I like that this review is written by someone who gave Math Academy a serious try. He mentions comments by a couple of other math educators (Michael Pershan and Dan Meyer), but I haven't seen any evidence that either of them has used Math Academy for enough time to evaluate it.
(At the time I recommended my son start doing Math Academy, I had done 3722 XP myself, which is about 60 hours' worth.)
It's true that there's a stronger emphasis on procedural fluency than on conceptual understanding. But honestly I think that's good: there's so much good material online for conceptual understanding.
Imagine you're studying linear algebra: use Math Academy for rigorous introductions to topics and exercises and feedback, and watch 3Blue1Brown videos for conceptual understanding.
For younger kids (my son is 8.5yo) I wish Math Academy had some animated or video explanations. For example, the prealgebra course includes the product rule for surds. It was hard for my son to grok based on reading the written explanation. Not because the explanation was bad, but because my son has the attention span of an 8.5yo. So I spent a few minutes walking through that same explanation with him, using a paper and pencil to guide his attention.
I also wish they had a mascot and a streak feature. Those features on Duolingo somehow help to motivate my son.
This tracks with my experience. I had to start from the very basics, but now I'm able to watch 3B1B and really understand what he's talking about. During lessons I occasionally reach to ChatGPT to help explain a concept or (rarely) a Khan Academy video.
I agree that the other articles mentioned didn't seem to give much of a chance. The Perhan thread has a comment chain between the author and the MA director [1] which is easy to miss. It's notable to me when users complain of negative XP, I've been working on MA every day for six months and got my first negative XP today! I've gotten 0 XP plenty, but if someone's getting negative XP often, they're half-assing it.
> I also wish they had a mascot and a streak feature.
I believe a streak calendar is coming based on the developers' Twitter. I dislike streaks but I imagine they will be optional as leagues are. I edited it out but my previous comment on this thread said "the day they get a cute mascot is the day I quit forever!" Maybe that was too harsh ;) But I do like the serious tone of the site as an adult learner.
One of the things mentioned in that comment thread is that, even with MA as it is today, some 4th graders can still benefit from it.
I can attest to this.
My son completed the 4th and 5th grade courses. They took about a year in total (5 months plus 7 months IIRC). During that time, I spent maybe 30 mins per week giving him some additional explanation. (At school he was studying Singapore Math 3A and 3B, which don't overlap much with these courses.)
If MA didn't exist, I would instead have my son speed run Beast Academy 4 and 5, to make sure he has no gaps, and then I would then need to tutor him 1:1 using the AoPS Prealgebra book, which is good, but a bit thick and intimidating for a child to self-study.
Honestly I’d do the BA. Later he’ll have memories of silly characters, funny pictures with hidden cultural references, and bad jokes, not just grinding XP. We still refer to Grogg and Captain Kraken.
Procedural fluency is the basis for conceptual fluency. Maths is very layered: every layer builds on the next. You can't do algebra if your arithmetic is slow, as every algebra problem has half a dozen arithmetical subproblems. And you can't do calculus if your algebra is slow, as every calculus problem has half a dozen algebraix subproblems. And every differential equations problem requires you do to a bunch of calculus.
You can't understand a topic without doing lots of examples, so that you can feel out what is incident to the problem and what is inherent to the class of problems: when do we do integration by parts? When do we do integration by substitution? Etc.
So as a result you need to build procedural mastery at every layer before moving to the next and being able to build that conceptual understanding.
This review reads very much like one written by someone who reluctantly is trying to find excuses to criticize it. Many of the things he says feel made up.
For example,
> Many of the questions I answered correctly were in areas that I don’t feel I understand well, but where I suppose I had drilled enough in the past for my procedural fluency to survive 20 years of atrophy.
The author is complaining that he felt like he should have done worse because emotionally he thought he knew it less than he actually did. This is an entirely fictional, made-up criticism. He obviously knew it better than he "felt" he did.
The entire review reads like this, trying to make up real-sounding criticisms that to me just don't make very much sense. The DAG criticism as well doesn't make sense. There is no "one way". The DAG approach probably works for many if not most students, but of course there are other students it won't work for. Exactly like how phonics works for many students but some need to be taught whole word reading. You can't criticize the DAG approach just because some students can't learn that way, but they have to pick one method. I would expect a teacher to have a better critcism than that.
