Cruncher: An implementation of Bret Victor's Scrubbing Calculator

If by algebra you mean "the transformation of one written notation to another by rule," that is only one way of finding general solutions to mathematical problems. There is a lot of writing about "how" people doing original math think, and many of them work in ways far divorced from algebraic notation. Some seem to develop graphical models they picture in their head, others report using other senses like sound and smell.

Freeman Dyson on Feynman: "The reason Dick's physics was so hard for ordinary people to grasp was that he did not use equations... Dick just wrote down the solutions out of his head without ever writing down the equations. He had a physical picture of the way things happen, and the pictures gave him the solutions directly with a minimum of calculation."

Einstein on his thinking: "The words of 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 of thought are certain signs ( symbols ) and more or less clear images which can be voluntarily reproduced and combined... The above mentioned elements are, in my case, of visual and some muscular type."

More interesting discussion here: http://www.sv.vt.edu/classes/ESM4714/methods/vizthink.html

The idea that "real math is bare symbols" isn't true. Not only isn't it true for doing groundbreaking math, it isn't true even for easy math. I would challenge you to teach children mathematics as a purely symbolic game, never showing them that numbers correspond to physical entities, or that derivatives correspond to tangents on a curve. Those of us who are "good" at algebra are good at it because we understand it in a way deeper than the symbols.

Yes, symbolic notation is a great form of communication and necessary for proof, but it is usually not the way to create the initial intuitive spark underlying a solution.

[0] http://worrydream.com/LadderOfAbstraction/