On Jul 11, 2013, at 1:10 AM, Joe Niederberger <firstname.lastname@example.org> wrote:
> Perhaps, too glib again. I suspect most people, even non mathematical types, could be taught to see the theorem, if its explained what one is looking for. Given an ordinary continuous everywhere differentiable curve, the tangent can be intuitively and visually conveyed. Once the chord between two points is illustrated, the notion that the mean value theorem could be violated would be seen to be impossible. Naturally, a guide is needed until one becomes a scout on their own. Its even more visually compelling with the intermediate value theorem.
So now you are talking something very different. You are talking about using visualizations to teach abstract things. I never complained about that. I wouldn't even try to teach without it. My complaint is when people start thinking that the visualization is the abstract thing and the rat-sense behind it is the same as thinking.
I was part of an interesting discussion on a physics teaching site having to do with electric circuits. It started out with the the usual reformist idiocy of "traditional teaching is wrong because..." The point was that teachers teach about current and voltage in circuits as if the electrons push each other down the wire like peas in a straw. The reformist view was that this is wrong because after a very very brief amount of time the excess charge in the wire arranges it self in a stationary configuration and it is these stationary excess charges doing the pushing on the electrons in the middle of the wire. I had no choice but to concede that the electrons in a wire are nothing like peas in a straw. It was Zeev that saw through the idiocy of this discussion that by this time had lasted a couple hours at least. After I conceded that in an actual circuit the electrons act nothing like peas in a straw, Zeev asked "But how does that help students learn the relationship between voltage, current and resistance in a circuit?"
That is when it hit me. We hold onto these intuitive models dearly, even the physically ridiculous ones, not based on how alike they are to the abstract concepts we are learning but based on how well they help us learn the abstract concepts we are learning. There is always some sort of analogy involved but that by no means implies equivalence.
You can really dig deep and learn a lot by listening to reformers.