American Institute of Mathematical Sciences

Advanced
  • Previous Article
    The basic reproduction number $R_0$ and effectiveness of reactive interventions during dengue epidemics: The 2002 dengue outbreak in Easter Island, Chile
  • MBE Home
  • This Issue
  • Next Article
    Some recent developments on linear determinacy
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

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, (1996).   Google Scholar

[2]

Fred Brauer and Carlos Castillo-Chavez, "Mathematical Models in Population Biology and Epidemiology,", Springer, (2000).   Google Scholar

[3]

Carlos William Castillo-Garsow, "Teaching the Verhulst Model: A Teaching Experiment in Covariational Reasoning and Exponential Growth,", Ph.D thesis, (2010).   Google Scholar

[4]

Carlos William Castillo-Garsow, Continuous quantitative reasoning,, in, (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.   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.  doi: 10.2307/749228.  Google Scholar

[7]

E. V. Glasersfeld, "Radical Constructivism: A Way of Knowing and Learning,", Falmer Press, (1995).   Google Scholar

[8]

Fernando Hitt and Orlando Planchart, Graphing discrete versus continuous functions: A case study,, In, (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.  doi: 10.3102/00346543060001001.  Google Scholar

[10]

T. R. Malthus, "An Essay on the Principle of Population,", Sixth edition, (1826).   Google Scholar

[11]

Kevin C. Moore, "The Role of Quantitative Reasoning in Precalculus Students Learning Central Concepts of Trigonometry,", Ph.D thesis, (2010).   Google Scholar

[12]

Kevin C. Moore, Coherence, quantitative reasoning, and the trigonometry of students,, in, (2012).   Google Scholar

[13]

L. P. Steffe and Patrick W Thompson, Teaching experiment methodology: Underlying principles and essential elements,, in, (2000), 267.   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, (2008).   Google Scholar

[15]

Patrick W. Thompson, Conceptual analysis of mathematical ideas: Some spadework at the foundation of mathematics education,, in, (2008), 45.   Google Scholar

[16]

Patrick W Thompson, In the absence of meaning,, in, (2013).  doi: 10.1007/978-1-4614-6977-3_4.  Google Scholar

show all references

References:
[1]

M. Begon, J. L. Harper and C. R. Townsend, "Ecology: Individuals, Populations and Communities,", Third edition, (1996).   Google Scholar

[2]

Fred Brauer and Carlos Castillo-Chavez, "Mathematical Models in Population Biology and Epidemiology,", Springer, (2000).   Google Scholar

[3]

Carlos William Castillo-Garsow, "Teaching the Verhulst Model: A Teaching Experiment in Covariational Reasoning and Exponential Growth,", Ph.D thesis, (2010).   Google Scholar

[4]

Carlos William Castillo-Garsow, Continuous quantitative reasoning,, in, (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.   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.  doi: 10.2307/749228.  Google Scholar

[7]

E. V. Glasersfeld, "Radical Constructivism: A Way of Knowing and Learning,", Falmer Press, (1995).   Google Scholar

[8]

Fernando Hitt and Orlando Planchart, Graphing discrete versus continuous functions: A case study,, In, (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.  doi: 10.3102/00346543060001001.  Google Scholar

[10]

T. R. Malthus, "An Essay on the Principle of Population,", Sixth edition, (1826).   Google Scholar

[11]

Kevin C. Moore, "The Role of Quantitative Reasoning in Precalculus Students Learning Central Concepts of Trigonometry,", Ph.D thesis, (2010).   Google Scholar

[12]

Kevin C. Moore, Coherence, quantitative reasoning, and the trigonometry of students,, in, (2012).   Google Scholar

[13]

L. P. Steffe and Patrick W Thompson, Teaching experiment methodology: Underlying principles and essential elements,, in, (2000), 267.   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, (2008).   Google Scholar

[15]

Patrick W. Thompson, Conceptual analysis of mathematical ideas: Some spadework at the foundation of mathematics education,, in, (2008), 45.   Google Scholar

[16]

Patrick W Thompson, In the absence of meaning,, in, (2013).  doi: 10.1007/978-1-4614-6977-3_4.  Google Scholar

[1]

Dalibor Pražák. Exponential attractor for the delayed logistic equation with a nonlinear diffusion. Conference Publications, 2003, 2003 (Special) : 717-726. doi: 10.3934/proc.2003.2003.717
[2]