He made the initial diagnostic test sound like it has a massive impact on what you'll study. It doesn't. Even if MA thinks you know a topic well based on the diagnostic test, that doesn't mean you won't encounter it again. He seems to have missed the fact that every time you answer a question, this affects the estimate that MA has about your comfort with one or more topics.
The excessive focus on procedural fluency is a reasonable criticism but, as I said in another comment, this doesn't bother me at all. There are plenty of other sources for this. And procedural fluency is a good enabler for developing conceptual understanding.
I've been using Math Academy for a couple of months now seriously (~3.5k XP). Initially I was reviewing topics / concepts that I already had an intuitive understanding of from college courses, and I believe for those it has helped take me to the next level in that I now can, 99% of the time, solve problems in those topics. As I start to approach topics that I haven't fully covered in college or have some intuitive understanding for I do find that I can mechanically carry out the operations, but I lack an understanding of "why" I'm doing this.
I think the author of this makes some good points to this effect. I believe that the best approach is a bit inverse of what the author is saying however. I think that Math Academy should be the base, where 80-90% of your time is spent. The remaining 10-20% should be used to supplement this. Either with lectures, textbook / edutainment reading or other ways to develop more of that intuition.
The Internet Archive lost a court case against the publishers, so many books (including all books that are still available for sale) have been (or are being) soft-removed :-/
as someone who's been using mathacademy for a few months now, I'm pretty happy with what it provides. Yes it focuses on "procedural fluency" and thats not a bad thing.
if learning mathematics was like body building, traditional classes would be the equivalent of going to a traditional classroom where you get a lecture on the various pathways to build muscle strength. its on you to spend your off ours going into the gym and doing the work. Progress might be slow becase your feedback on your "problem sets" comes back a few days later.
Mathacademy on the other hand is like having a trainer at the gym. every day they take a look at you and tell you what exercises to do and to set the difficulty. You might not know exactly how all the pieces fit together but thats ok. you're getting stronger. you are seeing progress in your ability to just power through problems.
I personally have found my algebra skills to have improved substantially. I remember factoring to be a struggle for me when I was in high school. after a few weeks of mathacademy its pretty easy now. Does it mean I have a great conceptual framework for solving real world problems? probably not but I'm in a much better position to learn that skill as a result of mathacademy.
BTW math academy guys, if you're reading this. Id LOVE a class on circuit analysis!
I had a longer comment but it was too all over the place. For me, ultimately, I think Math Academy has really helped me. That being said, I put a lot of time and effort into it, and I imagine it would be difficult for people with a lot of commitments. I sometimes reach for supplementary materials, sometimes get ChatGPT to re-explain something, but mostly I manage.
I think the article gives the impression that MA is like Duolingo, but it's not. There's no flashy animations, sound effects, or mascots, the interface is spartan like Hacker News. It can't be (effectively) done in five minutes a day. It requires total concentration and a lot of time commitment.
The most significant difference from Duolingo is that people get results with it. I’ve known literally hundreds of people who have put real time into Duolingo, over 10 hours a week for years in a few cases and zero of them have actually learned a language to a B2 level or higher through that massive time commitment. The best case I’ve seen is from people who accumulated a back of memorized word translations and grammar patterns through Duolingo and then spent time reading and listening to content that included them. Even those people learned slower than those who just focused on input from the beginning, though.
In contrast, I don’t know anyone who’s put 10 hours a week into Math Academy for multiple years but those I know who have done it for 6 months have made considerable progress, sometimes the equivalent of years of course work.
Duolingo’s flashy animations and casino-style gamification techniques are actually a negative signal for an educational product.
I agree totally, that was my point but perhaps it wasn't clear. I can't stand Duolingo or any flashy education tech, as soon as there's a sound effect and bubbly animations I'm out. I learned nothing from Duolingo (I ended up learning the language through other means) but I've learned a ton in my (coincidental) 6 months of Math Academy. MA's Justin Skycak put it best:
I would love to find a Mathacademy style language-learning app. Fluent Forever is good, but an app that drills mercilessly without any flash would be perfect.
