You are not saying anything different here than you were saying before, that rational thought has more than one point of origin. Does the geometer develop different instincts and intuitions than the algebrist? Absolutely. But mathematics requires rational thought, which derives from a sense of reason, which has one point of origin, and those instincts and intuitions are developed to serve that. In other words, even though the algebrist and geometer develop different instincts and intuitions, they do so under the same sense of, and requirements of, reason.
I am incorporating everything you say but you seem to not like my delineations. Is this an accurate assessment of our differences in opinion?
Remember, this discussion didn't start out to be about what is different between geometry and algebra and what is the same, even though that is a very good discussion to have. It started out about using visualizations in in the teaching of mathematics. I would never try to teach mathematics without using visualizations, but they must be used properly. Call them aids or call them analogies, but the purpose has to be tied to the subject you are teaching. For example, if I use visualizations to teach algebra, i must use them to teach algebra, not geometry.
For example, you have seen the visualization on completing the square using tiles, right? Have you ever thought deeply as to what pedagogical purpose this has in teaching algebra? I won't answer that. I am curious as to what you think that is.
 geometer: one versed in solving problems involving the elements of geometry, such as points, segments, lines, angles and shapes.  instincts: innate behavior such as how one approaches a problem.  intuitions: the ability to sense without rational thought such as when one decides on a solution before knowing that it is a solution.  algebrist: one versed in solving problems involving the elements of algebra, such as symbolic representation, equality and manipulation.  reason: that which appears logically correct.  pedagogical: of or relating to teaching.
On Jul 13, 2013, at 2:51 PM, Joe Niederberger <email@example.com> wrote:
> Let's try a different tack in order to break through the merry-g-round. > > Does the representation of a problem make a difference in how one thinks about it? > (Yes.) > > Can thinking about a problem be divorced from how its represented? > (No.) > > Are mental pictures one form of representation? > (Yes.) > > Can one think about multiple representations of the same problem and switch among them? > (Yes.) > > > Cheers, > Joe N