There’s someone who has been bugging me to post something for a while (you know who you are!), so I’ll actually post something substantial. I hope this will be worth the wait.
Studying for a Test
It’s very common in math class that the people who perform consistently well on math tests never study for those math tests specifically. Of course, they still do math often, and truly enjoy the subject. There are a couple exceptions, but in the end everyone knows that’s all there is; they’re just good at tests.
But this first class of people have a name, though this is a very uncommon description. We are known as the competition math kids, though we usually just get called “math kids.” It’s very well known that math is unique in the fact that competition math has nothing to do with school math; competition physics and school physics correlate, as does chemistry, computing, etc.
So why do the kids who do competition math perform so well on school tests, without those performing relatively well on school tests excelling at competition? What makes the difference? And why do people who study for tests for hours each day the week before usually fail, or barely “pass?”
While this is the natural order to ask the questions in for most people, the natural order to answer them in is the opposite. The way you ask your questions, in addition to the actual questions themselves, influence your answer. The reason people have been asking these questions in the wrong order while feeling that it is the right order is because they’ve been conditioned to think about it in the wrong way. The question I’m asked most often is, “How are you smart?” as opposed to “Why am I not?” which reflects the order these type of questions are asked in. (Some survivorship bias is also present when asking just the people who succeed. Often more than not, quite a couple strokes of luck are involved.)
You’ll notice that the lines of thinking for math tests go differently. Some people think, “I’ll study to get a good grade. It’d be nice to get smarter, but that’s not a priority.” Other people think, “I study to get smarter. It’s good to do well on tests, but that’s not a priority.” Surprisingly, it is often the latter type of person who ends up doing better on tests. However, most people think the former, which demonstrates a backwards thinking and a lack of understanding of math.
Most of us know people who study a week or so before every math test. Some end up only studying the night before, and others end up getting respectable grades. But even for those who believe grades are all that matter, people who study not for grades but to get smarter usually are the ones with the best grades.
But having the right mindset is not alone to succeed. I personally know some people who don’t study for grades but are still failing a particular subject. This doesn’t make their efforts worthless; this work ethic is arguably much more valuable than the specific success itself. (While I think some parents overstate the “genius” of competitive math kids, part of the reason they say this is because of the work ethic. After all, nobody gets a job doing high school math olympiads…)
The kids who do math for the sake of math become good at math, which is why they generally crush school tests, if they aren’t exceedingly stupid. (By “exceedingly stupid” I mean “show your work” or something of that sort.) The kids who do math for the sake of a grade become good at getting grades if they are really lucky, but they don’t get much better at math, meaning they have to do it again the next time there’s a test.
In general I think people understate how much they have done and overstate how much they can do. This makes sense from a motivational standpoint; the more you think you can do, the more you will try to do, and the less you feel you have gotten done, the more you feel you need to get something done.
But sometimes I feel this backfires. The feeling of “I’ve never done anything, and anything I try to do ends up sucking” can be very detrimental all the same. When no noticeable improvement occurs during the first few weeks/months of whatever you’re practicing, it can be very disconcerting. It’s very easy to beat yourself up over this, but we often forget how much we have accomplished. I’m certainly very guilty of this. Having your handouts written down somewhere though does help to give perspective, so I have it easy there.
Everything has to come with moderation, though; spending too much time thinking about what you’ve already done leads to a false sense of satisfaction and not doing anything else. Spending too much time thinking about how little you’ve done also leads to getting nothing done. I’m not sure how to quite accomplish this balance, but I’m vaguely aware it’s important.