On Jun 18, 2013, at 1:31 PM, Joe Niederberger <email@example.com> wrote:
> R Hansen says: >> You seem to claim it is the method. Can you give me an example of how, in your mind, you arrived at the proof you spoke of by "visualizing"? > > No, I meant to say, it is *one* of the tools available when working out math problems. > > As Pam has mentioned, using visualization this way is a fluid process, not a static picture. Any verbal description is bound to be somewhat dissatisfying, but for what its worth... > > For the specific problem, I began by forming several pictures in my mind, and asking myself if those pictures shed any light on the problem. Those pictures were formed, obviously, based on knowledge I had acquired earlier. As I said I formed several pictures, and sort of vaguely noted which ones I thought promising. A second pass I tried to form the pictures even more clearly in my mind, and note more clearly the various bits of information I could "extract" from that pictures. After sometime of this kind of concentration, I had a single, simple, and clear picture selected. I knew intuitively that picture had the key information "close to the surface". But I was satisfied and stopped the process, after forming the clearest and simplest picture in my mind that I could. If I had been asked to articulate on the spot a "proof", I would have had a hard time. It was a very simple matter though, at home, with pencil and paper, to translate the picture into an acceptable algebrai! c ! > form (with a couple key facts from calculus.) Only a few lines long. > > Cheers, > Joe N
I will give you an example of one that I did way back in an advanced calculus class. It has to do with this theorem...
My method has, I am certain, been done before but I thunk it up on my own back then. I thought of a crumpled piece of paper lying on top of an identical flat (un-crumpled) sheet of paper s.t. it was contained within the borders of the flat sheet. I then thought about the outline that the crumpled sheet made on the flat sheet and what this same outline now looks like on the crumpled sheet. I then thought about the outline of this crumpled outline and so on. Since the whole crumpled sheet lies within the border of the flat sheet, then each crumpled outline lies with the border of its associated non-crumpled outline on the flat sheet. If you continue this process then at the very least you would approach a point on the crumpled sheet that is lying directly above the same point on the flat sheet.
Even though this is visualized, there is still a lot of mathematical reasoning here. On the surface at least, the recursive construction and the notion of limit. And then there is instinct.
But let's forget fancy terms (lest Lou Webster jumps my case again:) You asked the following question in another post...
>  How do people think while solving mathematical problems?
My answer: To say that it is mathematical then there must be a chain of thought.
I don't care if it involves visualizations or symbolic analysis. I don't care if the student understands the principles of logic. Even if it involves trial and error it can't only be trial and error because that would be random guessing. Even trial and error requires a chain of thought in selecting the trial and error strategy.
Was there a chain of thought in your proof? You couldn't have been randomly picking curves. Is it possible that you are taking that chain of thought for granted?
And my original position wasn't that visualizations are bad math, it was that visualizations, the way most teachers use them at least, do not represent the chain of thought theme well. And a lot of the reason for this is that the teachers incorrectly believe or are led to believe that a visualization is some sort of alternative to a chain of thought. And this goes similarly for teaching strategies based on "data" or "modeling". Every time I review them, unless its an implementation in an advanced class, the first thing that strikes me is "Where is the math?" which I should from now on put as "Where is the chain of thought?"