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2013, 10(5&6): 1437-1453. doi: 10.3934/mbe.2013.10.1437

## The role of multiple modeling perspectives in students' learning of exponential growth

 1 Mathematics Department, Kingston Hall 216, Eastern Washington University, Cheney, WA 99004-2418, United States

Received  October 2012 Revised  May 2013 Published  August 2013

The exponential is among the most important family functions in mathematics; the foundation for the solution of linear differential equations, linear difference equations, and stochastic processes. However there is little research and superficial agreement on how the concepts of exponential growth are learned and/or should be taught initially. In order to investigate these issues, I preformed a teaching experiment with two high school students, which focused on building understandings of exponential growth leading up to the (nonlinear) logistic differential equation model. In this paper, I highlight some of the ways of thinking used by participants in this teaching experiment. From these results I discuss how mathematicians using exponential growth routinely make use of multiple --- sometimes contradictory --- ways of thinking, as well as the danger that these multiple ways of thinking are not being made distinct to students.
Citation: Carlos Castillo-Garsow. The role of multiple modeling perspectives in students' learning of exponential growth. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1437-1453. doi: 10.3934/mbe.2013.10.1437
##### References:
 [1] M. Begon, J. L. Harper and C. R. Townsend, "Ecology: Individuals, Populations and Communities," Third edition, Blackwell Science, 1996. Google Scholar [2] Fred Brauer and Carlos Castillo-Chavez, "Mathematical Models in Population Biology and Epidemiology," Springer, New York, 2000.  Google Scholar [3] Carlos William Castillo-Garsow, "Teaching the Verhulst Model: A Teaching Experiment in Covariational Reasoning and Exponential Growth," Ph.D thesis, Arizona State University, Tempe, AZ, 2010. Google Scholar [4] Carlos William Castillo-Garsow, Continuous quantitative reasoning, in "Quantitative Reasoning and Mathematical Modeling: A Driver for STEM Integrated Education and Teaching in Context" (eds. R. Mayes, R. Bonillia, L. L. Hatfield and S. Belbase), WISDOMe Monographs, Vol. 2, University of Wyoming Press, Laramie, WY, 2012. Google Scholar [5] Jere Confrey and Erick Smith, Exponential functions, rates of change, and the multiplicative unit, Educational Studies in Mathematics, 26 (1994), 135-164. Google Scholar [6] Jere Confrey and Erick Smith, Splitting, covariation, and their role in the development of exponential functions, Journal for Research in Mathematics Education, 26 (1995), 66-86. doi: 10.2307/749228.  Google Scholar [7] E. V. Glasersfeld, "Radical Constructivism: A Way of Knowing and Learning," Falmer Press, London, 1995. Google Scholar [8] Fernando Hitt and Orlando Planchart, Graphing discrete versus continuous functions: A case study, In "Proceedings of the 20th Annual meeting of PME-NA," ERIC, Columbus, 1998. Google Scholar [9] G. Leinhardt, O. Zaslavsky and M. K. Stein, Functions, graphs, and graphing: Tasks, learning, and teaching, Review of Educational Research, 60 (1990), 1-64. doi: 10.3102/00346543060001001.  Google Scholar [10] T. R. Malthus, "An Essay on the Principle of Population," Sixth edition, John Murray, London, 1826. Google Scholar [11] Kevin C. Moore, "The Role of Quantitative Reasoning in Precalculus Students Learning Central Concepts of Trigonometry," Ph.D thesis, Arizona State University, 2010. Google Scholar [12] Kevin C. Moore, Coherence, quantitative reasoning, and the trigonometry of students, in "Quantitative reasoning and Mathematical modeling: A driver for STEM Integrated Education and Teaching in Context" (eds. R. Mayes, R. Bonillia, L. L. Hatfield and S. Belbase), WISDOMe Monographs, Vol. 2, University of Wyoming Press, Laramie, WY, 2012. Google Scholar [13] L. P. Steffe and Patrick W Thompson, Teaching experiment methodology: Underlying principles and essential elements, in "Research Design in Mathematics and Science Education" (eds. R. Lesh and A. E. Kelly), Erlbaum, Hillsdale, NJ, (2000), 267-307. Google Scholar [14] April Strom, "A Case Study of a Secondary Mathematics Teacher'S Understanding of Exponential Function: An Emerging Theoretical Framework," Ph.D thesis, Arizona State University, 2008. Google Scholar [15] Patrick W. Thompson, Conceptual analysis of mathematical ideas: Some spadework at the foundation of mathematics education, in "Proceedings of the Annual Meeting of the International Group for the Psychology of Mathematics Education" (eds. O. Figueras, J. L. Cortina, S. Alatorre, T. Rojano and A. Sepulveda), Vol. 1, PME, Morelia, Mexico, (2008), 45-64. Google Scholar [16] Patrick W Thompson, In the absence of meaning, in "Vital Directions for Mathematics Education Research" (ed. K. Leatham), Springer, New York, 2013. doi: 10.1007/978-1-4614-6977-3_4.  Google Scholar

