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

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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

Carlos Castillo-Garsow ^1,

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

Received October 2012 Revised May 2013 Published August 2013

Abstract
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Under a Creative Commons license

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.

Keywords: logistic, continuity, covariation., geometric, differential equation, Exponential, teaching experiment.

Mathematics Subject Classification: Primary: 97M60, 97M30; Secondary: 92B0.

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.
[2]	Fred Brauer and Carlos Castillo-Chavez, "Mathematical Models in Population Biology and Epidemiology," Springer, New York, 2000.
[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.
[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.
[5]	Jere Confrey and Erick Smith, Exponential functions, rates of change, and the multiplicative unit, Educational Studies in Mathematics, 26 (1994), 135-164.
[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.
[7]	E. V. Glasersfeld, "Radical Constructivism: A Way of Knowing and Learning," Falmer Press, London, 1995.
[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.
[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.
[10]	T. R. Malthus, "An Essay on the Principle of Population," Sixth edition, John Murray, London, 1826.
[11]	Kevin C. Moore, "The Role of Quantitative Reasoning in Precalculus Students Learning Central Concepts of Trigonometry," Ph.D thesis, Arizona State University, 2010.
[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.
[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.
[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.
[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.
[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.

show all references

References:

[1]	M. Begon, J. L. Harper and C. R. Townsend, "Ecology: Individuals, Populations and Communities," Third edition, Blackwell Science, 1996.
[2]	Fred Brauer and Carlos Castillo-Chavez, "Mathematical Models in Population Biology and Epidemiology," Springer, New York, 2000.
[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.
[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.
[5]	Jere Confrey and Erick Smith, Exponential functions, rates of change, and the multiplicative unit, Educational Studies in Mathematics, 26 (1994), 135-164.
[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.
[7]	E. V. Glasersfeld, "Radical Constructivism: A Way of Knowing and Learning," Falmer Press, London, 1995.
[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.
[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.
[10]	T. R. Malthus, "An Essay on the Principle of Population," Sixth edition, John Murray, London, 1826.
[11]	Kevin C. Moore, "The Role of Quantitative Reasoning in Precalculus Students Learning Central Concepts of Trigonometry," Ph.D thesis, Arizona State University, 2010.
[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.
[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.
[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.
[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.
[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.

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