triangulation of collected mixed methods data (and its results) occurring before the incorporation of the intervention. The next subsection will examine the triangulation occurring after the intervention (or the implementation of practices from contemplative pedagogy) was included in the course. 4.7.2 Post-Intervention Triangulation of Mixed Method Results Analogous to Section 4.7.1 which considered pre-test (Stage 1) triangulation, the triangulation occurring post-test (Stage 2 or post-intervention) is considered herein. Based on the themes from the individual semi-structured interviews, the two post-test questionnaires, the final exam grades, and the full researcher field notes, we postulate that: 1. There was no noticeable improvement for the control group: Overarching Theme ISSI1C (Discovering negatives) shared the frustration and self-disappointment felt by the participants. In particular, feeling stuck in their predicament and the absence of change was at the root of those emotions. This theme is supported by results presented in Table 4 (Initial Coding Themes from the Math Origin Stories (Initial MOS)). No significant change occurred in the mindfulness facets, yet a significant increase in overall and math course anxiety was determined. Moreover, a significant decrease in course grades was attributed to the control group. The increase in math anxiety and the decrease in course grades both demonstrate a worsening and not an improvement. When comparing the 166 control and intervention groups through an analysis of covariance (ANCOVA), we further saw that intervention group saw various improvements (implying that the control did not). Finally, based on comments made by participants (Theme ISSI2C: Not giving up), some members of the control group wanted to try to improve their situation the next time around, compounding their awareness that they had not improved across the term. 2. The intervention group saw a decrease in math anxiety and its effects (consequences): Again, as shared in Section 4.4.2, Theme ISSI3Ia (Decreasing math anxiety) presented supporting quotations relating to a decrease in perceived math anxiety. This theme is supported by quantitative results emanating from Table 5 (Re-Focused Coding Themes from the Math Origin Stories (Re-Focused MOS)), which displays significant decreases in all three math anxiety components as well as overall math anxiety at the end of the term. Also, as mentioned in Section 4.4.2, Themes ISSI1Ic and ISSI1Id (Improving math attitude & Increasing math self-confidence) as well as ISSI2Ib (Decreasing math avoidance) demonstrated lessened effects (or consequences) from math anxiety for the intervention group. Furthermore, Theme ISSI2Ic (Positive physiological effects of practices) also described decreased physiological effects of math anxiety due to practices adopted by the intervention group. Moreover, as in written above, the pre- and post-test analysis for the intervention group (Table 5) demonstrated significant decreases in math anxiety levels (including all components); hence, the quantitative data supports the qualitative themes included here. Finally, when comparing the control and intervention groups through an analysis of covariance (ANCOVA), we further saw 167 that intervention group saw various significantly lower math anxiety levels for all MARS components. 3. The intervention group saw the building of community: As described in Section 4.4.2, Themes RFN2 (Community building in the intervention section), RFN3 (Vulnerability and voice were offered and respected), and ISSI3Ib (Making connections, feeling included, and building community) provided positive aspects of community in the classroom. Supporting quantitative results arise from the FFMQ post-test, as the intervention group experienced a significant increase in community-respecting qualities – overall mindfulness, non-judgement, non-reactivity, and awareness (see Table F4). 4. The intervention group experienced positive effects from its community: Again, as shared in Section 4.4.2, the intervention group not only built community but encountered positive effects in consequence. Themes ISSI2Ia (Decreased fear of trying, of making mistakes, of asking for help) and RFN3 (Vulnerability and voice were offered and respected) demonstrated this fact. Similar to what was stated in point 3 above, when consulting Figure 33 (Impact of Intervention Determined by Analysis of Covariance at Post-Test), we see that there were significant increases in overall mindfulness levels, as well as the non-judgement, non-reactivity, and awareness facets. 5. The intervention group underwent positive changes/transformation due to practices. Again, as seen in Section 4.4.2, the concept of positive change and transformation emerged in points 2 to 4 above, but also from a few other themes: Theme ISSI1Ia (Becoming more open-minded), Theme ISSI1Ib (Unlearning), Theme ISSI1Ie (Demonstrating acceptance and self-compassion), Theme ISSI2Ic (Positive physiological 168 effects of practices), Theme ISSI3c (Continuing practices & gratitude for flexibility), and Theme RFN5 (Transformation occurred and was felt by the intervention participants, physically and mentally). Finally, as will be connected in the next section when exploring the research question in depth, various quantitative tests supported this theme as well. In particular, the intervention group (the group that were exposed to the intervention) underwent a significant decrease in math anxiety levels and a significant increase in course grades, overall mindfulness as well as in two facets of mindfulness – awareness and non-judgement – when compared with the control group. These two past subsections have considered the triangulation of data occurring at two distinct moments in time – before the incorporation of the intervention and afterwards. The differentiation between these moments in time forms part of the basis of the convergent mixed-method approach, that of “convergence” of the data collection at two separate stages (Stage 1 – pre-test and Stage 2 – post-test). Based on the triangulations at the two stages as well as the conclusions drawn when comparing the two moments in time for both the control and the intervention group, the overarching research question How can contemplative pedagogy support a mandatory post-secondary mathematics course? will be explored along with its relevant sub-questions in the subsequent section. 4.8 Analysis of the Research Question This chapter has presented a wealth of results and triangulations, with a focus on the convergent mixed methods design in an effort to inform the overarching research question and 169 its relevant sub-questions How can contemplative pedagogy support a mandatory post-secondary mathematics course? Does it: a. Decrease math anxiety? b. Reduce math avoidance? c. Improve math attitudes? d. Increase math performance? After considering all the mixed method results shared in this chapter, we can state that there are many ways that contemplative pedagogy can (and did) support a mandatory post-secondary mathematics course. However, before elaborating upon this point, let us first address the related sub-questions individually in each of the subsections below. 4.8.1 Intervention Effect on Math Anxiety The incorporation of practices from contemplative pedagogy formed the “intervention” to which the intervention section was exposed while investigating the main research question How can contemplative pedagogy support a mandatory post-secondary mathematics course? The sub-question Does contemplative pedagogy decrease math anxiety? arose during the Re-Focused coding of the Math Origin Stories. The answer to this sub-question is a resounding ‘yes’, as not only did participant feedback during the individual semi-structured interviews demonstrate decreased math anxiety (e.g., Theme ISSI3Ia: Decreasing math anxiety, Theme ISSI2Ic: Positive physiological effects of practices), but the overall math anxiety level from the MARS decreased significantly for intervention group members (when compared to their pre-test values as well as when compared to the control group). In fact, even the analysis of 170 covariance (ANCOVA), which considers the overall effect of the intervention concluded that math anxiety decreased for the intervention group. 4.8.2 Intervention Effect on Math Avoidance The research instigated by the overarching research question How can contemplative pedagogy support a mandatory post-secondary mathematics course? Led to another sub-question, that of Does contemplative pedagogy reduce math avoidance? We can address this sub-question by basing our conclusions on statements made by participants in the intervention group. Referencing Theme ISSI2IB (Decreasing mathematics avoidance), we saw that participants from the intervention group not only desire to continue their path (which may involve mathematics), but some even gained confidence to consider STEM fields because of their newly acquired math confidence. Hence, we can confidently say that ‘yes’, contemplative pedagogy did reduce math avoidance. 4.8.3 Intervention Effect on Math Attitude When analysing the Math Origin Stories, another sub-question emerged: Does contemplative pedagogy improve math attitudes? The response to this sub-question is informed by themes arising from the individual semi-structured interviews. In particular, Themes ISSI1Ia (Becoming more open-minded), ISSI1Ic (Improving math attitude), and ISSI1Id (Increasing math self-confidence) all pointed toward having a more positive attitude in mathematics and all these themes were presented by the intervention group. 171 4.8.4 Intervention Effect on Math Performance Though intricately linked with math anxiety and math attitude, the sub-question of math performance emerged. In this context, math performance could be examined quantitatively through the change between the test 1 and final exam grades. Hence, in addressing the sub-question Does contemplative pedagogy increase math performance? we begin by consulting some of our figures and tables. In Figure 22 (Pre- and Post-Test Course Grades (Control and Intervention Groups)), we saw that the intervention group improved (significantly) while the control group decreased (significantly). Finally, when comparing the control and intervention groups at post-test (Table F6), we saw that a significant difference occurred between the two groups, in particular, a significant increase in final exam grades for the intervention group (based on an analysis of covariance and the t-test). Finally, when performing the analysis of covariance (ANCOVA), we recognized that intervention values were larger than control ones. All these results point to the intervention group performing better on the final exam, thereby improving math performance for the intervention group. Now that the sub-questions have been considered, let us return to the main research question How can contemplative pedagogy support a mandatory post-secondary math course? It is evident, based on the four subsections above, that contemplative pedagogy incorporated into a mandatory post-secondary mathematics course CAN decrease math anxiety, reduce math avoidance, improve math attitudes, and increase math performance. Arising from all the attempts to saturate the data, re-focus the themes, and let theories emerge, a few further observations arose. That is, when posing the overarching question How can contemplative pedagogy support a mandatory post-secondary math course? other 172 insights presented themselves. The intervention group participants saw positive transformations beyond what was detailed in the sub-questions (e.g., Themes ISSI1Ia: Becoming more open-minded, ISSI1Ib: Unlearning, ISSI1Ie: Demonstrating acceptance and self-compassion, Theme ISSI2Ia: Decreasing fear of trying, of making mistakes, of asking for help, Theme ISSI2Ic: Positive Physiological effects of practices). Furthermore, the intervention group participants not only experienced the building of community (e.g., Theme ISSI3Ib: Making connections, feeling included, and building community, Theme RFN2: Community building in the intervention section, Theme RFN3: Vulnerability and voice were offered and respected), but they also underwent positive effects from that community (e.g. Themes ISSI2Ia: Decreased fear of trying, of making mistakes, of asking for help and RFN3: Vulnerability and voice were offered and respected). These two supplementary theories extend beyond the research sub-questions and further inform the overarching question of the effect of the intervention. Figure 35 presents the culmination of data triangulation related to the intervention group as determined in the two previous sections. It further summarizes the response to the main research question and its related sub-questions. As a counterpoint to Figure 35, it is important to restate that the control group, based on qualitative data analysis, suffered from disconnection, from self, from others, and from mathematics (see Figure 5: Visualization of Individual Semi-Structured Interview Control Group Themes). Furthermore, the quantitative data supported either a stagnancy or worsening in math anxiety components and course grades. Hence, not only did the intervention group improve, when compared to the control group, it flourished. 173 Figure 35 Results of the Research All in all, the investigation of the overarching question How can contemplative pedagogy support a mandatory post-secondary math course? produced results, analyses, and triangulations which conclude that the incorporation of contemplative pedagogy into a mandatory post-secondary mathematics course had multifaceted positive effects for the intervention group. Discussions related to these statements are elaborated upon in the next chapter. How can contemplative pedagogy support a mandatory post-secondary math course? By increasing math performance By improving math attitudesBy decreasing math anxietyBy reducing math avoidanceBy building community, resulting in positive effectsBy creating positive transformation 174 Chapter 5: Analysis and Discussion Chapter 4 presented the results from the convergent mixed method study examining the impact of introducing contemplative pedagogy into a first-year, mandatory post-secondary mathematics course. As presented in Table 3 (Data Collection Types and Stages) as well as Figure 3 (Visualization of Data Types and Stages), a variety of mixed method data was collected for this research during two stages. This first data collection, the Math Origin Stories, along with its Re-Focused coding laid the foundation for future data collection and analysis. Further data was collected throughout the fall 2022 and winter 2023 terms, including researcher field notes (RFN, Appendix E), two pre- and post-test questionnaires (MARS, Appendix C & FFMQ, Appendix D), course grades (test 1 and final exam), and individual semi-structured interviews (ISSI, Appendix B). From this data, results emerged from different perspectives and were triangulated to emphasize the conclusions. This chapter discusses the results, establishes links with the existing literature, recognizes the limitations of this study, and introduces possible future directions. 5.1 Summary of Research Findings The questions guiding this research were How can contemplative pedagogy support a mandatory post-secondary mathematics course? and Does it [contemplative pedagogy]: a. Decrease math anxiety? b. Reduce math avoidance? c. Improve math attitudes? d. Increase math performance? 