Mathematical modeling and kinematics: a study of emerging themes and their implications for learning mathematics through an inquiry-based approach

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Carrejo, David John

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In recent years, emphasis on student learning of mathematics through “real world” problems has intensified. With both national and state standards calling for more conceptual learning and understanding of mathematics, teachers must be prepared to learn and implement more innovative approaches to teaching mathematical content. Mathematical modeling of physical phenomena is presented as a subject for new and developing research areas in both teacher and student learning. Using a grounded theory approach to qualitative research, this dissertation presents two related studies whose purpose was to examine the process by which in-service teachers and students enrolled in an undergraduate physics course constructed mathematical models to describe and predict the motion of an object in both uniform and non-uniform (constant acceleration) contexts. This process provided the framework for the learners’ study of kinematics. Study One involved twenty-three in-service physics and math teachers who participated in an intensive six-hour-a-day, five-day unit on kinematics as part of a professional development institute. Study Two involved fifteen students participating in the same unit while enrolled in a physics course designed for pre-service teachers and required in their undergraduate or graduate degree programs in math and science education. Qualitative data, including videotapes of classroom sessions, field notes, researcher reflections, and interviews are the focus of analysis. The dissertation presents and analyzes tensions between learner experience, learning standard concepts in mathematics and learning standard concepts in physics within a framework that outlines critical aspects of mathematical modeling (Pollak, 2003): 1) understanding a physical situation, 2) deciding what to keep and what not to keep when constructing a model related to the situation, and 3) determining whether or not the model is sufficient for acceptance and use. Emergent themes related to the construction of the learners’ models included several robust conceptions of average velocity and considerations of what constitutes a “good enough” model to use when describing and predicting motion. The emergence of these themes has implications for teaching and learning mathematics through an inquiry-based approach to kinematics.