DIGITAL RESOURCES AND MATHEMATICS: TEACHERS’ CONCEPTIONS AND NOTICING By Eryn Stehr A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of Mathematics Education – Doctor of Philosophy 2017 ABSTRACT DIGITAL RESOURCES AND MATHEMATICS: TEACHERS’ CONCEPTIONS AND NOTICING By Eryn Stehr Mathematics teachers at all levels are expected ever more frequently to integrate use of emergent technologies (e.g., calculators, software, online tools, device applets) in mathematics teaching and learning. Too often technology is used for technology’s sake, rather than in intentional ways to support mathematical reasoning, sense-making, and understanding. Teacher decision-making is necessarily complex as they consider the complex interactions of mathematics, technology, and learning, especially attending to affordances and limitations of technologies in relation to mathe- matical goals, rigor, representation, and interaction. It is critical to consider ways in which teachers can be supported in developing decision-making in locating, designing, adapting, and integrating technologies for effective mathematics teaching and learning. In this study, I explore teachers’ con- ceptions of the nature of mathematics, and mathematics teaching and learning, teachers’ noticing of characteristics of online tools and resources, and potential relationships between their concep- tions and noticings. I also describe implications for mathematics education and directions for future research. This study is a qualitative analysis of cases, with six focus teachers: Breanna, Melinda, Jana, Nicole, Heidi, and Alison. Breanna, Melinda, and Jana were grouped in a lower elementary grades team in the course, Nicole and Heidi in an upper elementary team, and Melinda in a middle grades team. Teachers varied in length of teaching experience: Breanna (2 years), Melinda (3), Jana (16), Nicole (5), Heidi (6), and Alison (4). Nicole, Heidi, and Alison had a mathematics credential, while Breanna, Melinda, and Jana had not. Regarding the three conceptions of the nature of mathematics described by Thompson (1992) and Ernest (1989b), findings of the study included the necessity of expanding meanings of aspects of conceptions and focusing expanded aspects to include school mathematics. Meanings of aspects of mathematical conceptions should reflect teachers’ meanings of those aspects as well as reflecting researchers’ meanings. All teachers showed some evidence of an Instrumentalist conception, but Breanna and Melinda most strongly showed alignment with an Instrumentalist conception. Jana, Nicole, Alison, and Heidi showed close alignment with a Platonist conception. Alison additionally indicated alignment with a Problem-Driven view. Teachers showed rich understanding of aspects of the nature of mathematics that are excluded from Ernest’s framework. In considering teachers’ noticing of mathematical aspects of online tools, patterns across levels of sophistication and alignment with conceptions emerged. Noticing patterns emerged that sup- ported past research on levels of sophistication in noticing structures of mathematical tasks and aspects of mathematics curricula. For example, teachers with fewer years of experience attended to general goals more easily than specific goals as described by M. G. Sherin and Drake (2009). Teacher with fewer years of experience also attended to non-mathematical aspects (e.g., visuals, reasons for engagement) of online tools more frequently than mathematical aspects (B. Sherin & Star, 2011; M. G. Sherin & van Es, 2009). All teachers expressed expectations for more online tools to provide feedback on correctness than to provide opportunities for interaction with mathe- matical consequences. Relationships between teachers’ conceptions of the nature of mathematics and their noticing of aspects of online tools emerged through teachers’ focus on three aspects of online tools: learning goals, strategies for learning, and knowledge structures. For each, language used by teachers was considered in tandem with descriptions of relationships between Instrumentalist and Platonist conceptions of the nature of mathematics and approaches to teaching (Thompson, 1992; Ernest, 1989b). Breanna’s noticing showed close alignment with an Instrumentalist conceptions, while Nicole’s and Alison’s showed close alignment with a Platonist conception. Jana, Melinda, and Heidi showed some variation in their alignments and showed some balance between alignment with Instrumentalist and with Platonist conceptions. Copyright by ERYN STEHR 2017 ACKNOWLEDGEMENTS I thank my advisor and dissertation co-chair, Dr. Jack Smith, who has served as advisor, boss, course professor, reference, and mentor over six years of (more or less) biweekly meetings. I