Alan Kay, Etoys, Drive a Car, and math education
In most demos, Alan Kay shows off a sequence of tutorials from Etoys. These are lessons in powerful ideas specifically designed for children, but I think they're equally powerful for adults too.
Drive a Car
If you want to play along, download a copy of etoys.
In this lesson children use a built-in paint program in Etoys to paint their own car. In the next tutorial, they write instructions to make the car drive around the screen in various ways. These become lessons in turtle graphics from logo. To an uninformed adult, it just looks like a funny way to draw geometric shapes. But closer examination of turtle graphics reveals something profound. They are a beautiful existence proof that powerful software tools enable elementary aged children to learn very high level mathematics, in this case, differential geometry of vectors. As with the example of geometric growth, there are also experiential lessons in functions and infinity. Students can learn how different combinations of very simple concepts (e.g. repeat, move forward a little, turn a little) can lead to larger forms. Moreover, they get direct and concrete experience with variables (a key ingredient of algebra), a powerful idea that can otherwise seem quite abstract and confusing.
In an exercise on Speed and Acceleration, students compare constant velocity, constant acceleration, and random acceleration in a fairly direct way.
There's another fun lesson in these activities that I haven't heard Alan Kay mention directly. When these lessons call for the car to leave a trail of dots behind, students are subtly introduced to the fundamental concept of graphing and visualizing mathematical behavior. Although the exercises in these examples don't go there, turtle graphics can reinforce lessons about the number line and introduce Cartesian coordinates.