Art of Problem Solving
Aug. 22nd, 2022 10:14 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
cahnI have threatened for years now to rant about how excellent Art of Problem Solving (AoPS) is, and I guess now is the time, because E has just finished her first math class with them (Introduction to Counting and Probability), which she took over the summer. (She had previously taken their two python classes and I'd been super impressed, but I wasn't sure how representative those were.) And because her school has run out of math classes for her, she will be taking her math classes with them for... at least the next two years, and hopefully beyond that. I now kind of wish that she'd taken all her math classes ever with them (though for E in particular I greatly prioritized her socializing with other kids in person, and she's adored the math teacher she had for the last two years), and if I can possibly do so I will make this happen with A.
AoPS was started by math contest geeks, and provides math curriculum and online math classes from Prealgebra through Group Theory, which go more deeply into the curriculum and have more challenging problems than your run-of-the-mill math class, and have as a core philosophy trying to teach problem-solving skills in general rather than just how to do specific problems. (Beast Academy is their curriculum for the lower grades, which A. is enrolled in at the second/third grade level and which I've talked about before.)
It's geared towards strong math students and also has a decided bent towards contest prep, although you don't have to be into math contests to take or to appreciate the classes. (The site does have, in addition to more "conventional" math classes, contest prep classes for Mathcounts as well as prep for the AMC/AIME/Olympiad route; I have no experience with any of those classes at this point.) It's really amazingly great for kids (like E) who need to be mathematically challenged and are having a hard time getting that from conventional school (I LOVE that she sometimes has difficulty solving AoPS problems immediately, which is not a use case that she has gotten at school very often), but also it's just a really strong and in-depth math curriculum that I'm kind of in awe of and wish I could have gotten when I was her age. I would honestly recommend it for any strong math student who would like to go more in-depth into topics, with the caveat the child would have to work rather harder in these classes than in most math classes at a conventional school.
The classes that E has done with them are text-based. There is no video component. There is no Zoom. (They do now have some classes that have a video component, but these were the first and are still the ones they are known for.) There is a synchronous component, where everyone basically can type into the same window, and the instructor presents a lecture in text, liberally peppered with questions that the students can answer. The students are moderated, so when the instructor asks questions, their answers go through teaching assistants before being released to the whole class (and kids can ask questions themselves privately).
I was really skeptical about this the first time E took a class with them. But it turns out we all LOVE this format. It means we don't have to worry about finding a quiet room for her, or tell her brother that he has to be quiet while she's in class. It means that she doesn't have to worry about being present in the zoom or looking at the teacher or something (which is something we all hate about zoom). It means that if she needs to take a five-minute break for a snack, she can do that without being disruptive, and she can easily rejoin because the transcript is right there in front of her. (I assume it also makes it very easy for the instructor -- I imagine quite a lot is preprogrammed in.) It means that if there's a part she needs to go over again, the transcript is right there -- or if there's a part that she knows and doesn't really need to pay a lot of attention to, she can kid around with her brother while watching to see whether something more challenging comes up. The only drawback is that it means that she has no personal connection to any of the other kids taking the class. (At least at this point, she doesn't seem to think that virtually interacting is really at all interesting on a personal level.)
These classes changed the way I thought about homework. My experience with homework in elementary school and half of high school was that it was a thing that (by and large) teachers made you do for essentially no reason but which was easy enough, so I would just do it in whatever class was right after the class the homework was assigned in (because it wasn't like I was paying attention to the class, usually) and that would be that. (At second high school, I had teachers who understood the function of homework pedagogically, but I am pretty sure I didn't understand it at the time -- it was more challenging and I enjoyed it more, and couldn't get away with doing it in class anymore, but that was as much as I thought about it.) And when E started going through school, she had just awful homework at her first school that annoyed her a lot (stuff she'd known how to do for years), and so I was definitely on the "homework is bad!" bandwagon.