175 As presented in Chapter 4, Figure 35 presents a visualization of the results from the intervention group. It is shown here again, labelled Figure 36. Figure 36 Results of the Research Reiterated Data analysis indicated that the incorporation of contemplative pedagogy into a mandatory mathematics course demonstrated a significant decrease in math anxiety (and all its components), a significant increase in overall mindfulness (as well as facets awareness and non-judgement) and the post-test course grades (final exam). Furthermore, a significant correlation between overall mindfulness (from FFMQ) and overall math anxiety (from MARS) was present. As determined in Section 4.8, the triangulation of all the mixed method data indicated that the control group did not have the same experience as the intervention group. The 176 intervention group demonstrated a decrease in math anxiety, a reduction in math avoidance, an improvement in math attitudes, an increase in math performance, a building of community, and the experience of positive transformation. The next section will interpret these results and include pertinent implications. 5.2 Interpretation of Results By picking up an old thread, my positionality played an important role when performing the research and analyzing the collected data. The rest of this dissertation will continue to consider that, though I am currently “re-storying” what participants have shared, it is their voices which need to be heard and amplified. In fact, the use of “we” in lieu of “I” will continue in this chapter as I carry and embody participant voices. The Math Origin Stories and the individual semi-structured interviews demonstrated that many participants have felt anxious, frustrated, or stuck because of their experiences involving mathematics and/or by mathematics courses for a long time (in the span of their lifetimes). Their voices and their quotations need to reach relevant and interested audiences. That is, those who can make a difference for them or others like them in the future. Moreover, according to Reid et al. (2017), a thoughtful consideration of the dissemination of results is important. In this context, both the interpretation and the dissemination were undertaken in an authentic and compassionate manner. Furthermore, it is of paramount importance that this last chapter is accessible to as many readers as possible. Part of being accessible means choosing language that is inclusive and not elitist. In fact, according to Meyer (2001), “there is an abundance of ’10-dollar words’ (p. 101) within academia” (p. 101, cited in Kovach, 2010, p. 41). In consequence, I would like to adhere to 177 Thomas’ “Uncle Paul/Auntie Doris” principle (cited in Kovach, 2018): “a good indicator of whether your research is relevant and makes sense to the community is the extent to which Uncle Paul and/or Auntie Doris get what you are saying about your research. Coming full circle, we refer back to Kovach’s three questions (see Section 4.1: Math Origin Stories) and use them as a framework for interpretation, that is: What are you saying? Why does it matter? Why are you presenting it the way you are?” (Kovach, 2018, p. 228, emphasis added herein). I neither have an Uncle Paul nor an Auntie Doris, but specific names are not of importance in this context. In the years while working on this doctorate, I have presented many times, in many situations, and in diverse contexts. The idea which has floated to the top of these actions relate to accessibility and inclusivity. It was always important to manage vocabulary and statements, to never sound pompous or condescending to the audience. Considering this approach and how it aligns with my positionality, the following section will present the interpretation of the research guided Kovach’s (2018) three questions. 5.2.1 What Are You Saying? After writing over four chapters, it is easy to get lost in the weeds. I am reminded of working through long math problems years ago. Once I was three-to-four pages into a solution, when I felt that I was floundering and had lost direction. What was I even looking for or trying to solve? When I taught Differential Calculus, this concept arose a number of times. During assessments, questions would have multiple parts in an effort to decrease this blockage and guide the student, especially in these stressful situations. Given the ideas surrounding 178 contemplative pedagogy, this very place and moment in the dissertation requires grounding. What am I saying? All of this began with a curiosity to support the mandatory post-secondary mathematics classroom, for students and educators alike – once in a class, you are stuck together, whether you want to be or not. So, what can be done to improve that experience for all? Driven by insights provided by extant literature (see the next section) as well as my own (second) master’s research, I truly wanted to improve the environment and the connection in the classroom. Beaten down by the sight of struggling students, I wanted to help. Mathematics is not a monster, yet many students saw it so. Given my own experience of pausing and checking in with myself, I wanted students to have that same opportunity. Instead of drowning in mathematical concepts, could we prevent the weight of mathematical overwhelm, and transform it into levity and positivity? Understanding full well the effect of math anxiety and its intimate connections (see Figure 2 from Section 2.1 presented again below, and laelled Figure 37). It was important to consider the 17 arrows demonstrating the links between the causes of math anxiety. Instead of considering cause or consequence, I believe that each arrow is an opportunity, a chance for improvement and to augment the life changes of students. Math anxiety has been studied for almost 70 years, primarily by psychologists and those in (math) education. Mathematicians have commented on the apparent connection between math anxiety and math avoidance or 179 math performance6. Few have suggested solutions. This research attempts to address this gap, that is, what are possible solutions? Figure 37 Causes and Consequences of Math Anxiety Revisited Returning to the results, both qualitatively and quantitatively, we concluded that the intervention group experienced a decrease in math anxiety. This means that participants felt more at ease in a post-secondary mathematics course; they underwent fewer physiological 6 Math avoidance and math performance are the two fundamental consequences of math anxiety, particularly through the lens of post-secondary math educators. 180 responses and less of the consequences discussed in Section 2.1 (Math Anxiety). The latter statement, which is explored in detail in Section 5.3, reminds us of the interconnective nature of math anxiety along with its causes and consequences (as demonstrated in Figure 37 above). Again, any opportunity to decrease math anxiety (or any of its causes or consequences) is worth investigating and amplifying as the ripples are large: for example, feeling better about oneself, not just in math, but also in life (Tobias, 1993). This transformation from negative (math) self-esteem to positive (math) self-esteem also involved a factor of acceptance of oneself. In Section 4.2, through the qualitative analysis of the individual semi-structured interviews, it was determined that some intervention group participants no longer wanted to purposefully avoid mathematics courses or fields, where avoidance is one of the main consequences of math anxiety (Hembree, 1990; Ma, 1999; Perry, 2004; Tobias, 1993). In fact, some participants desired greater exposure to mathematics in the future. This result is meaningful as less students avoiding mathematics fields and careers implies a greater likelihood of more of students being interested in STEM fields and careers. At a time when both the use of automation and artificial intelligence are increasing and solutions for the climate crisis are needed, satisfying the demand for more students enrolled in STEM courses and later working in STEM fields would be valuable to society as a whole (Penprase, 2020). From the individual semi-structured interviews, we learned that the participants of the intervention group improved their attitude toward mathematics. It is not difficult to imagine that having fewer negative attitudes in a classroom could foster a more productive and welcoming environment. Instead of feeling the weight of mathematical concepts, students may 181 be able to be more open-minded and flexible with course content asking less often “What do I need to learn this for?” (generally accompanied by derision). By attaching fewer negative thoughts to mathematics, one would be more likely be more tolerant of including mathematics in their lives. Having a more positive attitude implies, perhaps, being more willing to ask good questions or to take mathematical risks and attempt assigned problems. The more someone spends time doing mathematics, the more likely they will build mathematical skills (thereby decreasing math anxiety and improving math performance, as described in Figure 2). Finally, for the last sub-question, quantitatively, we determined that the intervention group performed better on their final exam. Overall, doing better in mathematics is interconnected with many factors. From Section 2.1 (Math Anxiety), we know that an increase in math performance is correlated with both decreased math anxiety as well as reduced math avoidance. Returning to the main research question and based on conclusions drawn at the end of Chapter 4, there are a multitude of ways that contemplative pedagogy can support a mandatory post-secondary mathematics course. The first ways are stated and interpreted above – decreasing math anxiety, reducing math avoidance, improving math attitudes, and increasing math performance. Beyond these previously discussed benefits, there was also the building of community as well as positive transformations being experienced by the intervention group participants. From the qualitative data analysis, a theme linked to the building of community, with both peers and educators, was discovered. This theme offers benefits as well, such as feeling more comfortable asking for help, feeling safe to be vulnerable with others, being less 182 judgemental with self and others, thereby decreasing fears of making mistakes alone or with others and increasing confidence to take risks in mathematics (such as attempting a problem or having less fear of being wrong when solving a problem), and being less reactive. The built community increased feelings of inclusion (and having similar perspective on mathematics to others), hence participating more in classroom conversations and asking questions more frequently as well as overall safety. Fievre (2021) defines safe spaces as “environments where students feel the freedom to make mistakes without lasting judgment or ridicule and where they can engage in critical, honest, civil, and challenging discussions about sensitive topics” (para. 2). This definition amplifies the importance of not judging others and not reacting to promote safe spaces and how increases in non-judgement and non-reactivity could support such environments. Another benefit connected to the research question, via both qualitative and quantitative data analysis, is that the intervention group underwent positive changes and/or transformation due to practices from contemplative pedagogy. The changes and/or transformations included becoming more open-minded, unlearning, demonstrating acceptance and self-compassion, decreasing fear of trying, of making mistakes, of asking for help, and positive physiological effects of practices. By being more open-minded, participants may be more patient with the content instructed, waiting before asking the ageless question “when will I ever use this?” so that they may answer it themselves. This characteristic also encourages possible group work or at least consulting others when stuck or desiring validation when work is complete. According to Oxford Languages (n.d.), to unlearn is “to discard (something learned, especially a bad habit or false or outdated information) from one's memory”. By unlearning, 183 participants demonstrated that they self-reflected and pondered their own beliefs surrounding mathematics, based on their own lived experiences. By shedding old beliefs, habits, or stereoptypes, together with being more open-minded, participants display the ability to transform and grow, perhaps fostering further positive changes or learning. All the benefits presented in this subsection, whether relating to the overarching research question or its related sub-questions, are intimately connected with the intervention group. These positive aspects are in stark contrast to the control group which carried its negativity throughout the term and felt stuck in their circumstances. Hence, this polarization of perspectives and outcomes in view of a mandatory post-secondary mathematics course supports the incorporation of contemplative pedagogy for the betterment of students. 