Areas,anti-derivatives,and adding up pieces: Definite integrals in pure mathematics and applied science contexts |
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| Institution: | Mathematics Education, Brigham Young University, 167 TMCB, Provo, UT 84602, United States;Norwegian University of Science and Technology, Department of Teacher Education, NO-7491, Trondheim, Norway;Department of Internal Medicine, Gastroenterology and Hepatology with Cardiology Unit and Centre for the Treatment of Heart Failure and Cardioncology, Clinical University Hospital in Olsztyn, Poland;University of Nebraska-Lincoln, United States;Michigan Technological University, 1400 Townsend Drive, Houghton, MI 49931, United States;Departamento de Matemática, ICE - Universidade Federal de Juiz de Fora, 36036-330 Juiz de Fora MG, Brazil;Department of Mathematics “F. Casorati”, University of Pavia (Italy), Via Ferrata, 5, 27100, Pavia, Italy |
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| Abstract: | Research in mathematics and science education reveals a disconnect for students as they attempt to apply their mathematical knowledge to science and engineering. With this conclusion in mind, this paper investigates a particular calculus topic that is used frequently in science and engineering: the definite integral. The results of this study demonstrate that certain conceptualizations of the definite integral, including the area under a curve and the values of an anti-derivative, are limited in their ability to help students make sense of contextualized integrals. In contrast, the Riemann sum-based “adding up pieces” conception of the definite integral (renamed in this paper as the “multiplicatively-based summation” conception) is helpful and useful in making sense of a variety of applied integral expressions and equations. Implications for curriculum and instruction are discussed. |
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| Keywords: | Calculus Definite integral Riemann sum Area Anti-derivative Physics and engineering |
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