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?

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).

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.

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.

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.

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.

What's odd here is the line of demarcation between mathematics (or technical subjects in general) and liberal arts -- two of the quadrivium are arithmetic and geometry. And while we can probably agree that the influential aspects of astrology are no longer worth serious study (except, perhaps, from a historical, psychological or sociological perspective), the periodicity -- the astronomy -- of astrology was also considered part of a well-rounded education. (Music was the fourth element of the quadrivium.)

Perhaps it would be best to refer to a completely non-technical education as trivial since it ignores more of the liberal arts than it incorporates.

This is completely shocking to me. Is it really allowed not to be able to do 12 * 9 in your head after completing school? I am serious. (Since this is not the case in South Korea.)

Nor around here when you're leaving primary school. So I assume it is a kind of hyperbole used by your parent. At the same time, however, not having any scientific literacy at all seems to be accepted or even glorifies in Western popular culture.

Then again, other troublesome behavior is quite accepted as well, like, for example drinking alcohol. How often haven't I heard a parent proudly tell that his or her child just had her/his first beer.

Some bad behavior seems just socially and culturally accepted. And changing culture is hard, so, even though there have been programmes stimulating STEM for decades now, there seem to almost no real improvement.

Well, anecdotally, since that's all I have to rely on, yes. I'm fairly certain that if you were to survey the basic maths skills on an average liberal arts college in most of Europe or the US, you'd get a result that would shock you.

My experience from the UK is that more kids can do 12 * 9 in their heads at the end of primary school than at the end of secondary. Possibly because by the end of secondary a good percentage of them are too drunk or off their heads on ketamine or methadrone (or whatever it is this week) to either understand the question, or to even understand which way up to hold the bit of paper with the question on it.