5.2.2 Why Does It Matter? The second question suggested by Kovach (2018) is Why does it matter? This is an excellent question, bringing reality to the forefront. Of course, it is natural to study what is of interest to us, but what is of interest to us personally may not extend outwardly. I believe that this question needs to apply to the greater good, not only our own interests. As described in the Introduction, I have seen first-hand the effects of mandatory post-secondary mathematics courses on students. My purpose is not simply to alleviate my own pain in observing these reactions, as I can no longer bear witness to this oppression. Instead, my desire is to support students so that they no longer need to experience this negativity and to lead other educators in this effort. Hence, the overarching research question found in this dissertation How can contemplative pedagogy support a mandatory post-secondary mathematics course? is based on the principle that some students have challenges when taking a mandatory post-secondary 184 mathematics course. The consideration was how best to incorporate contemplative pedagogy to support those same students. The goal was to investigate the research question via a convergent mixed method study regarding whether the incorporation of contemplative pedagogy could improve their overall mathematical experience. The triangulation of researcher field notes, the Math Origin Stories, the individual semi-structured interviews, the pre- and post-tests of the FFMQ, the MARS, and the course grades identified different ways that contemplative pedagogy supported mandatory post-secondary mathematics students, thanks, in part, to its related sub-questions, but went even beyond. As evidenced by the Math Origin Stories, most domestic participants (93% or 26 out of 28) had negative experiences in previous mathematics courses. These courses were taken primarily in the K-12 system or during adult upgrading in college. Due to differences in educational systems, international students did not necessarily have the same experience. In fact, some international participants reported positive experiences from their previous mathematics courses (67% or 12 out of 18). As written earlier, math anxiety research has been mostly focused on its causes and consequences (see Figure 2, Causes and Consequences of Math Anxiety). By providing interventions to address math anxiety, the results of this doctoral research fill a gap in the literature. In particular, this could be beneficial in supporting the ever-growing need in STEM fields due to the rise in digital technology, artificial intelligence, and the climate crisis (Penprase, 2020). The results determined in this dissertation demonstrated that contemplative pedagogy could act as a mitigating factor for math anxiety (and thereby, by transitivity, its consequences). Furthermore, the building of community and the transformative aspects of the practices could 185 make mathematics courses and, in consequence, STEM courses and fields more appealing and more accessible to students. For example, as uncovered in the individual semi-structured interviews, engaging in practices from contemplative pedagogy with peers (and instructor) alleviated feelings of isolation, anxiety, and discomfort. Furthermore, changes in math attitude supported confidence building, encouraging participants to take chances when problem solving (that is, attempting a problem with less anxiety of getting it wrong) and increases in non-judgement meant that participants felt more comfortable asking questions or offering answers in class. It felt as though it had become a safe and inclusive safe, where their thoughts and ideas were respected. That kind of environment is certainly appealing and could invite further study. As discovered through the Math Origin Stories, there were many reasons for discomfort in a mandatory post-secondary mathematics course, for example, feelings of anxiety due to the math content or feeling alone in class or in one’s perspectives surrounding mathematics. Unlike others who have studied math anxiety (e.g., scholars in math education or in psychology), Perry (2004), a mathematician, presented anecdotal evidence emphasizing the difference in intensity of student math anxiety that he observed in his own post-secondary mathematics courses: ranging from ‘moderate test anxiety’, to students having ‘mixed feelings’ toward math, to a student having an insensitive or incompetent math teacher at some point during their earlier education, to a student having limited mathematical understanding or ability, to extreme anxiety including physiological effects. From mild to extreme, these observations agree with Beilock’s (2020) statements related to the impact of math anxiety on post-secondary students, in particular, that more than 90% are affected by math anxiety and 11% require counselling due 186 to the extreme consequences. In the end, math anxiety and its effects not only command attention, but also call for solutions for those impacted. As discussed in Section 2.1 (Math Anxiety), comprehensive and evidenced-based studies in the area of math anxiety haved primarily focused on the causes and consequences math anxiety (Ashcraft, 2002; Choe et al., 2016; Hembree, 1990; Ma, 1999; Namkung et al., 2019; Sonnert et al., 2020; Shishigu, 2018; Tobias, 1993). The intervention incorporated in this research offers a possible support for students, to prospectively break the cycle between causes and consequences of math anxiety. Again, positively affecting (or minimizing the negative effect) of any of the arrows from Figures 2 and 37 can have positive reverberations on one or many of the factors (e.g., math anxiety, math attitude, math performance, math avoidance). Outside of the interconnected web of math anxiety and its causes and consequences (Figures 2 and 37), another reason to incorporate contemplative pedagogy would to counter the feelings of isolation or being different because of personal sentiments toward mathematics. Building community and increasing the feeling of inclusion in the classroom can no doubt impact all students. Finally, as mathematicians, math teachers, and academics in mathematics education alike recognize the importance to address math anxiety at an early age (if Namkung et al. (2019) is correct, by grade 4), it is never too late to affect change. In fact, as will be discussed in Section 5.3, the mature students in this research experienced heightened number anxiety at the beginning of the course. This statement means that even when exposed to simple 187 mathematical concepts which necessitate the use of numbers (such as counting), anxiety was induced. Based on the literature and my own lived experience, high math anxiety and negative math attitudes are a ‘chicken or the egg’ scenario. In this case, as was investigated in Appendix F, Section F.1, based on feedback from the Math Origin Stories, teachers are the single largest influence on participants. For 93% of domestic participants, that influence was particularly negative, while 67% of international participants experienced positive impacts. These results indicate that, even students at a later age, such as those enrolled in post-secondary courses, need to consider their anxiety and attitude toward mathematics. After all, these perceptions are carried into the classroom, along with the course material. Furthermore, Paulos (1988) believed that “Some of the blame for the generally poor instruction in elementary schools must ultimately lie with teachers who aren’t sufficiently capable, and who often have too little interest in or appreciation of mathematics. In turn, some of the blame for that lies, I think, with schools of education in colleges and universities which place little or no emphasis on mathematics in their teacher training courses” (p. 102). In the end, even this aspect – the reasons and experiences leading to negative attitudes and math anxiety in elementary school teachers – is multi-faceted. Hence, Why does it matter? It matters because we can help. As we have discovered throughout this research, ways to improve the mandatory post-secondary mathematics course experience are available. We also discovered that teachers are the single largest influence on student mathematical perceptions. We, as educators of mathematics, as well as educators of 188 future educators of mathematics (and schools of education) have a responsibility to defy this belief and improve the state of mathematical instruction for all. 5.2.3 Why Are You Presenting It the Way You Are? Following a discussion surrounding the significance of this research, the third and final question suggested by Kovach (2018) is Why are you presenting it the way you are? I believe that this question is grounded in our positionality and is an expression of our genuine intentions. As I truly do desire to support students and their instructors in mandatory post-secondary mathematics endeavours, the format of my presentation should align with those intentions. What this means is that the results from my doctoral research MUST be approachable, accessible, and implementable with minimal investments of time when understanding or using the intervention. I understand both Chapter 4 and Appendix F have presented a gluttony of statistics and numerical conclusions. Unfortunately, for many readers (if I have not lost them all by now), this may have caused the exact discomfort that my research is trying to avoid. An attempt to compartmentalize those numbers was made as they are needed to offer quantitative credibility to the work and results contained herein Instead, in Sections 5.1 and 5.2, I have attempted to present more streamlined and thought-provoking statements in lieu of numbers. To have been able to present results in this manner, I was facilitated by the trust and vulnerability offered as a gift from the participants. I bore witness to their challenges and oppression by mathematics through the Math Origin Stories (and later in interviews with the control group participants) and celebrated in their transformation and triumphs (for those in the intervention group) during the individual semi- 189 structured interviews. The qualitative data provided the jarring, heart-breaking, and elating results which I embodied. This experience of witnessing and embodiment was only possible by exploring grounded theory according to Charmaz (2006). Glaser (2002) claimed that the more interpretivist and subjective interview techniques required by Charmaz is an “unwarranted intrusion of the researcher” (Glaser, 2002, para. 8), making the “researcher’s interactive impact on data more important than the participants’” (Glaser, 2002, para. 20). I disagree with Glaser’s (2002) claim, as I feel that the relationality during the interview process supported greater communication (and connection) between researcher and participant as well as the divulging of deeper truths. Though I, as the researcher, may have had an impact, I believe that it only strengthened my will to empower the participants’ voices, not overshadow it. This section interpreted the results of the study, guided by the three questions posed by Kovach (2018) and based on the overarching research question How can contemplative pedagogy support a mandatory post-secondary mathematics course? The next section will discuss the effects of these conclusions in conjunction with the extant literature. 5.3 Connections to Extant Literature The two previous sections summarized the main findings of this doctoral research and interpreted them. In this section, we compare the results and conclusions from this dissertation with the extant literature. We begin with statements related to math anxiety, followed by ones relating to the intersection of mathematics and contemplative pedagogy (at the post-secondary level). 190 5.3.1 Math Anxiety Literature Comparisons To compare the results found herein with the extant literature, we re-consider factors relating to math anxiety presented in Figures 2 and 37 (Causes and Consequences of Math Anxiety). To simply the discussion, an abridged version of those figures appear below, labelled Figure 38, with only eight of the 17 original factors being included. Notice that the numbering is not sequential but based on the original numerical labels from Figures 2 and 37. Figure 38 Abridged Version of Figures 2 and 37: Causes and Consequences of Math Anxiety 191 As the primary focus of this figure is Math Anxiety, we consider our comparisons with the relevant extant literature through that lens and not elaborate on all points originally presented in Figure 2 (then again in Figure 37). Also, for the purpose of concision, references made to specific authors and articles are not repeated here unless it was absolutely required (see Section 2.1: Math Anxiety for detailed references). As we obtained both qualitative and quantitative methods data surround math anxiety, there are a number of statements that can be made to support the applicable points listed in Chapter 2, specifically: • Factors 1 and 13: Using feedback from the Math Origin Stories, for example, Theme Re-Focused MOS3 (Changing attitude and/or confidence in math due to one person or experience), we confirm that adults and peers as well as instructor behaviours influence participants view of mathematics. As is demonstrated in Section F.1, not only are teachers and family/friends sources of influence on participant’s mathematics attitude (which is intimately connected to math anxiety), but teachers of domestic participants have a had particularly negative influence, when compared to international participants. Qualitatively, we posited that the themes related to not feeling included or belonging or having a fear of asking for help are related to instructor behaviours and attitudes. • Factor 2: In this case, through the Math Origin Stories, we discovered high math anxiety and negative mathematics attitudes occurred in conjunction, as expressed by Theme Re-Focused MOS1: Struggling with math due to: a) math anxiety and c) having a negative math attitude and/or feeling stuck in math. Furthermore, after the intervention, Theme ISSI1Id (Improving math attitude) and Theme ISSI2Ib (Decreasing math anxiety) presented themselves in tandem. 192 • Factor 5: The effect of math anxiety on working memory was demonstrated when the concept of blanking was used in the Math Origin Stories (especially Re-Focused MOS1: Struggling with math due to… d) blanking out when writing a test or exam). The connections, after the intervention, was made through Theme ISSI2Ic (Physiological effects of practices), where one participant stated that they no longer “blanked” after breathing. • Factor 7: The relationship between math anxiety and math performance has been well-documented and is summarized as follows for the meta-analyses considered, where n represents the number of included studies, r, the sample linear correlation coefficient, and R2, the effect size. a. Hembree (1990), post-secondary, n = 122 studies, r = -0.27 and R2 = 0.07. b. Ma (1999), grade school, n = 26, r = -0.27. (No R2 was provided.) c. Namkung et al. (2019), school aged, n = 131 studies, r = -0.34, R2 = 0.115. As was presented in Chapter 4, though this was not a meta-analysis, we also determined a negative correlation between math anxiety and mathematics performance. When measured at the pre-test stage (that is, the MARS pre-test value along with the test 1 grade), we obtained, with n = 68 participants, r = -0.6531 and R2 = 0.4266. At the post-test (that is, the MARS post-test value along with the final exam grade), we obtained, with n = 68 participants, r = -0.5621 and R2 = 0.3160. • Factor 9: The effect of math anxiety on math avoidance presented itself in the Math Origin Stories as it was clear that participants had avoided certain math and STEM programs and careers. Moreover, the direct relation in the opposite direction arose 193 from the intervention group with Theme ISSI2Tb (Decreasing avoidance). In particular, CPMM49 even quoted “I feel more confident, so I need to rethink my major. I’d really like to become a doctor. I think I can do it now.” When math anxiety is minimized, then math avoidance can be reduced as well. • Factor 11: The concept that math anxiety is not simply test anxiety was validated when examining the average values of the pre-test MARS (Table F3 for the control group and Table F4 for the Intervention group). Consulting these two tables, we see that the math test anxiety (MTA) value is higher than the two other categories (number anxiety, NA, and math course anxiety, MCA). However, the linear correlation performed on the test 1 grade vs MTA pre-test value had a smaller effect size (R2 = 0.3706) and a weaker linear correlation coefficient (r = -0.6088) than when including the overall MARS pre-test value (n = 68 participants, r = -0.6531 and R2 = 0.4266). Emphasis also arises from the qualitative data analysis as well as Themes Re-Focused MOS4 (Having a fear of making mistakes or of asking for help) and MOS5 (Avoiding math and math-related fields) relate to regular course/classroom actions, not just tests. • Factor 14: As students age or time passes since their last mathematics course, math anxiety, especially number anxiety increases. In Appendix F, the number anxiety levels (from the MARS) of mature students were compared with those of non-mature students (see Section F.2). This comparison resulted in a significant difference between the number anxiety levels, with mature students having greater number anxiety. In fact, there was an increase of between 1.73 and 2.44 on the 5-point Likert scale for the mature student group, a substantially larger value than the non-mature group. We 194 further note that when considering the overall math anxiety level and not simply number anxiety, the mature participants scored between 0.36 and 1.47 higher on average on the 5-point Likert scale when compared to the non-mature average. This subsection considered the most relevant points from Figures 2 and 37 (Causes and Consequences of Math Anxiety), abridged above in Figure 38, and compared results emanating from this research. Even though there are a number of factors demonstrating the interconnectivity of math anxiety and its causes and consequences, it is important to note that a positive or even simply a mitigating effect on one of the arrows can greatly impact a student. The following subsection will consider conclusions drawn from the intersection of contemplative pedagogy and math anxiety. 