R Hansen says: >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... http://en.wikipedia.org/wiki/Brouwer_fixed-point_theorem My method has, I am certain, been done before but I thunk it up on my own back then...[etc.]
That seems more to me like an intuitive justification - but not sure how it leads to a proof.
R Hansen says: >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.
You seem to be looking for the ghost in the machine. All the parts of thought work together - visualizations, interpretations thereof, acquired stratagems, abstraction (in the sense I gave earlier), recognition of the places where previously acquired abstractions can be employed, tactile senses, auditory perhaps, language, simple facst that can be recalled...
I wouldn't expect to find "mathematics" any more in "chain of thought" than I would in any other part, taken in isolation.