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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 re

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
  - edanm 8y ago
    No, no chaining. I just do the equivalent of having two cards. They would be: Q1: "Prove that condition A iff condition B [Prove forward direction]" A1: "Assume A. Then [...] implies condition B." Q2: "Prove that condition A iff condition B [Prove backward direction]" A2: "Assume B. Then [...] implies condition A." Just that instead of generating this manually, I have a note type that allows me to put in the question once, then add the two answers, and it generates these two cards. In reality it's a bit more complicated, because I can have more than 2 proofs, and can give each a title. E.g. sometimes you have theorems of the form "All of the following conditions are equivalent: 1) 2) 3) 4)." And then the proofs can be something like "1 implies 2, 2 implies 1. 3,4 imply 2, 1 implies 3" or some complicated thing like that. But these are pretty rare, it's usually just at most a 2 part backwards and forwards proof.
    - Kagerjay 8y ago
      can you make a screenshot sample of one note generating two cards? I know how this is setup on anki, but I'm curious to see the actual math content you write there. at least for the common 2part backwards and forward proof, not the really obscure case ones. Are you essentially solving the same question in two different ways? (forward proof, and backward proof) Or are they two unique questions each asking different things? (in the same related family of topics) I'm working on improving my ability to write math proofs atm so I would find this helpful

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
- edanm 8y ago
  No, no chaining. I just do the equivalent of having two cards. They would be: Q1: "Prove that condition A iff condition B [Prove forward direction]" A1: "Assume A. Then [...] implies condition B." Q2: "Prove that condition A iff condition B [Prove backward direction]" A2: "Assume B. Then [...] implies condition A." Just that instead of generating this manually, I have a note type that allows me to put in the question once, then add the two answers, and it generates these two cards. In reality it's a bit more complicated, because I can have more than 2 proofs, and can give each a title. E.g. sometimes you have theorems of the form "All of the following conditions are equivalent: 1) 2) 3) 4)." And then the proofs can be something like "1 implies 2, 2 implies 1. 3,4 imply 2, 1 implies 3" or some complicated thing like that. But these are pretty rare, it's usually just at most a 2 part backwards and forwards proof.
  - Kagerjay 8y ago
    can you make a screenshot sample of one note generating two cards? I know how this is setup on anki, but I'm curious to see the actual math content you write there. at least for the common 2part backwards and forward proof, not the really obscure case ones. Are you essentially solving the same question in two different ways? (forward proof, and backward proof) Or are they two unique questions each asking different things? (in the same related family of topics) I'm working on improving my ability to write math proofs atm so I would find this helpful