> But should you learn integration before differentiation? I’d never considered this approach until I read David M. Bressoud’s fantastic book Calculus Reordered,
Ha, I learned calculus integration first from Ivan Niven's "CALCULUS An Introductory Approach," copyright 1961, that as a kid I found in the attic in my dads undergrad engineering books. When I got to high school and took calc, I was able to do all the stuff they were teaching, but in weird orders and methods compared to more modern books.
Now, 40+ years later, I still have that book on my bookshelf in the calculus section :)
Till now, it's it only book I've seen that teaches it this way.
> So why do some learning resources designed for autodidacts—such as Math Academy, or the generally very good Execute Program—rely so heavily on dependency graphs?
I went to a university where the upper-division undergraduate math courses didn't have much in the way of prerequisites.
A foreseeable consequence of this approach is that the first several weeks of each ten-week class are spent covering material that's shared with other classes, because that material is relevant to both classes but isn't included in the lower-division prerequisites. This is very bad as a matter of curriculum design, but good if you're more interested in making sure students never have scheduling conflicts.
Another example of dependency graphs is that when my sister signed up for Portuguese classes, she took "Portuguese 1 for Spanish speakers" rather than "Portuguese 1". You can learn Portuguese from an English perspective, or you can learn it from a Spanish perspective, and those are both approaches that can work, but they're not approaches you'd want to combine. In this case, one of the approaches is clearly superior - if you can relate Portuguese to your knowledge of Spanish, that will work better than relating it to English - but even where no particular dependency structure is preferable to another, it remains true that the plan for going from A to B isn't the same as the plan for going from B to A, or the same as the plan for going from nothing to B. So the curriculum needs to rely on a dependency tree.
I like that this review is written by someone who gave Math Academy a serious try. He mentions comments by a couple of other math educators (Michael Pershan and Dan Meyer), but I haven't seen any evidence that either of them has used Math Academy for enough time to evaluate it.
(At the time I recommended my son start doing Math Academy, I had done 3722 XP myself, which is about 60 hours' worth.)
It's true that there's a stronger emphasis on procedural fluency than on conceptual understanding. But honestly I think that's good: there's so much good material online for conceptual understanding.
Imagine you're studying linear algebra: use Math Academy for rigorous introductions to topics and exercises and feedback, and watch 3Blue1Brown videos for conceptual understanding.
For younger kids (my son is 8.5yo) I wish Math Academy had some animated or video explanations. For example, the prealgebra course includes the product rule for surds. It was hard for my son to grok based on reading the written explanation. Not because the explanation was bad, but because my son has the attention span of an 8.5yo. So I spent a few minutes walking through that same explanation with him, using a paper and pencil to guide his attention.
I also wish they had a mascot and a streak feature. Those features on Duolingo somehow help to motivate my son.
This tracks with my experience. I had to start from the very basics, but now I'm able to watch 3B1B and really understand what he's talking about. During lessons I occasionally reach to ChatGPT to help explain a concept or (rarely) a Khan Academy video.
I agree that the other articles mentioned didn't seem to give much of a chance. The Perhan thread has a comment chain between the author and the MA director [1] which is easy to miss. It's notable to me when users complain of negative XP, I've been working on MA every day for six months and got my first negative XP today! I've gotten 0 XP plenty, but if someone's getting negative XP often, they're half-assing it.
> I also wish they had a mascot and a streak feature.
I believe a streak calendar is coming based on the developers' Twitter. I dislike streaks but I imagine they will be optional as leagues are. I edited it out but my previous comment on this thread said "the day they get a cute mascot is the day I quit forever!" Maybe that was too harsh ;) But I do like the serious tone of the site as an adult learner.
1. https://pershmail.substack.com/p/math-academy-wants-to-super...
One of the things mentioned in that comment thread is that, even with MA as it is today, some 4th graders can still benefit from it.
I can attest to this.
My son completed the 4th and 5th grade courses. They took about a year in total (5 months plus 7 months IIRC). During that time, I spent maybe 30 mins per week giving him some additional explanation. (At school he was studying Singapore Math 3A and 3B, which don't overlap much with these courses.)
