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Comment by PandaRider | Hacker News Reader

> From this perspective, fields that require deep understanding, like math, require memory just as fields with a breadth of shallow knowledge do, though in different ways.

I'm interested in understanding how others use Anki for conceptual subjects like pure math or physics. I believe many fundamental rules in Spaced Repetition (e.g. like keeping cards concise) are thrown out the window for conceptual subjects.

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Yeah most of the advance assumes you have the data ready at hand and just need to phrase the cards right, get the number of words right. Whereas for conceptual domains the biggest problem is: how do I encode this as question-answer pairs at all? What I want to read more of is people sitting down and writing in the first-person perspective how they go about it, like Michael Nielsen does here: https://cognitivemedium.com/srs-mathematics

I wrote a bit more about this problem here: https://borretti.me/article/the-applicability-of-spaced-repe...

hey fernando, I read your article a lot and it's helped me a lot in my own spaced repetition so thanks from me!

a note on your request, have you seen this video before? Andy has some custom PDF reader he built with flashcards built-in, and it's two hours of tacit flashcard creation centered around quantum mechanics: https://www.youtube.com/watch?v=OFuu4pesKf0

Perhaps "Using spaced repetition systems to see through a piece of mathematics " [1] might be of interest for you. I have read author's "Augmenting Long-term Memory" [2] and have incorporated a lot of his advice into my Anki practice.

For me, it's quick access recipes (breakfast pancakes for kids), what was the name of the glacier that we hiked to last year, behavioral prompts etc.

1: https://cognitivemedium.com/srs-mathematics

2: https://augmentingcognition.com/ltm.html

Great blog posts. Exactly on point.

I took my first real analysis course last semester, and I made flashcards with pen and paper for every single non trivial definition, theorem, lemma, and corollary that we covered in lecture.

Analysis definitions and theorems get really complicated with intricate and difficult to follow logical chains, and there are a lot to remember.

These definitions and results don’t mean much on their own without exploring their neighbourhoods by proving relevant things, and I could have learned these definitions and results by just doing proofs. But being absolutely sure I could recite every theorem and definition definitely helped me on the final exam.

I think if you’re learning algorithms (like find the area under a curve) in a calculus course for example, flashcards might have more limited value, as in that case problems are relatively short and you’re better off just running through your set of algorithms a ton of times by doing problems.

I also took a group theory course last semester and I memorized every definition and result from lecture via flashcard, but didn’t practice using them enough by writing proofs. I ended up with like 2 or 3 out of 10 complete proofs and the rest half finished on the final exam because I had the right starting points, but not enough practice using what I knew in unexpected ways. Still passed somehow.

> I took my first real analysis course last semester,

> These definitions and results don’t mean much on their own without exploring their neighbourhoods

Were these epsilon-neighborhoods?

I couldn’t resist :)

> I ended up with like 2 or 3 out of 10 complete proofs and the rest half finished on the final exam because I had the right starting points, but not enough practice using what I knew in unexpected ways.

I feel ya. I memorised definitions for my algorithms course as well and also experienced diminishing rewards of using SRS flashcards (especially when the conceptual questions get more novel).

Like what you say, we have to practice using the factual knowledge enough by writing proofs.

The next generation of flashcards would probably use AI to generate concise questions on writing proofs.

[1] A Little Randomness May Not Be Enough - https://www.scotthyoung.com/blog/2014/11/07/srs-for-concepts...

Thank you for sharing your experiences!

I am a fan of Anki for lots of things but it always puzzled me people trying to learn analysis and group theory by memorising lots and lots of parts of their lecture notes.

I would test myself on the main steps of some of the proofs before exams but that was the nearest I got to memorisation. It felt like if you needed to memorise definitions then you hadn't used them enough. Even 30 years later, not having done maths for most of that time, I could still tell you the most of the definitions and could still do an epsilon-delta proof - they feel like conceptual things rather than memory things.

Perhaps I was in a way doing spaced repetition just by trying to solve lots of problems and looking things up if I didn't know them but it didn't feel like a memorisation process, more a process of trying to really really understand the mathematics.

Depends on how you use the flashcards. You can use them to memorize definitions and equalities, and you can also use them as quiz questions which excercise your reason and not simply your memory. For example, you make a flashcard for each excercise question in your textbook. Once you identify what you're struggling with, make more flashcards of that same problem type to avoid remembering the solutions. This will take you from a shaky understanding to much firmer ground pretty quickly.

Honestly just making the flashcards and elaborating on/modifying problems you're struggling with will take you a very long way.

It is a bit more challenging, but it is workable.

Same caveat as in the article: Spaced repetition is just one (minor) part of learning math/physics. It alone won't get you anywhere.

For math - particularly higher level math, the most obvious use case is definitions. There are so many!

You can put theorems in there, but it is a bit challenging on how to phrase it. A single theorem could result in several smaller flash cards.

I think what works better is taking a theorem, finding a representative problem that is solved via that theorem, and make the problem statement the question. The downside of this approach is each card takes longer to process as this is not just plain recall, but actively solving a problem. For this reason, I keep such cards in a separate deck and review them only when I have time I can dedicate (e.g. spending well over a minute per card).

I actually tend to keep my cards super concise. I treat Anki as a way to practice fundamentals, like memorizing certain formulas. Anytime I try to add conceptual stuff to cards I feel like I'm only memorizing one specialized version of the thing and it doesn't feel super useful.

I kinda feel like using memorization techniques for things that require deeper understanding probably isn't the most efficient way to learn.

IMO you want to be actively trying to map the new concepts to things you already understand, and constantly working to update your mental model.

I think of it like drills in a sport. If your practice is 100% drills, you'll be pretty bad. But drills give you an awesome foundation to do the really complex stuff intuitively.

> IMO you want to be actively trying to map the new concepts to things you already understand, and constantly working to update your mental model.

It's not an either-or.

Where SRS comes in handy is when you have to take long breaks between your study sessions (due to job + family). Have you ever tried learning an advanced math topic where you get to work on it for a few days, then may have to stop for a few weeks (or even months), then resume, and repeat over and over?

Chances are, no matter how intense you study during those few days, you'll likely forget important definitions/theorems in the periods you don't.

SRS takes care of those gaps.

Case in point - many years ago I put a lot of my intro to statistics course in flashcards and actively reviewed them. I hadn't done actual statistics for over a year, and then made a (false) claim here on HN. Someone gave me a counterexample using the chi-squared distribution. And it was amazing that I could recall the basic properties of the chi-squared distribution, and enough other theorems to verify what he said without consulting any book.

I've never used the chi-squared distribution for anything before or after.

(Sadly, I stopped using those cards years ago so I've forgotten the material!)

Familiarity through repetition is 95% of learning any subject. Mathematics isn't special.

You can take advantage of spaced repetition just for scheduling the review of proofs and problems you've solved before.

By review, I mean attempting to solve them like you're seeing the problem statement for the first time.

Just put every definition, theorem, and exercise in a textbook into a card in an anki deck.

There are no "rules" for how flashcards should work.

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