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The role of multiple modeling perspectives in students' learning of exponential growth
1.  Mathematics Department, Kingston Hall 216, Eastern Washington University, Cheney, WA 990042418, United States 
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 CastilloChavez, "Mathematical Models in Population Biology and Epidemiology," Springer, New York, 2000. 
[3] 
Carlos William CastilloGarsow, "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 CastilloGarsow, 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), 135164. 
[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), 6686. 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 PMENA," 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), 164. 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), 267307. 
[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), 4564. 
[16] 
Patrick W Thompson, In the absence of meaning, in "Vital Directions for Mathematics Education Research" (ed. K. Leatham), Springer, New York, 2013. doi: 10.1007/9781461469773_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 CastilloChavez, "Mathematical Models in Population Biology and Epidemiology," Springer, New York, 2000. 
[3] 
Carlos William CastilloGarsow, "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 CastilloGarsow, 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), 135164. 
[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), 6686. 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 PMENA," 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), 164. 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), 267307. 
[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), 4564. 
[16] 
Patrick W Thompson, In the absence of meaning, in "Vital Directions for Mathematics Education Research" (ed. K. Leatham), Springer, New York, 2013. doi: 10.1007/9781461469773_4. 
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