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handoflixue comments on Proofs, Implications, and Models - Less Wrong
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"I will remark, in some horror and exasperation with the modern educational system, that I do not recall any math-book of my youth ever once explaining that the reason why you are always allowed to add 1 to both sides of an equation is that it is a kind of step which always produces true equations from true equations."
I can now say that my K-12 education was, at least in this one way, better than yours. I must have been 14 at the time, and the realization that you can do that hit me like a ton of bricks, followed closely by another ton of bricks -- choosing what to add is not governed by the laws of math -- you really can add anything, but not everything is equally useful.
E.g., "solve for x, x+1=5"
You can choose to add -1 to the equation, getting "x+1+-1=5+-1", simplify both sides and get "x=4" and yell "yay", but you can also choose to add, say, 37, and get (after simplification) "x+38=42" which is still true, just not useful. My immediate question after that was "how do you know what to choose" and, long story short, 15 years later I published a math paper... :)
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.
And yet so many do. Numbers are horribly scarce in this area, though. Sometimes I get desperate.
Anecdotally, I do remember saying something very similar to high school peers, in a manner that assumed they already understood it, and seeing their face contort in exactly the same manner that it would have had I suddenly metamorphosed into a polar bear and started writing all the equations for fourier and laplace transforms using trails of volcanic ash in mid-air.
This was years after basic algebra had already been taught according to the curriculum, and we were now beginning pre-calculus (i.e. last year of secondary / high school here in québec, calculus itself is never touched in high school level courses).