> if I have to learn hundreds of pages of math, there is no other way than creating my own lecture notes, basically

Well, that is what I actually did with Anki in university. For any lecture, I would go over the material, identify the important snippets, and make them into Anki cards. Creating the deck itself is very productive work because to identify the cards and make them small enough, you have to have understood the material. If you just copy the entire page-long proof in, you haven't understood it. Anki forces you to break it down because a card like that is useless.

So if you're already making your own notes why would you want flashcard software?

- It optimizes reviewing, ensuring you look at forgotten cards every day, and easy ones only once a year, instead of randomly poring over the entire set of notes and hoping to stumble upon the forgotten ones.

- It allows higher card quality, e.g. I can paste screenshots of visualizations instead of scribbling them down. My language decks have audio pronounciation, a paper medium does not support that.

- It allows higher card volume, I can't fit 10000 cards in a notebook but I can do so on my phone. Especially not high-quality cards with images in them. Of course nobody even considers fitting 10k cards on paper because reviewing such a collection is not feasible without spaced repetition software (see two paragraphs above).

- Its much easier to pull out my phone and do some reviews in the car or public transit than to do the same with a notebook.

So going far afield, I teach karate and use this all the time though I was unaware that there were typical time constants across the population. I may start using that to teach better.

In karate we teach some forms we call kata. When I teach these, I like to cover the details slowly at the beginning, break for other stuff, and then come back to it at the end to test their memory because it's long enough that short-term memory will start to degrade and they'll have to actually work at remembering it (and thus identify the parts they didn't really quite get).

It's very obvious to watch in realtime. It's a waste of time to just work on teaching the kata for 2 hours straight because they get exhausted and lose focus. It's just as effective to teach it for a half hour at the beginning as long as I have a 10 minute review at the end to solidify what they learned. I usually think of it as three separate practices -- warming up the form, practicing the form, and executing the form.

Warming up means going at half speed with the intent being to refresh the memory. Practicing means going at 70% of full speed to help get it into active memory more solidly. Executing means going at 80% of full speed and focusing on checking muscle memory (video works great) under conditions of some mental stress, and also actually developing the techniques and theory further.

To get back to your point, I see no reason why you couldn't make your card require all three types of reviewing something. But I am not sure if the time constants on forgetting practiced skills is different than memorized facts. In karate your deeply practiced skills are in muscle memory (or for you perhaps in your most accessible memory with the math details you don't even realize you're using which are not known to most people). Having stuff in muscle memory is the bare minimum required to actually move further.

You can't learn karate from reading a bunch of articles and watching karate videos. You have to actually practice it to reinforce the memory and build new connections between different concepts. Does that sound familiar to how you think about learning math?

This sounds like a good strategy for mastering and memorizing a pattern of behavior, which is the demand in most grade school and college math courses. It's nothing like what higher mathematics demands, which is much less about rote memorization and much more about deep understanding of the intricate interconnections between a comparatively small number of concepts.

Put another way: math isn't really about memorization, either of behavior patterns or of facts. That's why I've been confused and somewhat skeptical about the utility of flash cards for learning higher mathematics.

> Put another way: math isn't really about memorization, either of behavior patterns or of facts.

I mean, on the surface level that's true. But obviously we could reductio ad absurdum this claim - would a mathematician be able to work if they lost all their memories every day? Clearly not!

It's something of a matter of how good is your memory, plus how actively you use it. How many people who complete PHDs can still prove the PHD 10 years later? Not many. Which is not to say that you can or should use Anki to learn something as complicated as a PHD dissertation, that clearly won't work. But when learning a new topic, it's a very helpful tool, and it makes sure that even if you now focus only on linear algebra, you'll still recall at least the basics of, say, set theory, which most mathematicians who don't actively study probably don't use much.

> It's nothing like what higher mathematics demands, which is much less about rote memorization and much more about deep understanding of the intricate interconnections between a comparatively small number of concepts.

Well Anki certainly won't help you actually do higher math :) But I find it is surprisingly good at making these weird connections, because in the middle of learning say modern algebra, you'll suddenly need to recall things from linear algebra, and suddenly see interesting new connections. Or things from real analysis, which will make you go "hmmm, so that's why a field is defined this way" or something.

