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Not the parent, but: I use Anki constantly to study Math (or really any technical stuff, but Math is what I mostly study). I would never study without it again

by edanm 8y ago

Not the parent, but: I use Anki constantly to study Math (or really any technical stuff, but Math is what I mostly study). I would never study without it again - I'm still unhappy that I did an entire 1st degree in CS before I knew about Anki, and barely remember most of it. The more I use Anki to study Math, the more cards I tend to make. I usually make a few different kinds of cards (usually as I'm studying a textbook): 1. Definitions - usually something like "What is Normal operator? [Define]", with the back being something like "A Normal operator is an operator that commutes with its adjoint". I might add a bit more detail and flair (depending on the context). Sometimes it's a more complex definition, e.g. "What is a group [Explanation, 3 axioms]", with the back being an overview plus the group axioms. 2. Basic theorems/properties - no proofs, just something like "What kind of matrices always have property X", "Orthogonal matrices". These are often hard to phrase right, since it might be far too ambiguous to remember 2 years from now. I try to use these semi-sparingly. An easier example: "What is the project of vector U on vector V", answer: "<u,v>/<v,v>v". (Again, I'm omitting details here like saying "in an inner product space", etc.) 3. Proofs - I tend to use these a lot these days. I'll often either just make a card for a proof "Prove that a compact set in a metric space is closed". I'll also often err on the side of adding a proof, instead of using a "thereom result" card like in my point 2 above. E.g. instead of asking what is the project of u on v, I actually have a card saying "what is the projection of u on v? Answer and prove". That way I can always reprove it in my head. I usually do this when the proof or explanation is short, if not, I might just split it into two cards (one asking the answer, one asking for the proof). 4. Calculations - I have cards for base calculations if the subject warrants it. E.g. "what is the result of this matrix multiplication". Or even simpler things - I realized while studying math that I forgot a lot of even trivial stuff from high school, e.g. how to factor quadratics using "complete the square" - so I have cards with a few examples of this that I need to solve. 5. I also have some basic facts that I want to memorize, e.g. "derivative of ln(x)" or stuff like that. Note that while I use Anki extensively for Math, I actually use it for lots of other stuff too.