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##### References:
 [1] M. Begon, J. L. Harper and C. R. Townsend, "Ecology: Individuals, Populations and Communities," Third edition, Blackwell Science, 1996. Google Scholar [2] Fred Brauer and Carlos Castillo-Chavez, "Mathematical Models in Population Biology and Epidemiology," Springer, New York, 2000.  Google Scholar [3] Carlos William Castillo-Garsow, "Teaching the Verhulst Model: A Teaching Experiment in Covariational Reasoning and Exponential Growth," Ph.D thesis, Arizona State University, Tempe, AZ, 2010. Google Scholar [4] Carlos William Castillo-Garsow, Continuous quantitative reasoning, in "Quantitative Reasoning and Mathematical Modeling: A Driver for STEM Integrated Education and Teaching in Context" (eds. R. Mayes, R. Bonillia, L. L. Hatfield and S. Belbase), WISDOMe Monographs, Vol. 2, University of Wyoming Press, Laramie, WY, 2012. Google Scholar [5] Jere Confrey and Erick Smith, Exponential functions, rates of change, and the multiplicative unit, Educational Studies in Mathematics, 26 (1994), 135-164. Google Scholar [6] Jere Confrey and Erick Smith, Splitting, covariation, and their role in the development of exponential functions, Journal for Research in Mathematics Education, 26 (1995), 66-86. doi: 10.2307/749228.  Google Scholar [7] E. V. Glasersfeld, "Radical Constructivism: A Way of Knowing and Learning," Falmer Press, London, 1995. Google Scholar [8] Fernando Hitt and Orlando Planchart, Graphing discrete versus continuous functions: A case study, In "Proceedings of the 20th Annual meeting of PME-NA," ERIC, Columbus, 1998. Google Scholar [9] G. Leinhardt, O. Zaslavsky and M. K. Stein, Functions, graphs, and graphing: Tasks, learning, and teaching, Review of Educational Research, 60 (1990), 1-64. doi: 10.3102/00346543060001001.  Google Scholar [10] T. R. Malthus, "An Essay on the Principle of Population," Sixth edition, John Murray, London, 1826. Google Scholar [11] Kevin C. Moore, "The Role of Quantitative Reasoning in Precalculus Students Learning Central Concepts of Trigonometry," Ph.D thesis, Arizona State University, 2010. Google Scholar [12] Kevin C. Moore, Coherence, quantitative reasoning, and the trigonometry of students, in "Quantitative reasoning and Mathematical modeling: A driver for STEM Integrated Education and Teaching in Context" (eds. R. Mayes, R. Bonillia, L. L. Hatfield and S. Belbase), WISDOMe Monographs, Vol. 2, University of Wyoming Press, Laramie, WY, 2012. Google Scholar [13] L. P. Steffe and Patrick W Thompson, Teaching experiment methodology: Underlying principles and essential elements, in "Research Design in Mathematics and Science Education" (eds. R. Lesh and A. E. Kelly), Erlbaum, Hillsdale, NJ, (2000), 267-307. Google Scholar [14] April Strom, "A Case Study of a Secondary Mathematics Teacher'S Understanding of Exponential Function: An Emerging Theoretical Framework," Ph.D thesis, Arizona State University, 2008. Google Scholar [15] Patrick W. Thompson, Conceptual analysis of mathematical ideas: Some spadework at the foundation of mathematics education, in "Proceedings of the Annual Meeting of the International Group for the Psychology of Mathematics Education" (eds. O. Figueras, J. L. Cortina, S. Alatorre, T. Rojano and A. Sepulveda), Vol. 1, PME, Morelia, Mexico, (2008), 45-64. Google Scholar [16] Patrick W Thompson, In the absence of meaning, in "Vital Directions for Mathematics Education Research" (ed. K. Leatham), Springer, New York, 2013. doi: 10.1007/978-1-4614-6977-3_4.  Google Scholar
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