5.3.2 Intersectionality Comparisons The previous subsection considered the interconnectivity of math anxiety with its causes and consequences. Here, we will examine conclusions made in relation to the intersection of math anxiety and contemplative pedagogy in the extant literature. Galla and Wood (2012) stated that the effects of math anxiety on math performance could be mitigated if students regulated their emotions. In the context of this research, as contemplative pedagogy (including mindfulness) has been linked with emotional regulation (Mackenzie, 2015; Shapiro et al., 1998), we determined the connection between math anxiety levels and overall mindfulness levels from the FFMQ as well as correlation between math anxiety and math performance for the intervention group. Hence, our results agree with the statements from Galla and Wood (2012). 195 Other research examining the intersection of contemplative pedagogy and math anxiety was Brunyé et al. (2013). They determined that short-term contemplative pedagogy (in particular, guided breathing) did minimize math anxiety, a result which was confirmed and extended in this context. Bellinger et al. (2015) examined the relationship between mindfulness and math performance (particularly in stressful situations) in a lab setting. Math performance was not measured by grades, instead it was related to math accuracy. Though we did investigate the direct correlation between mindfulness and math performance (based on FFMQ and course grades), none of our results were significant. We speculate that mindfulness levels, as measured by the FFMQ, indirectly affect math performance, in part, due to its effect on math anxiety (as stated above). However, when considering the effect of the intervention (or contemplative pedagogy, including mindfulness) on math performance at the post-test stage (the final exam) through the analysis of covariance, we obtained significant results. Samuels and Warner (2019) determined that their intervention of growth mindset videos and mindfulness techniques decreased math anxiety, when compared to their control group. Their intervention group also reported greater self-reported confidence in mathematics as a result of their intervention. Though we did not work with growth mindsets specifically, Shapiro et al. (2011a) expanded upon the concept of neuroplasticity and its connection with contemplative pedagogy, specifically mindfulness techniques. When analysing this mixed method data from this research, we obtained similar results, where our intervention group experienced decreased math anxiety levels (demonstrated both quantitatively and quantitatively). 196 Finally, Shayer (2020) represents the last of the intersecting research and was used as a foundation for this doctoral research. Shayer (2020) focused on a case study involving a small, gendered sample where contemplative practices were introduced outside of post-secondary mathematics courses. Shayer concluded that contemplative practices did reduce math anxiety at the post-secondary level, in that local and specific context, based on changes in the FFMQ pre- and post-tests as well as participant individual semi-structured interviews. Unfortunately, no quantitative measures of math anxiety or math performance were collected, a noticeable absence in the field of math anxiety literature. This doctoral study extended the results from Shayer (2020) to a larger and more diverse sample. Furthermore, extra qualitative, but especially quantitative data was collected to be able to generalize as well as deepen the conclusions. In this context, we determined that practices from contemplative pedagogy did reduce math anxiety at the post-secondary level, extending and compounding the results speculated therein. As was shared in Section 4.5 and Appendix F, we have also elaborated upon the significant effects of the intervention on math anxiety and math performance, but also the effects on math attitude, math avoidance, community building, and positive transformation. This section completes the comparison of the results determined in this research with those found in other studies, emphasizing results such as the importance of decreasing math anxiety and the diversity of positive effects that contemplative pedagogy may have at the post-secondary level, especially in mathematics courses. The subsequent section will present limitations encountered in this research. 197 5.4 Limitations So far in this chapter, we have shared the summary and interpretation of the results produced by asking the question How can contemplative pedagogy support mandatory post-secondary mathematics courses? as well the comparison of these results with existing literature. To provide balance as well as credibility to our statements, it is important now to divulge limitations (perceived as well as real). The limitations arose in different contexts, in particular, institutional and social barriers as well as limitations that emerged as a result of the data collection and data analysis process. 5.4.1 Institutional and Social Barriers The first set of limitations are linked to ones encountered simply by performing the research at an institution. To effectively incorporate contemplative pedagogy in any post-secondary course, a ‘buy-in’ from colleagues as well as administration is important. Furthermore, the fact that the incorporation of contemplative pedagogy was not advertised meant that students who did not wish to participate still found themselves in a course offering contemplative pedagogy, while students who did want to participate could not join the intervention section once the study was announced. As the researcher on this project as well as the course instructor, I did not know who was participating, to minimize undue influence. However, this did limit the way that certain practices from contemplative pedagogy were presented, especially in the classroom. Further institutional barriers included low (in person) attendance, especially by international students. Until November 2022, the number of allowable hours worked by an international student doubled, from 20 to 40. Even during the month of October 2022, it was 198 clear that students were applying for supplementary work (see Appendix E). Increases in absenteeism is not simply a local issue. In fact, in the 2022-2023 academic year, absenteeism more than doubled in the U.K. when compared to pre-pandemic levels (Public First Report, Sep 2023). Furthermore, connections have been maded between anxiety levels and school absenteeism (Finning et al, 2019). Seeing as I am a mathematics educator, my department is filled with mathematicians and statisticians who may not be interested in contemplative pedagogy and may have concerns themselves of someone including it in the department. It can be difficult to convince the utility of contemplative pedagogy to those in mathematics, as self-reflection, journaling, and meditating may not align with a more mathematical and logical approach. Moreover, Grace (2011) recognized that contemplative pedagogy is not for everyone, students and educators alike. Hence, even if contemplative pedagogy does support a mandatory post-secondary mathematics course, there may not be much interest by the post-secondary mathematics community. This will certainly be a hurdle that will need to be addressed, with a focus on the scientific evidence, for both students and educators. As mentioned in the Researcher Field Notes (Appendix E), negative attitudes from colleagues, including department members, not believing that contemplative pedagogy is valuable7, were occasionally present. Unfortunately, this reaction from colleagues could negatively impact future participants in similar research, and definitively changed the warm and welcoming environment to a more judgemental one. 7 Is not interesting that “others” (including educators) effect students with their negative mathematics attitudes, but mathematicians do similar things with contemplative pedagogy? 199 Based on the Researcher Field Notes (Appendix E), I perceived the discomfort of students at the idea of participating in practices from contemplative pedagogy, especially in front of new classmates, the instructor, or strangers to the classroom, such as students or instructors before or after the class. Moreover, from CPMM67, from the intervention group, as shared in Section 4.2.2.3 (Theme ISSI3I): “Posting things on Moodle [the College’s Learning Management System] really helped me, especially at the beginning. I wasn’t sure what others would think of me if I tried stuff in class. (…)” (emphasis added). This discomfort can be perceived as a limitation for the prospective participant, but particularly, the instructor as the latter has the responsibility to try to set the best conditions for inclusion in the classroom. For example, being exposed to negative attitudes relating to contemplative pedagogy from other instructors or peers could cause apprehension in engaging in practices. To mitigate this limitation, posting practices on the learning management system quickly affected the classroom dynamic in a positive manner. The first few classroom discussions helped as well. Before then, however, I did make sure that those who did not wish to participate in view of others always had a pertinent math problem to work on in class during practices or discussions (see Appendix E). Some students may have been uncomfortable at the thought of contemplative pedagogy being introduced in the course, especially with the previously described concerns relating to commodification of contemplative pedagogy (especially mindfulness). It was, therefore, very important for the teaching assistant to present evidence-based information when introducing the research and reiterated these ideas when consent was explained. 200 The following subsection will present limitations related to the data collection as well as the analysis of that data. 5.4.2 Data Collection and Data Analysis Limitations Limitations due to data collection often occur in research, and they did here as well. From a statistics perspective, the sampling used in the research, though it was purposeful and specific, was equivalent to Volunteer Sampling. This sampling type affected, particularly, the individual semi-structured interviews, as only 10 out of 36 control group members participated. In statistics (Moore et al., 2017) especially with volunteer sampling, only participants with strong opinions/emotions will participate. This means that 26 out of 36 may have had indifferent (not necessarily negative) opinions or were apathetic in their participation. We do not know if any of the control group participants experienced any positive results but, based on Moore et al. (2017), feel that if someone had a notable positive experience, something would have been shared. Finally, as quantitative data may be generalized, connected qualitative data, such as decreases in math anxiety by the intervention group, may be as well. The rest of the qualitative results, however, cannot officially be generalized, and in theory, should only be limited to this local and specific time and place. The fact that the participation rates were high, and that triangulation occurred amongst qualitative data does offer greater confidence in extensions. Unlike quantitative methods involving experiments and hypotheses to prove or disprove, qualitative methods cannot assign causation or correlation with numerical accuracy, nor can they associate the size of a p-value or an effect size. Instead, the qualitative researcher may use casual connection words, that is, merely hinting, not implying. Moreover, “the qualitative 201 researcher usually tries to assure the reader that the purpose has not been to attain generalization but to add situational examples to the readers’ experience” (Stake, 2010, p. 23). In this instance, this study gained insights into the lived experience of the participants and prospective ways to support them in a mandatory, post-secondary mathematics course. Any ameliorations can only be used as a basis for further study, to see if the successes (if any) apply to others. Sharing results and determining the interest of other individuals to change their perspectives will, therefore, be a very natural step after completing the research. As such, possible future directions are included in the following section. 