A Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy

Arizona State University May 2007

MODELING TEACHERS’ WAYS OF THINKING ABOUT RATE OF CHANGE by Edward Ellis Coe

has been approved March 2006

Graduate Supervisory Committee: Patrick W. Thompson, Chair Marilyn P. Carlson Alfinio Flores James A. Middleton Glenn H. Hurlbert

ACCEPTED BY THE DIVISION OF GRADUATE STUDIES ii

ABSTRACT This study sought to develop and implement a method to explore high school teachers’ ways of thinking about rate of change and affiliated ideas in function-based situations. Previous research has shown that rate of change is an integral idea in understanding functions, covariational reasoning, and topics of calculus. It is necessary, then, for teachers to have a well-connected system of meanings for rate. Although existing research has focused on the mathematical concept of rate, there is little information as to how teachers think about different types of rate of change and how those types interconnect in their thinking. The aim of the study was to move beyond definitions and procedures to develop a method of modeling systems of meanings for how teachers think about rate. A constructivist approach provided the grounding for this study. Specifically, the researcher used Glasersfeld’s idea of conceptual analysis to develop a method of generating models to describe how the teachers may have been thinking about rate of change. Three secondary mathematics teachers participated in two interviews approximately six months apart to establish stability in their ways of thinking. The interviews incorporated a number of rate of change tasks that were classified in type as definition, constant rate, average rate, or changing rate. For each teacher the researcher proposed ways of thinking for each type of rate and considered how each teacher interconnected those meanings. The results indicate that the framework and method served as useful tools to describe teachers’ ways of thinking about rate. The results evidenced that iii

each teacher had unique systems for thinking about rate. One teacher primarily employed definitions, the second teacher thought about rate graphically in terms of steepness, and the third teacher thought in terms of comparisons. Each teacher’s system of meanings contained inconsistencies, and it was often during the average rate tasks that the teachers revealed them. Additionally, the teachers evidenced disconnected ways of thinking about constant, average and changing rate.

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Dedicated to my wife and best friend, Christa, for her unending support and encouragement.

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ACKNOWLEDGMENTS I wish to thank my committee members Alfinio Flores, Jim Middleton, and Glenn Hurlbert for their time and professional direction. I particularly wish to thank Marilyn Carlson for believing in me from the very beginning and for motivating me when I needed it. I especially thank Pat Thompson for his insight and ability to spur me to new ways of thinking. I have spent many years on this journey of learning, and I could not have completed it without the help of friends and colleagues at Scottsdale Community College. I specifically thank Jim Vicich for his daily advice and Sally Jacobs for her insight and helpful feedback. I also thank my family, and especially my wife, Christa. I can only begin to imagine the sacrifices you have made. I hope, though, that we have gone through this process in such a way that our five children will someday appreciate one key reason why it took so many years: that they should never feel less important than my work. Finally, I acknowledge the support given by the Maricopa Community College District and Arizona State University’s Center for Education in Science, Mathematics, Engineering, and Technology (CRESMET). I also wish to acknowledge support for this