On Jul 10, 2013, at 4:40 PM, Joe Niederberger <email@example.com> wrote:
> That the mean value theorem is a highly *visually* intuitive fact, makes its connection with that particular picture absolutely compelling.
You are confusing our concrete senses (what I will hereafter call our "rat-sense" in praise of Dehaene) with mathematical sense. That rat-sense gets in our way more than it helps, when it comes to higher level thinking. It's getting in the way right now.:) Yes, you can connect the MVT to a picture, IF you are familiar with the MVT. Let me ask you this. If I plot f(x) = x^2, that visualization has a distinctive signature to it. That quadratic acceleration of slope, etc. But how is that "mathematical"? Without any explanation or further mathematical understanding, it isn't even a mapping. Not even an intuitive mapping. It is just a curve with a distinctive signature, but not so distinctive as not to be confused with several other curves.
You say that the connection of the MVT to that picture is compelling. I agree, but I claim that this feeling of "compelling" is exactly what I warned against. Teachers think that there is something much more there than there actually is because it is so damn "compelling". In this context, compelling means that you know the MVT and it is just a damn neat feeling when you can THEN see it through your rat-sense. We all have that feeling. Every time we learn or discover something new in mathematics the first thing we try to do is see it through our rat-sense. We love our rat-sense. We trust our rat-sense. That phenomena doesn't work in reverse. Not without a great deal of reasoning. Which requires a different sense.