I do sometimes read online forums of educators, particularly math educators, since it’s fun to have somewhere to talk shop, and the topics of conversation are constant enough you don’t have to spend much time getting the flavor of a particular group before participating. If you suppose the students are lazy, the administrators meddling, the community unsupportive, and the public irrationally terrified of mathematics you’ve covered most forum threads. I had no luck holding forth my view on one particular topic, though, so I’ll try fighting again here where I can easily squelch the opposition.
The argument, a subset of students-are-lazy (as they don’t wish to understand mathematics), was about a mnemonic technique called FOIL. It’s a tool to help people multiply binomials. Binomials are the sum (or difference) of two quantities, for example, (a + 2) or (b + 5). Here a and b are numbers whose value I don’t care about; I don’t care about the 2 or 5 either, but by picking specific values I avoid having too much abstraction in my paragraph. The product of (a + 2) with (b + 5) is the sum of all the pairs made by multiplying one term in the first binomial by one term in the second. There are four such pairs: a times b, and a times 5, and 2 times b, and 2 times 5. And therefore the product (a + 2) * (b + 5) will be a*b + a*5 + 2*b + 2*5. That would usually be cleaned up by writing 5*a instead of a*5, and by writing 10 instead of 2*5, so the sum would become a*b + 5*a + 2*b + 10.
FOIL is a way of making sure one has covered all the pairs. The letters stand for First, Outer, Inner, Last, and they mean: take the product of the First terms in each binomial, a and b; and those of the Outer terms, a and 5; and those of the Inner terms, 2 and b; and those of the Last terms, 2 and 5.
Here is my distinguished colleague’s objection to FOIL: Nobody needs it. This is true.
You need to find all the pairs made of one term in the first binomial and one term in the second (so that a times 2 and b times 5 are right out), but there’s no need for an extra tool for that. More, if one has, say, a trinomial — three terms added together, such as (a + π [pi] + 2) — then FOIL breaks down altogether. There should be terms representing π [pi] * b and π [pi] * 5 which are neither First, Outer, Inner, nor Last, and no one has found a mnemonic which helps matters any.
There are similar problems if one wants to multiply two trinomials, or something with a monomial (a single term, such as b alone), or more complicated terms. And it does nothing if one needs to multiply three or more sets of binomials. Better, the argument goes, to teach students that this is all a consequence of the distributive law and not bother with a tool only good for multiplying two binomials.
I agree with the complaints that a tool to help people correctly multiply two binomials does nothing to help them multiply non-two non-binomials. And I agree that if one is comfortable with the reasoning behind why FOIL works where it does, then one does not need it. But I disagree that’s a reason not to teach it at all. Here’s why.
For one, FOIL does help people do a particular type of problem correctly. That shouldn’t be overlooked. There are many reasons to study mathematics, and the beauty of the field is a powerful and romantic reason. But one at least as potent must be that it is a practical field, letting one do things one wishes to do, and if one wishes to do something, one surely wishes to do it correctly. Simple tools which help that goal should not be lightly thrown away.
There is a bias in mathematical thinking: tools which work for only a small set of problems, like FOIL, look like clutter compared to general tools of broad applicability. I’m sympathetic to that bias. But there is value in tools which work for a very specific problem. An Allen wrench is a much less general tool than an adjustable screwdriver would be, but it can be much better for assembling flat-pack furniture. It’s simple and obvious to use, and I can use it without my father complaining I’ve put everything back in his toolbox wrong.
Also, I feel that it is harder to multiply two trinomials together than it is to multiply two binomials. Likely you agree. But the idea that one can use the distributive law to do both kinds of multiplications is a general application of the distributive law, the idea that a times (b + 5) must necessarily be equal to a times b plus a times 5, and that (a + 2) times b must be equal to a times b plus 2 times b.
There are some general ideas that people can understand at a glance. Usually, in my experience, these are ideas close to ones they have already seen. More often, one needs practice, on multiple examples which demonstrate the thing being studied without being too complicated. And what are the simplest cases of using the distributive law? … Well, a single number times a binomial, yes, but after that we get to binomials multiplied together.
I don’t see how one can reasonably expect a student to get comfortable with the general principle of the distributive law without exercising on the simple case of multiplying binomials together. And that exercise has to work out the problems correctly, at least most of the time. (It can be instructive to get a problem brilliantly wrong; at least, it can enlighten the student who goes on to learn how things went wrong, and makes needed relief for the person doing the grading.) FOIL is a tool to get that exercise done correctly. Not all students will need it, and not all will need it for long; but I’m hesitant to throw away a simple and useful tool just because a more powerful yet more abstract tool exists.