A balanced review of Math Academy
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newsletter.ozwrites.com
(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.
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...
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.
But the exercises in the practice books and BA online aren't as good as the MA exercises. And there's no spaced repetition.
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.
In general it's very very difficult to impart conceptual knowledge to someone else because of the nature of human brains. We don't understand how we understand, so to speak, so we can't directly explain how to understand something to someone else.
We teach through examples because that is the best we can do. You're exposing someone to any idea over and over again until it clicks, but that click is what's important, not the examples. If you could somehow perfectly model how a student's brain works and know the exact combination of words to say and models to draw that would make it instantly click for them, then that's all that would be needed, no procedural mastery. But we don't know how to do that.
That's because that's not how it works. From brains to LLMs [1], training creates a dense network of pathways that encode information and procedures. I lack many of the pathways to speak Swahili, and no mythical combination of words will change that. If I want to learn Swahili, I'll have to deliberately train myself to do so, building up those pathways incrementally. Then comes the click.
You are comparing a subject near one of the spectrum (learning a language) to one that is probably as far as you can get to the other end of the spectrum (learning mathematics).
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.
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.
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 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.
https://archive.org/details/differentialequa0000simm/page/n6...
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 :-/
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 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.
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.
https://www.justinmath.com/why-is-the-edtech-industry-so-dam...
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.