repeated the process on a smaller scale, filling in point by point, and describing a process of using each point to find
Teaching the Verhulst Model: A Teaching Experiment in Covariational Reasoning and Exponential Growth by Carlos Castillo-Garsow
A Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy
ARIZONA STATE UNIVERSITY August 2010
©2010 Carlos Castillo-Garsow All Rights Reserved
Teaching the Verhulst Model: A Teaching Experiment in Covariational Reasoning and Exponential Growth by Carlos Castillo-Garsow
has been approved July 2010
Graduate Supervisory Committee: Patrick Thompson, Chair Marilyn Carlson Michael Oehrtman Luis Saldanha Sergei Suslov
ACCEPTED BY THE GRADUATE COLLEGE
ABSTRACT Both Thompson and the duo of Confrey and Smith describe how students might be taught to build “ways of thinking” about exponential behavior by coordinating the covariation of two changing quantities. However, these authors build exponential behavior from different meanings of covariation. Confrey and Smith advocate thinking about discrete additive and multiplicative changes as the foundation for covariational reasoning and exponential functions learning, while Thompson advocates continuous variation as the foundation for covariational reasoning. This work describes a teaching experiment, composing a series of task-based exploratory teaching interviews that covered linear functions, compound interest, phase plane, exponential growth, and the logistic differential equation, in which I investigated the consequences of different types of covariational reasoning for exponential functions. The purpose of the experiment was to identify the operations of covariational reasoning that the students actually used, and the consequences of that reasoning for mathematics involving exponential growth. In the analysis of this teaching experiment, I identified two ways of thinking about change that differ from the discrete/continuous dichotomy above: thinking about “chunky” completed changes, or a “smooth” change in progress. With smooth and chunky as a basis, I also identify five different ways of understanding exponential growth: Geometric, compound, differential, harmonic, and stochastic. Lastly, I suggest that powerful understandings of exponential growth come not from the mastery of any one way of thinking, but from a rapid and fluent shifting amongst several ways of thinking.
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ACKNOWLEDGMENTS Without the support of a great number of people, this dissertation could never have come into existence. First and foremost, I would like to thank Patrick Thompson for his incredible mentorship and uncompromising standards of scholarship. His comments and questions have always been both terrifying and of immeasurable value. Special thanks also to Debra Castillo, for volunteering her editing expertise to make sure that this work approaches legibility. All remaining errors are mine, not theirs. At a more personal level, I would like to thank Carlos Castillo-Chavez, Wolfgang Wölck, Justin Boffemmeyer, and Uri Treisman, who at various points in my life, all helped me get back up after I had stumbled. Without the support of these excellent individuals, I never would have started a dissertation study at all. Lastly, I would like to thank Marilyn Carlson for keeping me in food, rent, and good humor while I wrote this. Research reported in this dissertation was supported by National Science Foundation Grant No. EHR-0353470, with Patrick Thompson as principal investigator. Any conclusions or recommendations stated here are those of the author and do not necessarily reflect official positions of NSF.
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TABLE OF CONTENTS Page LIST OF TABLES ............................................................................................................... viii LIST OF FIGURES................................................................................................................ ix CHAPTER 1
STATEMENT OF THE PROBLEM ............................................................... 1
2
STUDIES OF COVARIATION AND THE VERHULST MODEL PRIOR TO THE DESIGN OF THE TEACHING EXPERIMENT......................... 8 A Brief History of Covariation....................................................................... 8 Covariation and Exponential Growth .......................................................... 15 The Verhulst Model...................................................................................... 22 Covariation and the Verhulst Model............................................................ 31 Synthesis: Choosing a Framework for the Teaching Experiment............... 34 Summary....................................................................................................... 42
3
TEACHING EXPERIMENT DESIGN ......................................................... 44 Teaching Experiment ................................................................................... 45 Hypothetical Learning Trajectory................................................................ 66 A Day in the Life.......................................................................................... 88 Scheduling .................................................................................................... 88
4
THE STORY OF TIFFANY .......................................................................... 90 Simple Interest.............................................................................................. 90 Compound Interest ....................................................................................... 96 Constant Per-capita Interest........................................................................ 112
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CHAPTER
Page Compound Interest on the Phase Plane...................................................... 118 Phase Plane Analysis.................................................................................. 133
5
THE STORY OF DEREK ............................................................................ 145 Simple Interest............................................................................................ 146 Compound Interest ..................................................................................... 147 Constant Per-capita Interest........................................................................ 157 Compound Interest on the Phase Plane...................................................... 161 Phase Plane Analysis.................................................................................. 168 The Malthus Model .................................................................................... 175 The Food Model ......................................................................................... 181 Logistic Growth.......................................................................................... 185
