Just wanted to take a moment and point out a point that often gets overlooked:

It doesn’t matter how many times you’ve been exposed to a particular concept if you don’t actually understand that concept.

Nothing, nothing annoys me like the idea that doing well on the SAT or ACT is just a matter of “getting familiar with the test.”

It does not matter how many times you look at a vocabulary word and say, “Yeah, I’ve seen that before” if you do not actually know what the word means. Ditto for functions, Venn diagrams, dangling modifiers, and pretty much everything else that could get tested.

And by the way, it’s really not enough to go over a given concept once of twice. Just because you learn something on Saturday doesn’t mean you’ll still understand it on Tuesday, or that you’ll be able to recognize when it’s being tested the opposite way around, especially when you’ve been up since 6am and can’t stop listening to the kid in the next row tapping his pencil against his desk.

I’ve had students with whom I spent weeks going over comma splices. They were certainly very familiar with the idea of comma splices, and they could even spit back the correct definition of them (well, most of the time).

What they could not do, however, was either consistently recognize or correct them. And why could they do neither of these things? Because they had never learned to recognize what a sentence was, and thus had no idea when they needed to put a period or a semicolon rather than a comma between statements — something they should have mastered in elementary school. (Yes, I am actually suggesting that elementary school students be explicitly taught to recognize sentences — the horror!)

The problem had nothing whatsoever to do with the test itself; it showed up in their actual writing as well. The test was simply catching the problem, not creating it. In other words, it was doing exactly what it was designed to do. And ultimately there was no way to truly compensate for 10+ years of not knowing in a handful of sessions. We’d go over the concept, do 10 or 20 or 30 examples, they’d seem to get it just fine, and the next week we were back at square one. These were, incidentally, students scoring in the high 600s/low 700s — not the sort of kids who are typically thought of as needing remediation.

So to be clear:

“Familiarity” means being familiar with something. That’s it. It’s often related to understanding, but it does not by itself lead to understanding.

“Mastery” means understanding something at such a deep level that you pretty much can’t get it wrong, no matter how tired or stressed you are. It means you can roll out of bed and nail it, even if you haven’t really studied it for a while and it’s presented in a slightly different way than you’re used to seeing it.

It’s possible to have very little familiarity with the SAT or the ACT and still do extremely well on them; it’s also possible to be extremely familiar with those tests and still do very poorly.

Mastery is what ultimately leads to improvement, but it takes a lot more work to achieve.