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: 2022-08-26 11:27 am (UTC)no subject
Date: 2022-08-27 12:55 am (UTC)(I was not given scaffolding writing assignments during my university years. We had freshman writing -- it's maybe possible that one could have found a technical writing class to fill that requirement, but it certainly wasn't what most people did -- and then everyone just assumed you knew how to write, both for humanities and for science.)
no subject
Date: 2022-08-30 11:05 am (UTC)no subject
Date: 2022-08-27 02:33 pm (UTC)In my math classes, on homework assignments, we were not expected to write sentences. In fact, I had one prof who, if you wrote words in the English language rather than just numbers and symbols, would cross them out with his red pen. I don't think he deducted points? But we quickly got the message that words had no place in math assignments.
I had one (1) prof who was shocked and horrified that we did not write our homework assignments as though for a mathematical journal, and we were shocked and horrified that we were being penalized for not doing what we had never been told to do. It was even worse because a number of us had had "no English words, ever" prof the semester before! We protested! That was an interesting day in class.
The worst part, of course, was that said "write like this is a journal" prof never told us about LaTeX, and I didn't want to write out all those words by hand in my terrible handwriting, so I was stuck trying to do linear algebra in Word, with no instruction in using software for math, and it was SO PAINFUL. Matrices! With manual spacing! (Ugh, I'm having
war flashbacksfaint memories of adding the multi-line braces on the printout afterwards. *facepalm*)Only in grad school did I learn about LaTeX, FROM A FELLOW STUDENT.
But with the exception of "write like this is a journal" and "no English words ever" profs at the two extremes, our homework and exams were done by sticking connector words like "thus" and "because x is even" into the middle of a series of numbers and symbols. Definitely nothing you'd submit to a journal or that would prepare you for a thesis.
no subject
Date: 2022-08-30 11:10 am (UTC)no subject
Date: 2022-08-30 11:47 am (UTC)no subject
Date: 2022-09-02 04:38 am (UTC)