Sunday, December 17, 2006

Education metaphors

Just thinking about how to describe abstraction and the infinitesimal (as in calculus) to young students.

1. Abstraction is familiar to programmers. We use it for indirection. It allow one thing to dynamically refer to multiple other things. You get to the target thing through dereferences. OK, so that sounds complicated. But Robert's Rule of Order is an example that can be easily understood. The Rules descrribe how to run a meeting. But they also describe a system for conducting debate and settling differences which are in fact used to create rule. Robert's Rules is therefore a set of rules to create rules. Seems to me with a little more time (than I have right now,) one could develop this example into a concrete examply of abstraction. Cool!

2. The notion of the infinitesmal is at the core of calculus. But how to make that idea come alive for a student is problematic. I think I found a way: Everyone is familiar with AM and PM. It's often seemed odd to me to call 12 Midnight 12:00AM. I don't disagree with it at just brushes up against my curious bone.

The number 12 is the last number on the clock and follows 11 PM. If 11 you can say 11PM it seems that you ought to be able to say 12PM follows. But it doesn't. The next number is 12AM. As a programmer, I've done time math lots of times and have had to include qwerky logic to account for that. You might be able to have a student focus on that infinitesimal fraction of time that exists around 11:59:59.999999 and 12:00:00.000001. What happens there? There is no time that begins with 12 after midnight that ISN'T PM. (I use that argument when someone tries to tell me that 12:00AM should be 12:00PM but 12:01AM is obviously AM.) No matter how small you slice time up just before the 12, it's PM and no how small the increment is past it, you end up at AM. Thus, there is something very profound that happens at the imaginary gap between the two times. It's real, yet it's ethereal. It could be that it's infinitesimally small. And that's the idea. A real 'thing' that happens twice a day that exposes something strange yet familiar.

From this understanding, calculus seem a little more approachable.