5.5 Future Directions Based on the limitations discussed in the previous sections as well as realizations occurring during the research, a number of future directions are possible. Beginning with one of the challenges arising from limitations, obtaining the support of the institution to advertise a mandatory post-secondary mathematics course with contemplative pedagogy would be valuable. This way, students may choose the course on purpose, knowing that an intervention which may alleviate their anxiety related to mathematics is included, and therefore prospectively find themselves among kindred spirits. To obtain a deeper understanding of math avoidance, following participants after the course could prove to be useful. If possible, another round of individual semi-structured interviews or a focus group discussion a year or two after the completion of the intervention could be used for comparison with previously collected qualitative data. This follow up interview or discussion could also determine whether the improvements and/or 202 transformations were transitory or long lasting and whether the participants are still incorporating practices from contemplative pedagogy into their lives. Also, I would inquire as to their major and/or career aspirations. Furthermore, with participant consent, registrar data would be examined to investigate which courses are taken. Did they enroll in other mathematics or STEM courses? If so, how was their performance in those courses? To address math avoidance quantitatively, this data could offer insights along with the qualitative results we have already determined. Finally, an investigation into the causes of absenteeism in the mandatory post-secondary mathematics course should be undertaken, along with the question Why are students skipping classes? Is their lack of attendance due to socio-economic concerns, family obligations, cultural beliefs, high anxiety levels, or something else? By determining the root of the absences, more appropriate supports may be offered to minimize the impact of absenteeism. Since missed classes can further increase math anxiety (indirectly, by decreasing math skills, see Figure 2: Causes and Consequences of Math Anxiety) and thereby decrease math performance, time spent on this investigation would not be wasted. As we learn more about math anxiety (see Section 2.1), we see that the correlations are many, and teachers are not the only sources of influence on students. Even something as simple as a family member or friend with a negative attitude about mathematics can greatly affect a student’s anxiety and attitude regarding mathematics. Hence, we, as a society, need to be informed of our impact, so that we may actively promote the positive influences and minimize the negative ones. Unfortunately, as we have seen herein, the most negative impact 203 on the domestic participants emanated from their teachers in K-12 and their adult upgrading mathematics courses. Many studies note (Hembree, 1990; Namkung et al., 2019) that elementary teachers have the highest level of math anxiety, with a particular focus on female elementary teachers. In fact, female elementary teachers can have a noticeable negative affect on their female students’ math achievement (Beilock et al., 2009). Wood (1988) further noted that even if elementary (all, not just female) teachers have the same amount of math anxiety as the population at large, that is still too much anxiety. “Elementary teachers are charged with an extremely important role and that role is to engender an excitement for learning in all subject areas, including mathematics” (Wood, 1988, p. 11). Mihalko (1978) added that elementary teachers “cannot be expected to generate enthusiasm and excitement for a subject for which they have fear and anxiety” (p. 36), reinforcing the need to affect change when future teachers are in school, supporting them in ways so that they may generate an excitement for learning mathematics. Hence, in referencing the previous section, the empowerment of teachers in their pupal stage and gifting them with tools for themselves and their future students would also be a significant impact of the research found herein. Therefore, another future direction would be to incorporate contemplative pedagogy into education services courses, that is, courses for pre-pre-service (math for future elementary educators, MFEE) and pre-service teachers (B.Ed. programs). This ‘contagion’ of math anxiety and negative math attitudes needs to stop, the sooner the better. Generally, mathematics is considered an apolitical subject (Doolittle, 2020), giving mathematicians and students of mathematics at the post-secondary level the impression that 204 they may bury their heads in the sand of systemic racism, instead of taking a part in its active elimination. A different approach must be taken, holding all accountable, whether it be the marginalization in mathematics due to gatekeeper roles or other of the many inequities which may appear in the classroom. Transformation is difficult and intimidating. Hence, we all need to start taking those steps. As Kendi (2019) wrote “[o]ne either allows racial inequities to persevere, as a racist, or confronts racial inequities, as an antiracist” (p. 11). Differences in mathematics performance have been studied extensively, especially achievement gaps, or gaps presented between marginalized groups. According to Gutiérrez, “gap-gazing” is used by “faculty of colour and people who specialize in equity research” (p. 357), whose research differs greatly from those studying “achievement gaps”. The latter research re-marginalizes students (Gutiérrez, 2008) instead of empowering them and ignores the possibility that there could be factors within a certain group (class, race, or gender) that affect achievement just as much as between two different groups. Given this conclusion, finding ways to improve the overall mandatory post-secondary mathematics experience for all, irrespective of class, racial group, or gender, is of paramount importance. Statistical analysis performed on experiments involving control groups can assist to differentiate the effects attributable to within and those to between groups (Creswell & Creswell, 2018). Mighton (2008), a Canadian mathematician who studies numeracy and math performance, wrote that math is “a powerful tool for social justice” (p. 32). He stated that “the intellectual poverty that societies have imposed on the majority of children, out of ignorance of their real potential, is the deepest source of material poverty because it perpetuates social inequity” (Mighton, 2008, p. 32). Mighton’s recognition of the imbalance in intellectual 205 potential is further emphasized using mathematics as a gatekeeper subject in post-secondary programs, limiting career options and earning potential (Gutiérrez & Rogoff, 2003). This role emphasizes the power that mathematics has when it comes to equity in both the classroom and in society and complements Dewey’s (1916) desire to use education to create a “more equitable and enlightened social order” (p. 319). Another future direction following this doctoral research would be to study the use of contemplative pedagogy in conjunction with culturally sustaining pedagogy (Paris & Alim, 2018) as a tool to empower all students in mathematics, so that everyone can find their own success and fulfill their intellectual potential. Furthermore, the incorporation of Indigenous Knowledges, led by a local and place-based knowledge keeper, as a form of contemplation, self-reflection, and relationship building could also be investigated to improve the overall mandatory post-secondary mathematics course experience. Indigenization is becoming more important and more relevant at all grade levels as well as the post-secondary. This significance is not only due to the Truth and Reconciliation Commission’s Calls to action (TRC, 2015), but also because of the transformation into a more inclusive classroom when Indigenous Ways of Knowing and Being are incorporated (Cohen, 2010; Ragoonaden, 2017b; Ragoonaden & Mueller, 2017). Being a mathematician, an educator, and now a doctoral student in education has had its benefits. Weaving these three perspectives together supported the investigation of the question: How can contemplative pedagogy support a mandatory post-secondary mathematics course? Based on the study and its findings, student supports for those anxiously or uncomfortably enrolled in mandatory post-secondary mathematics courses will be shared with any or all who will listen. While also supporting overall student well-being, the 206 contemplative skills developed in such a research program could act as an important component of both the classroom and home environment. 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Lindisfarne Books. Zajonc, A. (2013). Contemplative pedagogy in higher education: Toward a more reflective academy higher education. In Olen Gunnlaugson, Edward Sarath, Charles Scott & Heesoon Bai (Eds.) Contemplative Learning and Inquiry across Disciplines. 15-29. State University of New York Press. 223 Appendices Appendix A: Math Origin Story Assignment Share your Who, What, Why, How, and When regarding mathematics. How you share is up to you – an essay, an in-class presentation, a podcast, a video, a blog, a song, a dance, etc. You are invited to think about your mathematical journey and to ponder how you got to where you are today. Your assignment should involve: • How you perceive/feel about mathematics today, • What types of challenges or successes you may have had (stressed/anxious when writing tests, “blanking” on test or exam days, distracted in class, disappointed when hearing others talking about grades, happy to hear applications or connections of course material), • Who has had an influence (positive or negative) on your perception/attitude of mathematics, • When these challenges or influences occurred, and • Why you think you feel the way you do about mathematics. 224 Appendix B: Questions for Individual Semi-Structured Interviews 1. Can you tell me about your academic background and major? 2. How about math? When I say that word, what springs to mind? 3. Why did you enroll in this course? 4. How did you feel when you registered for this course? a. What were your thoughts? b. Your physical sensations? c. Your feelings? 5. How did you feel when you walked into class, on the first day of classes? a. What were your thoughts? b. Your physical sensations? c. Your feelings? (Can you remember?) 6. How did you feel during your last class? a. What were your thoughts? b. Your physical sensations? c. Your feelings? 7. How would you classify how you feel/think/react to mathematics? For how long do you feel that this has been going on? 8. Do you feel/think/react differently to mathematics now? If yes, what do you think brought about this change? 225 Let’s discuss some of the practices that were introduced in class (For those in the Intervention Group only): 9. Did you recognize any of the practices? If so, which ones? 10. Do you feel one practice helped more than others did when it comes to how you feel/think/react about/to math? If so, which one? 11. Is there a practice that you see yourself continuing to use going forward? Why? 12. Is there a practice that you never want to do again? Why? 13. If you met a person who struggles learning math, would you recommend trying some of these practices? Can you be specific? 14. Is there anything else that you would like to add? 226 Appendix C: The Math Anxiety Rating Scale (MARS) For each situation, choose ‘Not at all’, ‘A little’, ‘A fair amount’, ‘Much’, or ‘Very much’, based on how you feel most of the time. Note: This is a modified MARS, taken from Alexander and Martray, 1987. How much anxiety do you have when… 1. Studying for a math test. 2. Taking a math section of a college entrance exam. 3. Taking a quiz in a math course. 4. Taking an exam in a math course. 5. Picking up a math textbook to begin working on a homework assignment. 6. Being given homework assignments of many difficult problems that are due the next class. 7. Thinking about an upcoming math test one week before. 8. Thinking about an upcoming math test one day before. 9. Thinking about an upcoming math test one hour before. 10. Realizing that you have to take a certain number of math classes to fulfill requirements. 11. Picking up a math textbook to begin a reading assignment. 12. Receiving your final math grade in the mail or via email. 13. Opening a math or stat book and seeing a page full of problems. 14. Getting ready to study for a math test. 227 15. Being given a “pop” quiz in a math class. 16. Reading a cash register receipt after your purchase. 17. Being given a set of numerical problems involving addition to solve on paper (no calculator). 18. Being given a set of subtraction problems to solve on paper (no calculator). 19. Being given a set of multiplication problems to solve on paper (no calculator). 20. Being given a set of division problems to solve on paper (no calculator). 21. Buying a math textbook. 22. Watching a teacher work on an algebraic equation on the board. 23. Signing up for a math course. 24. Listening to another student explain a math formula. 25. Walking into a math class. 228 Appendix D: The Five Facet of Mindfulness Questionnaire (FFMQ) This instrument is based on a factor analytic study of five independently developed mindfulness questionnaires. The analysis yielded five facets that appear to represent elements of mindfulness as it is currently conceptualized. The five facets are observing, describing, acting with awareness, non-judging of inner experience, and non-reactivity to inner experience. Please rate each of the following statements using the scale provided. Choose the value that best describes your own opinion of what is generally true for you. • never/very rarely true • rarely true • sometimes true • often true • very often/always true 1. When I’m walking, I deliberately notice the sensations of my body moving. 2. I’m good at finding words to describe my feelings. 3. I criticize myself for having irrational or inappropriate emotions. 4. I perceive my feelings and emotions without having to react to them. 5. When I do things, my mind wanders off and I’m easily distracted. 6. When I take a shower or bath, I stay alert to the sensations of water on my body. 7. I can easily put my beliefs, opinions, and expectations into words. 8. I don’t pay attention to what I’m doing because I’m daydreaming, worrying, or otherwise distracted. 9. I watch my feelings without getting lost in them. 229 10. I tell myself I shouldn’t be feeling the way I’m feeling. 11. I notice how foods and drinks affect my thoughts, bodily sensations, and emotions. 12. It’s hard for me to find the words to describe what I’m thinking. 13. I am easily distracted. 14. I believe some of my thoughts are abnormal or bad and I shouldn’t think that way. 15. I pay attention to sensations, such as the wind in my hair or sun on my face. 16. I have trouble thinking of the right words to express how I feel about things 17. I make judgments about whether my thoughts are good or bad. 18. I find it difficult to stay focused on what’s happening in the present. 19. When I have distressing thoughts or images, I “step back” and am aware of the thought or image without getting taken over by it. 20. I pay attention to sounds, such as clocks ticking, birds chirping, or cars passing. 21. In difficult situations, I can pause without immediately reacting. 22. When I have a sensation in my body, it’s difficult for me to describe it because I can’t find the right words. 23. It seems I am “running on automatic” without much awareness of what I’m doing. 24. When I have distressing thoughts or images, I feel calm soon after. 25. I tell myself that I shouldn’t be thinking the way I’m thinking. 26. I notice the smells and aromas of things. 27. Even when I’m feeling terribly upset, I can find a way to put it into words. 28. I rush through activities without being really attentive to them. 29. When I have distressing thoughts or images I am able just to notice them without reacting. 230 30. I think some of my emotions are bad or inappropriate and I shouldn’t feel them. 31. I notice visual elements in art or nature, such as colors, shapes, textures, or patterns of light and shadow. 32. My natural tendency is to put my experiences into words. 33. When I have distressing thoughts or images, I just notice them and let them go. 34. I do jobs or tasks automatically without being aware of what I’m doing. 35. When I have distressing thoughts or images, I judge myself as good or bad, depending what the thought/image is about. 36. I pay attention to how my emotions affect my thoughts and behavior. 37. I can usually describe how I feel at the moment in considerable detail. 38. I find myself doing things without paying attention. 39. I disapprove of myself when I have irrational ideas. 