I still think that there are at least some kids/family situations (including both of mine) for whom school-assigned homework makes very little sense in the early elementary years. But at E's age (upper elementary/middle) it can be used a tool for skill-learning. AoPS is very clear on their philosophy that problem sets are how mathematics and technical skills are learned and practiced and automaticity is gained, and AoPS emphasizes homework and de-emphasizes class -- to the extent that for a given course, you could skip all the classes and do all the problem sets and that would be a perfectly acceptable way of completing the course if you were able to learn the material on your own. (I think class participation counts for maybe 5%?) They don't particularly recommend that for most kids, but they also don't see a problem with it if it's a good way for you to learn. And the problems are great; a lot of them do come from math competitions, so are not necessarily as straightforward as what you'd find in a conventional course. They specifically say that they don't expect students to be able to do all of the problems, and certainly not all of them easily/at first glance -- and if you can do all the problems easily, maybe you should be taking a harder class. They want you to have the opportunity to sit with a hard problem for a while, maybe taking a while to make progress, maybe asking for help. And I certainly wish I'd had more experience with this kind of thing before going for a Ph.D., which is all about sitting with hard problems and making incremental progress, and maybe asking for help :P
They also have just thought out a lot of little but important things. For example, every class has "Are you ready?" and "Do you need this?" links, the former of which goes to a pre-test and the latter of which goes to a post-test, which the student can take and reassure themself (or often more importantly, their parent, haha) that (in the former case) they know the prereqs they need to know to take the class, or that (in the latter case) they know all the material in the class and taking it would just be review, and perhaps they should consider taking the class the next step up instead. (But it's always left up to the kid/parents; they don't care if you take a class that's "rated" as too hard for you or too easy for you, which I think is great.) This is so sensible that I don't know why all classes, especially online classes, don't do this! It is so much more of a better system than what I've always had to do for everything else, which is flat-out guess based only on the course description. (Hi there, trying to pick freshman math class!) They also have a publicly available syllabus so you can check and see what the class covers week-by-week, which is also super useful! and would be so nice if other places did this!
It's all just very well set up, much better set up than my rambly discussion of its merits, haha. Everything has clearly been thought out in such a way to prioritize learning. One of the things I've been really impressed by is the class forums -- each class has a message board where the kids can post questions they have about problems, ask for help, etc. The genius is that the kids are encouraged to post (in E's class there is a "discussion question" every other week for which they're required to post) and encouraged to ask questions publicly (and answer other kids' questions publicly, for that matter, although my perception has been that it's more likely for one of the teaching assistants to answer than for another kid to answer). (There are "office hours" for a semi-private venue when necessary, and you can also ask questions completely privately during class hours.) Asking questions has been something I've been encouraging E to do -- of course when she has difficulty she'd much rather ask D for help (she hates asking me for help in general, and I think we're now at the point where I'm not entirely sure I could always help her with math!), but as a life skill it's pretty important to be able to ask people for help who aren't your parents, and I'm impressed at how much AoPS encourages and facilitates that. (Our other online class experience, with CTY, has not been set up that way at all. I'm too lazy to tell you how terrible CTY was, but although the content of her class was okay, the interface/logistics/presentation of the class was all just annoying and awful. E has also taken a couple of classes from Outschool, but those have more been enrichment activities than actual classes.)
I'm also impressed by their commitment to teaching kids how to communicate mathematically in terms of doing things like encouraging writing proofs. This is something they're committed to -- every other week, in E's class, they have had a writing problem, which is graded both on technical merit and on style. This has been far and away the most frustrating part of the class for E. She has worked on math communication for years at school, but even so, pretty much every writing problem in this class she has said, "I know what the answer is, but I don't know how to write down how I got it!" Because she's worked on it at school, she does know quite a lot, actually, about how to articulate how she got it. Part of the issue is that a lot of her schoolwork on this has been oral, so she can speak about it but hasn't practiced writing as much (but which has a simple solution -- we ask her to talk to us about her answer and she can usually articulate it that way.) -- and part of the issue is sometimes that she's read enough math writing at this point that she can tell that the way she's writing it isn't quite right but she doesn't know how to say it the "proper" math way. But she's getting practice! They're even giving them some elementary LaTeX! (I should note here that I do think that a weakness of the Beast Academy curriculum for the earlier years is that they don't emphasize mathematical communication, which as I've said her school has actually been really good about.)