If MA didn't exist, I would instead have my son speed run Beast Academy 4 and 5, to make sure he has no gaps, and then I would then need to tutor him 1:1 using the AoPS Prealgebra book, which is good, but a bit thick and intimidating for a child to self-study.
Honestly I’d do the BA. Later he’ll have memories of silly characters, funny pictures with hidden cultural references, and bad jokes, not just grinding XP. We still refer to Grogg and Captain Kraken.
Oh yeah, we have all the BA books and he's read most of comics in the guide books. He really likes them.
But the exercises in the practice books and BA online aren't as good as the MA exercises. And there's no spaced repetition.
Valid, my younger kid forgets a lot of stuff without that kind of reinforcement.
Procedural fluency is the basis for conceptual fluency. Maths is very layered: every layer builds on the next. You can't do algebra if your arithmetic is slow, as every algebra problem has half a dozen arithmetical subproblems. And you can't do calculus if your algebra is slow, as every calculus problem has half a dozen algebraix subproblems. And every differential equations problem requires you do to a bunch of calculus.
You can't understand a topic without doing lots of examples, so that you can feel out what is incident to the problem and what is inherent to the class of problems: when do we do integration by parts? When do we do integration by substitution? Etc.
So as a result you need to build procedural mastery at every layer before moving to the next and being able to build that conceptual understanding.
This review reads very much like one written by someone who reluctantly is trying to find excuses to criticize it. Many of the things he says feel made up.
For example,
> Many of the questions I answered correctly were in areas that I don’t feel I understand well, but where I suppose I had drilled enough in the past for my procedural fluency to survive 20 years of atrophy.
The author is complaining that he felt like he should have done worse because emotionally he thought he knew it less than he actually did. This is an entirely fictional, made-up criticism. He obviously knew it better than he "felt" he did.
The entire review reads like this, trying to make up real-sounding criticisms that to me just don't make very much sense. The DAG criticism as well doesn't make sense. There is no "one way". The DAG approach probably works for many if not most students, but of course there are other students it won't work for. Exactly like how phonics works for many students but some need to be taught whole word reading. You can't criticize the DAG approach just because some students can't learn that way, but they have to pick one method. I would expect a teacher to have a better critcism than that.
He made the initial diagnostic test sound like it has a massive impact on what you'll study. It doesn't. Even if MA thinks you know a topic well based on the diagnostic test, that doesn't mean you won't encounter it again. He seems to have missed the fact that every time you answer a question, this affects the estimate that MA has about your comfort with one or more topics.
The excessive focus on procedural fluency is a reasonable criticism but, as I said in another comment, this doesn't bother me at all. There are plenty of other sources for this. And procedural fluency is a good enabler for developing conceptual understanding.
I've been using Math Academy for a couple of months now seriously (~3.5k XP). Initially I was reviewing topics / concepts that I already had an intuitive understanding of from college courses, and I believe for those it has helped take me to the next level in that I now can, 99% of the time, solve problems in those topics. As I start to approach topics that I haven't fully covered in college or have some intuitive understanding for I do find that I can mechanically carry out the operations, but I lack an understanding of "why" I'm doing this.
I think the author of this makes some good points to this effect. I believe that the best approach is a bit inverse of what the author is saying however. I think that Math Academy should be the base, where 80-90% of your time is spent. The remaining 10-20% should be used to supplement this. Either with lectures, textbook / edutainment reading or other ways to develop more of that intuition.
Author mentions Simmons' Differential Equations with Applications and Historical Notes (1972), which is at Internet Archive:
https://archive.org/details/differentialequa0000simm/page/n6...
This shows as “Borrow Unavailable” for me.
The Internet Archive lost a court case against the publishers, so many books (including all books that are still available for sale) have been (or are being) soft-removed :-/
Thank-you, you're right. Hopefully we can borrow it in the future.
as someone who's been using mathacademy for a few months now, I'm pretty happy with what it provides. Yes it focuses on "procedural fluency" and thats not a bad thing.
if learning mathematics was like body building, traditional classes would be the equivalent of going to a traditional classroom where you get a lecture on the various pathways to build muscle strength. its on you to spend your off ours going into the gym and doing the work. Progress might be slow becase your feedback on your "problem sets" comes back a few days later.