greatly appreciate your support and advice in my research and growth as I have struggled to keep up with the pack and find my way in the “ocean.” I thank my dissertation co-chair, Ralph, who has served as boss and co-instructor, course pro- fessor, reference, and mentor over these past six years. I appreciate the support you have provided me as I learn about technology in mathematics teaching and learning so many frameworks and perspectives. I appreciate the value you showed as we worked on the online course described in such detail in this study. I thank Dr. Sharon Senk. She told me the first day I met her that she would always be available and she would always support me, but I had to tell her what I needed and ask for time or support and she would be there. I have had countless conversations with her about courses, research, and personal struggles, and she has been a strong model and counselor for me. I thank faculty and mentors: Dr. Brin Keller, Dr. Tonya Bartell, and Jill Newton. For each of my faculty mentors, you have all treated me as a colleague. I have appreciated the advice and concern you have given me in our interactions. I look forward to continuing our relationships and friendships. I thank my parents, Lanny and Kathryn Monson, who listened as I struggled with my devel- oping ideas over the past five years, and before that. They gave me unconditional support. They listened any time that I needed to talk ideas out and really took the time to try to understand what I was working on, despite their own expressed discomfort with mathematics. They helped me find ways to move forward, over and over again. I must thank my whole family, immediate and extended, who have continued to support me through each high and every low. I thank all of my friends and colleagues on the Preparing to Teach Algebra project. Thank you, P.I. Dr. Sharon Senk and Co-P.I.s Dr. Jill Newton and Dr. Yukiko Maeda. I especially thank those v friends and colleagues – fellow graduate students and now doctors – that I have continued to work with over the past six years: Jia He, Alexia Mintos, Hyunyi Jung, and Vivian Alexander. I have reached out to each of you for support as I think through ideas for my dissertation, and you have always been there for me. I thank all of my friends and colleagues on the Strengthening Tomorrow’s Education in Mea- surement project. Thank you, P.I. Dr. Jack Smith. I especially thank those friends and colleagues – fellow graduate students and now doctors – that I have shared with through writing groups and con- ference presentation preparation, especially Dan Clark, Jia He, Nic Gilbertson, Funda Gon¨ ulates¸,¨ and Lorraine Males. I thank all of my special graduate program friends and buddies, you know who you are! In additon to everyone else, I especially thank Eun Mi Kim, Joanne Philhower, Kate Johnson, Dan Clark, Julie Hanch, Julie Nurnberger-Haag, and Linda Leckrone who have been my rocks over the past six years. You listened to me when I was sad, frustrated, angry, joyful, excited, etc. You gave me feedback through countless hours of brainstorming, practicing, sound-boarding, etc. I would not have completed this dissertation without you. I look forward to our futures as colleagues and friends. I also thank my non-mathematics education writing group, McKinley Green and Karin Ahlin, who likewise have given me ongoing feedback, encouragement, and sympathy on my dissertation. Finally, I must thank each of my research participants for giving me the opportunity to learn about their conceptions and noticing and my own, through exploring their thinking through the statements they made in discussions, reflections, and evaluations, and of course in our conversation. TABLE OF CONTENTS LIST OF TABLES . xii LIST OF FIGURES . xiv CHAPTER 1 INTRODUCTION . 1 Purpose . 3 Research Questions . 4 Overview of the Study . 5 Conceptual Issues . 6 Organization of the Dissertation . 7 CHAPTER 2 REVIEW OF LITERATURE . 10 Meaning of Technology . 10 Conceptions . 12 Beliefs and Conceptions in Mathematics Education . 13 Nature of Mathematics and Mathematics Teaching and Learning . 15 School and Disciplinary Mathematics: A Gap . 16 Aspects of the Nature of Mathematics . 18 Categorizing Conceptions of the Nature of Mathematics . 20 Instrumentalist Conception . 22 Platonist Conception . 23 Problem-driven Conception . 23 Impact of Three Conceptions on Teaching Beliefs . 24 Conclusion . 27 Noticing . 28 Mathematical Tasks . 28 In-the-Moment Decision-Making . 29 Adaptation and Construction of Curricula . 29 Instructional Resources . 30 Online Instructional Resources . 30 CHAPTER 3 METHOD . 32 Study Design . 32 Data Collection . 33 Mathematics, Technology, and Education Course . 34 Participants . 37 Interview Structure