R Hansens says; >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.
If one could substitute "deductive logic" with "abstract thought" in your statement, I could probably agree with it in part. But most people don't learn math in grade school by rigorously meeting the demands of deductive logic, nor would such demands have much meaning to them even if they did.
If a third grader "understands" multiplication by seeing a picture of a 3x5 grid of dots, he/she "abstracts" from it to realize it works for any size grid. Where that power resides in the brain, and how it operates, is not known currently. But for that third grader, that abstracted picture (a sort of NxM grid) is key to the understanding, in the same way, that later on, axioms and theorems and definitions become key. We acquire many such pictures and other facts, sometimes in conflict with one another (that must be sorted out and modified to make progress.)
So perhaps you are talking about that power of the brain to abstract from the specific 3x5 grid to the notion of a general NxM grid, and also to connect it to other completely different pictures, such as a picture of 5 baskets, each with 3 apples in it. Call it "the power of abstraction". It works seamlessly *with* our ability to visualize, or our language ability, or probably any other grounded sense. Note that power *is* what the Wikipedia article is talking about -- abstracting features from populations of concrete individuals to generic concepts.
If that's what you are talking about, then I still find myself disagreeing with how you talk about it. I'd say the process needs *grounding* to operate -- and pictures are one kind of grounding. You dispute the need for any kind of grounding and claim the process is a thing apart that develops on its own.