AoPS was started by math contest geeks, and provides math curriculum and online math classes from Prealgebra through Group Theory, which go more deeply into the curriculum and have more challenging problems than your run-of-the-mill math class, and have as a core philosophy trying to teach problem-solving skills in general rather than just how to do specific problems. (Beast Academy is their curriculum for the lower grades, which A. is enrolled in at the second/third grade level and which I've talked about before.)
It's geared towards strong math students and also has a decided bent towards contest prep, although you don't have to be into math contests to take or to appreciate the classes. (The site does have, in addition to more "conventional" math classes, contest prep classes for Mathcounts as well as prep for the AMC/AIME/Olympiad route; I have no experience with any of those classes at this point.) It's really amazingly great for kids (like E) who need to be mathematically challenged and are having a hard time getting that from conventional school (I LOVE that she sometimes has difficulty solving AoPS problems immediately, which is not a use case that she has gotten at school very often), but also it's just a really strong and in-depth math curriculum that I'm kind of in awe of and wish I could have gotten when I was her age. I would honestly recommend it for any strong math student who would like to go more in-depth into topics, with the caveat the child would have to work rather harder in these classes than in most math classes at a conventional school.
The classes that E has done with them are text-based. There is no video component. There is no Zoom. (They do now have some classes that have a video component, but these were the first and are still the ones they are known for.) There is a synchronous component, where everyone basically can type into the same window, and the instructor presents a lecture in text, liberally peppered with questions that the students can answer. The students are moderated, so when the instructor asks questions, their answers go through teaching assistants before being released to the whole class (and kids can ask questions themselves privately).
I was really skeptical about this the first time E took a class with them. But it turns out we all LOVE this format. It means we don't have to worry about finding a quiet room for her, or tell her brother that he has to be quiet while she's in class. It means that she doesn't have to worry about being present in the zoom or looking at the teacher or something (which is something we all hate about zoom). It means that if she needs to take a five-minute break for a snack, she can do that without being disruptive, and she can easily rejoin because the transcript is right there in front of her. (I assume it also makes it very easy for the instructor -- I imagine quite a lot is preprogrammed in.) It means that if there's a part she needs to go over again, the transcript is right there -- or if there's a part that she knows and doesn't really need to pay a lot of attention to, she can kid around with her brother while watching to see whether something more challenging comes up. The only drawback is that it means that she has no personal connection to any of the other kids taking the class. (At least at this point, she doesn't seem to think that virtually interacting is really at all interesting on a personal level.)
These classes changed the way I thought about homework. My experience with homework in elementary school and half of high school was that it was a thing that (by and large) teachers made you do for essentially no reason but which was easy enough, so I would just do it in whatever class was right after the class the homework was assigned in (because it wasn't like I was paying attention to the class, usually) and that would be that. (At second high school, I had teachers who understood the function of homework pedagogically, but I am pretty sure I didn't understand it at the time -- it was more challenging and I enjoyed it more, and couldn't get away with doing it in class anymore, but that was as much as I thought about it.) And when E started going through school, she had just awful homework at her first school that annoyed her a lot (stuff she'd known how to do for years), and so I was definitely on the "homework is bad!" bandwagon.
