New directions in differential equations: A framework for interpreting students' understandings and difficulties |
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| Institution: | 1. Eunice Kennedy Shriver National Institute of Child Health and Human Development, National Institutes of Health, NICHD Pregnancy and Perinatology Branch, 6710B Rockledge Drive, 2321B, Bethesda, MD 20817;2. Division of Neonatology, University of Alabama at Birmingham, Birmingham, AL;3. Department of Pediatrics, Childrenʼs Mercy Hospital, University of Missouri-Kansas City School of Medicine, Kansas City, MO;4. Division of Neonatal and Developmental Medicine, Department of Pediatrics, Stanford University School of Medicine and Lucile Packard Childrenʼs Hospital, Palo Alto, CA;5. Department of Pediatrics, University of Texas Southwestern Medical Center, Dallas, TX;6. Division of Neonatology, Nationwide Childrenʼs Hospital, Columbus, OH;7. Social, Statistical and Environmental Sciences Unit, RTI International, Rockville, MD;8. Division of Neonatology and Developmental Biology, University of California—Los Angeles, David Geffen School of Medicine, Los Angeles, CA;1. Department of Computer and Information Sciences, Fordham University, New York, NY 10023, United States;2. Department of Computer Science, Columbia University, New York, NY 10027, United States;3. Institute of Applied Mathematics, University of Warsaw, Poland |
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| Abstract: | The purpose of this paper is to offer a framework for interpreting students' understandings of and difficulties with mathematical ideas central to new directions in differential equations. These new directions seek to guide students into a more interpretive mode of thinking and to enhance their ability to graphically and numerically analyze differential equations. The framework reported here is the result of investigating in depth six students' understandings through a series of task-based individual interviews and classroom observations. The two major themes of the framework, the function-as-solution dilemma theme and students' intuitions and images theme, extend previous research on student cognition at the secondary and collegiate level to the domain of differential equations and reflect the increased recognition of situating analyses of student learning within students' learning environment. For new areas of interest such as differential equations, mapping out students' understandings of important mathematical ideas can be an important part of curricular and instructional design that seeks to refine and build on students' ways of thinking. |
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