I apologize for the obvious misplacement of this comment and for the fact that I, a stranger, am commenting on your sacred log, but I thought that this insight might be interesting enough to be worth your attention. The insight of which I speak regards your kinaesthetic balloon metaphor for algebra.

You stated in your entry that when solving problems you feel the need to feel the balloons to try to see whether the equations are getting smaller... In fact, this seems in a vague way to suggest a potential algorithm for solving mathematical problems computationally! Work by G. Chaitin (http://www.cs.umaine.edu/~chaitin) and others allows definition of the "complexity" or what you call "balloon size" of a mathematical problem in terms of the smallest computer program that can describe it. Additionally, there is an important algorithm in bioinformatics called the Monte Carlo algorithm (see http://en.wikipedia.org/wiki/Monte_Carlo_method) that allows one to try to minimize a value by trying to move parameters in different directions usually in ways that decrease the value but sometimes in ways that increase it...

Anyway, I think maybe the synergy of these two ideas could lead to some good new algorithms. Good luck with your writing!