non-nonstandard calculus, and the Carnival

Matt at The Everything Seminar talks about his experiences in teaching "non-nonstandard calculus". Formally, this is calculus done over the ring R[dx]/((dx)²=0); informally, this is calculus where dx is treated as a number whose square is zero. He writes:

By treating infinitesimals on the same footing as finite numbers, the approximation schemes of calculus become more intuitive. I think every mathematician has discovered this on their own, in their own private language. Why not make the language commensurable with the computations that we do?

I agree with this -- the reason we do whatever computations we do in a certain way is often because it's the easiest way we know. Why make the students suffer more than we have to?

As people have pointed out in the comments to Matt's post, this approach seems to have nothing to say about integration. But does the usual approach to teaching differentiation have anything to say about integration?

In general, it seems to me that mathematics and mathematics education are more separated from each other than they need to be. Obviously, some differences are necessary -- but why erect an artificial wall? (I feel this somewhat keenly because I am at the moment in my life -- two years into grad school -- where I am expected to jump over this wall.) John Armstrong has written about the Carnival of Mathematics and how it is becoming split between people who do lower-level mathematics and people who do higher mathematics, with the lower-level people winning. But it strikes me that they're claiming that the difference is one of kind, not one of degree. There should be a smooth continuum from 1+1=2 to, say, the Riemann hypothesis -- everything currently taught to students was research once -- but there isn't. It seems, though, that the Carnival of Mathematics welcomed submissions at all levels and it just turned out that there are more people writing at the lower levels.

This blog goes back and forth between levels quite often, so you'd expect me to think this. And that's intentional. One of my goals here is to make the point that mathematics -- well, at least some parts of it -- is not all that divorced from everyday experience.

(Oh, and by the way, yes I changed the layout. People have said for a while they didn't like it; the Adsense ads were making me only a pittance; and the old layout was orange and blue which are the Mets' colors.)