I still think that there are at least some kids/family situations (including both of mine) for whom school-assigned homework makes very little sense in the early elementary years. But at E's age (upper elementary/middle) it can be used a tool for skill-learning. AoPS is very clear on their philosophy that problem sets are how mathematics and technical skills are learned and practiced and automaticity is gained, and AoPS emphasizes homework and de-emphasizes class -- to the extent that for a given course, you could skip all the classes and do all the problem sets and that would be a perfectly acceptable way of completing the course if you were able to learn the material on your own. (I think class participation counts for maybe 5%?) They don't particularly recommend that for most kids, but they also don't see a problem with it if it's a good way for you to learn. And the problems are great; a lot of them do come from math competitions, so are not necessarily as straightforward as what you'd find in a conventional course. They specifically say that they don't expect students to be able to do all of the problems, and certainly not all of them easily/at first glance -- and if you can do all the problems easily, maybe you should be taking a harder class. They want you to have the opportunity to sit with a hard problem for a while, maybe taking a while to make progress, maybe asking for help. And I certainly wish I'd had more experience with this kind of thing before going for a Ph.D., which is all about sitting with hard problems and making incremental progress, and maybe asking for help :P
They also have just thought out a lot of little but important things. For example, every class has "Are you ready?" and "Do you need this?" links, the former of which goes to a pre-test and the latter of which goes to a post-test, which the student can take and reassure themself (or often more importantly, their parent, haha) that (in the former case) they know the prereqs they need to know to take the class, or that (in the latter case) they know all the material in the class and taking it would just be review, and perhaps they should consider taking the class the next step up instead. (But it's always left up to the kid/parents; they don't care if you take a class that's "rated" as too hard for you or too easy for you, which I think is great.) This is so sensible that I don't know why all classes, especially online classes, don't do this! It is so much more of a better system than what I've always had to do for everything else, which is flat-out guess based only on the course description. (Hi there, trying to pick freshman math class!) They also have a publicly available syllabus so you can check and see what the class covers week-by-week, which is also super useful! and would be so nice if other places did this!
It's all just very well set up, much better set up than my rambly discussion of its merits, haha. Everything has clearly been thought out in such a way to prioritize learning. One of the things I've been really impressed by is the class forums -- each class has a message board where the kids can post questions they have about problems, ask for help, etc. The genius is that the kids are encouraged to post (in E's class there is a "discussion question" every other week for which they're required to post) and encouraged to ask questions publicly (and answer other kids' questions publicly, for that matter, although my perception has been that it's more likely for one of the teaching assistants to answer than for another kid to answer). (There are "office hours" for a semi-private venue when necessary, and you can also ask questions completely privately during class hours.) Asking questions has been something I've been encouraging E to do -- of course when she has difficulty she'd much rather ask D for help (she hates asking me for help in general, and I think we're now at the point where I'm not entirely sure I could always help her with math!), but as a life skill it's pretty important to be able to ask people for help who aren't your parents, and I'm impressed at how much AoPS encourages and facilitates that. (Our other online class experience, with CTY, has not been set up that way at all. I'm too lazy to tell you how terrible CTY was, but although the content of her class was okay, the interface/logistics/presentation of the class was all just annoying and awful. E has also taken a couple of classes from Outschool, but those have more been enrichment activities than actual classes.)
I'm also impressed by their commitment to teaching kids how to communicate mathematically in terms of doing things like encouraging writing proofs. This is something they're committed to -- every other week, in E's class, they have had a writing problem, which is graded both on technical merit and on style. This has been far and away the most frustrating part of the class for E. She has worked on math communication for years at school, but even so, pretty much every writing problem in this class she has said, "I know what the answer is, but I don't know how to write down how I got it!" Because she's worked on it at school, she does know quite a lot, actually, about how to articulate how she got it. Part of the issue is that a lot of her schoolwork on this has been oral, so she can speak about it but hasn't practiced writing as much (but which has a simple solution -- we ask her to talk to us about her answer and she can usually articulate it that way.) -- and part of the issue is sometimes that she's read enough math writing at this point that she can tell that the way she's writing it isn't quite right but she doesn't know how to say it the "proper" math way. But she's getting practice! They're even giving them some elementary LaTeX! (I should note here that I do think that a weakness of the Beast Academy curriculum for the earlier years is that they don't emphasize mathematical communication, which as I've said her school has actually been really good about.)
no subject
Date: 2022-09-25 03:37 pm (UTC)Fair!