231 Appendix E: Researcher Field Notes Week 0 • Assignments prepared for both sections and a few formal practices for students in the intervention group are organized. • The Math Origin Story assignment is posted to both sections, to be handed in at the beginning of the third week of classes. • Due to my experience with anti-racist pedagogy and decolonizing the classroom, students are provided a variety of submission options, not simply written. • The department is made aware of my planned research. Some members seem sceptical. Concerns regarding whether they would need to implement something similar as well as whether they think these ideas are a hoax. Week 1 • First week of classes. • Notes of student composition taken. On the surface, demographics seem fairly similar. [Details of differences and similarities between participants are provided in Chapter 3.] • A department member teaches in that same class following mine. I feel looked down upon based on my research. I feel as though I am considered a joke. If I was not one of the most senior members of my department, I am not certain what I would do. It takes a lot of courage even being senior. 232 Week 2 • Same concepts are taught to both groups. • Despite the numbers remaining high in the course, absenteeism has already occurred in person, especially with international students. • All class notes [written in One Note and converted to PDF] are posted for both sections going forward. Week 3 • The teaching assistant presents the research in both sections of the course, when students could no longer transfer from one section to the other (as requested by the College for ethics as well as disruption). • Some students communicated with the teaching assistant regarding participation and consent. Due to undue influence concerns, I have no idea what “some” means. Week 4 • Math Origin Stories are due from all students in both sections. All are written, but some include drawings, photos, and/or online images. • The Math Anxiety Rating Scale (MARS) and the Five Facet Mindfulness Questionnaire (FFMQ) pre-tests are posted. 233 Week 5 • Test 1 occurs this week. • Concerns regarding a bus strike arise. Fears linked to the ability to attend on test day cause stress to students. • Attendance is becoming inconsistent in both sections, especially with international students, and during first two classes of this week. • Test review occurs midweek for both sections, with notes posted to try to alleviate the stress for those who cannot attend (in both sections). • Initial data analysis of Math Origin Stories occurs searching for common challenges. Week 6: *Every meeting of the intervention section now begins with three deep and intentional breaths* • A discussion of the test takes place in both sections during the first class of the week. • It is shared that practices from contemplative pedagogy will be offered in class and posted on the Learning Management System for the intervention section. • Conversations surrounding the Math Origin Stories ensue with the intervention section midweek. I share with the students my gratitude for their trust and their vulnerability in this situation. • The concepts of mindful listening and speaking in dyads (Barbezat & Bush, 2014; Stress Management and Resilience Training, 2015, p. 24-25) are introduced at the beginning of class at the end of the week, where students share some (what they feel comfortable with) of their story with a peer. 234 • Audible statements of familiarity (“me too” or “yea, that happened to me”) were heard during their sharing. When group discussions occurred, a few students commented that they had no idea that others felt the same way. • Goals versus intentions are discussed at the end of the week (Stress Management and Resilience Training, 2015, p. 26-27). Instead of expecting a 90% the course or to love math by the end of the course, consider small bites. For example, aim to try a couple of questions after class or print up the formula sheet and use it every time you do homework. Freely writing (Hart, 2004, p. 42) on the topic, then mindful listening and speaking in dyads occur afterwards. • A mathematics problem is posted on the board at the beginning of class and the end of the class for those who may not want to participate in the practices, or for those who do not wish for me to know that they are (or are not) participating in the research. Week 7 • Early in the week, the teaching assistant and I meet. As an international student, they share their own story and ask. “What happens if the group we are trying to support isn’t here for learning?” [This train of thought is picked up in Discussions.] • International students in both sections begin to ask for references for employment as well as some direction when applying. I am informed that the change to the maximum number of hours that an international student may work (from 20 to 40 hours) will be effective in November. 235 • To be as accepting and accessible as possible, I begin to incorporate practices from contemplative pedagogy, but also make reference to them being posted online. For those who do not wish to be seen by either myself or their peers can always try outside of class. Flashcards for some practices, but mostly audio files created by me are posted for relevant practices. • Discussions relating to Math Origin Stories arise. In particular, the positive aspects are emphasized during this first discussion. • Where are you now? (Hart, 2004, p. 36) is introduced as a regular check-in and incorporated with speaking in dyads. • A prompt for speaking in dyads (and mindful listening) this week is rising above challenges and being the hero of your own story. We discuss how stories are more interesting when the protagonist encounters challenges, but then overcomes them. Some of the student challenges were shared with the entire group (by the students themselves). I continue to offer my gratitude for their trust and vulnerability. • In the middle of the week, mindful movement Stress Management and Resilience Training, 2015, p. 34, 117) is performed in class, especially with graphs. We get up and physically act out the behaviour of functions (parabolas opening up “up like a cup” – arms up in the air in the shape of a U; parabolas opening down “down like a frown”, bending at the waist and holding an upside-down U shape). 236 • We talk about the graphs of polynomials being akin to adult Connect the Dots. The sketching is smooth, tranquil, and continuous. We take deep breaths before attending to these. Polynomials are predictable and expected. There are not any surprises. They are reliable, they are grounding. • At the end of the week, we discussed the 5-4-3-2-1 Practice along with the course Formula Sheet. The formula sheet can behave as an anchor. Students may add positive affirmations and/or encouraging drawings. • Three things you can hear brought the discussion of distractions and students were encouraged to bring ear plugs if they have difficulties with the sounds around them. Discussions surrounding quiet snacks during tests/exams were also made (for both hearing and tasting). The regular use of the formula sheet was encouraged to support grounding experiences and to remove the stress of blanking/forgetting. [Note: the control section was also allowed to bring earplugs and snacks and did have access to the same mathematical formulas during tests.] Week 8 • A conversation surrounding Diwali occurs in the class unexpectedly. Some students ask permission to share their culture, the meaning of Diwali, and how they celebrate. Some are dressed for celebration already. Even at 8 am, all students were engaged in lively discussions. I feel honoured to be a participant in these conversations. Prompt for discussion in dyads was how they could bring light into the lives of others. Group answers included by smiling, by helping others, and by random acts of kindness. 237 • The next class brought further discussions about traditions and different forms of meditation and self-reflection. The students discussed in dyads different kinds of practices they have tried and talked about the continued challenges of finding time and sticking with them. • The sharing of cultures and different forms of meditation and self-reflection led to a conversation about shared humanity. Freely writing on the topic was encouraged and time provided for speaking in dyads. • Kindness and compassion are discussed (Stress Management and Resilience Training Curriculum, 2015, p. 86, 91, 139-144) and the Loving Kindness Practice (Barbezat & Bush, 2014, p. 178-179) is presented midweek. • At the end of the week, the Pause Practice (Stress Management and Resilience Training Curriculum, 2015, p. 2-3) is introduced, along with the Triangle of Awareness (Stress Management and Resilience Training Curriculum, 2015, p. 23). The flexibility of the pause was appreciated as well as its accessibility. Taking a positive perspective — turn waiting for someone/something into a moment to pause – instead of getting frustrated. Some felt it was a “productive” use of time. Week 9 • Practices are reinforced (repeated and referenced online). A Sitting Practice, Awareness of Sensations, Emotions, and Thoughts ((Stress Management and Resilience Training Curriculum, 2015, p. 109) is posted on the Learning Management System for thos interested. 238 • Guided breathing, including 4-7-8 breathing, are introduced. • With the intervention group, a discussion relating to “blanking” (forgetting everything or low working memory) and anxiety arose, especially as Test 2 approached. Bellinger et al. (2015) and Shapiro et al. (2011) were referenced on how practices from contemplative pedagogy could minimize this occurrence. • Heavy snow fell overnight, buses are late and unexpected things arose due to weather on test day. [Anecdotally, many students from the intervention section seem calmer than I expected in this situation. It will be worthwhile to see which were participants after the last day of grade appeal. Note from after the last day for grade appeal: the calmer students were ALL participants in the intervention group.] Week 10 • Further bad weather leads to the first class of the week to be offered via Zoom. Some students are particularly grateful as note only the notes are posted but also the entire lecture. • The intervention group explores Visualization using the Three Questions Visualization, with a focus on test intentions, (Stress Management and Resilience Training Curriculum, 2015, p. 27-28). “Fake it until you make it” is a prompt for discussion in breakout rooms, then returned to the group at large. Believe that you can have success (discussed what success looks like for them), then try a question. • Midweek, feeling comfortable with being uncomfortable when learning/growing. The concept of turning toward instead of away becomes another prompt for discussion. Getting out of one’s comfort zone and reflecting upon that discomfort. 239 • Students discuss in dyads learnings and unlearnings. The process of learning and acquiring versus unlearning and letting things (e.g., biases, concepts) go. • At the end of the week, self-compassion and shared humanity are shared along with treating yourself as you would a dear friend (Neff, 2009). Freely writing and speaking in dyads ensued. Week 11 • This week less than ten students attended each class. To support confidentiality of participation, prompts are still shared for dyads and a course-related mathematics problem is always on the board for those who do not wish to participate. Practices are posted online (to intervention group) to use if or when they desire. • Discussion surrounding concentration occurs and Mindful Eating (Stress Management and Resilience Training Curriculum, 2015, p. 12-15) is referenced (and shared online). • Repeating patterns (rituals) to gain confidence. I point out how I start the class (after breathing and mindful dyads), reviewing concepts from the previous class and inform what will be discussed that day. Furthermore, I explain how new ideas fit with previous concepts and end the class previewing the next class. This ritualistic behaviour is done in both sections, as it is part of the math course. Pointing it out as a ritual and discussing rituals is only for the intervention section. Repeat it often enough and it flows more easily – there's less resistance. [One of the greatest challenges in math is to take the first step. It requires courage and confidence to do so. I acknowledge their courage every time they try.] Working on math (self-) confidence becomes a repeating pattern, from different perspectives for the next few weeks. 240 • Building community involves providing the language and vocabulary to communicate. Sometimes it also means me, the instructor, reflecting and listening to the language and vocabulary provided to me. For example, when teaching trigonometric concepts, instead of opposite, adjacent, and hypotenuse, I included perpendicular, base, and hypotenuse. No one had the courage to tell me before. Week 12 • A prompt surrounding acceptance occurs and of taking (mathematical risks). The latter discussion encourages students to attempt math questions/problems by treating problem solving according to the improv situation “yes and…”. Acknowledge the question, and then write something, anything. Students are encouraged to not leave anything blank and to try. • Another discussion ensues, this time considering the consequences of trying and questions and getting it “wrong”. In groups of two, discussions surrounding “what’s the worst that can happen?” also arise. • The concept of ambiguity and the challenge of having many options to answer a question (as well as many practice options) also arises. For some, this is empowering. For others, this is not. In test situations, ideally the number of decisions required should be minimized to decrease anxiety and improve working memory. Plan ahead by reflecting on which techniques and practices work for you, then repeat those patterns (rituals) of behaviour so they become natural (and harder to forget). 241 Week 13 • Last of the content is covered, and Test 3 takes place. In essence, everything was tested by one of the three tests. • Prompts for discussions are shared: o Visualize success and fain confidence – believe you can do it and take the first step. (Linked to “yes and…” and well as The Three Questions Visualization from previous weeks.) o Give names to things that look scary – normalize them, accept them. Then proceed. o Take three deep breaths, then reassess. • Previously discussed practices are repeated: o The 5-4-3-2-1 Practice (including added a positive affirmation on the formula sheet and/or bringing in an artefact to support present moment awareness and/or confidence) o The Pause Practice and guided breathing o Loving-Kindness. Week 14 • This week only has a single review class, then office hours for students. • MARS and FFMQ post-tests are posted. • Strategies for successful test-taking are discussed with both sections. 242 • A discussion surrounding Not Doing (Hart, 2004, p. 35) and Doing vs Being (Stress Management and Resilience Training Curriculum, 2015, p. 23, 167) occurs this week. Encouragement to “just be” is made as part of the final exam preparation. • A gratitude practice (Lai, 2017, p. 13) is shared and posted online for the intervention group. Week 15 • The final takes place this week, with a lot of nerves. • Anecdottaly, most of the intervention section was noticeable calmer. 