Kagerjay 8y ago
This sounds really cool. I'm trying to use anki for more math oriented concepts. The 1,2,4,5 points you make are very small and simple, but I have a question regarding point 3: Aren't proofs going to be absurdly long? You might have a one sentence question, but it demands a 5+ line proof. That takes the longest to review the card though. I almost want to say a multicloze statement is better here, to prove individual parts of the proof. But then you get way too many cards, etc. It seems to contradict best practices of what most anki practioners emphasize. https://math.stackexchange.com/questions/2111662/prove-compact-subsets-of-metric-spaces-are-closed https://math.stackexchange.com/questions/2111662/prove-compa... "Prove that a compact set in metric space is closed" seems to be 5 or so major steps involved. I suck at proofs and find that math is a really big weakness I have. I tried using anki for mostly math, but I still haven't reached that pivotal point where everything just makes sense.
- edanm 8y ago
  Good question! The short answer is yes, proofs are long. But it's complicated. For one thing, there are a few reasons to formulate things with smaller answers - one is that there's not always a logical connection between different things, one is to know them separately, one is that if you fail part of a card, you want to only fail that part - and you can't do that, so you need it separately. The thing is, with proofs, they are logically part of a whole. They do flow logically, one part after the other, at least once you know the big idea - given the big idea, you should be able to do the rest of the steps by simply applying previous steps (usually). That said, reviews of proofs do tend to take me longer, which is not ideal. And I do let myself "cheat" a bit with proofs - if I know the main idea behind a proof, and am fairly confident I can develop the rest of it (especially if it's a calculation only), then I might consider it answered. But I am fairly strict that I really do think I could develop it. For example, the compact sets question was something I recently failed. Here's why: the main idea behind it is to prove that the complementary set is open, and the way to do that is to pick each point, define a ball around it and around each point in the compact set, then show how that proves every point has a ball around it. Now I know all those big-level steps, but when I tried to remember how to develop it further, I realized that I wasn't sure why I needed the original set to be compact - why doesn't this work with any set? So I failed the card. One other point - I do use a few custom note types for some math things. I sometimes use clozes, but I also have a note type for "Multipart Proof", which allows me to split up a proof into parts. I almost always actually use it to separate the "forward" and "backward" directions of an "if and only if" proof, since they are relatively separate anyway, but I do want to keep the card logically together.
  - Kagerjay 8y ago
    Ah okay I see where you are going with proofs. I format my anki cards similarly for longer expanded question: E.g. - Question - Short Answer (e.g. the conceptual idea and starting point for solving the proof) - Long answer (the full proof) What I tried to do instead, is reword each question slightly to ask a different part of the proof. Keep the long-answer the same, but modify the short-answer slightly. I find from my experience the anki question/answer needs to be a one-liner though. I use a yellow highlighted background for the short-answer portion. Its a basic front/back question/answer with some specialized stylesheets I made. ------------------------------------------------------------------ which allows me to split up a proof into parts. I almost always actually use it to separate the "forward" and "backward" directions of an "if and only if" proof, since they are relatively separate anyway, but I do want to keep the card logically together. Are you chaining multiple questions / answers in one card? Something similar to glutanimate's video here? https://www.youtube.com/watch?v=ag4Hdnkig8U https://www.youtube.com/watch?v=ag4Hdnkig8U
DavidSJ 8y ago
Thanks for telling me your approach! For all the examples you gave, I feel like those are the sort of thing that become so familiar in the process of working with those objects (e.g. spending long hours seeking proofs of theorems when I first studied them) that they’d be hard to forget, and I have trouble seeing how such cards would be helpful. I suppose if I just read through the textbooks and sought to memorize the definitions and proofs by rote then this would make sense, but I’ve always been told, and practiced, that the best way to learn the material is to find the proofs myself rather than just reading them straight through. And having done that, the key ideas of the proof, and especially the definitions (which became burned into my brain in the process of working with them) seem to last for months or years without explicit recall. With language learning, it feels much more to me like rote memorization of an arbitrary mapping between labels in one language and labels in the other, so I find spaced repetition much more helpful. I’m not trying to be dismissive here, just genuinely trying to understand. I would love to find a more efficient method for studying math, as it’s the main thing I do right now with my time and I expect at least several more years of it. If I’m missing a better way then I’d really like to know. Thanks very much.
- edanm 8y ago
  Look, everyone is different. But I did a CS degree about 10 years ago. I learned calculus, and even had to memorize a bunch of proofs for the course. Now I'm not saying that I don't remember anything from the course... but I definitely remember extremely little. Even some of the "base objects" that we studied. (It's not like I'm a mathematician who actively uses this stuff). > I’ve always been told, and practiced, that the best way to learn the material is to find the proofs myself rather than just reading them straight through. Let's be clear, I completely agree with this. I don't just copy into Anki - I study, and then I put into Anki. In fact, just the fact of putting something into Anki tends to make me think about it much more clearly, as I decide what needs to be in there, how to rewrite the proof to be more concise but still understandable by me, etc.
  - DavidSJ 8y ago
    > I learned calculus, and even had to memorize a bunch of proofs for the course. I'm curious about this. Did your class require students to find many of their own proofs of key results, or was it more about regurgitating proofs from the textbook?
btrettel 8y ago
Great comment. I am the parent commenter, but I don't think I could improve this answer. "Property" cards can be very useful, even if only as a reminder. I can recall a card I had (back when I used Anki regularly) which asked you to draw a diagram showing how different types of convergence are related to each other (e.g., L2 convergence implies...). Just the exercise of constructing that diagram was very educational by itself.

by edanm 8y ago

Kagerjay 8y ago

This sounds really cool. I'm trying to use anki for more math oriented concepts. The 1,2,4,5 points you make are very small and simple, but I have a question regarding point 3: Aren't proofs going to be absurdly long? You might have a one sentence question, but it demands a 5+ line proof. That takes the longest to review the card though. I almost want to say a multicloze statement is better here, to prove individual parts of the proof. But then you get way too many cards, etc. It seems to contradict best practices of what most anki practioners emphasize. https://math.stackexchange.com/questions/2111662/prove-compact-subsets-of-metric-spaces-are-closed https://math.stackexchange.com/questions/2111662/prove-compa... "Prove that a compact set in metric space is closed" seems to be 5 or so major steps involved. I suck at proofs and find that math is a really big weakness I have. I tried using anki for mostly math, but I still haven't reached that pivotal point where everything just makes sense.