2) I mean, I stand by what I said, but if you are wanting to point out that what I said is not AT ALL relevant for E herself and may be detrimental if she takes it to heart
Exactly; after all, the question was "bad for E in particular" and not "highly inaccurate." ;)
So, first, thank you for the additional context. That definitely helps. And you may be entirely right that that's what the designers of the platform were thinking.
If so, I still disagree strongly with them. I would say (and, if it were me, explain to E), that the difference isn't "child vs. adult" but "learning vs. mastery." Adults have mastered more skills than children, but the learning process is exactly the same at any age: it involves making a ton of mistakes. And it's necessary to feel safe making mistakes, and to be able to emotionally regulate around things not working on the first or even tenth try, or you won't get nearly as far in life (and you'll be much more miserable while doing so).
I would talk to E about this difference. I would point out that she has already mastered a bunch of skills, and that the value of mastering skills, which she already knows, is that it feels good, it allows you to master new skills (can't do algebra if you're still unable to add and subtract!), it allows you to achieve whatever the goal of that skill is to achieve (it will take longer to leave the house if you need thirty tries to tie your shoes every time), and it allows you to be entrusted with responsibilities involving those skills. Which will be lucrative as an adult who has mastered a lot *more* skills than a child, but is still useful even as a kid.
And I would tell her that we need to treat people very very differently depending on whether they've mastered a skill or are still learning the skill. That pharmacist calculating dosages from your example in a previous conversation wasn't--I hope!--someone still learning the skill. I would hope that job would be given to someone who had mastered it (which is what I meant above by mastery allowing you to be entrusted with responsibilities). When you yourself submit finished projects to clients, you're a highly paid analyst because you've *already* mastered the skills you need to submit good work, and you're entrusted with responsibilities that depend on that mastery. You're not asking the clients to evaluate your work as part of teaching you how to do the thing.
But it took me 30 tries to figure out how to set up my wifi extender a couple weeks ago, because I was still learning. I am *not* getting paid for my mastery of this skill.
A student, similarly, is still learning. I would spend most of my time (as you seem to) talking to E about the importance of mistakes in the learning process, saying that mistakes are like when your Python code prints out an error message. You're not going to get very far in learning without error messages, and so instead of fearing or hating mistakes, you can hope for them just like you do with error messages, and make use of them the same way.
One thing Alfie whats-his-name says that I went "YES!" at, because I'd independently come up with that as part of my dreams of reforming education, is that the only two "grades", if there must be grades, should be "A" and "incomplete." And I would say that's because one means mastery and one means still learning. Forcing kids to stop halfway through the learning process and tell them they got a "C" and they have to move on makes no damn sense. (It make make sense to move on and return to it later. But that's what "incomplete" means.)
And likewise, 2 tries to get the right answer seems absurdly low to me. Especially if we're talking about going one problem at a time to identify your mistakes, learn from them, and apply them to getting later problems right, and if there are a finite number of problems. (One problem at a time, btw, was the correct answer, for the exact reasons you described. :P E just needs help emotionally regulating here.)
Someone like certain of E's classmates needs emphasis on valuing mastery more highly, so they don't just keep not caring about getting wrong answers indefinitely. (As you know, I would not start with punishments and rewards; I would start by emphasizing the intrinsic consequences of mastery I listed above and try *really hard* to get the child to understand and value those intrinisic consequences.)
Someone like E needs emphasis on emotionally regulating through the learning process. And part of that, speaking as someone who struggled with that part and not with the valuing mastery part, is realizing that this learning process, with lots of mistakes, continues forever, and that it's part of the lives of those people above you, called adults, who've mastered many more skills and thus may seem like they always know what they're doing (and who probably live in a culture that's taught them to have difficulty admitting to their mistakes).
especially since she will almost certainly go to a less progressive high school than the school she attends now, and therefore be subject to traditional tests and so on, and she will need to be able to deal with that)
Ugh. I was hoping she'd be able to stay in progressive schools or home school or something. Yeah, in that case, if it were me, I'd be emphasizing the ways in which traditional schools are the product of some uninformed beliefs and traditions that go back to at least the Middle Ages, and that we've learned some things about cognitive science in recent decades, and that life will get better again as an adult, especially if she knows what to look for and how to end up in a healthier environment. (There are plenty of jobs and bosses that go back to the Middle Ages. Knowing how to identify them and being able to externalize as "doesn't know any better" and "there are better options out there" is worlds better than buying into the system.)