243 Appendix F: Calculations This appendix contains extra calculations and figures related to the results (Sections 4.5) and discussions (Chapter 5) surrounding the quantitative data and its analysis. F.1 Chi-Squared ( ) Test for Independence – Impact of Influence As detailed in Section 4.1, the qualitative data arising from the Math Origin Stories was both thick and rich. When delving into the details of these stories, the influence from one person or experience emerged. This section will investigate the source of that influence as well as its impact by breaking down the Math Origin Stories according to the source of influence (teacher, friend/family, or other), the nature of influence (positive/negative), and certain demographics. As written earlier, since the Math Origin Stories occurred before the incorporation of contemplative pedagogy, no differentiation is made between the groups. The first investigation, summarized in Table F1, presents general results from the Math Origin Stories for all participants (control and intervention groups combined) and all demographics. The focus of this query was the nature of influence (positive or negative) and the source of the influence (teacher, friend/family member, or ‘other’). The ‘other’ category includes influences of the self (self-talk) and the media. 244 Table F1 Breakdown of Source and Nature of Influence, based on the Math Origin Stories (Control and Intervention Groups Combined) ALL Teacher Friend/Family Other Total Positive 14 17 2 33 Negative 32 2 1 35 Total 46 19 3 68 Inspecting Table F1, we see that teachers represent 67.65% (or 46 out of 68) of the source of influence and 69.57% (or 32 out of 46) of the teachers’ influence is negative. This observation leads us to believe that not all sources and nature of influence are the same. To investigate this suspicion, we performed a (chi-squared) test of independence, with the following specific hypotheses: H0 (null hypothesis): Source and Nature of Influence are independent (or = 0). HA (alternative hypothesis): Source and Nature of Influence are dependent (or > 0). The (chi-squared) test of independence resulted in a test statistic of (2) = 19.18, with p = 0.000069 < 0.05 (statistically significant), and Cramer’s V = 0.53 (medium effect size). Hence, there is a dependence between the nature of the influence and the source of the influence in Table G1. This means that the nature of the influence (positive or negative) depends on the source of influence (teachers, family or friends, or other). More specifically, not all influences (positive or negative) are treated the same. To dig deeper into the sources of influence, one demographical piece was chosen to explore – that of participant status (domestic versus international participant). This piece was explored as part of grounded theory as fundamental differences between domestic and 245 international participants arose while rereading the Math Origin Stories. The other demographical pieces did not have any noticeable codes or categories which emerged. Furthermore, note that all international participants were enrolled in their first math course in Canada; hence, all their previous teacher influences occurred outside of Canada. Hence, focusing solely on teachers as a source of influence, Table F2 presents the breakdown of the nature of that influence on domestic and international participants. Table F2 Nature of Influence for Domestic and International Participants, With Teachers As the Sole Source of Influence Teacher Domestic International Total Positive 2 12 14 Negative 26 6 32 Total 28 18 46 Analogous to the analysis performed above on Table F1, a (chi-squared) test of independence was undertaken to determine whether there was a dependence between the nature of the influence and the status of the participant (domestic or international), that is, the observed values seen in Table F2. The hypotheses in this situation are: H0 (null hypothesis): Nature of Influence and Participant Status are independent (or = 0). HA (alternative hypothesis): Nature of Influence and Participant Status are dependent (or > 0). 246 The test produced a chi-squared test statistic (1) = 18.34, with p = 0.000019 < 0.05 (statistically significant), and = 0.63 (medium effect size). Hence, the nature of the influence (positive or negative) depends on whether the participant is a domestic or international. More specifically, not all influences (positive or negative) are treated the same. As discussed in Chapter 5, this analysis supported the fact that international participants reported positive teacher influences (66.67%) more often than domestic ones (7.14%). Furthermore, domestic participants reported more negative teacher influences (92.86%) than the international ones (33.33%). This section examined statements made by participants with respect to sources and nature of influence. As was described in Chapter 5, this investigation was particularly pertinent when reminding us, as educators, of our possible impact on the mathematics perception of our students. The next section explores the t-tests that were used when comparing the pre- and post-tests for both groups separately, as well as when comparing the control and intervention groups at Stage 1 (pre-test) and Stage 2 (post-test), separately. F.2 Paired and Independent Sample t-Tests As expressed in Section 3.7.1 (Quantitative Data Analysis), two types of t-tests were performed on the quantitative data: 1. Paired t-tests to determine the significance of the difference between the pre-test (Stage 1) and post-test (Stage 2), for the FFMQ and the MARS questionnaires as well as the course grades, for the control and intervention groups separately. 247 This analysis is considered a within group examination, as the lens is focused on changes happening within a specific group over a certain period of time. 2. Independent two sample t-tests to compare the FFMQ, MARS, and the course grade averages for two distinct moments in time – Stage 1 (pre-test) and Stage 2 (post-test) – for the control and intervention groups. This investigation is labelled between as a reference to studying the difference between the two groups. All the executed paired t-tests were comprised of data which were approximately Normal. Paired t-tests were used to determine the significance between the pre- and post-test values (or Stage 1 and Stage 2). Table F3 presents the results of the pre- and post-tests comparison for the control group, while Table F4 provides insights for the intervention group. Both tables first display descriptive statistics – the average and the standard deviation (SD) – for each variable (all the five facets of mindfulness as well as the overall mindfulness level, the three components of math anxiety as well as the overall math anxiety level, and both course grades – test 1 and the final exam). In both Tables F3 and F4, a paired t-test was performed to determine whether the difference between the pre- and the post-tests average for each variable is significant (or whether there is significant difference within each group for a respective variable). The result of this test is found under the “t-statistic” column, with significance levels attributed based on the calculated p-values, with one asterisk (*) denoting a p-value less than 0.05 and two asterisks (**) less than 0.01. Finally, as effect size is an important consideration when performing a statistical test, Cohen’s d is provided in the last column. These effect sizes have been labelled as 248 large (a), medium (b), and small (c), according to Sullivan and Feinn’s (2012) scale. An effect size that does not have a label is considered to be zero or too small to notice. All the data, connected to F3 and F4 satisfied the assumptions of the paired t-tests, including being approximately normal. When examining Table F3, we see that there is a significant increase in overall math anxiety levels and math course anxiety for the control group with large and medium effect sizes, respectively. Furthermore, a significant decrease in course grades for the control group is seen with small effect size. Discussions surrounding these statements were presented in Section 5.1. Table F4 presents the pre- and post-averages and standard deviations as well as the results of the paired t-tests applied to the intervention group. Upon examination of Table F4, we note that the intervention group experienced significant increases in three facets of the mindfulness questionnaire (non-judgement, non-reactivity, and awareness) as well as the overall mindfulness level, all with large effect sizes. Also, the intervention group underwent significant decreases in overall math anxiety as well as in each component of math anxiety, with either medium or large effect sizes. Finally, a significant increase in course grades, with small effect size, was demonstrated by the intervention group. Further discussions on the implications of these statements were included in Section 5.1. 249 Table F3 Descriptive Statistics, Paired t-Test Significance, and Effect Sizes for the FFMQ, MARS, and Course Grades (Control Group) Variable Pre average (SD) Post average (SD) t-statistic Cohen’s d FFMQ (n = 27) Observe 3.20 (0.72) 3.20 (0.86) 0.05 0.02 Describe 3.22 (0.69) 3.14 (0.85) -0.65 -0.20c Non-Judgement 3.07 (0.64) 2.95 (0.74) -1.29 0.51b Non-Reactivity 2.98 (0.65) 3.07 (0.61) 0.94 -0.35c Awareness 2.75 (0.77) 2.65 (0.75) -0.66 -0.17 Overall 3.05 (0.48) 3.01 (0.53) -0.68 -0.36c MARS (n = 36) Math Test Anxiety 2.85 (0.86) 2.91 (0.89) 1.03 0.44c Number Anxiety 2.33 (1.22) 2.51 (0.97) 1.47 0.33c Math Course Anxiety 2.05 (0.80) 2.24 (0.88) 2.07* 0.62b Overall 2.57 (0.84) 2.68 (0.58) 2.32* 1.29a Course Grades (n = 36) Test 1/Final Exam 78.42 (11.83) 72.22 (12.45) -8.86** -0.35c Note. alarge effect size; bmedium effect size; csmall effect size; * p < 0.05; ** p < 0.01. The participation rate for the control group was 75% (or 27 out of 36) for the FFMQ and 100% (or 36 out of 36) for the MARS and the course grades. 250 Table F4 Descriptive Statistics, Paired t-Test Significance, and Effect Sizes for the FFMQ, MARS, and Course Grades (Intervention Group) Variable Pre average (SD) Post average (SD) t-statistic Cohen’s d FFMQ (n = 29) Observe 3.37 (0.67) 3.43 (0.57) 0.78 0.33c Describe 3.43 (0.64) 3.32 (0.65) -1.46 -0.69b Non-Judgement 2.87 (0.71) 3.10 (0.69) 2.51* 0.94a Non-Reactivity 2.95 (0.51) 3.11 (0.43) 2.36* 1.15a Awareness 2.87 (0.77) 3.13 (0.75) 3.02** 1.19a Overall Mindfulness 3.11 (0.33) 3.23 (0.37) 2.62* 1.97a MARS (n =32) Math Test Anxiety 3.24 (0.82) 2.46 (0.67) -6.83** -1.88a Number Anxiety 2.64 (0.98) 1.93 (0.84) -4.58** -0.91a Math Course Anxiety 2.46 (0.94) 1.71 (0.58) -5.89** -0.75b Overall Math Anxiety 2.94 (0.78) 2.18 (0.58) -7.70** -2.37a Course Grades (n=32) Test 1/Final Exam 74.25 (8.62) 75.59 (8.42) 3.29** 0.25c Notes. alarge effect size; bmedium effect size; csmall effect size; * p < 0.05; ** p < 0.01.The participation rate for the intervention group was 90.63% (or 29 out of 32) for the FFMQ and 100% (or 32 out of 32) for the MARS and the course grades. 251 Unlike Tables F3 and F4 (above), Tables F5 and F6 (below) compare the control and intervention groups at two different moments in time (or Stages), that is, for the pre-test as well as the post-test. Again, this is considered a between investigation. The pre-tests occurred before the implementation of the intervention, while the post-tests transpired afterward. Since the control and the intervention groups are independent from one another, Tables F5 and F6 computed independent sample t-tests (and not the paired t-tests calculated in Tables F3 and F4). All the independent two sample t-tests performed used data that satisfied the assumptions of the test, that is, approximately normally distributed data for each group (control and intervention) and homogeneity of variances (satisfying Levene’s test for homogeneity of variances). Consulting Table F5, we see that there is no difference in mindfulness levels (including all facets) between the control and intervention groups at the pre-test stage. However, the intervention group demonstrated significantly higher levels of math anxiety, in particular, overall math anxiety, math test anxiety, and math course anxiety, at the pre-test stage (with small effect sizes). The test 1 grades are significantly higher in the control group at the pre-test stage, with a small effect size. Elaborations upon these statements were presented in Chapter 5. 252 Table F5 Descriptive Statistics, Independent Sample t-Test Significance, and Effect Sizes for the Difference between the Control and Intervention Groups for the FFMQ, MARS, and Course Grades (Pre-Tests) Variable Pre average (SD) t-statistic Cohen’s d FFMQ (control n = 27, intervention n = 29) Observe -0.17 (0.69) -0.94 -0.25c Describe -0.21 (0.66) -1.18 -0.12 Non-Judgement 0.20 (0.68) 1.09 0.29c Non-Reactivity 0.03 (0.58) 0.21 0.06 Awareness -0.12 (0.77) -0.59 -0.16 Overall Mindfulness -0.06 (0.41) -0.51 -0.14 MARS (control n = 36, intervention n = 32) Math Test Anxiety -0.39 (0.84) -1.91* -0.46c Number Anxiety -0.32 (1.11) -1.17 -0.28c Math Course Anxiety -0.41 (0.87) -1.92* -0.47c Overall Math Anxiety -0.38 (0.81) -1.91* -0.46c Course Grades (control n = 36, intervention n = 32) Test 1 4.17 (10.45) 1.64* 0.40c Notes. alarge effect size; bmedium effect size; csmall effect size; * p < 0.05; ** p < 0.01 253 Table F6 is the last table presenting descriptive statistics, statistical significance, and effect sizes for this research. It is also the most meaningful table so far as it explores the differences between the control and intervention groups in conjunction with the post-tests. When examining the FFMQ post-test results, only the awareness facet and the overall mindfulness level are significantly higher for the intervention group (with medium effect size). Moreover, when investigating the MARS post-test results, the math anxiety levels including all components are significantly higher for the control group (with medium effect sizes). Deeper discussions and connections were explored in Section 5.1. The remaining assumptions for both t-tests above, that is, of continuity of data, independence of observations, and absence of outliers, were also met. The results of these tests we presented and triangulated in Chapter 4 and then discussed in Chapter 5. A separate independent two sample t-test was also performed to compare the pre-test overall math anxiety as well as the number anxiety levels for mature participants with non-mature participants, as reported in Chapter 5. As described in Chapter 3, a mature student at this Western Canadian College is a student who is at least 19 years of age and has not attended high school full-time for at least a year. The specific hypotheses for this test are: H0 (null hypothesis): Mature participants and non-mature participants have the same overall math anxiety (or number anxiety): HA (alternative hypothesis): Mature participants have higher overall math anxiety (or number anxiety) than non-mature participants: 254 Table F6 Descriptive Statistics, Independent Sample t-Test Significance, and Effect Sizes for the Difference between the Control and Intervention Groups for the FFMQ, MARS, and Course Grades (Post-Tests) Variable Post average (SD) t-statistic Cohen’s d FFMQ (control n = 27, intervention n = 29) Observe -0.23 (0.72) -1.20 -0.32c Describe -0.18 (0.75) -0.89 -0.24c Non-Judgement -0.16 (0.71) -0.82 -0.22c Non-Reactivity -0.04 (0.52) -0.28 -0.07 Awareness -0.48 (0.75) -2.37* -0.63b Overall Mindfulness -0.22 (0.46) -1.83* -0.49c MARS (control n = 36, intervention n = 32) Math Test Anxiety 0.45 (0.80) 2.32* 0.56b Number Anxiety 0.59 (0.91) 2.65** 0.64b Math Course Anxiety 0.54 (0.76) 2.93** 0.71b Overall Math Anxiety 0.50 (0.70) 2.96 ** 0.72b Course Grades (control n = 36, intervention n = 32) Final Exam -3.37 (10.75) -1.29 -0.31c Notes. alarge effect size; bmedium effect size; csmall effect size; * p < 0.05; ** p < 0.01 255 This independent two sample t-test had 9 mature participants and 59 non-mature participants. When testing the overall math anxiety level, Levene’s test for homogeneity of variances was satisfied and resulted in a test statistic of t(66) = 3.314, with p < 0.001 (which is less than 0.05, statistically significant), and Cohen’s d = 1.54 (large effect size). On average, the mature participants scored between 0.36 and 1.47 points higher on the 5-point Likert scale on overall math anxiety. When considering only the number anxiety component, which was investigated due to the statement that mature students were afflicted even number simply working with numbers, the independent sample t-test was undertaken yet again comparing mature participants to non-mature participants. Unlike the independent t-tests referred to above, this one did not satisfy the homogeneity of variances: Levene’s test for homogeneity was used and the null hypothesis of equal variances was rejected. Hence, the two-sample t-test with unequal variances was used. This test resulted in a test statistic of t(25.958) = 12.105, with p < 0.001 (which is less than 0.05, statistically significant), and Cohen’s d = 2.79 (large effect size). Furthermore, not only is there a significant difference between the number anxiety of mature and non-mature participants, but the mature participants also have a significantly higher number anxiety, scoring between 1.73 and 2.44 points higher, on a 5-point Likert scale. This section considered all the independent sample and paired t-tests used in the quantitative data analysis. The next section extends the results found herein to consider the overall effect of the intervention. 