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I had to google "FOIL mnemonic" to learn what it is. My reaction would be "why in ???? would you teach your pupils such a trick, given that it breaks down when generalizing to e.g. (x+3y+z)(2x+5y+z)? (anybody who could say "that's just ((x+3y)+z)((2x+5y)+z), so I'll just apply it multiple times would be able the much more general "just add all pairs (something from the left, something from the right; there ar #items on the left times #items on the right such pairs"

Do they really teach that somewhere?

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It applies to the period in algebra instruction devoted specifically to quadratic equations. It's a first step towards the concept of "multiply all combinations and combine like terms", taught at a time when solving and factoring quadratic equations are brand new subjects.

I don't think FOIL should be given much weight as some sort of eternal mathematical truth, but it's a useful stop on the journey.

Why is it useful? Isn't it simpler to just say "multiply everything with everything"? This is the way I was taught.

They teach that everywhere. Just graduated high school here, most of my peers still use it. They even taught a special method for three term polynomials. Algebra II is the worst taught class of them all. People spend years after it still learning to factor. But yeah, FOIL is incredibly stupid, I was the only one that refused to call it that, and called it distributing instead (which it was).

Same reaction here. It also does not generalize to (x+1)(x+2)(x+3). It especially does not generalize to (x+y)^n, let alone (x+y+z)^n, which I think is more important. I think these kinds of specialized tricks are to be avoided at all cost. Math is about abstraction after all.

Yes. It's a common teaching method (and not just in math) to teach simplified methods so that people can see them in action, and once students have some experience experimenting with them, delve into the underlying theory. I think it's a perfectly valid teaching method. Students should probably be past the point where they really think "Now I'll use FOIL" by the time they finish a class where it's used as a teaching method. But in any class, there are going to be students who don't really end up grokking the theory.

What is the underlying theory? I've taken a lot of math courses significantly past the point where I learned FOIL and I don't remember it being explained.

Multiplication distributes over addition.

But what is simplified about it? It isn't easier than

    (a plus b) times 'whatever'
          equals
     a times 'whatever' plus b times 'whatever'

Which you can easily show geometrically, and which helps you generalize to products of sums of three terms, etc.

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