Student application of the fundamental theorem of calculus with graphical representations in mathematics and physics.

摘要

One mathematical concept frequently applied in physics is the Fundamental Theorem of Calculus (FTC). Mathematics education research on student understanding of the FTC indicates student difficulties with the FTC. Similarly, a few studies in physics education have implicitly indicated student difficulties with various facets of the FTC, such as with the definite integral and the area under the curve representation, in physics contexts. There has been no research on how students apply the FTC in graphically-based physics questions.;This study investigated student understanding of the FTC and its application to graphically-based problems. Our interest spans several aspects of the FTC: student difficulties, problem-solving strategies, and visual attention.;Written and interview findings revealed student difficulties common to mathematics and physics, e.g., confusion between the antiderivative difference and the function difference. Three problem-solving strategies were identified: algebraic, graphical, and integral. For a deeper analysis of problem-solving strategies, we applied the perspectives of epistemological framing (student expectations/perceptions) and epistemic games (problem-solving games). While most observed frames and epistemic games were somewhat modified versions of those previously reported, we identified one new game: the equation-based analytical game. In addition, a novel eye-tracking study was conducted to explore students' visual attention to different parts of graphically-based FTC questions. Results indicated that students' visual behavior was affected by the representations in the questions, such as the presence or absence of certain equation(s) and/or graphical feature(s), as well as context (math vs. physics). Because student responses seemed to be both conceptually and salient-feature driven, the results were explained using the cognitive perspectives of top-down (conceptually driven) and bottom-up (feature-driven) processes.;Eye-tracking results provided support for interview findings about problem-solving strategies. For many students, the absence of specific visual cues led to a particular framing of the problem that was associated with inappropriate e-games for that problem. Minor interviewer prompting often enabled students to reframe a problem and invoke relevant knowledge and strategies, suggesting that students possess knowledge of individual facets of the FTC, but this knowledge may not be elicited by a particular problem representation(s). Additionally, specific difficulties can be seen as due to inappropriate problem framing.