This is a long overdue update on my math learning project. The short of it is that I’ve mostly been using Math Academy to fill in gaps in my background. I’ve also been following Terence Tao’s book Analysis I. This post is about Math Academy. I’ll plan to write about what led me to Tao’s book in a future post.
Math Academy is a great and quite comprehensive platform for everything from 4th grade math all the way to advanced undergraduate topics like Linear Algebra, Discrete Mathematics, Multivariable Calculus, and Differential Equations (with even more advanced topics, such as Abstract Algebra, to come). The courses don’t hold back, either; having completed both Linear Algebra and Multivariable Calculus, I can testify that both are more comprehensive than any of my university courses in those subjects.
Math Academy is especially good at identifying and remediating weak points. It does this first by having you take a placement exam and then adapting as you get problems right or wrong. I started with its Mathematical Foundations I, II, and III, which are a streamlined version of the standard algebra, precalculus, and calculus curricula. I was a bit rusty, but (as one might expect based on my background) I tested out of most of the material, but it was still great to review. After finishing the sequence, I was well-prepared for Linear Algebra, which as noted was quite comprehensive (covering such advanced topics as the singular value decomposition, QR factorization, and Jordan canonical form).
I enjoyed Math Academy’s material enough that I decided to go back and fill in all the gaps, starting with Algebra II. I tested out of nearly all of that course, though it did identify some weaknesses in conic sections. Precalculus was similar, identifying some gaps in my understanding of hyperbolic functions. I then plowed through its excellent Calculus I and Calculus II courses, before returning to the main sequence and going through Multivariable Calculus. Unusually, this course requires Linear Algebra as a prerequisite, which pays great dividends when covering topics such as series expansions for functions of more than one variable. As of this writing, I’m doing the Discrete Mathematics course, which is old hat for computer science majors but includes a lot of material (including number theory, graph theory, and algorithms) that I’ve had little exposure to as a physicist.
My gripes with Math Academy are few. It’s brilliant both at covering new material and at reviewing old material, but learners will have to find the “magic” of mathematics in other sources; Math Academy works best when you are already motivated to learn some mathematics. It also follows standard curricula quite closely, which means that, to the extent that the standard material isn’t ideal, neither is Math Academy. But of course this is hardly Math Academy’s fault. Math Academy is also strongest at applied mathematics, with some limited introduction to proof-writing but with no ability (as of this writing) to evaluate student-written proofs. Finally, a recent change in the UI removed the multiple-choice questions from the sections when accessed through the table of contents (reserving them for the main dashboard interface), making it harder for learners to remediate their weaknesses directly.
Altogether, I highly recommend Math Academy both for high-school students and for adult learners. It’s especially good for anyone who needs to fill in some gaps before moving on to more advanced topics. Since this happens in the privacy of your own computer, students are spared the embarrassment of having to cover material that might seem “too easy” for their grade level. And for adult learners it means they can get up to speed even if it’s been some years since they last looked at the corresponding subjects.