Secondary Mathematics Teachers’ Pedagogy Through the Tool of Computer Algebra Systems by Candace Pearl Terry A Dissertation Submitted in Partial Fulfillment of the Requirements for the Degree of Doctorate of Philosophy in Mathematics and Science Education Middle Tennessee State University August 2018 Dissertation Committee: Dr. Angela Barlow, Co-Chair Dr. Michaele Chappell, Co-Chair Dr. Nancy Caukin Dr. Mary Martin Dr. Jeremy Strayer This dissertation project is dedicated to my three brilliant children, Leah, Patsy, and Ruth Ann, and to my husband, Alton, the love of my life. These four have given me love, advice, support, challenge, joy, encouragement, and the best family life imagined. As a special memorial to Leah Christine Terry, it is with great honor that I acknowledge the first philosophy degree in the family to you. Your literary piece inspired me to continue without ceasing to research, write, and complete. You are missed. We Were Writers We want immortality, so we try for it. Better if we can do it by not dying, but we’ll take what we can get. So we have to write, as insurance that we will continue forever and ever or until the world stops because that’s the only way we will. We want to be original. We want to write something new, we must. Because when we write, we are translating ourselves onto paper. We pour the contents of our being into every stroke of every pen and every tap of every key. We believe that we are wasting ourselves, and nothing we write is ever enough. But we want it to be. Because it is the vessel in which we hold ourselves, so it is the only way we can know if we are enough. We were writers, we thought, but writing is not an occupation for the living. It is a speaker from the grave. We were writers. ii ACKNOWLEDGEMENTS Foremost, I give glory to the God of all creation. With Him all things are possible. I placed this verse where I could see it as a daily reminder. “Do not be anxious about anything, but in every situation, by prayer and petition, with thanksgiving, present your requests to God. And the peace of God, which transcends all understanding, will guard your hearts and your minds in Christ Jesus” Phillipians 4:6-7. I am forever grateful to my advisors Michaele Chappell and Angela Barlow. You both challenged me to look deeper, reach farther, and persist to produce flourishing research. The behind the scenes efforts are concealed by all barring me. Like the wind that flutters through the leaves, your operative skills and energies are manifested in this work. You two are my heroes of excellence and integrity in mathematics education. This research would not have been possible without the stories from Shasta and Springer. I am thankful for the investment of your time. The majestic and fresh quality of your pedagogical practice was surely evident. May you continue to produce innovative lesson designs with exemplary skill of CAS-rich instruction. I acquired many new acquaintances through interviews, informal conversations at conferences, presentations, and through my literature review. I am thrilled to have built new understandings about mathematics teaching and learning. Thank you to Kaye Stacey and Robyn Pierce for the provision of a viable framework to enlighten my research question. And to my committee members Nancy Caukin, Mary Martin, and Jeremy Strayer I am appreciative of your ideas, critiques, and recommendations. To Patsy, thank you for creating the graphic design model. It is perfection. iii This PhD journey began with my dear friend, Sam, and our MTF family. I adore all of you and when I think of the MTF, I burst into laughter because of the joy from so many adventures. Rick, Kyle, and Michaele, you created an environment for all of us to branch out and grow as educators. The safety net was that at the end of a hard-days work, we would laugh, cry from laughter, and laugh some more. Our MTF season ended, but Sam and I persisted in our terminal degrees. It is an honor that we did this together. Learning styles are so varied. I always embrace opportunities to communicate and collaborate as a way to process my thinking. As a result, many people have partnered with me on projects over the years, or have just been available to confer about ideas. As well, an entire team of study buddies was essential to work through the independent requirements. My valued members are listed in order from preliminary exam groups to writing groups: Kyle, Derek, Ameneh, Tasha, Jan, Shari, Kristin, Matt, and Amber. You shared your workspace and work products with me. Together we cleared the pathways throughout the journey. That bond will always be strong—check back with me from time to time. I also benefited as a result of in-class and out-of-class discussions from such insightful MSE peers. To my dear friends in the MSE program not already mentioned: Teresa, Angeline, Wes, Jennifer, Brandon, Rachel, Jeffrey, Chris, and countless others, it was an amazing journey together. iv ABSTRACT Computer algebra systems (CAS) have been available for over 20 years and yet minimal CAS-rich opportunities present themselves formally to high school students. CAS tools have become readily accessible through free or inexpensive versions. Educators are emboldened to integrate essential mathematical tools in the reasoning and sense making of mathematical knowledge for students. It is the teacher that is at the heart of technology instruction, creating authentic environments for all learners. This study investigated two secondary teachers pedagogy in classes that exploited CAS in the development of mathematical knowledge. A qualitative within-site case study design was used to explore each teacher’s instructional practices. Teachers that exemplified qualities of CAS-infused instruction were purposively selected. Rich descriptive lesson vignettes as captured from classroom observations, written reflections, and interviews revealed participants’ pedagogy. The pedagogical map framework guided the identification of participant pedagogical affordances of the utilization of CAS. Eight opportunities were observed as exploited by the participants that included subject level adjustments; classroom interpersonal dynamics with students; and mathematical tasks. Data revealed several emergent themes in operation as the teacher participants oriented their mathematics instruction: viewing CAS as a mathematical consultant, verifying answers, applying multiple representations, regulating access, providing guidance, and outsourcing procedures. The components interlock with one another to form a cohesive depiction of pedagogical decisions in the presence of CAS-rich classroom instruction. The schema of CAS-oriented instruction serves as a methodology for educators to create opportunities that enrich the development of mathematical content knowledge. v TABLE OF CONTENTS Page LIST OF TABLES……………………………………………………………………. ... xiv LIST OF FIGURES…………………………………………………………………… .. xvi CHAPTER I: INTRODUCTION……………………………………………………….....1 Introduction ..........................................................................................................................1 Background of Study ...........................................................................................................2 Technology Position Statements .....................................................................................3 Technology Concerns ......................................................................................................4 Cognitive Tools and Technologies .................................................................................6 CAS Technology Background ........................................................................................7 CAS as an Essential Tool ................................................................................................8 CAS Literature Overview ................................................................................................9 Summary........................................................................................................................ 12 Theoretical Framework..................................................................................................... 13 Statement of the Problem ................................................................................................. 16 Statement of Purpose and Research Questions ............................................................... 17 Significance of Study ........................................................................................................ 18 Definition of Terms........................................................................................................... 19 Computer Algebra Systems (CAS) .............................................................................. 19 Symbolic Algebra ......................................................................................................... 19 Dynamic......................................................................................................................... 20 CAS Platforms............................................................................................................... 20 vi Orientation ..................................................................................................................... 20 Pedagogical Opportunities............................................................................................ 21 P-Map............................................................................................................................