256 F.3 Analysis of Covariance Section F.2 discussed the use of different t-tests when analysing the quantitative data. As explained in Chapter 3, t-tests are useful when comparing pre- and post-values for one group at a time (within a group, or paired t-tests) or when comparing two different groups at one moment in time (between two groups, or independent sample t-tests). To determine the effect of the incorporation of contemplative pedagogy into a mandatory post-secondary math course, we would need to perform both tests at once, that is, to determine the change between pre- and post-values as well as the difference between the control and intervention groups. To achieve this result, we performed a few analyses of covariance (ANCOVA). Tables F3 through F6 above displayed descriptive statistics (averages and standard deviations) along with significance and effect sizes related to t-tests (paired t-tests and independent sample t-tests). The significance level provided was related to the difference between the various means – either pre- and post-test averages, or control and intervention group averages. What these tables could not investigate was the overall contribution of the intervention to mindfulness levels and to math anxiety levels. This section explores the effects of the intervention by using a one-way analysis of covariance (ANCOVA). As provided in Chapter 3, hypotheses relating to ANCOVA are: H0 (null hypothesis): There is no post-test difference in the mindfulness/math anxiety/test grades between the control and the intervention groups (that is, ), while controlling for the pre-test values 257 HA (alternative hypothesis): There IS a post-test difference in the mindfulness/math anxiety/test grades between the control and the intervention groups (that is, ), while controlling for the pre-test values. The following subsections provide supplementary details for various ANCOVAs related to the results shared in Section 4.5 (Quantitative Data Analysis). F.3.1 Intervention and Mindfulness This subsection considers a one-way analysis of covariance (ANCOVA) to consider the effectiveness of the intervention (incorporation of contemplative practices) while controlling for the pre-test overall mindfulness level (and to determine whether the differences are significant). Assumptions were met, such as Levene’s Test of equality of variances and normality assumptions. The result of the test was that there was a significant difference between the control and intervention groups, with test statistic F(1,53) = 4.718, p = 0.034 < 0.05 (statistically significant), partial = 0.082 (small effect size). Based on the post-hoc pairwise comparison of the two means, we saw that the intervention group had higher values than the control group by between 0.014 and 0.340 points on average, on the 5-point Likert scale. As discussed in Chapter 5, we are led to believe that the intervention did cause an increase in the mindfulness level, as reported by the FFMQ. Investigations using ANCOVA were used to determine the effect of the intervention, in particular where significant differences arose in Table 7 (Descriptive Statistics, Significant, and Effect Sizes for the Difference Between the Control and Intervention Groups (Post-Tests)) or 258 where the sign changed for the t-statistic (from positive to negative or vice-versa) between Table 6 (Descriptive Statistics, Significant, and Effect Sizes for the Difference Between the Control and Intervention Groups (Pre-Tests)) & Table 7. For all the tests below, ANCOVA assumptions were met. The ensuing results are: 1. Awareness facet of FFMQ: The ANCOVA produced a significant difference, with a small effect size: F(1,53) = 7.055, p = 0.010 < 0.05, partial = 0.117. Furthermore, the intervention group obtained between 0.097 and 0.695 points higher in post-test awareness on average, based on a 5-point Likert scale. 2. Non-judgement facet of FFMQ: The ANCOVA resulted in a significant difference with a small effect size: F(1,53) = 6.409, p = 0.014 < 0.05, partial = 0.108. Moreover, the intervention group scored between 0.064 and 0.552 points higher on average in post-test non-judgement, on average, based on a 5-point Likert scale. 3. Non-reactivity facet of FFMQ: The ANCOVA resulted in an insignificant different, with a very small effect size: F(1,53) = 0.279, p = 0.600 > 0.05, partial = 0.005. In consequence, the control and intervention group score similarly in post-test non-reactivity, on average, based on a 5-point Likert scale. The results provided above related to the effect of the intervention on overall mindfulness levels as well as certain facets of the FFMQ, based on the data produced from the FFMQ pre- and post-tests. The next subsection will proceed similarly to investigate the effect of the intervention on math anxiety levels, as determined by the MARS pre- and post-tests. 259 F.3.2 Intervention and Math Anxiety Analogous to the previous section, one-way analyses of covariance (ANCOVAs) were undertaken to consider the effectiveness of the intervention (incorporation of contemplative practices) while controlling for the pre-test math anxiety levels, and to determine whether the differences are significant. For the ANCOVAs below, all assumptions were met, such as Levene’s Test of equality of variances and normality assumptions. The results are: 1. Overall math anxiety: The ANCOVA produced a significant difference, with a medium effect size: F(1,65) = 65.294, p < 0.001, partial = 0.501 (medium). The intervention group scored between 0.584 and 0.968 less (almost a full point) on post-test overall math anxiety than the control group, on average, based on a 5-point Likert scale. 2. Math test anxiety (MTA) component of MARS: The ANCOVA resulted in a significant difference and a small effect size: F(1,65) = 38.224, p < 0.001, partial = 0.370. The intervention group obtained between 0.504 and 0.985 less (almost a full point) on the post-test MTA component than the control group, on average, based on a 5-point Likert scale. 3. Number anxiety (NA) component of MARS: The ANCOVA produced a significant difference, with a small effect size: F(1,65) = 22.074, p < 0.001, partial = 0.254. The intervention group scored between 0.440 and 1.090 less (sometimes a full point) on the post-test NA component than the control group, on average, based on a 5-point Likert scale. 260 4. Math course anxiety (MCA) component of MARS: The ANCOVA resulted in a significant difference and a small effect size: F(1,65) = 33.150, p < 0.001, partial = 0.338. The intervention group obtained between 0.512 and 1.056 less (sometimes a full point) on the post-test MCA component than the control group, on average, based on a 5-point Likert scale. This subsection presented the Analysis of Covariance (ANCOVA) results linked to the effect of the intervention on math anxiety and its three components. The next and final subsection on ANCOVA will explore the effect of the intervention on course grades. F.3.3 Intervention and Course Grades This last subsection uses the Analysis of Covariance (ANCOVA) tests to examine the effect of the intervention on course grades, in particular the final exam. As was done in the two previous relating to the analysis of covariance, the assumptions of the ANCOVA were verified and satisfied in this case as well. This ANCOVA resulted in a significant difference and a medium effect size: F(1,65) = 75.301, p < 0.001, partial = 0.537. The intervention group scored between 5.72 and 9.14 higher on the post-test course grade (final exam) than the control group, on average. This section has considered the effect of the intervention, that is, the incorporation of contemplative pedagogy, on the post-test FFMQ facets, the post-test MARS components, and the final exam grade using the Analysis of Covariance (ANCOVA). The next section will explore the correlation between different pertinent variables. 261 F.4 Correlation and Linear Regression Analyses Section 4.5.3 (Correlation and Regression) addressed the last of the quantitative analysis, by exploring correlations between variables, with hypotheses as follows: H0 (null hypothesis): There is NO significant correlation between the variables. HA (alternative hypothesis): There IS a significant correction between the variables. As expressed in Chapter 3, statistical analyses examined the correlation between the following variables: 1. Mindfulness and math performance (Bellinger et al., 2015; Galla & Wood, 2012), 2. Math anxiety and math performance (Hembree, 1990; Ma, 1999; Namkung et al., 2019), and 3. Mindfulness levels and math anxiety (relating to working memory, Ashcraft, 2002, Bellinger et al., 2005). Each of these analyses are performed using Regression Analysis of Variance (ANOVA) will be considered over the subsequent subsections. Notice that the effect size in this context is given by R2, the coefficient of determination. F.4.1 Mindfulness and Math Performance As discussed in Chapter 5, the correlation between mindfulness and math performance was examined in Bellinger et al. (2015). In this context, we investigated the correlation between the overall mindfulness level (from the FFMQ) and math performance at both pre-test (Stage 1) and post-test (Stage 2). We used a Regression ANOVA with the hypotheses stated above to evaluate the correlation, with the following results: 262 1. Pre-test overall mindfulness level (from FFMQ) and test 1 grade: The regression ANOVA resulted in an insignificant correlation and a very small effect size: F(1,54) = 2.295, p = 0.136, R2 = 0.041. The (sample) linear correlation coefficient, r, was calculated to be 0.202. 2. Post-test overall mindfulness level (from FFMQ) and final exam grade: The regression ANOVA resulted in an insignificant correlation and a very small effect size: F(1,54) = 2.938, p = 0.092, R2 = 0.052. The (sample) linear correlation coefficient, r, was calculated to be 0.227. Discussions surrounding the lack of a significant correlation between the mindfulness and math performance variables were found in Chapter 5. The next subsection will investigate the correlation between math anxiety and math performance. F.4.2 Math Anxiety and Math Performance Similarly to the analyses performed in the previous subsection, we consider the correlation between math anxiety and math performance variables. Using the Regression ANOVA as well as the hypotheses listed at the beginning of this section, we discovered the following: 1. Test 1 grades and pre-test overall MARS: The regression ANOVA produced a significant result with a medium effect size: F(1,66) = 49.101, p < 0.001, R2 = 0.4266. The (sample) linear correlation coefficient is given by r = - 0.6531. 2. Test 1 grades and pre-test math test anxiety (MTA) component from the MARS: The regression ANOVA produced a significant result with a small effect size: F(1,66) = 263 38.855, p < 0.001 < 0.05 (statistically significant), R2 = 0.3706 (small effect size). The (sample) linear correlation coefficient is given by r = - 0.6088. 3. Final exam grade and the post-test overall MARS: The regression ANOVA produced a significant result with a small effect size: F(1,66) = 17.721, p < 0.001, R2 = 0.2210. The (sample) linear correlation coefficient is given by r = - 0.5520. 4. Final exam grade and the post-test overall MARS (intervention section only): The regression ANOVA produced a significant result with a small effect size: F(1,30) = 13.857, p < 0.001, R2 = 0.3160. The (sample) linear correlation coefficient is given by r = - 0.5621. 5. Final exam grade and the post-test overall MARS (control section only): The regression ANOVA produced a significant result with a very small effect size: F(1,34) = 6.445, p = 0.016, R2 = 0.1590. The (sample) linear correlation coefficient is given by r = - 0.3987. The five points above details the results from the Regression ANOVAs which were performed to determine the correlation between two variables. The meaning behind the results shared above were discussed in Chapter 5. The next subsection considers the last of the investigated correlations, that between mindfulness and math anxiety. F.4.3 Mindfulness and Math Anxiety The two previous subsections examined the correlation between mindfulness and math performance as well as math anxiety and math performance. This subsection investigates the correlation between mindfulness and math anxiety, following the same procedure as was used above with the same hypotheses provided at the beginning of Section F.4: 264 1. Pre-test overall mindfulness (from the FFMQ) and pre-test overall math anxiety (from the MARS): The regression ANOVA produced a significant result with a small effect size: F(1,54) = 5.237, p = 0.26, R2 = 0.088. The (sample) linear correlation coefficient is given by r = - 0.297. 2. Post-test overall mindfulness (from the FFMQ) and post-test overall math anxiety (from the MARS): The regression ANOVA produced a significant result with a small effect size: F(1,54) = 14.07, p < 0.001, R2 = 0.207. The (sample) linear correlation coefficient is given by r = - 0.455. This subsection completed not only the section on correlation and regression, but also Appendix F, which served to provide further details relating to the calculations summarized in Chapter 4 and discussed in Chapter 5. All the graphs and visuals presented and Chapter 4 as well as the interpretation and impacts described in Chapter 5 are grounded in the analysis performed herein.
Thesis/Dissertation
eng
Interdisciplinary Studies
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Ragoonaden, Karen
Mathematics anxiety and contemplative pedagogy at the post-secondary level
Text