Calculus, as we know it today, emerged from a historic struggle to systematically understand the motion of objects, especially as it is described in terms of position, velocity and acceleration. This goes back to the Greeks and culminates in the work of Galileo, Newton, and Leibniz. For this reason, calculus can be seen as a highly formalized version of the mathematics of motion. The purpose of this course is to help students gain an understanding of calculus - or to deepen the understanding they already have - by giving them an opportunity to develop and extend their knowledge of the world of motion. They will do this by working in a computer based lab with a variety of hands-on mechanical devices on the kinds of problems that calculus was developed to answer. One such device consists of a pair of cars that run along parallel tracks controlled by a velocity-time graph drawn by the students on the computer screen. The problem of accurately predicting the motion of the cars from their velocity-time graphs, a fascinating one in itself, is a version of the standard calculus problem of describing a function from information about its derivative.
The course is aimed at two kinds of students: those who have always wondered what calculus is really about and have been unable to find out because of its formal and symbolic language, and those who have learned calculus in math courses but would like to have a deeper grasp of the conceptual core of the subject. These two kinds of students should be able to learn from one another. This course should be of particular interest to teachers who are interested in the question of how deep and complex mathematics can be made accessible to a wide variety of students through alternative educational environments. It should also provide teachers with an opportunity to reflect on the process of their own and others' learning as a way to better understand their own students' experience and thereby improve their teaching.