KT, Please stop bullying me 😡
I haven’t been posting in a while.
We may have another IGP soon. However, we’re also planning to make an entrance exam for next year’s kids (more details on this when we’ve actually started planning some things), so any new problems we create will probably go on the test. However, I usually create some new problems for every IGP, so I’ll probably be using my new problems for IGP and for the Entrance Exam. This causes an issue because I can’t post the IGPs up on geometryexplorer.xyz because I don’t want the exam questions to be findable, even if they aren’t explicitly marked. (For my kids, they will be on the IGP Handouts, if I decide to do it next Tuesday.)
Because of this, the IGPs from now may not be public or only a selection of the IGPs will be public from now on. I apologize for this, but I think the Entrance Exam will be worth the wait.
(P.S. To my kids: Please start working on writing problems! Most of you are very far behind, unless you’ve been working behind my backs, which I hope is the case right now…)
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.
On Thursday I took the PSAT 8/9. Since I’m contractually bound from discussing any questions, I’ll be talking about the SAT instead and using past questions. Regardless, the point is the same; the SAT, and standardized tests in general, are garbage.
Skill Floor and Skill Ceiling
In theory, it would be nice if we could perfectly tell apart how good people are at something. In practice, this is not possible, but we do want to strike a reasonable balance. The three things we desire are accuracy, a low skill floor, and a high skill ceiling. This means that we’d like our evaluations to be correct within a reasonable range.
Consider video games as an example. Two of my favorite games growing up were Super Smash and Super Mario. Despite being in different genres, the underlying philosophy behind them is the same. Both games reward proficiency in a consistent manner, and are enjoyable for everyone, including those who are not so experienced at it.
Now replace “games” with “the AMC series” and you have a good idea of what makes them so unique. While approachable to beginners in problem-solving, the AMC series is able to accurately distinguish between those who are merely the best in their class and those who have invested their time into getting better at math. Just as I invest time into improving my platforming and platform fighter skills by playing Mario and Smash respectively, I invest time into improving my problem solving abilities by taking the AMCs.
The SAT fails at every count here. Its skill ceiling is the same as its unbearably low skill floor, the accuracy of the test is almost as bad as blind guessing (which, of course, makes sense when that perfectly describes the English section of the SAT), and it’s boring as hell. There’s a very good reason we barely care when someone we regard as smart got a 1600, and we don’t care at all when they don’t.
But I think there ends up being something more important than the inaccuracy and non-existent skill range of the SAT. As major as these flaws are, there is a far larger issue at hand. The SAT is completely boring. There’s no reason to want to take it, the only reason to take it and do well is because “college,” and the questions don’t make people think.
When the AMC or MATHCOUNTS/mathleague tests are over, kids immediately begin discussing the questions.
“What did you get for this?”
“How did you do this problem?”
“There’s actually a nicer way to do the problem; just draw this line, and notice the similarity ratio is 2/3.” (Bonus points for whoever knows which problem I’m referencing!)
“Notice that 13-14-15 triangles have an altitude of 12.”
“How did I get this one wrong?”
“Yes! I got it right!”
These quotes are not word for word, but these get the idea across. Let’s see what people had to say about the PSAT.
“It was really boring.”
“I slept so late last night!”
“You were sleeping during the test? How?”
“I was so close to finishing Mario on my calculator, but I died!” (This last quote was said by me.)
Not so flattering an image, is it?
But MATHCOUNTS suffers many of the problems the SAT does, albeit to a far smaller scale. There is an emphasis on speed over deeper thinking (Countdown Round, anyone?) and the entire Team Round can be described as “unlegit.” So what makes MATHCOUNTS any better than the SAT? MATHCOUNTS is responsible for some of the best times of people’s lives, while the SAT is responsible for some of the worst. (The questions from MATHCOUNTS are also miles ahead of the quality of the SAT questions, but that’s like saying the sky is blue.)
I believe the AMC series and MATHCOUNTS are much more enjoyable than the SAT. This is because the AMCs and MATHCOUNTS build futures; the SAT tears them down.
