Please Excuse My Dear Aunt Sally, SOCATOA, All Students Take Calculus, FOIL, ...
It seems math education is riddled with acronyms and mnemonics. Students love them because they afford an easy way to remember what otherwise might be a challengingly complex concept. What is a little more surprising is that many teachers are also very found of these tricks. Last semester I had a student who seemed to have a mnemonic for everything, and she claimed that her high school math teacher showed her dozens of them. What a shame, I thought.
Now I understand that students will not be able to advance very far in mathematics without being able to remember the order of operations, but there must be a better way. Consider the "All Students Take Calculus" example. If you are not familiar with it, this is the trick to remember which trigonometric function is positive in which quadrant. Starting in quadrant I, all three trig functions (sine, cosine, tangent) are positive. In the second quadrant, sine is the only positive one (Sine = S = Students), in the third tangent is the positive one, in the fourth cosine is. Okay, so this definitely works, and I'm sure there are students who know that sine and tangent are negative in quadrant IV but cosine is positive because of this, if for no other reason. But there's the problem: if for no other reason!
Look, let's be honest. Students do not need to know which quadrant tangent is positive in. It is unlikely they will every need that fact on the job, nor be asked about it in a job interview. There are no exclusive math parties where that particular piece of knowledge is required to gain entry. This of course does not mean we shouldn't teach the topic. Determining the sign of a trig function based on the quadrant of its angle is a perfect exercise in understanding the meaning of the trig functions. Students should already know which of x and y are positive in a given quadrant. Students should already know the definition of the trig functions in terms of x and y. Putting these pieces together is exactly why we teach trigonometry at all. Its to develop that kind of thinking.
On the other hand, there are some formulas or concepts which simply need to be memorized. For example, I would not expect my students to be able to derive the quadratic formula each time they need it. If they can think up a little jingle to help them remember which constant belongs where, more power to them. Additionally, I must admit I was surprised to hear that some schools are avoiding mentioning FOIL. I would probably not teach students that method of multiplying two binomials at first, but if they already know the trick, it seems a waste not to use that simply label to remind them what's going on.
So how should we approach mnemonics such as these? Depending on the particular example, my answer might change, but in general, I avoid introducing these tricks completely. I might ask my class how they plan on remembering a particular formula or concept. If a student suggests a mnemonic, I act surprised, as if I have never heard that one before. I then go through it and "check" that it works. This reinforces the original, conceptual basis for the fact, as well as challenges the students to think creatively about the subject. Once the class knows a mnemonic, I tend not to be a hard-ass about it. I would never respond to a student suggesting we FOIL an expression with, "what's FOIL? do you mean doubly distribute?"