Wow, thanks for the comments guys!
@Paul: You got it – we were essentially integrating the equation for circumference. But if you call it that from the outset, and define it rigorously, people’s eyes will glaze over :).
And as you said, the use of x (input) and y (output) are conventions. So the regular way would be to say the equation is really 2 * pi * x, where x is the radius (never mind that we always learned it as 2 * pi * r). dy/dr is a perfectly fine way of saying it too.
One interesting thing about integration is seeing how something that doesn’t “look” like a curve (a bunch of rings) can be twisted into a format that does.
@Mike: Thanks for the awesome comment! You really nailed it, there are such beautiful ideas buried in math, which could really encourage people, but don’t have a chance because we jump into the details.
Conceptual discussions & drills have their place. It may be like listening to fun music (rock, rap, etc.) and being inspired to play. Then you start learning an instrument and memorize scales (doing drills). Drills are much more manageable when you have an appreciation for why you’re doing them.
Those side discussions you mention can be awesome – it highlights the discovery side of math. For every equation, there was someone seeing it for the first time and saying “whoa”.
@Jesse: That’s a very good point. I see it similar to teaching Physics: we start with Newtonian mechanics, which are “intuitive” to a degree. Then, as people advance, we teach them about the exceptions: strange things happen at the speed of light (relativity) and when you get really small (quantum mechanics).
But if we started off with relativity and quantum we’d lose everyone along the way.
@Prateek: Thanks for the kind words! Just a curious learner here. I know what you mean – I’ve taken many math classes, but the formulas just seemed to stay there, and didn’t really change how I viewed the world.
I’m usually in the Boston or Seattle area, and if you’re around feel free to drop me an email (email@example.com).
@Justin: Thank you for the kind words, that really means a lot. Yeah, I wish I posted more frequently too :).
The articles can be time consuming (10-15 hours) but I think my brain is the bottleneck – procrastination, perfectionism, and sometimes it’s a struggle to have a “good enough” insight (I don’t want to rewrite what’s already on wikipedia). Maybe I can find a way to trick myself into writing more :).
@James: That would be awesome. Unfortunately I don’t have any animation skills either.
@Rodrigo: I agree – math would be a boring place if it was only about pushing numbers around :).