The Future of Proof-Based Courses on Math Academy
And why we refer to ourselves as still being "in beta."
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There is sometimes confusion about people thinking Math Academy is stopping at its current level of depth/difficulty, when in reality, we are still building out the curriculum. It is nowhere near finished.
Yes, currently, our only proof-based course out is Methods of Proof. Does it cover epsilon-delta proofs? Yes. Does it cover papa Rudin? No, because that’s well out of scope. Does that mean we’re stopping short of papa Rudin? No, it just means we haven’t built up to that yet.
Another example: we have a computation-based Linear Algebra course that will be a prerequisite for a proof-based “Advanced Linear Algebra” or “Abstract Linear Algebra” course later down the road. That second Linear Algebra course will go deeper into the theory and proofs that one might encounter while taking a linear algebra course at an elite university that uses, say, Axler’s book.
Unfortunately, people sometimes move the goalposts and say “your [first] Linear Algebra course is not as intense as Axler,” when this isn’t even an apples-to-apples comparison. That’s like pointing at a high school calculus class and complaining that it’s not as intense as a real analysis course – of course it’s not! It’s a completely different course; in fact, a prerequisite course; and it’s not meant to cover the same material.
Axler is really a second course in linear algebra, even if some universities throw students into it as their first course (which ends up causing a lot of unnecessary struggle). I often joke that Axler’s book “Linear Algebra Done Right” should really be called “Linear Algebra Done a Second Time.”
(Some concrete examples: Proving the projection formula goes beyond the scope of a first computational course in linear algebra. That would go in a second proof-based course, which is on our roadmap. It’s kind of like how you learn a bunch of derivative rules in calculus but real analysis is where you actually learn to prove most of them. Or like how you learn the rational roots theorem in precalculus and prove it in abstract algebra.)
All of these proof-based courses are on our roadmap, but we’re building our curriculum from the ground up, scaffolding everything to the max, and it’s mastery-based (students are only asked to learn things after having mastered the prerequisites) – so, naturally, we are going to be doing computation-based versions of courses before proof-based versions. The proof-based courses are ultimately just different courses, and we are getting the prerequisite courses in place to build up to them.
Why We're In Beta
The situation right now is we have this unfinished system that is useful enough that a lot of people are getting value out of it.
People are ripping it out of our hands, and we’re like “it’s not finished!!!”
Many people’s reactions are “I don’t care, it solves my problems in its current state, I want to use it!”
Other people’s reactions are “this could be better.”
… to which all we can say is “OMG, we know, we know! Shit’s hard! We’re working on it! Try to understand our vision and give us the necessary time to achieve it!”
This is one of the reasons we refer to ourselves as being “in beta.”
And it’s not just more courses that we need to roll out, but also a ton of additional core software functionality (e.g., behavior coaching, streaks, notifications, goals, user groups, progress reports, tons of other stuff to help students stay motivated and consistent, custom courses, flashcard-style math facts automaticity training, an “I already know this” button on lessons where we ask a couple questions from the last knowledge point and the student can get credit if they solve both correctly, …).
And it’s also our grand plan to augment the learning experience to include interesting stories about the history of math, connections to other topics, notes about cool applications, projects actually carrying out some cool applications, etc. (of course without cluttering things up for students who want to keep it as streamlined as possible).
We’ve just been focusing on core functionality to start, but our long-term vision is for this system to eventually include every single dimension that you would expect from an amazing human tutor, not only structured / scaffolded / adaptive / efficient pedagogy but also behavior / motivation coaching and interesting tidbits that add more “flavor” to the learning experience.
That’s why the interface is still rough around the edges in places – we are still focusing on getting core functionality in place, and there is a LOT of core functionality that remains to be built. The list could go on for pages, it’s a long road and we’re well aware of that. That’s why we’re in beta.
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