Students' solution strategies to differential equations problems in mathematical and non-mathematical contexts
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Abstract
The present study investigated undergraduate students' understanding of slope fields and equilibrium solutions as they solved problems in differential equations. The following questions were addressed: 1) Does performance on complex problems vary by context (mathematical, non-mathematical)? 2) When considering a complex problem in a mathematical and a non-mathematical context, are participants who answer the problem in one context correctly more likely to answer the corresponding problem in the other context correctly? 3) Does performance on simple problems predict performance on complex problems?
A written test, Differential Equations Concept Assessment (DECA), was designed and administered to 91 participants drawn from three introductory differential equations courses. Of those participants, 13 were interviewed. DECA consists of four complex problems, two in mathematical contexts and two in non-mathematical contexts, and six simple problems that assess aspects of slope fields and equilibrium solutions.
The data obtained from DECA and the interviews showed that participants performed significantly better on complex problems in non-mathematical contexts than on complex problems in mathematical contexts. There was a significant relationship found between performance on a problem in a mathematical context and performance on the isomorphic problem in the context of population growth, but a significant relationship was not found between a different pair of isomorphic problems, one in a mathematical context and the other in the context oflearning. However, for all the complex problems, participants illustrated a preference for algebraic rather than geometric methods, even when a geometric approach was a more efficient method of solution. Although performance on simple problems was not found to be a strong predictor of performance on complex problems, the simple problems proved to elicit difficulties participants had with aspects of slope fields and equilibrium solutions. For example, participants were found to overgeneralize the notion of equilibrium solution as being any straight line and as existing at all values where a differential equation equals zero. Participants were also found to identify slope fields as determining only equilibrium solutions.
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Thesis (Ed.D.)--Boston University
PLEASE NOTE: Boston University Libraries did not receive an Authorization To Manage form for this thesis or dissertation. It is therefore not openly accessible, though it may be available by request. If you are the author or principal advisor of this work and would like to request open access for it, please contact us at open-help@bu.edu. Thank you.
PLEASE NOTE: Boston University Libraries did not receive an Authorization To Manage form for this thesis or dissertation. It is therefore not openly accessible, though it may be available by request. If you are the author or principal advisor of this work and would like to request open access for it, please contact us at open-help@bu.edu. Thank you.