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Defining Mathematical Modeling for K-12 Education DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degr Defining mathematical modeling for K-12 education DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Kimberly A. Groshong Graduate Program in Education Teaching & Learning The Ohio State University 2018 Dissertation Committee: Patricia Brosnan, Ph.D., Chair Theodore Chao, Ph.D. Kathy Malone, Ph.D. Copyrighted by Kimberly Ann Groshong 2018 Abstract This dissertation reports the research methodology and findings of three studies focused on mathematical modeling as a structured problem-solving process where the student translates relevant details from a situation, event, or phenomena in daily life into mathematical language with mathematically-defined structures and operations, which form the mathematical model. This model can be manipulated to yield a solution, description, explanation, or prediction, which are then interpreted within the context of the real-life event. Part of this process requires determining when the mathematical model can be adapted to new situations and identifying the model’s boundaries and limitations in providing descriptions, explanations, and predictions. Constructing, testing, and applying mathematical models meet the needs of science, technology, engineering, and mathematics (STEM) educational goals for advancing students’ knowledge and skills and for increasing students’ understanding of the world around them. To capture the complexities of mathematical modeling, this research defined the Systems Theory in Mathematical Modeling Education (STiMME) theoretical framework to investigate the interactions that influence students’ mathematical modeling experiences. Many stakeholders, e.g. researchers, educators, students, etc., work at various levels to deliver content standards, develop materials, implement lessons, engage in learning, and participate in a myriad of ways that affect outcomes. The STiMME framework probes these interrelated elements and their impact on students in hopes of revealing new ways of thinking about mathematical modeling teaching and learning practices and theories. ii This research defined three STiMME subsystems. The first study scrutinized 241 STEM literature artifacts distinguishing between models and representations, models and mathematical models, and modeling and mathematical modeling to yield definitions for laying a united foundation for investigating mathematical modeling teaching and learning in all STEM education subjects. Similarities in mathematical modeling, engineering and technology design, and scientific methodological processes were identified that may improve students’ transfer of knowledge and skills within different STEM courses. The second study scrutinized 342 resources to create a taxonomy for the 14 different types of mathematical models described in professional and educational writing. The final study examined student work from the same mathematical modeling activity presented in three math and physics classes to determine the impact that student question posing, tracking, and reviewing had on the types of mathematical models they constructed. The students asked exploratory, consolidating, clarifying, and elaborating questions. It was noted that students with more exposure to mathematical modeling activities and more experience with question strategies were able to construct more sophisticated mathematical models. iii Acknowledgments “It is the supreme art of the teacher to awaken joy in creative expression and knowledge.”-Albert Einstein While trying to find the words to might capture my gratitude I feel towards my family and towards my dissertation advisor, Dr. Patricia Brosnan, and the members of my dissertation committee, Dr. Theodore Chao and Dr. Kathy Malone, a friend reminded me of this quote from Albert Einstein. Some may find it cliché, but it speaks appropriately to my feelings as I finish this academic journey. I was truly surrounded by great teachers. Family: Through the patience, love, and support of my husband and children, I was able to find the time to study, research, and write. Many times they patiently and attentively listened to me talk about my dissertation. Their words of counsel helped me write more clearly and concisely. Dr. Patricia Brosnan: My work would have come to a grinding stop without your support. You stood by me through my grievous losses. You were there and offered guidance when I struggled to find focus, direction, and voice, and you restored my joy for this work. Words are insufficient to express my gratitude for the hours you set aside to listen to my ideas and my challenges as well as the overwhelming hours you spent reading and commenting on my many drafts. Dr. Theodore Chao: Your endless optimism and enthusiasm brought energy and creativity to my thinking. Your ability to quickly connect different ideas opened my thinking and helped me better understand the many complexities that exist as we strive to understand how students learn. Dr. Kathy Malone: You were always there when I needed feedback on the many drafts I sent to you. I am very grateful for the time and thought you put into each comment and suggestion that helped me complete this work. You were an ever present and constant force of encouragement. Your enthusiasm for staying on-top of field of modeling offered knowledgeable insight into every question I posed to you. iv Vita 1994................................................................A.A.S. Aeronautical Electrical Engineering Technology, Community College of the Air Force 2001................................................................B.S. Mathematics, Physics, Secondary Education Integrated Science, Ashland University 2008................................................................M.S. Applied Mathematics, University of Akron 2012 to 2015 .................................................Graduate Teaching Associate, Department of Education, The Ohio State University Publications Groshong, K. (2016). Different types of mathematical models. In C. R. Hirsch & A. R. McDuffie (Eds.), Mathematical modeling and modeling mathematics (pp. 17-24). Reston, VA: National Council of Teachers of Mathematics. Groshong, K. A., & Park, J. Y. (2017). Examining secondary mathematics teachers' mathematical modeling knowledge for teaching. In E. Galindo & J. Newton (Eds.), Proceedings of the 39th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. Indianapolis, IN: Hoosier Association of Mathematics Teacher Educators. v Fields of Study Major Field: Education Teaching & Learning Science, Technology, Engineering, and Mathematics (STEM) Education vi Table of Contents Abstract .......................................................................................................................................... ii Acknowledgments ........................................................................................................................ iv Publications ....................................................................................................................................v Fields of Study .............................................................................................................................. vi Table of Contents ........................................................................................................................ vii List of Tables ............................................................................................................................... xii List of Figures ............................................................................................................................. xiii Chapter 1: Introduction to Mathematical Modeling in Education ..........................................1 Study Design .............................................................................................................................. 3 Study rationale ......................................................................................................................... 3 Study structure ......................................................................................................................... 3 Background ................................................................................................................................ 4 Distinguishing between the Real World and the Mathematical World ................................... 4 Classifying Mathematics Activities with Real-World Narratives ........................................... 5 Problem-solving in mathematical modeling .......................................................................... 13 vii Mathematical modeling in K-12 education standards ........................................................... 14 Defining a Theoretical Framework........................................................................................ 15 General Systems Theory (GST) ............................................................................................ 16 Systems Theory in Mathematical Modeling Education Framework (STiMME) .................. 19 Dissertation Structure ............................................................................................................. 24 Chapter 2 (Article 1): Defining mathematical modeling in STEM education ...................... 26 Chapter 3 (Article 2): Classifying different types of mathematical modeling ..................... 27 Chapter 4 (Article
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