Symbolizing motion, flow, and contours: The experience of continuous change

This is a study of how six high school students thought and learned about the relationship between a function and its derivative situated in three different contexts: motion (position vs. time and velocity vs. time), air flow (volume vs. time and flow rate vs. time), and contours (height vs. distance and slope vs. distance). Each student worked in a learning environment that included tools with which to experience the phenomena. Functions in these environments reflected kinesthetic actions. In the motion environment, the students used a computer-based motion detector to create real-time graphs as they moved a toy car with their hands. The air flow environment included a device to control the air flow in and out of a cylindrical bag; a special sensor connected to the computer allowed students to generate the functional graphs. With the Contour Analyzer the students could traced contours of objects producing graphs on the computer screen. The analysis revealed differences and commonalities between learning environments and how students can learn qualitative integration and differentiation by exploring the relationship between physical and symbolic behavior.