Visual thinking in mathematics, to my way of thinking, is not that my mind generates slick animations or beautiful typeset equations, but that simply there is subtle visual thinking, subtle images that fly through my mind as an aid to what I'm doing on the paper, or as an aid to understanding a concept. The subtle images *support* the effort to count, manipulate algebraic equations, or whatever. They aren't the whole thing, but I suspect they *are* an important ingredient.
I wasn't aware of the extent to which I have subtle mental imagery until I started tutoring and paying more attention to what I'm doing when I solve an equation, or a tough AMC problem. It can be below the level of easy, direct awareness, but it is there, and it is critical.
I found that providing animations (which I did for several topics) for my student was a tremendous *aid* to understanding (not the whole story, but critical).
What I am wondering about now, is whether people who don't do much visual thinking are going to have a hard time with algebra. If so, then one goal of videos or math educational software would be to stimulate their visual thinking.
The other possibility is that people who aren't good at visual thinking (in a mathematical way) should use the mode of thinking they *are* good at. What I'm tossing over in my mind right now is whether that is a good compromise, or whether visual thinking is so integral to algebra that it is simply more effective to train their visual thinking. If they aren't natural visual thinkers, this can only go so far. But maybe it is the best solution in practical terms. (An example of a practical goal would be passing the GE requirements in college.)