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moshez comments on Proofs, Implications, and Models - Less Wrong
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My mat class provided the simple "how to choose" heuristic that you want X to be alone. So if you have "x+1" on one side, you'll need to subtract 1 to get it by itself. X+1-1=5-1. X=4.
I can see how this wouldn't get explicit attention, since I'd suspect it becomes intuitive after a point, and you just don't think to ask that question. I can't see how one could get through even basic algebra without developing this intuition, though o.o
Yes, clearly, a bit after I asked, I learned how to use intuition, and at some point, it became rote. But the bigger point is that this is a special case -- in logic, and in math, there are a lot of truth-preserving transformations, and choosing a sequence of transformations is what doing math is. That interesting interface between logic-as-rigid and math-as-something-exploratory is a big part of the fun in math, and what led me to do enough math that led to a published paper. Of course, after that, I went into software engineering, but I never forgot that initial sensation of "oh my god that is awesome" the first time Moshe_1992 learned that there is no such thing as "moving the 1 from one side of the equation to the other" except as a high-level abstraction.