In the age of computing devices like calculators that are essentially computers and websites like Wolfram Alpha, what's important to teach? Is it important to teach the process of long division? The process of factoring a polynomial? u-substitution for integration?

DPatrick and I revisited this conversation yesterday. We've had it a zillion times. In fact, we don't even have the conversation anymore -- one of us starts on a topic and then we both realize that we're having "that conversation" again, and we stop. My position is essentially that none of these processes are important to know, but that all of them should be taught, because it is the process of learning that is important, not the facts being learned. I waver on that position sometimes, but in the absence of a better way to learn how to learn, I'll stay with it. That said, I wouldn't be surprised if educational best practices (as practiced by top educators, not as practiced by the school systems) look very different in 30 years. But at this point, it's still not clear how best to shift the focus of education from accumulating tools to the process of developing tools. We try to do that at AoPS, but I expect there might be a better way. However, I think many of the most important changes would have to take place at an even younger age than we currently work with at AoPS.

For me, the difficult part is finding something meaningful for students to do to help them develop these skills. For many students on AoPS, MATHCOUNTS and AMC provide that sort of impetus, but that's a pretty small portion of students nationwide. This may be an essentially impossible problem to solve, because I think that "meaning" is a very individual decision, and what's important enough to make one student learn how to learn won't necessarily work in a room of 20 people sorted by geography and age. And this, I think, is where internet education can be so powerful -- it can sort by interest, intellectual level, and learning style, which I would guess is a much more effective grouping method. Of course, students also need time to pursue their passions, which is probably part of why homeschoolers appear at the top of more and more pursuits. If only schools could find a way to break open a window of time for students to engage their passions. Very hard for schools to do, of course -- personalized attention is hard to scale in a school. At least there's the summer....