On Jun 17, 2013, at 2:23 PM, Joe Niederberger <firstname.lastname@example.org> wrote:
> I remember how I solved the old chestnut "what is greater, e^pi or pi^e?" I did it "in my head" without paper while walking my dog. Basically through examining curves in my imagination, plus a few elementary facts.
Are we not right back to visualization again? How is a visual proof such as yours any different than a visualization in general? When I go to the board and draw something I do so generally to make a point, which is to say "to prove something". The original animation showing that multiplication can result in smaller or larger is a proof. All visualizations are essentially point something out. Even if I go to the board, draw a circle and label its parts, that is a proof in a way. It is a proof that the diameter, radius and circumference exist and can be identified.
Remember, I am not saying that you are not using mathematics to create these visualizations. I am saying that mathematics is not a visual sense. Recognizing visually that two things are equal or different is not mathematics. Producing a visualization showing that two abstract things are equal or different is. Showing it to someone else doesn't transfer that mathematics to them.
You said that you imagined "in your head" these curves. When I do that I associate a whole bunch of abstract and intuitive qualities with them. Continuity, curvature, concavity, connectedness, etc. etc. I am sure that you do as well. Without all that abstraction, intuition and instinct it's nothing but a picture.
The concern I raise is with teachers that think that visualization is some sort of easier path to mathematical enlightenment. That visualization somehow escapes the demands of abstract thought. They contrast it (incorrectly) to symbolic and analytical mathematics. There is no contrast. At the root of all mathematics is abstract thought. Whether your end product is a visual proof or an analytical argument, it is the acts of abstraction and reason that are key. I can't imagine teaching deductive (synthetic?) geometry without drawing all over the board. But last time I checked, students have more trouble with deductive geometry than they do with algebra.
I believe that the root cause of all of these teacher myths is simply the difficulty in teaching abstract thought to most students. I am not even sure it can be taught. Maybe just coached. The algebra teacher that complains about useless symbols and factoring is really just complaining that they are useless to his or her students. So they stop teaching it. What they do next is just bizarre and unreasonable. They adopt a different approach and to prove to us that this new approach is better than the old way, they provide a rhetorical argument. They don't show us advanced students. They don't even show us students that have broken through the abstraction barrier. They show us a rhetorical argument. I am sorry, I would rather try and fail to teach algebra than to give up altogether. And if I did give up, I would be honest about it. And most teachers actually are honest. It's rather rare that someone like Dan Meyer turns this angst into a self serving activity to enrich himself.