6
RETROSPECTIVE ANALYSIS.................................................................. 194 Chunky and Smooth ................................................................................... 194 Rate ............................................................................................................. 204 Per-capita Rate of Change.......................................................................... 214 Exponential Growth ................................................................................... 223 Contrasting Derek and Tiffany................................................................... 226
7
ANALYSIS OF THE DESIGN.................................................................... 230 Confrey and Smith...................................................................................... 230 Thompsons’ Continuous Variation ............................................................ 234 The Design of the Tasks............................................................................. 237
8
CONCLUSION............................................................................................. 239
vi
CHAPTER
Page Chunky and Smooth ................................................................................... 239 The Geometric Exponential ....................................................................... 240 The Phase Plane Exponential..................................................................... 243 The Harmonic Exponential ........................................................................ 244 The Stochastic Exponential........................................................................ 245 Multilingualism .......................................................................................... 245
REFERENCES .................................................................................................................. 250 APPENDIX A
ORIGINAL TASK PROTOCOLS ............................................................ 255
B
SAMPLE ASSENT FORMS ..................................................................... 283
C
HUMAN SUBJECTS EXEMPTION ........................................................ 288
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LIST OF TABLES Table
Page 1.
Schedule of Teaching Episodes ...................................................................... 89
viii
LIST OF FIGURES Figure
Page
1.
An investment compounded annually at 8% per year.......................................20
2.
Per-capita rate of change as a function of population with r=2 and K=10000 ......................................................................................................... 28
3.
A phase plane diagram of the logistic model. The vertical axis measures rate of change. The horizontal axis measures population.............................. 41
4.
Behavior of the Verhulst model (r=2, K=10,000) for various initial conditions including N(0)=K (green) and N(0)=0 (x-axis). .......................... 42
5.
“Tiffany's” solution to an informal integration problem.................................. 53
6.
"Derek's" sketch of the “original function” given a step-wise linear "rate of change function."............................................................................................ 53
7.
A portion of Derek's graph of the sum of the blue and red functions.............. 56
8.
Tiffany's solution to the sums of function problem (black dotted line), and Tiffany’s solution (red dotted line) to a “challenge problem” of finding the difference between the red and blue functions......................................... 56
9.
Tiffany's solution to graphing the composition of functions h and g. ............ 57
10.
Derek's solution to the composition problem (cropped for space). ................ 57
11.
Derek's (dotted line) solution to the "challenge problem," in graphing the sum of two given functions (solid lines)........................................................ 58
12.
Phase plane graph of the PD8 account........................................................... 78
13.
One possible representation of growth in the phase plane approximated in discrete time.................................................................................................... 79
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Figure 14.
Page Approximation graph of the account over time, showing the rate being held constant for a period of time before being updated. .............................. 80
15.
Per-capita rate of change as a function of population for the Malthus model (purple), and the Verhulst model with a carrying capacity of seven billion (red). .................................................................................................... 84
16.
Phase plane graph of the Verhust model......................................................... 86
17.
The qualitative behavior of a population over time in the Verhulst model.... 87
18.
Tiffany's function (formula) for the value of Phil's account at any time........ 93
19.
Tiffany's function for the value of an EZ8 (simple interest) account with any initial investment. .................................................................................... 94
20.
Tiffany's solution to the value of Phil's account after a quarter of a year....... 98
21.
Tiffany's calculation for the value of Phil's account at the end of half a year (blue) contrasted with Tiffany's calculation for the value of Patricia's account at the end of the second quarter (green). .......................................... 98
22.
Tiffany's rewritten calculations for the value of Patricia's account after one quarter (top) and two quarters (bottom) after some instruction in the distributive property. ...................................................................................... 99
23.
Tiffany's calculation for the value of Patricia's account at the end of the second quarter................................................................................................. 99
24.
Tiffany's calculation for the value of Patricia's compound interest account at the end of the second quarter.................................................................... 102
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Figure 25.
Page Tiffany's calculation for the value of Patricia's compound interest account at 0.1 years. The box around .1 was added later.......................................... 103
26.
Tiffany's calculation for the value of Patricia's account after .6 years. ........ 106
27.
Tiffany's function pieces for the value of Patricia's account after x years. Top: On a domain of 0
