>I have students do little projects in which they are forced to grapple with >situations underlying difficult concepts *before* the concepts are defined >-- after all, that's where the concepts come from.
>I suspect that this is true to some extent at every level.
>This is not saying that we should always go from s. to g.; just that >sometimes it really works.
The last two years after the AP exam I've used a little book from the MAA and the U. of New Mexico, entitled "Research Problems in Calculus" (or something like that. It's at school.) I've had the kids do short papers on these interesting problems. I wouldn't say it's been totally worthless, because I'm giving them all one of the more interesting problems about Houdini and a strait jacket and a flask full of water to do over the weekend. But I do think we might have better spent our time. Yes, they do end up producing nice little papers that will look good if I apply for another job, but I know they didn't learn nearly as much as when we were working towards the AP exam.
This year we're working out of "Fractals for the Classroom". More structure, more learning.