TRACING STUDENTS’ UNDERSTANDING OF INSTANTANEOUS CHANGE BY WILLIAM BRADLEY HALIEN A dissertation submitted to The Graduate School of Education Rutgers, The State University of New Jersey in partial fulfillment of requirements for the degree Doctor of Education Graduate Program in Mathematics Education Approved by ____________________________ Carolyn A. Maher, Chair ____________________________ Alice S. Alston, Committee ____________________________ Robert Speiser, Committee New Brunswick, New Jersey May 2011 Acknowledgements Completion of this dissertation would have been impossible without the support, encouragement and understanding of the love of my life, my wife Phyllis. I am grateful also of the love and support of my family, specifically my children Mary and Paul. I also owe thanks to dear friends Tom and Gayle DeVoe, and Charyl and Aaron Nietfeld for their patience, understanding and support. I owe a debt of gratitude to Dr. Carolyn Maher for her unending patience, kind words of support and encouragement, and for the unending amount of research she does in the name of improving the way mathematics is taught and learned. Thank you to Dr. Alice Alston who was the first professor at Rutgers to challenge the way I looked at the way I taught math, for providing an unending library of research on math education, and for being a part of this endeavor. Thank you to Dr. Robert Speiser for allowing me to impose on his talents, knowledge, and experience long distance. Dr. Speiser provided me with valuable insights into what it means to justify an answer to a rich mathematical problem, as well as research material into what it means when we experience math. I am grateful to Marjory Palius for always being a patient resource to turn to, and to Robert Sigley who provided every bit of data contained in this dissertation. Colleagues who have been patient and understanding about my time and to Dr. Ralph Pantozzi who said this would be easy, thank you. I would also like to acknowledge the hard work and dedication of all of the researchers, cameramen, Romina, Magda, Aquisha, Brian, Jeff, Victor and Benny as well as the rest of the students involved in that institute. I am also grateful for the Rutgers Longitudinal Study and funding provided by NSF Grant REC9814846 which continues to provide rich opportunities to understand how mathematics is learned and improve the way mathematics is taught. ii Abstract of the Dissertation Tracing Students’ Understanding of Instantaneous Change By: William Bradley Halien Dissertation Director: Carolyn A. Maher Students are often taught higher level math the same way their teachers were taught, using textbook models, function, and graphical representations. Modern research in mathematics education encourages teachers to create situations where students are able to experience math to develop understanding. The Catwalk problem was designed to develop a students’ understanding of a fundamental concept of calculus, instantaneous change. This research will address the following questions, 1. What physical and mathematical knowledge and strategies do students use to solve this problem? 2. What mathematical arguments do students use to support their solutions to the problem? 3. How do, if at all, students distinguish between instantaneous and average rates of change, and 4. If so, what methods do they use? This problem asks that students find the speed of a cat in two particular still photographs out of a series of 24. The problem solution called for the development of a conceptual understanding of instantaneous change as opposed to average change. Because the difference in the cat’s velocity before and after the frame was dramatic, some students were opposed to representing the change as an average. The students used mathematical models including data sets, graphical evidence, and velocity calculations to argue knowledge they developed from experiencing the catwalk firsthand. The students iii developed a physical model that allowed them to experience kinesthetically what they witnessed the cat do in the photographs. This physical experience guided the mathematical and verbal arguments of the students, both for and against the use of average velocity to solve this task. The Summer Institute was fundamentally a research institute (Maher 2005). The students were given specific problems so that the researchers were able to gain insights into how the students thought about solving the problems as well as what the students learned from the problems. Given a learning situation instead of a research situation, the same outcomes are desirable (development of physical and mathematical understandings of instantaneous change). With minor encouragements educators can extend this problem to include the conceptual understandings of limits and continuity. We also learn that higher mathematics can be experienced, and internalized through that experience. iv Table of Illustrations Figure 1 Catwalk Photographs ............................................................................................ 1 Figure 2 Library Tape Line ............................................................................................... 18 Figure 3 Hallway Tape Line ............................................................................................. 19 Figure 4 Aquisha's frame to frame measurements ............................................................ 20 Figure 5 Aquisha's final line and proportional computation ............................................. 21 Figure 6 Speed/Distance Graph recreated by Romina in response to Victor’s statement. 28 Figure 7 Victor's Proportion Calculation .......................................................................... 31 Figure 8 Romina's First Table ........................................................................................... 33 Figure 9 Romina's Second Table ...................................................................................... 34 Figure 10 Romina Presenting a Distance v.Time Graph .................................................. 36 Figure 11 Romina presenting a Speed v. Distance Graph ................................................ 37 Figure 12 Romina's Velocity v. Time Graph .................................................................... 39 Figure 13 Victor's Model .................................................................................................. 47 Figure 14 Romina's Model ................................................................................................ 47 Figure 15 Visual Representation of Angela's and Victor's Argument .............................. 49 Figure 16 Visual Representation of Romina's Argument ................................................. 50 Figure 17 Romina's 7/16/19 Presentation ......................................................................... 52 Figure 18 Romina's Second Table using estimated distances ........................................... 56 Figure 19 A Portion of Romina's First Table .................................................................... 59 Figure 20 A Portion of Romina's Second Table ............................................................... 60 Figure 21 Romina’s Distance v. Time Graph ................................................................... 62 Figure 22 Romina’s Speed v. Distance Graph .................................................................. 63 Figure 23 Romina's Velocity v. Time Graph .................................................................... 64 v Figure 24 Illustration of Romina's interpretation of Frame 10 ......................................... 75 vi Table of Contents Acknowledgements ............................................................................................................. ii Abstract of the Dissertation ............................................................................................... iii Table of Illustrations ........................................................................................................... v Table of Contents .............................................................................................................. vii Chapter 1 – Introduction ..................................................................................................... 1 1.1 Statement of the Problem ..................................................................................... 1 1.1.1 The Problem - Cat Walk .......................................................................................... 1 1.2 Instantaneous Change ........................................................................................... 2 1.3 The Purpose of the Study ..................................................................................... 5 1.4 Research Questions ................................................................................................... 5 1.5 Necessary Conditions for the Research .................................................................... 6 1.6 Background ............................................................................................................. 7 Chapter 2: Methodology ..................................................................................................... 8 2.1 Data Source .......................................................................................................... 8 2.2 The Setting ........................................................................................................... 8 2.3 Data