Well, I'm not trying to pitch it or anything. I haven't tried it either for this. I do have a PhD in Materials Science, but it sounds like your experience is different than mine. For me, my experience is like this:

Step 1: Read about stuff people are doing.

Step 2: Read between the lines to understand how it fits into other things I already know about.

Step 3: Evaluate based on my intuition whether the fit is reasonable. Since the data is presumed real, if the fit is bad it usually means I didn't understand the details of what they did. See if I can make the fit with other things I know coherent.

Step 4: If I can explain the fit coherently, consider what problems might be solved by using that connection.

Step 5: Research how that problem is normally solved and why.

Step 6: Go back to Step 1 and keep going until I find something that is actually solved better using my weird idea than it is solved currently.

I am terrible at memorizing, and if I am in Step 3 or Step 4 or Step 5 it's a real roadblock if I'm trying to understand why I can't harmonize the reported data and my understanding and it's because I've forgotten which sites are interstitials in a fluorite lattice. I know the information I need, but now I have to go look it up. Of course this is why people have reference materials, but it's definitely a speed bump.

The important aspects of the work up there clearly aren't about memorization, but it sure helps me actually do it in practice. I really wish I were better at it.

Another thing that this made me think of was using it to remember student/coworker names. Remembering my unreasonably numerous cousins' kids' names would also be nice.

What a weird coincidence, since I actually am doing karate too... And I totally get what you're saying. There is a tremendous amount of knowledge hidden in each kata (in addition to having to remember the kata itself, and there are 26 of them) and without repeating how you describe it there is just now way to make progress in shorter time intervals. But I would not have thought to make the parallel between math and karate, I always viewed them as requiring distinct learning approaches.

That's great! The standard practice advice I give my students (and use myself) is that it is best to practice each of your kata twice each day. The first time you go from Heian Shodan on up, at half speed, just to get it into memory. Then you go in reverse order at normal speed so that you don't get bad muscle memory from always doing it in the same order.

Of course, that's half for exercise instead of just memorization. And obviously it's less necessary from a memory perspective pretty quickly, I can go at least many months without doing a kata and still rapidly pull it back into working memory from a video review (during those times in my life when I couldn't practice regularly) and a few times through myself.

But if it's not in active memory I definitely won't be improving it. Just reinforcing the memory and getting a bit of exercise.

You can read my other comments of how I use Anki to learn math, especially about how I put in proofs and stuff.

But specifically: > [...] I'm curios, if you cram a piece of knowledge that is so interconnected as math is in flashcards and you rote learn all the definitions, theorems, proofs and examples - are you then capable of seeing the connections between them?

Well, I wouldn't think of using Anki as rote learning things. A basic principle is that you must understand (at least in something like math) before you put it in Anki. I wouldn't make a card without first understanding what it says. In fact, the very fact of making a card will often make me understand the proof much better, just by trying to write a more minimal and easy flowing version of the proof, or trying to put in a mnemonic or visualization that will make the proof clearer. (Btw, even after doing this process, I still find that I often can't recall the proof easily even 4 days later! Such is my memory, at least. I can sometimes reprove, but not always.)

Hopefully, if the exam asks to prove a new lemma, having lots of examples in your memory of how to prove things or how to solve exercises in e.g. linear algebra, will make it much easier to do it.

Look, I'm no scientist, but the science here is pretty solid - active recall helps a lot. It's essentially what lots of students are doing when studying for tests, only (scientifically shown to be) more efficient - solving example problems/proofs, etc. And it has the benefit of being inside a system that will hopefully keep the knowledge alive even in 10 years, not just for the exam and that's it.

> are you then capable of seeing the connections between them?

This is probably the best hidden feature of Anki. Because I often will get basic cards surfacing a few months after learning them, I will often find connections that I didn't think about (or couldn't know) the first time! I've even had some cases where I will recall a definition or theorem that I wrote from the beginning of the study, and realize that actually, I got part of it wrong! And now that I know a lot more, I easily see "wait, that can't be right" and dig deeper to discover what was my misunderstanding!