Tracing students' understanding of instantaneous change

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 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.Ed.D.Includes bibliographical referencesby William Bradley Halie