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Out of curiosity, what sort of cards do you use for math? For language learning it's obvious -- words, phrases, etc. -- but for math, is it like prompts for pro

Out of curiosity, what sort of cards do you use for math? For language learning it's obvious -- words, phrases, etc. -- but for math, is it like prompts for proofs of theorems, or what?

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.

Out of curiosity, what sort of cards do you use for math? For language learning it's obvious -- words, phrases, etc. -- but for math, is it like prompts for proofs of theorems, or what?

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.