Theorizing Teaching Practices in Mathematical Modeling Contexts Through the Examination of Teacher Scaffolding Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Stephen T. Lewis Graduate Program in the School of Education: Teaching and Learning The Ohio State University 2018 Dissertation Committee Azita Manouchehri, Advisor Theodore Chao, Co-Advisor James Fowler 1 Copyrighted by Stephen T. Lewis 2018 2 Abstract The purpose of this study was to examine the reflexive ways in which tensions between desired academic perspectives and life experiences, which are brought forth by learners when dealing with modeling contexts, are addressed by teachers so to advance learners’ modeling capacity. In particular I examined the activities of one 11th grade pre- calculus teacher as she attempted to establish a classroom culture of mathematical modeling. Of interest was unpacking how learners were enculturated into a mathematical modeling practice, and ways that the teacher’s enacted view of modeling advanced the development of both modeling capacity and curricular knowledge. The multiple lenses of ethnomathematics (D’Ambrosio, 1985) and in particular ethnomodeling (Orey & Rosa, 2010) coupled with linguistic discourse analysis (Bloome et al, 2010) were utilized to more precisely identify the local views of mathematical modeling that the participating teacher held. The analysis of data revealed 23 specific components indicative of the teachers’ mathematical modeling practice. Results of this study indicated that the teacher’s view of mathematical modeling was not as a vehicle for the learning of new mathematical concepts, nor was it that of a content in its own right, but rather a bridge between these two perspectives which served the opportunity to both advance modeling capacity and meet curricular demands. ii Dedication To my wife, Annie, and my children, Lindsey, Michael, and Jimmy, without whom this would not have been possible. iii Acknowledgments Foremost, I would like to express my sincerest gratitude to my advisor, Dr. Azita Manouchehri for all of her time in developing me as a mathematics education researcher. I am grateful to have had the opportunity to learn from her experience and wisdom. Her sustained support helped better me as a scholar and a researcher. I am truly grateful to have worked with her. I am grateful for my co-advisor Dr. Theodore Chao for his insights into equitable teaching practices in mathematics education. I am grateful for my committee member Dr. Jim Fowler for providing the voice of academic mathematics. Our discussions have helped me recognize how student ideas connect to the boundary of human knowledge in mathematics. I am grateful for Dr. David Bloome who provided me with substantial training in discourse analysis. Without him, I would not have been equipped to notice those small details in human interactions that signal important ideas. I am grateful for Dr. Milton Rosa for his insights into ethnomathematics and ethnomodeling. Without he and Dr. Daniel Orey, the theoretical framework used in this study would be significantly lacking and this study incomplete. I am grateful for my family, especially my wife, Annie, and my children, Lindsey, Michael, and Jim. It was their love and support that allowed me to persevere through this iv doctoral program. I am grateful for my mother, Linda, who taught me to never give up and to keep smiling no matter what. I am grateful for my friends, especially Adam, Dug, Micky, and Nick, who helped remind me to take a break and play a game. It was in those moments of rest that insight into this work happened. I am grateful for my academic brothers and sisters, in particular James, Azin, Ding Ding, and Ayse. Without that community, their questions, and insights, my work would not be of the quality that it is today. Finally I am grateful for my participating teacher Mae and her students for welcoming me into their classroom, allowing me to ask many questions, and share with me their insights into mathematics. Without this group, this study would not have been possible. v Vita 2005...................................................................................... B.S. Mathematics 2008...................................................................................... B.S. Psychology 2009...................................................................................... M.Ed. Mathematics Education Publications Manouchehri. A., & Lewis, S. T. (2017). Reconciling intuitions and conventional knowledge: The challenge of teaching and learning mathematical modelling. In G. Wake, G. Stillman, W. Blum & M. North (Eds.) Crossing and Researching Boundaries in Mathematical Modelling (pp. 93-100). Springer: New York. Noteman, N., & Lewis, S. T. (2017). Postscript: Support your side. Teaching Children Mathematics, 24(1), 64. Fields of Study Major Field: Education: Teaching and Learning Specialization: Mathematics Education vi Table of Contents Abstract ............................................................................................................................... ii Dedication .......................................................................................................................... iii Acknowledgments.............................................................................................................. iv Vita ..................................................................................................................................... vi List of Tables ...................................................................................................................... x List of Figures .................................................................................................................... xi Chapter 1: Introduction ....................................................................................................... 1 Chapter 2: Literature Review and Theoretical Framework ................................................ 5 2.1 Mathematical Models and Modeling ........................................................................ 5 2.2 Theoretical Perspectives on Mathematical Modeling in Educational Settings ....... 15 2.2.1 Modeling as a Vehicle ..................................................................................... 15 2.2.2 Modeling as Content ........................................................................................ 23 2.3 Ethnomathematics ................................................................................................... 29 2.4 Empirical Summary ................................................................................................ 37 2.5 Theoretical Framework ........................................................................................... 41 2.5.1 Discourse Analysis in Educational Research ................................................... 41 2.5.2 An Ethnomodeling Cycle ................................................................................. 56 Chapter 3: Methods ........................................................................................................... 64 3.1 Research Design...................................................................................................... 64 3.2 Orientations Towards Data Generation .................................................................. 66 3.2.1 Modeling Competence and Competencies....................................................... 67 3.2.2 Sociomathematical Norms and Social Practices .............................................. 70 3.2.3 Ethnomathematics and Ethnomodeling ........................................................... 73 3.2.4 Event Mapping ................................................................................................. 73 3.2.5 Transcribing ..................................................................................................... 80 vii 3.3 Study Procedures .................................................................................................... 81 3.4 Description of Research Site ................................................................................... 86 3.4.1 Background and Description of Cooperating Classroom Teacher .................. 87 3.4.2 Description of Classroom Contexts ................................................................. 90 3.4.3 Researcher Positionality................................................................................... 91 3.5 Data Analysis Process ............................................................................................. 92 3.6 Report on Pilot Study ............................................................................................ 109 Chapter 4: Data Analysis and Results ............................................................................. 113 4.1 Mae’s Short-Term Goals, Long-Term Goals, and Orientation Towards Mathematics ................................................................................................................ 114 4.2 Outline of Modeling Sessions ............................................................................... 120 4.2.1 The Answer is Task ....................................................................................... 121 4.2.2 Measure the Height Task ............................................................................... 127 4.2.3 Shortest Distance to the Cafeteria Task