So I have started asking my students — Algebra II and Calculus — how they remember things. If they have mnemonics, or if they have funny methods, or what? I don’t know why I didn’t ask this before. It seems so obvious.
How they remember which graph is a sine graph and which is a cosine graph? How they remember certain formulas? How they remember know how to solve absolute value inequalities?
I’ve gotten some pretty interesting answers, things I wouldn’t have come up with on my own.
For example, when doing absolute value inequalities, you might have 
How do you know which is an AND statement, and which is an OR statement. Well, one student (who probably got it from a tutor, or another student), said: “You can remember that because the absolute value is less thAND a number, or the absolute value is greatOR than a number.” Love it!
So when you’re doing something particularly new and challenging, remember that students come up with seemingly inscrutible methods all the time. It helps to ask them what they’re thinking. Not only to see if they’re on track, but also because their thoughts might be super valuable to others.
Continuous Everywhere but Differentiable Nowhere 
How much should we care whether their mnemonic is tied to the actual meaning?
On the one hand, I’m glad these things work for them, but on the other, pragmatism may not be optimal long-term, if they’re just memorizing random statements to spew out on tests.
What do you think? I’m honestly not sure.
– Matt (a colleague of Meera’s)
I think mnemonics and tricks to remember things are good, as long as they don’t replace understanding. I mean, do you actually derive the quotient formula in calculus, each time you use it, or do you sing a little ditty: “low de high less high de low, and down below, denominator squared goes.” (A new teacher in my school learned it: “ho de high minus high de ho over ho ho” — I say no talking about “ho”s in my classroom!)
Or when doing trigonometry, using “All Students Take Calculus” to know which trig functions are positive in which quadrants. They learn *why* and then they learn the mnemonic.
Or when it comes to the quadratic formula, don’t they memorize it (I use the quadratic rap to teach it). It’s not that we don’t talk about *why* that’s the formula and where it comes from, but once they get that, it’s time to move on.
Another way to think about it: do you use any tricks when you do math? I do. And do you know them any less well? I don’t. Sometimes, and I agree with your worry, these tricks replace understanding. But I think they can be used once understanding has been achieved.
I have started putting concept questions on my exams, where students have to write out explanations of why things are so, or what things mean. My sister even had the great idea to put a solution to a problem on a test and tell students to decide if the solution is wrong or right, and if it is wrong, where and why is it wrong. I like that.
So that might be one way to see if they “get” it.
True, it’s not an “either-or” situation.