EDIT: I now consider these example cards to be too long, although somewhat functional (pun intended). I recommend using cloze deletions instead, with small cards that are a couple of sentences long. The overlapping cloze add-on is also useful for remembering proofs in order, or sequential definitions which are hard to break into several cards.
The idea to use Anki for math is something I’ve considered but never figured out a good way to do because the natural fact unit of advanced math is more complicated than a the definition of a natural language word. I would be happy to hear details more about how you did this.
I didn’t really have trial and error when I started doing this in November, I just dove in and it worked. I had, after all, experience making well over ten thousand Anki cards. The key is to only make new cards composed of concepts which you already understand. I’ll try to unpack how I make math cards, though, by making a new card and introspecting on what my instincts are.
Let’s say I’m trying to make a card for “what is a mathematical function?” (this isn’t “advanced math”, but I find that my card-entry approach remains effective as the material gets harder). I’m first going to show the card I made for this just now.
In writing the flip-side, I notice the following mental events, in order:
I want to motivate why I cared about this concept in the first place. I want to think of an intuitive situation representing a mathematical function.
Different types of cards may not need motivations, but I generally like to motivate definitions. Why on earth did I even put in the effort to click ‘new card’ to enshrine this concept? What new things does this definition let me compactly discuss?
I want to find a picture illustrating a function—some quantity being “transformed” into some other quantity.
If I’m a computer scientist, I notice that I want to clarify that mathematical functions don’t have dynamics—they’re just input-output pairs. See my first textbook review ever:
”Functions f:X→Y are juststatic sets of ordered pairs {(x,f(x)):x∈X} . They are not dynamic indexing functions, they do not perform efficient lookup, please do not waste an hour of your life trying to figure out how you could do such a thing within the simple machinery afforded by set theory up to that point.”
I want to write the formal definition.
I want to check that I’ve done a good job with the card—the card contains a new insight but explains it using concepts, theorems, and formalisms which I already understand.
In this case, if I didn’t understand sets, I wouldn’t have used sets. Instead, I’d just say “to each value x in its domain X, a fassociates to xa single value in its range Y. For example, f(x):=2x associates the number 2x to each real number x in its domain.”
Once I’m done with the definition card, I now want to add in a few more example problems. I would look up “exercises to understand mathematical functions” and put the best problem / answer pairs in.
After that, I want to make sure I didn’t make any obvious conceptual mistakes. I’m comfortable that I didn’t in this situation, but that’s because the real me already understands functions; I generally don’t already deeply understand concepts for which I make cards. I might do something like search “most common misconceptions which students have about mathematical functions.” (I often don’t do this step, but it can be a good idea)
Also, I have a “conflicting cards” deck for cards which are either too complicated for now (I need to learn what a ring is before learning what a ring ideal is) or too similar to another concept I’m presently learning (don’t try to learn accuracy vs recall vs specificity vs F1 score all at once). I’ll add those cards to the main deck later, after I’ve fully learned the current set of concepts.
More advanced math
Here’s a card I have that’s a little longer and less atomic:
The point to this card is that if I want to be able to walk through the concepts in the right order, with the understanding that I could expand the concepts into full formal detail if needed (since I understand the subconcepts of e.g. the Lebesgue measure, simple functions, measurable sets, etc). I have the prereqs, and I just need to remember how to put them together.
Then I just add philosophical concepts and clarifications at the end, in case I get confused later / want to just add notes.
I’m really surprised at how big your cards are! When I did anki regularly, I remember getting a big ugh-feeling from cards much smaller than yours, just because there were so many things that I had to consciously recapitulate. It was also fairly common that I missed some little detail and had to choose between starting the whole card over from scratch (which is a big time sink since the card takes so much time at every repeat) or accept that I might never remember that detail.
I’m super curious about your experience of e.g. encountering the function question. Do you try to generate both an example and a formalism, or just the formalism? Do you consciously recite a definition in words, or check some feeling of remembering what the definition is, or mumble something in your mind about how a function is a set of ordered pairs? Is the domain/range-definitions just there as a reminder when you read it, or do you aim to remember them every time? Do you reset or accept if you forget to mention a detail?