edanm 8y ago
Good question! The short answer is yes, proofs are long. But it's complicated. For one thing, there are a few reasons to formulate things with smaller answers - one is that there's not always a logical connection between different things, one is to know them separately, one is that if you fail part of a card, you want to only fail that part - and you can't do that, so you need it separately. The thing is, with proofs, they are logically part of a whole. They do flow logically, one part after the other, at least once you know the big idea - given the big idea, you should be able to do the rest of the steps by simply applying previous steps (usually). That said, reviews of proofs do tend to take me longer, which is not ideal. And I do let myself "cheat" a bit with proofs - if I know the main idea behind a proof, and am fairly confident I can develop the rest of it (especially if it's a calculation only), then I might consider it answered. But I am fairly strict that I really do think I could develop it. For example, the compact sets question was something I recently failed. Here's why: the main idea behind it is to prove that the complementary set is open, and the way to do that is to pick each point, define a ball around it and around each point in the compact set, then show how that proves every point has a ball around it. Now I know all those big-level steps, but when I tried to remember how to develop it further, I realized that I wasn't sure why I needed the original set to be compact - why doesn't this work with any set? So I failed the card. One other point - I do use a few custom note types for some math things. I sometimes use clozes, but I also have a note type for "Multipart Proof", which allows me to split up a proof into parts. I almost always actually use it to separate the "forward" and "backward" directions of an "if and only if" proof, since they are relatively separate anyway, but I do want to keep the card logically together.
- Kagerjay 8y ago
  Ah okay I see where you are going with proofs. I format my anki cards similarly for longer expanded question: E.g. - Question - Short Answer (e.g. the conceptual idea and starting point for solving the proof) - Long answer (the full proof) What I tried to do instead, is reword each question slightly to ask a different part of the proof. Keep the long-answer the same, but modify the short-answer slightly. I find from my experience the anki question/answer needs to be a one-liner though. I use a yellow highlighted background for the short-answer portion. Its a basic front/back question/answer with some specialized stylesheets I made. ------------------------------------------------------------------ which allows me to split up a proof into parts. I almost always actually use it to separate the "forward" and "backward" directions of an "if and only if" proof, since they are relatively separate anyway, but I do want to keep the card logically together. Are you chaining multiple questions / answers in one card? Something similar to glutanimate's video here? https://www.youtube.com/watch?v=ag4Hdnkig8U https://www.youtube.com/watch?v=ag4Hdnkig8U

DavidSJ 8y ago

Thanks for telling me your approach! For all the examples you gave, I feel like those are the sort of thing that become so familiar in the process of working with those objects (e.g. spending long hours seeking proofs of theorems when I first studied them) that they’d be hard to forget, and I have trouble seeing how such cards would be helpful. I suppose if I just read through the textbooks and sought to memorize the definitions and proofs by rote then this would make sense, but I’ve always been told, and practiced, that the best way to learn the material is to find the proofs myself rather than just reading them straight through. And having done that, the key ideas of the proof, and especially the definitions (which became burned into my brain in the process of working with them) seem to last for months or years without explicit recall. With language learning, it feels much more to me like rote memorization of an arbitrary mapping between labels in one language and labels in the other, so I find spaced repetition much more helpful. I’m not trying to be dismissive here, just genuinely trying to understand. I would love to find a more efficient method for studying math, as it’s the main thing I do right now with my time and I expect at least several more years of it. If I’m missing a better way then I’d really like to know. Thanks very much.

edanm 8y ago
Look, everyone is different. But I did a CS degree about 10 years ago. I learned calculus, and even had to memorize a bunch of proofs for the course. Now I'm not saying that I don't remember anything from the course... but I definitely remember extremely little. Even some of the "base objects" that we studied. (It's not like I'm a mathematician who actively uses this stuff). > I’ve always been told, and practiced, that the best way to learn the material is to find the proofs myself rather than just reading them straight through. Let's be clear, I completely agree with this. I don't just copy into Anki - I study, and then I put into Anki. In fact, just the fact of putting something into Anki tends to make me think about it much more clearly, as I decide what needs to be in there, how to rewrite the proof to be more concise but still understandable by me, etc.
- DavidSJ 8y ago
  > I learned calculus, and even had to memorize a bunch of proofs for the course. I'm curious about this. Did your class require students to find many of their own proofs of key results, or was it more about regurgitating proofs from the textbook?

btrettel 8y ago

Great comment. I am the parent commenter, but I don't think I could improve this answer. "Property" cards can be very useful, even if only as a reminder. I can recall a card I had (back when I used Anki regularly) which asked you to draw a diagram showing how different types of convergence are related to each other (e.g., L2 convergence implies...). Just the exercise of constructing that diagram was very educational by itself.