An Integrated Learning Environment for Developing "Function Sense"(2)-From Velocity to First Steps in Calculus Using Spreadsheets-

Abstract: We would like to present a pilot study for exploring how to develop a "function sense " in an integrated learning environment (Kakihana et al., 2000) with spreadsheets and experiments using a graphing calculator. In this pi-lot study, high school students and college students were examined to see how they used their "function sense " so that they could better understand the relation between a velocity and the concept of differential calculus. At first, they tried to understand the velocity and the shape of a function using a graphing calculator while actually walking. Then, they simulated many walking situations in an integrated environment using spreadsheets. They input the walking data (ve-locity or distance) or the algebraic expression of the velocity of this duration. Graphs of the distance and velocity were drawn automatically. On the worksheets, students were asked what kinds of relations were shown on the graphs, the in-putted numeric data, the walking situations and the concept of velocity. As the result of these activities students were able to correlate the walking situations and the velocity. These experiences, we believe, will lead students to effectively understand calculus. 1. Background 1.1 First steps in calculus. In high school calculus, the concept of a rate of instantaneous change is used as a derivative of a function. In textbooks, a derivative of a function is defined as follows