I’m really surprised at how big your cards are! When I did anki regularly, I remember getting a big ugh-feeling from cards much smaller than yours, just because there were so many things that I had to consciously recapitulate. It was also fairly common that I missed some little detail and had to choose between starting the whole card over from scratch (which is a big time sink since the card takes so much time at every repeat) or accept that I might never remember that detail.
First, the Lebesgue integration card is one of my is one of my largest; I selected it to show Isusr how I might handle a non-decomposable question. I recommend that new Anki users be careful not to do this kind of thing: find a way to decompose, whenever possible!
I generally tolerate missing details that I would have worked out were I actually expanding the mental sketch into formal writing; for example, if I make an arithmetical error but perform the correct kinds of calculations, I count that. If I make a sign error that I wouldn’t make if I had been writing it out pencil and paper, I’ll usually count it. If I think momentum is a scalar quantity or something crazy, though, I fail that card.
I want to penalize the kind of error that I’m unlikely to fix with unaided reflection, or instinctive errors (like my gut sense for a concept just being wrong).
I guess I use Anki to recall the “felt sense” of a proof sketch, and I don’t sweat the details. Here’s another long card I have:
Believe it or not, I can decompose this proof into a sequence of ~five “urges”, each of which I can (but generally don’t) unpack into actual math if pressed.
That said, this card was a PITA to learn, even with my level of experience. It took a lot of ’fail’s to get right consistently. Again: if one is thinking about using Anki for the first time, do not do this kind of thing! It’s hard to pull off at first, knowing if your mistake was worth redoing; if you make yourself recite the whole card above, you’ll be reviewing it forever.
I’m super curious about your experience of e.g. encountering the function question. Do you try to generate both an example and a formalism, or just the formalism? Do you consciously recite a definition in words, or check some feeling of remembering what the definition is, or mumble something in your mind about how a function is a set of ordered pairs? Is the domain/range-definitions just there as a reminder when you read it, or do you aim to remember them every time? Do you reset or accept if you forget to mention a detail?
Imagining that card, I first imagine the motivation before superimposing the formal definition “on top” of it to capture the relevant part of the situation. I remember the phenomenon of apple exchange, and then overlay the function to describe what happens. Then I recite the formal definition.
I don’t make myself remember to label the “domain” or “range”, because that’s not fair game—it wasn’t on the prompt. Those should go on separate cards. (I’d only learn one of those at a time, for similarity reasons)
Interesting. I have used multiple different approaches with Anki. Last year I had to learn all pre-university math to enter uni, and I decided to use Anki which worked out pretty well. For a lot of the problems it’s about remembering what steps to take in general. So I put in practice problems so I could remember the steps. The numbers don’t matter, they could’ve been anything.
I also tend to put in definitions, rules, theorems, standard and harder derivatives and primitives.
Now I’m in uni and I used Anki for learning linear algebra, I mostly tended towards clozes.
For linear algebra I tended to put in all theorems and facts, tried to keep them short so they’re easier to remember and then do complementary exercises handed out by the course. Being an autodidact myself, I must say that if the course is done right it can make learning a lot easier. Sadly many of the courses weren’t as good as the linear algebra course.
The definition I gave is the standard one, but that’s an interesting question. I’m not sure I know enough measure theory to find a hole in this right away; presumably there is one, because this would be a much more succinct definition.
EDIT: TheMajor pointed me to this SE question; looks like this can work if you can prove that the aforementioned set is measurable in the product sigma-algebra. I like this definition a lot more.
This area definition is equivalent to the standard definition, although this was (to me) not immediately obvious.
Some statements (linearity of integrals, for example) are obvious from the one definition, while others (the Monotone Convergence Theorem) are obvious from the other definition. Unfortunately, proving that the two definitions are equivalent is pretty much the proof for these statements (assuming the other definition).
The general approach of “given a claim, test it on indicator functions, then simple functions, then all integrable positive functions, then all integrable functions, then (if desired) integrable complex functions” is called the standard machine of measure theory, so there is educational benefit to seeing it.
