It sounds like he's just talking about using variables to solve equations. I think the point he's making in that section is just about learning a good process so you can solve harder and harder problems, e.g.
Problem 1: Two numbers add to 100, one is 20 larger.
Smart student: oh, I see, 60 and 40.
Dumb student Knuth: x + y = 100 and x = y + 20, solves to x=60, y=40.
..
Problem 2: Four numbers sum to 1024, one is half the sum of the other three less 17, one of the others...
Smart student: uh, I don't see the answer.
Dumb student Knuth: w + x + y + z = 1024, w = (x + y + z)/2 - 17, ... solved it.
Algebra goes so much deeper than this. I had so much trouble with my compilers class because my professor expressed all the ideas and principles of compilation using algebra.
So I second OP's comment - I wish there were a course one could take on algebra itself - not merely numeric expressions with variables.
> I wish there were a course one could take on algebra itself
Depending a little on just what you mean, that's likely the upper-division undergraduate math department course called "Algebra" at most universities. Groups and rings and such.
> the upper-division undergraduate math department course called "Algebra"
And be careful what you wish for - the way eighth-grade algebra hurt your brain when you were 13 will be nothing compared to the way abstract algebra will hurt your brain when you're 20.
I thought of that, but it sounded like it was just studying particular mathematical structures -- not learning "how to read arbitrary algebraic expressions".
Or perhaps algebraic expressions in different mathematical specialties are differentiated enough to make knowledge about how to read a given expression mostly non-transferable?
Yes of course it does and Knuth is well aware (as am I to a lesser extent), but I’m just saying I don’t think it’s relevant to the point he’s making here.
After reading the relevant article section, I'd guess he's talking about the idea that problems can be mapped onto mathematical structures, allowing use of pre-known rules within said structure, for transformation and identity and so on, such that the problem can then be solved. He's saying that for simple problems if you're clever you can intuit the solution without that mapping/manipulation/solve process, but as a result you can never see how to solve more complex problems. Implication being that if you had been slightly less smart, you'd end up understanding mathematical structure earlier in life, with associated benefits in terms of success in certain fields.
Like: https://en.wikipedia.org/wiki/Abstract_algebra
Problem 1: Two numbers add to 100, one is 20 larger.
Smart student: oh, I see, 60 and 40.
Dumb student Knuth: x + y = 100 and x = y + 20, solves to x=60, y=40.
..
Problem 2: Four numbers sum to 1024, one is half the sum of the other three less 17, one of the others...
Smart student: uh, I don't see the answer.
Dumb student Knuth: w + x + y + z = 1024, w = (x + y + z)/2 - 17, ... solved it.