The Interaction of Geometric and Spatial Reasoning: Student Learning of 2D Isometries in a Special Dynamic Geometry Environment Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Leah M. Frazee, B.S., M.A.T. Graduate Program in Education: Teaching and Learning The Ohio State University 2018 Dissertation Committee: Michael T. Battista, Advisor Patricia Brosnan Theodore Chao Copyright by Leah M. Frazee 2018 Abstract The relationship between spatial reasoning and success in mathematics has been well established by both educational researchers and cognitive psychologists. However, the ways in which spatial reasoning is used in mathematics learning is not clearly understood. Many current studies investigating spatial reasoning in mathematics quantitatively correlate aspects of students’ mathematical proficiency with standardized measures of spatial reasoning. To extend the research that connects spatial ability and the learning and doing of mathematics, there is a need for studies that elaborate the ways in which students use spatial reasoning while learning specific mathematics content. The present study examines the relationship between spatial reasoning and geometric reasoning in the context of 2D isometries, specifically reflections and rotations. Rotations and reflections are often included in spatial reasoning assessments. Moreover, isometries are important formal mathematical concepts, which have properties, specific parameters, and lend themselves to analytic reasoning. Thus, isometries are foundational topics for studying the link between spatial reasoning and the learning of mathematics. The present study investigated student learning of reflections and rotations using a special dynamic geometry environment called Individualized Dynamic Geometry Instruction, or iDGi. Drawing on a constructivist view of learning, the iDGi isometry environment integrates dynamic motion with a grid in which students can learn the geometric properties of isometries while using visualization and analytic reasoning. The methodology of the study is one-on-one teaching iii experiments with three students who had different levels of spatial ability at the start of the study. The research questions motivating the study are the following: (1) What types of spatial reasoning do students use while learning about reflections and rotations in the iDGi dynamic computer environment? (2) How do students’ analytic and mental imagery strategies interact while learning about reflections and rotations in the iDGi dynamic computer environment? (3) In the context of learning about reflections and rotations in iDGi, how is property-based knowledge related to analytic and spatial reasoning? The results of this study show that spatial visualization and spatial-analytic reasoning are critical to students' learning and reasoning about reflections and rotations, and how these two processes were differentially interrelated, depending on the student and task. Students used both static and dynamic mental models to support their visual reasoning with varying levels of success depending on the problem type. Students also showed evidence of implicit and explicit property knowledge, at times using properties to correctly solve problems, but not always expressing the properties in formal mathematical statements. The present study describes the mental models students used when solving problems about reflections and rotations and proposes four new configurations describing the inter- relationship between spatial visualization and spatial analytic reasoning. The present study also outlines how the processes of coordination and integration were used as students constructed and employed their mental models for solving problems about reflections and rotations. The results provide new models and theory that can describe more clearly how students come to understand new ideas in geometry, as well as the difficulties they may face. iv Dedication For my Mom and Dad, Your patience, encouragement, and food gift cards have sustained me. You are and always have been my biggest supporters, and I am forever grateful. v Acknowledgments I wish to thank Dr. Michael Battista, my advisor, for his patient guidance, for sharing his depth of insight, and for allowing me to use his dynamic geometry curriculum, Individualized Dynamic Geometry Instruction (iDGi), to study how students come to understand concepts in geometry. I would like to thank my committee members, Dr. Patricia Brosnan and Dr. Theodore Chao, for all of their encouragement and advice throughout my time at OSU. I would also like to thank to Dr. Karen Irving for serving on my exam committee and helping to focus my work in student thinking. A special thanks to my family and friends whose patience, encouragement, understanding, and unconditional support has made this journey possible. vi Vita 1997................................................................Ironton High School 2001................................................................B.S. Mathematics, Cedarville University 2002................................................................M.A.T. Mathematics, Miami University 2003-2014 ......................................................Secondary Mathematics Teacher, Wheaton Academy 2015 to present ..............................................Graduate Teaching & Research Associate, Department of Teaching and Learning, The Ohio State University Fields of Study Major Field: Education: Teaching and Learning Special Area: STEM (Mathematics) Education vii Table of Contents Abstract .......................................................................................................................................... iii Dedication ....................................................................................................................................... v Acknowledgments .......................................................................................................................... vi Vita ................................................................................................................................................ vii List of Tables ................................................................................................................................. xi List of Figures .............................................................................................................................. xiv Chapter 1: Background and Significance ....................................................................................... 1 Spatial and Geometric Reasoning ........................................................................................... 2 Research Questions and Methods ........................................................................................... 5 Definitions of Important Terms .............................................................................................. 6 Results and Significance of Study .......................................................................................... 9 Chapter 2: Literature Review ........................................................................................................ 11 Learning Theory: Constructivism ......................................................................................... 11 Review of Research Relating Geometric and Spatial Reasoning ......................................... 18 Spatial Reasoning and Mathematical Reasoning .................................................................. 27 Geometric and Spatial Reasoning with Isometries ............................................................... 28 Spatial Analytic Reasoning and Properties of Isometries ..................................................... 42 viii Chapter 3: Methodology ............................................................................................................... 48 Data Collection ..................................................................................................................... 48 Clinical Interviews and Teaching Experiments in Mathematics Education Research ......... 50 Conceptual Analysis ............................................................................................................. 59 Models of Student Thinking and the Development of a Dynamic Learning Environment .. 60 Data Analysis ........................................................................................................................ 74 Chapter 4: Results ......................................................................................................................... 77 Reflections: Calvin ................................................................................................................ 83 Rotations: Calvin ................................................................................................................ 107 Reflections: Amber ............................................................................................................. 140 Rotations: Amber ................................................................................................................ 160 Reflections: Brad ................................................................................................................ 181 Rotations: Brad ................................................................................................................... 197