Mathacademy on the other hand is like having a trainer at the gym. every day they take a look at you and tell you what exercises to do and to set the difficulty. You might not know exactly how all the pieces fit together but thats ok. you're getting stronger. you are seeing progress in your ability to just power through problems.
I personally have found my algebra skills to have improved substantially. I remember factoring to be a struggle for me when I was in high school. after a few weeks of mathacademy its pretty easy now. Does it mean I have a great conceptual framework for solving real world problems? probably not but I'm in a much better position to learn that skill as a result of mathacademy.
BTW math academy guys, if you're reading this. Id LOVE a class on circuit analysis!
I had a longer comment but it was too all over the place. For me, ultimately, I think Math Academy has really helped me. That being said, I put a lot of time and effort into it, and I imagine it would be difficult for people with a lot of commitments. I sometimes reach for supplementary materials, sometimes get ChatGPT to re-explain something, but mostly I manage.
I think the article gives the impression that MA is like Duolingo, but it's not. There's no flashy animations, sound effects, or mascots, the interface is spartan like Hacker News. It can't be (effectively) done in five minutes a day. It requires total concentration and a lot of time commitment.
The most significant difference from Duolingo is that people get results with it. I’ve known literally hundreds of people who have put real time into Duolingo, over 10 hours a week for years in a few cases and zero of them have actually learned a language to a B2 level or higher through that massive time commitment. The best case I’ve seen is from people who accumulated a back of memorized word translations and grammar patterns through Duolingo and then spent time reading and listening to content that included them. Even those people learned slower than those who just focused on input from the beginning, though.
In contrast, I don’t know anyone who’s put 10 hours a week into Math Academy for multiple years but those I know who have done it for 6 months have made considerable progress, sometimes the equivalent of years of course work.
Duolingo’s flashy animations and casino-style gamification techniques are actually a negative signal for an educational product.
I agree totally, that was my point but perhaps it wasn't clear. I can't stand Duolingo or any flashy education tech, as soon as there's a sound effect and bubbly animations I'm out. I learned nothing from Duolingo (I ended up learning the language through other means) but I've learned a ton in my (coincidental) 6 months of Math Academy. MA's Justin Skycak put it best:
https://www.justinmath.com/why-is-the-edtech-industry-so-dam...
I would love to find a Mathacademy style language-learning app. Fluent Forever is good, but an app that drills mercilessly without any flash would be perfect.
> But should you learn integration before differentiation? I’d never considered this approach until I read David M. Bressoud’s fantastic book Calculus Reordered,
Ha, I learned calculus integration first from Ivan Niven's "CALCULUS An Introductory Approach," copyright 1961, that as a kid I found in the attic in my dads undergrad engineering books. When I got to high school and took calc, I was able to do all the stuff they were teaching, but in weird orders and methods compared to more modern books.
Now, 40+ years later, I still have that book on my bookshelf in the calculus section :)
Till now, it's it only book I've seen that teaches it this way.
Calculus by Thomas Apostol (also from the sixties) teaches integration before differentiation as well. Great textbook!
> So why do some learning resources designed for autodidacts—such as Math Academy, or the generally very good Execute Program—rely so heavily on dependency graphs?
I went to a university where the upper-division undergraduate math courses didn't have much in the way of prerequisites.
A foreseeable consequence of this approach is that the first several weeks of each ten-week class are spent covering material that's shared with other classes, because that material is relevant to both classes but isn't included in the lower-division prerequisites. This is very bad as a matter of curriculum design, but good if you're more interested in making sure students never have scheduling conflicts.
Another example of dependency graphs is that when my sister signed up for Portuguese classes, she took "Portuguese 1 for Spanish speakers" rather than "Portuguese 1". You can learn Portuguese from an English perspective, or you can learn it from a Spanish perspective, and those are both approaches that can work, but they're not approaches you'd want to combine. In this case, one of the approaches is clearly superior - if you can relate Portuguese to your knowledge of Spanish, that will work better than relating it to English - but even where no particular dependency structure is preferable to another, it remains true that the plan for going from A to B isn't the same as the plan for going from B to A, or the same as the plan for going from nothing to B. So the curriculum needs to rely on a dependency tree.