I'm not trying to be waspish, but are you suggesting we just be like Feynman? I will say I'm rather dubious we can achieve that with a simple interface change.
There is certainly merit and utility to graphical representation; we all use it via matlab, python+scientific packages, and so on. And, of course, the faster we can change something and see the result the better. It is not always an incremental change, but a pretty fundamental change, to see something change in real time.
Yet, this only really works in tiny cases. I recall Victor demonstrating using scrubbing variables to change how a game reacts. Okay, but the vastest amount of the programming I do (and I do fairly numerical things in the computer vision line) I either don't see the utility, or practicality. I can't variable scrub my way to a better algorithm.
I've seen, for example, graphical demonstrations of things like binary tree algorithms, sorting, and the like. It's momentarily interesting, but I can't claim that it really made any difference in my understanding of the subject. I think this is a case where people vary. I am not terribly visual, and most of Victor's arguments go over like a lead balloon with me. Yet, I cannot deny the excitement of others. I find it far more commodious to work with, for example, linear algebra. Certainly there are times when I am drawing 3D arrows and the like to work out a problem, but by-and-large the symbolic representation serves far better.
There is a reason we use text for communication on HN, and not pictures. It is not because we lack vision or foresight (no pun intended), but because it is the most expressive and flexible way to communicate our ideas. And, of course, sometimes we labor away, trying to communicate in words what would be so easy to illustrate with a single image. Neither is 'best'; we just use the best tool for the job.
None of that is to argue against one of your points - that math is more than symbolic representation. Once you are reduced to applying symbolic rules to an equation you are often quite lost, and that is often not taught in schools. Yet so much of math really cannot be done graphically at all, yet we succeed. A lot of people stall out in math (I think) at after the algebra/calc I,II,III level because beyond that there is often no recourse to graphics and physical representations; graphics can be as much a crutch as a help.
Anyway, visual thinking is certainly a form of thinking that should not be downplayed, but it is not how some to many brains work, nor is it even possible in many forms of important thought (I leave savants to one side, we can not make decisions based on exceptional/unique people).