Anki for Math
Metric/Topological/Banach/Hilbert/Complete/Linear/Normal/Probability/Sub-Spaced Repetition Software
I am not the first to write about using Anki for Math, nor am I the best. I recommend quantum physicist Michael Nielsen’s introduction to Anki and using Anki to understand a piece of math. Justin Skycak has made optimal teaching of math and CS with spaced repetition his career, and he's shredded to boot. On the other hand, I am likely the closest person to Adin Pepper Fox to opine on this subject. Adin Pepper Fox, I hope this piece inspires you to employ spaced repetition to great effect.
A note of warning: I include examples of cards from my own decks. That means they are mostly from courses which typically come after multivariable calculus and linear algebra. Following the meaning of the math hopefully won’t be necessary to get the spirit of the notes.
A penultimate note of warning: This is exactly what it sounds like. It doesn’t have much humor, there are no cute anecdotes, and if I manage to trace the shape of the human condition through my contemplation of memory and the mundane then I certainly did not notice. It’s just notes. Notes about using Anki for math.
A final note of warning: This is not about using spaced repetition to learn math. I’ve never used spaced repetition to learn math. Instead, I’ve learned a piece of math with pencil and paper and textbook and all that. Then, I’ve used Anki to forget less of it.
Why I use Anki (as opposed to nothing, not as opposed to some other software)
I am an undergraduate student studying math. About a year ago, at the start of my Junior year, I began using Anki for math classes. For that reason I remember much more about complex analysis and differentiable surfaces than I do about cryptography and multivariable calculus. That’s a shame, because now I will never be able to encrypt all the data I’ve been storing in the divergence of my vector field. On the other hand, the contrast was evidence that this spaced repetition stuff works. I don’t remember all the proofs from my post-Anki classes. I don’t even remember everything that I tried to store in my cards. Still, I remember a whole lot more than I would otherwise.
Types of cards I make
What type of thing?
When I see a map Phi in the context of Brad’s Big Theorem, it is way easier to comprehend if I instantly know that Phi is a function from the integers to the reals, or an operator that acts on simplexes, or whatever else. Cards for this that I use looks like:
where I fill in the range and domain
Often I know about some property and have a fuzzy mental model of the vibes behind it. It is super helpful to have instant recall of what exactly the property applies to.
I have several cards like the above, ensuring I instantly know what some small gear means when I encounter it in a larger piece of machinery.
Examples and Counter Examples
Knowing canonical examples of an object, examples that are perverse in some way, and snappy counter examples for plausible false statements is great. It helps me build intuitions while avoiding false assumptions. This kind of card is especially useful for propositions where the converse isn’t true, but both directions feel plausible (A—>B but not B—>A). Often, these cards don’t tell me that I will necessarily be providing an example on the front, since often counterexamples are used to answer broader questions like “is an everywhere continuous function necessarily differentiable?” Cards that I use for this look like:
Q: Prove or give a counter example that all solvable groups are abelian:
Q: Is a map with a locally isothermal chart always conformal?
A: Yes
Q: Does a conformal map always have a locally isothermal chart?
A: No
It would probably be good to add a counter example to go along with the second question as well.
Proofs
I find it difficult to memorize whole proofs. I have not memorized many whole proofs through Anki. Largely, I’ve shied away from this type of card due to the effort required. I still firmly believe they can be quite effective, which is what the aforementioned Nielsen post is mostly about. I sometimes add proofs if they are very short, or break them down into small components as best I can, or include a hint in the card. Some examples:
Q: “If for some power series we have
then the radius of convergence is 1/L. Give a heuristic of the proof.
A: The limit test yields |z-z_0|/L, so when |z-z_0|<1/L we have that the limit test is less than 1 and the function converges
Q: What is the only thing we need to show to prove that DN_p is self adjoint?
A: It is sufficient to prove that DN_p is self adjoint on a basis of the tangent space
Intuition Cards
The rare and scintillating intuition card is something of a unicorn in my math decks. I am unsure if it is helpful but am leaning towards yes. Sometimes I earn a new intuition through exercises or play, or come across someone else’s clever way of conceptualizing a piece of math-y content. “When I think of it like that, it’s obvious! I will never forget this ever again.” But then I do. So, I try making cards like:
Q: Give an intuitive explanation of a ruled surface
A: A surface that can be made by the ‘sweeping of a line.’
Q: What are two ways to think about the Augmented Homology?
A: Homology with a copy of Z removed from the zeroth homology, or homology such that the point’s homology is entirely trivial.
Short Answer Section
Short answer cards are similar to multiple choice questions on a speedy quiz. Math, famously, requires cognitive work, but many questions are almost-instantaneous-level-easy if you understand the key info. Make sure these “trivial” exercises are very, very trivial. If you make a deck full of cards that are easy after just a little thought, then a whole day’s review will require tons of thought, and that’s nem jó. Still, these quiz style cards are fun and ensure you can apply facts you’ve learned. Often, I use multiple quiz questions for one idea: the combined effect of cards “is 42 even,” “is 67 even,” and “is “-1 even” is a pretty good proxy for knowing what even means. Of course, I could just memorize the specific answers, and if I notice myself doing this I reassess those cards. Here are some examples of short answer cards in my deck:
A: Yes
If I have a good memory of what the graph of x|x| looks like, then this is obvious. If I’ve lost that mental model, it’s a tossup.
Q: How many edges are in the 3 uniform complete graph on 4 vertices?
A: 4
A: ω
It was hard to find examples of these cards. That’s not surprising, given that definitions and proofs are provided in textbooks and short form questions to quiz my understanding aren’t. Still, I love ‘em when I see ‘em so I plan to start making more.
Open Questions
How often should I memorize proofs?
The rule of thumb for Anki is usually “if its worth 5-10 minutes of your time to know it forever.” With a proof, however, it requires many cards, and I might have to modify them later as I realize there are small errors, and maybe I want to change the structure of the proof because—this has become more than 10 minutes. So given a proof X, what function f(X) gives how much time it is worth to me to memorize the proof?
Should I spaced repeat time consuming exercises?
If I can’t solve a problem about some material now, and I could when I learned it, then I haven’t maintained my understanding. Still, this seems like a quite strenuous practice that should be separate from general Anki memorization. So, what’s the best option? I could build a deck of exercises that I study separately when I’m extremely locked in, or let my “exercise ability” atrophy and hope I it comes back quickly when the material became relevant, or wait for Math Academy to expand their catalogue soon to do all my work for me.
When does information become “useless?”
Math is known to be shockingly interconnected. I have taken classes that have been irrelevant for a semester, and then suddenly helpful in a subject I thought I would be unrelated. Being a broad mathematician is both fun and helpful. Still, these math decks can be quite heavy, a real time commitment. If I haven’t used my graph theory knowledge for 3 years, should I still be reinforcing graph theory? I don’t know how to decide this cutoff.
What are the best scheduled practices to playfully explore math while maintaining accountability?
At the end of the day, the spirit of math is not memorization. I can use Anki to stabilize my recall, the framework of knowledge that I stand on. That’s good, but I don’t just want to stand on the framework; I want to jump about, do somersaults, hit my head and hang upside down. One might work through textbooks/course notes, do puzzle style problems, or pause and ponder.
Smash that like button and leave a comment if you have any thoughts to share on using SRS for math.
Kinda fun how your math development goes from memorization to not and back again for you! Love it Benji!
You might not be the first but by God you are the best!