Farid Tari. Geometric properties of the integral curves of an implicit differential equation. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 349-364. doi: 10.3934/dcds.2007.17.349
[3]

Michael Scheutzow. Exponential growth rate for a singular linear stochastic delay differential equation. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1683-1696. doi: 10.3934/dcdsb.2013.18.1683
[4]

Luis Caffarelli, Serena Dipierro, Enrico Valdinoci. A logistic equation with nonlocal interactions. Kinetic & Related Models, 2017, 10 (1) : 141-170. doi: 10.3934/krm.2017006
[5]

Dilip B. Madan, Wim Schoutens. Zero covariation returns. Probability, Uncertainty and Quantitative Risk, 2018, 3 (0) : 5-. doi: 10.1186/s41546-018-0031-1
[6]

Ulrike Kant, Werner M. Seiler. Singularities in the geometric theory of differential equations. Conference Publications, 2011, 2011 (Special) : 784-793. doi: 10.3934/proc.2011.2011.784
[7]

Maika Goto, Kazunori Kuwana, Yasuhide Uegata, Shigetoshi Yazaki. A method how to determine parameters arising in a smoldering evolution equation by image segmentation for experiment's movies. Discrete & Continuous Dynamical Systems - S, 2019  doi: 10.3934/dcdss.2020233
[8]

J. García-Melián, Julio D. Rossi. A logistic equation with refuge and nonlocal diffusion. Communications on Pure & Applied Analysis, 2009, 8 (6) : 2037-2053. doi: 10.3934/cpaa.2009.8.2037
[9]

John A. D. Appleby, John A. Daniels. Exponential growth in the solution of an affine stochastic differential equation with an average functional and financial market bubbles. Conference Publications, 2011, 2011 (Special) : 91-101. doi: 10.3934/proc.2011.2011.91
[10]

Antonio Suárez. A logistic equation with degenerate diffusion and Robin boundary conditions. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1255-1267. doi: 10.3934/cpaa.2008.7.1255
[11]

Jumpei Inoue, Kousuke Kuto. On the unboundedness of the ratio of species and resources for the diffusive logistic equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020186
[12]

Jian-Wen Sun, Wan-Tong Li, Zhi-Cheng Wang. A nonlocal dispersal logistic equation with spatial degeneracy. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 3217-3238. doi: 10.3934/dcds.2015.35.3217
[13]

Eugenio Montefusco, Benedetta Pellacci, Gianmaria Verzini. Fractional diffusion with Neumann boundary conditions: The logistic equation. Discrete & Continuous Dynamical Systems - B, 2013, 18 (8) : 2175-2202. doi: 10.3934/dcdsb.2013.18.2175
[14]

Anna Doubova, Enrique Fernández-Cara. Some geometric inverse problems for the linear wave equation. Inverse Problems & Imaging, 2015, 9 (2) : 371-393. doi: 10.3934/ipi.2015.9.371
[15]

Anne-Sophie de Suzzoni. Continuity of the flow of the Benjamin-Bona-Mahony equation on probability measures. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 2905-2920. doi: 10.3934/dcds.2015.35.2905
[16]

Samia Challal, Abdeslem Lyaghfouri. Hölder continuity of solutions to the $A$-Laplace equation involving measures. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1577-1583. doi: 10.3934/cpaa.2009.8.1577
[17]

Nguyen Huy Tuan, Donal O'Regan, Tran Bao Ngoc. Continuity with respect to fractional order of the time fractional diffusion-wave equation. Evolution Equations & Control Theory, 2020, 9 (3) : 773-793. doi: 10.3934/eect.2020033
[18]

Nikolay A. Gusev. On the one-dimensional continuity equation with a nearly incompressible vector field. Communications on Pure & Applied Analysis, 2019, 18 (2) : 559-568. doi: 10.3934/cpaa.2019028
[19]

Qingxu Dou, Jesús Cuevas, J. C. Eilbeck, Francis Michael Russell. Breathers and kinks in a simulated crystal experiment. Discrete & Continuous Dynamical Systems - S, 2011, 4 (5) : 1107-1118. doi: 10.3934/dcdss.2011.4.1107
[20]

Shouwen Fang, Peng Zhu. Differential Harnack estimates for backward heat equations with potentials under geometric flows. Communications on Pure & Applied Analysis, 2015, 14 (3) : 793-809. doi: 10.3934/cpaa.2015.14.793

2018 Impact Factor: 1.313

Tools

Metrics

  • PDF downloads (60)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences