Why Not Just Learn from a Textbook, MIT OpenCourseWare, Khan Academy, etc.?

I learned from those kinds of resources myself, and while I came a long way, for the amount of effort I put into learning, I could have gone a lot further if my time were used more efficiently. That's the problem that Math Academy solves.

Want to get notified about new posts? Join the mailing list and follow on X/Twitter.

I self-studied a bunch of math subjects on MIT OpenCourseWare (OCW) when I was in high school.

OCW is a good resource and I came a long way with it, but for the amount of effort that I put into learning on OCW, I could have gone a lot further if my time were used more efficiently.

Just to name a handful of inefficiencies in OCW:

• not super scaffolded $\to$ you periodically run into situations where you bang your head on a wall thinking "how the heck did they get from here to there?" and it takes a long time to figure out what kind of logical leap is happening (if you figure it out at all)
• doesn't track your knowledge / make sure you've mastered the prerequisites for anything new you're supposed to learn $\to$ you often feel a large gap between your level of knowledge and the new material, which leads to more banging your head on a wall trying to figure out what prerequisite knowledge you're missing and how to learn it
• no spaced review $\to$ you quickly get rusty on a lot of what you learn, which not only means you come out of the course having forgotten a lot of content, but even during the course, you're constantly forgetting prerequisites
• doesn't adapt to your level of performance $\to$ you waste a lot of your time doing the wrong amount of work. Sometimes you grasp a topic quickly and end up doing way more practice problems than you need; other times you struggle with a topic and don't do enough practice problems to reach mastery
• leaves the definition of "mastery" open to interpretation by the learner $\to$ as a learner, it's hard to know when you've mastered something well enough to continue moving forward. You often think you've learned something well enough, when you actually haven't -- but you won't know unless there's an expert who is evaluating your knowledge. On the flipside, you can also take things too far being a perfectionist, spinning your wheels on the same topic for a week over some minor point that doesn't make perfect intuitive sense to you, when it would be more productive to just keep moving forward and solidify your understanding by building on top of it.
• not enough successful problem-solving experiences $\to$ in a typical college course, you might solve 50-100 homework problems all semester. There's so much educational friction that it takes you 20+ minutes to struggle through each problem, often not even getting it right until you give up and look at a solution or get help at a TA session. But if the problems were broken down and presented in a more finely scaffolded sequence, and each problem only presented once you've mastered the prerequisites, then you could get through problems much faster, say, just a couple minutes per problem on average. That's what a good tutor would do: scaffold your learning experience so that you're solving a problem every couple minutes. They would build up your learning in bite-size increments. You could get through 10x as many problems that way, with a much higher success rate.
• not enough knowledge audits $\to$ most college courses have only a handful of exams throughout the entire semester (and even grade school classes seldom have more than one quiz per several weeks). But quick, frequent timed quizzes -- say, 15 minutes every couple days -- are a powerful way to engage in retrieval practice and build automaticity while simultaneously identifying weaker areas in need of additional practice.
• not enough targeted remediation $\to$ it's rare to find a resource that gives you additional practice after you miss a question on an assessment. At best, you might take it upon yourself to review the questions you missed. But it would be far better to complete a battery of additional problems like each one you missed, until you're able to successfully and consistently solve those problems -- and then you'd want to evidence that knowledge on a quiz retake (with different problems of the same types).

I could keep going with this list, but by now you probably get the point:

All of these things introduce unproductive friction into the learning process, leading you to make less educational progress per unit time/effort that you put towards learning.

That’s one reason why I’ve been so motivated to help build Math Academy. We take away as much of this learning friction as possible and maximize your learning efficiency.

That’s our main value proposition: sure, it’s possible to learn math elsewhere, but it’s way more efficient with us.

Despite it being possible to learn math elsewhere, most learners don’t actually do it because there’s so much friction in the learning process.

And that’s the real kicker: efficiency is important not only because you make faster progress, but also because you’re less likely to quit.

In practice, people get off the train and stop learning math once it begins to feel too inefficient. In anything you do, once the progress-to-work ratio gets too low, you’re going to lose interest and focus on other endeavors where your progress-to-work ratio is higher.

Efficiency keeps that progress-to-work ratio as high as possible, keeping you on the math learning train as long as possible.

So far, this post has talked about remedying sources of friction in the learning process.

But I also want to point out that these remedies can be seriously exploited to increase learning speed beyond the status quo.

For instance, it’s a problem when classes don’t review previously-learned material. Students constantly forget things to the point of continually having to re-learn them almost from scratch, which introduces lots of friction into the learning process.

You can reduce that friction by, well, reviewing previously-learned material. Any teacher worth their salt knows that.

BUT there is still plenty more room to improve!

Review is better than no review… but what’s BEST is to optimize the review process so that

1. you are reviewing only what you absolutely need to, and
2. you are selecting learning tasks that minimize the amount of time you have to spend reviewing, to knock out all the review you need to do.

Ideally, you want to spend as much time as possible learning new material while simultaneously getting practice on the things you’ve previously learned.

As I’ve detailed here, Math Academy achieves this ideal by leveraging the fact that advanced mathematical topics often “encompass” simpler ones. The idea is that we are often able to have students knock out review by learning something new instead.

For instance, if a student learned how to solve $ax=b$ equations yesterday, and they’re due for a review today… let’s just learn the new topic $ax+b=c$ equations instead!

Solving $ax+b=c$ “encompasses” solving ax=b as a component skill, so it provides the review that’s needed – all while the student is learning something new.

(And whenever we can’t “knock out” all a student’s due reviews with new material, we can still compress them into a much smaller set of review tasks. Instead of having to review 10 topics, you might just have to review 2 topics that collectively encompass all those 10.)

There are numerous other instances where you can take a remedy, lean into it further, and turn it into an exploit. For instance, instructional scaffolding: some is better than none, but more is better!

In all the sources of friction I described above – and many others which I didn’t mention – the remedy can be seriously exploited, turning what was once a massive slowdown into a massive speedup.

Now, I just want to end by emphasizing that I think it’s great that there are free resources available.

Different students have different needs; some are willing to invest in higher learning efficiency while others prefer a free resource even if there’s more friction in the learning process. And that’s perfectly okay!

It’s kind of like how, in the fitness industry, there’s a variety of options. Some people want personal training, other people just want full gym access, other people just want a couple pieces of minimalist equipment that they can use to work out from their home.

But like I said, while (e.g.) OCW is a great resource, and I came a long way with it, and it was totally life-changing…

for the amount of effort that I put into learning on OCW, I could have gone a lot further if my time were used more efficiently.

Want to get notified about new posts? Join the mailing list and follow on X/Twitter.