> On Jun 15, 2013, at 3:00 PM, Joe Niederberger > <firstname.lastname@example.org> wrote: > > > Of course I can. I take it *you* cannot, nor can > you apparently widen the scope of your imagining > beyond yourself. > > Then prove it. Surely you accept that everyone on > this list understands to some degree and acknowledges > abstract analytical (non visual) thinking. I think > the task is up to you to prove that this mysterious > non abstract visual thinking exists and is up to the > task of mathematics. I have found no evidence of it. > Sure, there are plenty of classrooms where teachers > stand in front of students and display a visual, ask > the students if they "see" and the students nod their > heads. But they aren't seeing math. Visualization is > a lot of things to mathematics but the source of > generalized abstraction in thought it is not. > > Bob Hansen
"The words or the language, as they are written or spoken, do not seem to play any role in my mechanism of thought. The psychical entities which seem to serve as elements in thought are certain signs and more or less clear images which can be 'voluntarily' reproduced and combined. .... This combinatory play seems to be the essential feature in productive thought before there is any connection with logical construction in words or other kinds of signs which can be communicated to others". Albert Einstein in a letter to Jacques Hadamard.
Mike, another article you may find helpful:
Increasing Math and Science Learning by Improving Spatial Thinking