I wish I could find some theoretical remedy to fix the SAT. After all, if it gets fixed before I take the real one, I have no reason whatsoever to complain. But the SAT is one of the few things beyond saving. It really doesn’t matter how good the SAT becomes, and it’s not like anyone’s going to be trying to save the SAT. If someone thought of a good math problem, there are plenty of places to put it. People could submit to the AMCs, to MATHCOUNTS, or make their own competition. (This is one of the reasons Revenge of NIMO was so popular!) One of the last places it would be put is the SAT.
So as of now, my stance is that the SAT should be put down like an old dog. It’s far outlived its supposed purpose, and there is no real need or reason for the SAT to exist.
Today for class I gave a problem that appeared as #21 in a Mock AMC 10 this year.
The problem is available here.
Non-spoiler comments follow below. However, I still recommend you try the problem before reading if you’re interested.
When I assigned it, I thought that this problem was very misplaced for its difficulty; it is literally just a bunch of given angles with a nice observation and some algebra. Nevertheless, it was a good problem, which is why I gave it.
I still think it was misplaced, but now I think it was misplaced in the opposite direction.
Let’s take a look at 2017 AMC 10A #21, which is recent, placed in the same location, and a geometry problem following the same vein (looking at angles). However, this one is very straightforward. The crucial observation could be made almost instantly, and the application is straightforward. The crucial observation in the mock is much more difficult to make. Though the way the problem is written gives a hint to the crucial step, some cleverness in algebraic manipulations is required to actually find the answer.
Even though it’s a tad too hard for the test, and none of my kids managed to solve it in 30 minutes, I think they’ll enjoy the problem and its solution, which in my eyes was pretty clever.
(Clarification: The above post refers to the mock problem.)
I assigned kids in my class homework. Most kids are usually good about homework, but some kids don’t care and never do it at all.
Recently, I decided to make kids who don’t do homework do pushups instead. We have 8 kids in our class, and 5 problems for homework. Let’s take a look at my homework policy:
“If you do your homework, you get 5 pushups for every problem you get wrong.
If you do not do your homework, you get 2 pushups for every problem other kids get right, and 5 pushups for each extra credit problem the class does (if two people from the class do #34 independently or with each other, it counts as 1 problem regardless…)”
Also, I am considering making the minimum # of pushups for people who don’t do homework be what you would’ve got if you submitted them all wrong.
Let’s see who doesn’t do homework now.
Disclaimer: Much of this post doesn’t talk about grades and the reasoning only ties in to reflect grades near the end. The reason the title of this post talks about grades is because the abolition of grades will signify that some major problems plaguing the public school system have been addressed.
Recently I’ve had quite a lot of homework. This mostly is my fault due to procrastination; things I had weeks to work on, I chose to save for the last minute. But I’m not the only person I know who doesn’t procrastinate; nearly every kid who goes to school tends to procrastinate their homework, and this is the first time I really mean all schools when I say school, not just public schools. This is mostly because people don’t view homework as something worth doing.
When I see something worth doing, I will usually start on it pretty soon and worry about the nitty-gritty later. This is because the things I want to do are usually worth doing; I would not be so eager to start doing something I didn’t want to. Conversely, I worry more about the details for things I don’t want to do (see homework). I will usually postpone it and get ideas for it (if it requires and usage of brain cells) without putting them into action. So what does this mean for homework? It’s one of the things we’d like to have done, but that we never do until we absolutely have to. Usually, the reason we do homework is because we have to; and for those who say “You can choose not to do your homework,” that’s akin to saying “You can choose to abandon your future by pushing your car off a cliff.” That’s not a choice. Wrecking your car would make your wallet unhappy, and wrecking your grades would make your future unhappy.
But school should be a time to experiment with what matters to you; children don’t know what they want to do in the future (usually), so they have a place that is supposed to be convenient to experiment. Instead we get the opposite; experimentation is discouraged, and conformity is a must. There are many problems with school, including the extremely short periods we have to work with, stupid spirit days, “core” classes being mandatory, and this comes together to penalize motivation.
However, examples of schools that are starting at better situations and have fewer problems exist; the example I use is the CACC of Pleasanton. Nothing’s perfect, but doing something like that – or doing anything, really, is a long shot better than what we have now.