And I read a lot of books and articles -- which is why sometimes you are like "growth mindset!" and I'm all "yup, got it!"
Lol! That definitely makes these conversations easier and better, that we have at least some part of our language in common. :)
I mean, I'm doing a lot of things wrong too that I don't talk about.
Yeah, this is what I mean about learning vs. mastery, and learning being something that continues as an adult! And for something like parenting, it's non-deterministic, unlike calculating dosages, so you never fully master it, it's always a learning process. And the emotional regulation skills needed for non-deterministic, especially people-oriented, learning are the same as but even harder than for deterministic skills, which is why E needs to keep developing those skills around her deterministic learning, because wow is she going to need them for her non-deterministic learning!
of course I'm working on it too (not always successfully, but trying) but :( )
And this is why you need more than 2 tries, and it's good that real life gives you that. ;)
(And yeah, patients die, because medicine is non-determnistic and thus always a learning process*, and sometimes life *doesn't* give you more than 1 or 2 tries. Bodies are not wifi extenders, alas.)
* Although many professionals have confused their mastery of a bunch of individual skills with mastery of medicine, and society's tendency to send "as an adult there's definitely a premium to be able to do things correctly" messages does not help with this.
there's something about my matter-of-fact/intense approach to certain things that codes to E like I'm yelling at her
AAAHHH, my wife has something very like this! Anything that isn't super positive (and many things that are) codes to her as like I'm yelling at her. So any time I'm even mildly confused or disappointed or concentrating on something else or anything that comes out with even slight negative tones in my voice codes as "You're doing it wrong!"
And, I mean, I *know* I have tone of voice struggles, but I can use the exact same tone of voice at work to go, "Is there a reason the username was changed?" as I use at home to go "Is there a reason the chair is on the porch?", and I can get a totally routine "No, that was an accident, please put it back/Yes, it's because of X" answer at work, while at home I get a huge "OH NO she's mad because the chair is on the porch!!" reaction, and I'm like, "I just want to know if I should put it back or leave it!"
It's because she grew up in an environment where people wouldn't tell you what they were thinking and there were huge consequences for not figuring out if they were mad at you, so she learned to read minds and that it was safer to assume the worst. Whereas I grew up in a home where I couldn't *imagine* not knowing what my mother was thinking; there was a constant running commentary on her state of mind from minute to minute! (My wife's face when I said this was like my mother was from Mars.)
I actually speculated to my wife recently that maybe one of the reasons I was so delayed in learning to read body language and facial expressions was because I never needed to! It wouldn't have added anything. Not until I met her did I really feel like there was this whole second language that I needed to know, and that it might contradict the words that were coming out of her mouth. And that was when I learned to read faces. (Or at least one face; it remains to be seen if it translates, since I don't interact with anyone else face-to-face.)
I suspect E has one thing that's similar to my wife: not the upbringing, but the neuroendocrinology. The fact that her amygdala is pumping out glucocorticoids at a truly prodigious rate, and that this causes her to put an exaggeratedly fearful interpretation on neutral or slightly negative stimuli. Some people are simply born with higher levels of anxiety, and E seems to be one of them*. (My stepdaughter is an example of what we call "marinating in glucocorticoids": her amygdala will cause her to put a negative spin on pretty much any stimulus she encounters, hence the near-paralysis.)
* It's often comorbid with autism. After meeting my stepdaughter, I realized that the word for what one of my autistic brothers has is agoraphobia.