Vol.2-882
PME-NA 2006 Proceedings
TEACHING INNOVATIONS FOR PROBLEMS INVOLVING RATES IN CALCULUS
Nicole Engelke California State University, Fullerton
[email protected] Related rates problems in first semester calculus are a source of difficulty for many students. These problems require students to be able to visualize the problem situation and attend to the nature of the changing quantities. I have developed a sequence of teaching activities that employs a computer program designed to foster the students’ exploration of related rates problems in a covariational context. I investigated the impact of these activities on students’ abilities to understand and solve rate of change and related rates problems. I present the results of the first two activities which focus on rate of change here. Background Little research has been published on how students understand and solve related rates problems in first semester calculus. The research to date suggests that students have a procedural approach to solving related rates problems (Martin, 2000; White & Mitchelmore, 1996). When solving a related rates problem, students tend to focus on using an algorithm that essentially consists of the following steps: draw a diagram, choose a geometric formula, differentiate it, substitute in values, and solve (Engelke, 2004). A student may need to engage in covariational reasoning to construct a mental model that accurately reflects the problem situation and that may be manipulated understand how the problem situation works (Carlson, Jacobs, Coe, Larsen, & Hsu, 2002; Engelke, 2004; Saldanha & Thompson, 1998). The Study I conducted a teaching experiment consisting of six teaching sessions with a group of three students from my calculus class in the Fall 2005 semester. The participants were chosen from a group of volunteers and met for these teaching episodes outside of the regular class sessions. They did not attend the regular class periods in which related rates were taught to the complement of the class. The students were paid for each session they attended, and each teaching episode was videotaped and transcribed for analysis. Results In the first session of the teaching experiment, the students used a custom computer program to investigate the average rate of change and instantaneous rate of change for some common geometric problem situations. For example, the students were asked to consider the following problem: Suppose we have a plane that is flying over a RADAR tower, TA, and is on course to pass over a second RADAR tower, TB. Let u be the distance between the plane and TA, and let v be the distance between the plane and TB. What is the rate of change of u in relation to v? The computer program allowed the students to have a visual representation that may be manipulated to observe what happens as they make the plane move. The students decided that it would be Δu Δv helpful to have time given so that they may compute average velocities: Δt and Δt . To allow the students to do this, another version of the plane problem was opened in the computer program that allows the students to observe what happens to each variable, including time, as _____________________________ Alatorre, S., Cortina, J.L., Sáiz, M., and Méndez, A.(Eds) (2006). Proceedings of the 28th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. Mérida, México: Universidad Pedagógica Nacional.
Technology
Vol.2-883
they moved the plane and allows them to generate a table of values for the variables. After computing the average velocities, the students thought that they could relate u and v by flipping and multiplying to cancel out the Δt ’s. The concept of rate as one quantity as opposed to the ratio of two independent quantities appeared to be difficult for students to grasp. Throughout the first and second sessions the students struggled with whether the delta t’s really cancel in the above plane problem. Ali chose to open the second teaching session with the question: “I know I can relate u to time and v to time but I don’t know how to relate them to each other.” Amy and Ben referred back to the previous meeting saying that you just “flip and multiply” to cancel out the Δt ’s, suggesting that they may not have internalized the notion of rate. In the second teaching session, during a discussion of the chain rule, the students began to think about rate as one quantity versus two. The students argued about whether they could really cancel the deltas when multiplying rates. This shift in thinking allowed the students to begin relating rates in other situations. This exploration allowed the students to choose time as a common variable through which they may relate variables, a common practice in related rates problems which were the focus of subsequent teaching sessions. Conclusion Time as a variable was student generated in these problem situations. Students’ interactions with the computer program likely cultivated the students’ thinking about how each variable changes across time and may have helped them internalize the notion of rate. The ability to imagine each variable as it changes across time and as a function of time may also foster students’ understanding of the chain rule and its application to related rates problems. The data suggests that the use of the computer program to visualize problem situations and measure quantities can aid students’ development of mental models in future problem situations and their understanding of the concept of rate. References Carlson, M., Jacobs, S., Coe, E., Larsen, S., & Hsu, E. (2002). Applying covariational reasoning while modeling dynamic events: A framework and a study. Journal for Research in Mathematics Education, 33(5), 352-378. Engelke, N. (2004). Related rates problems: Identifying conceptual barriers. In D. McDougall (Ed.), 26th Annual Conference of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 455-462). Toronto, Ontario, Canada. Martin, T. S. (2000). Calculus students' ability to solve geometric related-rates problems. Mathematics Education Research Journal, 12(2), 74-91. Saldanha, L. A., & Thompson, P. W. (1998). Re-thinking covariation form a quantitative perspective: Simultaneous continuous variation. Paper presented at the Annual Meeting of the Psychology of Mathematics Education - North America, Raleigh, N. C.: North Carolina State University. White, P., & Mitchelmore, M. (1996). Conceptual knowledge in introductory calculus. Journal for Research in Mathematics Education, 27(1), 79-95.