I haven’t thought it through, but I suspect the theorems you want to prove about integrals are probably easier to prove with the definition TurnTrout gave than with that one.
What process do you use to review cards? Do you look at a prompt until you can say exactly what is on the card? Or if not verbatim what tolerance do you have for missing details/small mistakes?
Not verbatim. I mark the card ‘correct’ if I conjured the concept in the right way. If I was supposed to do a physics calculation, I don’t pass it if I could recite the answer, I pass it if I went through the correct reasoning steps. “Calculate the gravitational force of a .1kg apple on a 70kg person, when they’re 1m apart” → ”.1 * 70 * G / (1^2) = ?” is marked correct.
How do you find doing problems/exercises from these textbooks when you have prepared using Anki? And are you finding that earlier material seems obvious when reread?
Sorry if this is all coming across as critical and /or doubtful. I’ve tried to use Anki for theory before and dismally failed; the success you claim is very exciting and I’m trying to understand where I was going wrong.
So far I think I have focused too much on creating cards that can be memorised exactly (formulae and what-not), rather than having general concept cards that are used to develop fluency and familiarity (and later understanding), which sounds like what you are doing.
EDIT: I now consider these example cards to be too long, although somewhat functional (pun intended). I recommend using cloze deletions instead, with small cards that are a couple of sentences long. The overlapping cloze add-on is also useful for remembering proofs in order, or sequential definitions which are hard to break into several cards.
I didn’t really have trial and error when I started doing this in November, I just dove in and it worked. I had, after all, experience making well over ten thousand Anki cards. The key is to only make new cards composed of concepts which you already understand. I’ll try to unpack how I make math cards, though, by making a new card and introspecting on what my instincts are.
Let’s say I’m trying to make a card for “what is a mathematical function?” (this isn’t “advanced math”, but I find that my card-entry approach remains effective as the material gets harder). I’m first going to show the card I made for this just now.
In writing the flip-side, I notice the following mental events, in order:
I want to motivate why I cared about this concept in the first place. I want to think of an intuitive situation representing a mathematical function.
Different types of cards may not need motivations, but I generally like to motivate definitions. Why on earth did I even put in the effort to click ‘new card’ to enshrine this concept? What new things does this definition let me compactly discuss?
I want to find a picture illustrating a function—some quantity being “transformed” into some other quantity.
If I’m a computer scientist, I notice that I want to clarify that mathematical functions don’t have dynamics—they’re just input-output pairs. See my first textbook review ever:
”Functions f:X→Y are just static sets of ordered pairs {(x,f(x)):x∈X} . They are not dynamic indexing functions, they do not perform efficient lookup, please do not waste an hour of your life trying to figure out how you could do such a thing within the simple machinery afforded by set theory up to that point.”
I want to write the formal definition.
I want to check that I’ve done a good job with the card—the card contains a new insight but explains it using concepts, theorems, and formalisms which I already understand.
In this case, if I didn’t understand sets, I wouldn’t have used sets. Instead, I’d just say “to each value x in its domain X, a f associates to x a single value in its range Y. For example, f(x):=2x associates the number 2x to each real number x in its domain.”
Once I’m done with the definition card, I now want to add in a few more example problems. I would look up “exercises to understand mathematical functions” and put the best problem / answer pairs in.
After that, I want to make sure I didn’t make any obvious conceptual mistakes. I’m comfortable that I didn’t in this situation, but that’s because the real me already understands functions; I generally don’t already deeply understand concepts for which I make cards. I might do something like search “most common misconceptions which students have about mathematical functions.” (I often don’t do this step, but it can be a good idea)
Also, I have a “conflicting cards” deck for cards which are either too complicated for now (I need to learn what a ring is before learning what a ring ideal is) or too similar to another concept I’m presently learning (don’t try to learn accuracy vs recall vs specificity vs F1 score all at once). I’ll add those cards to the main deck later, after I’ve fully learned the current set of concepts.
More advanced math
Here’s a card I have that’s a little longer and less atomic:
The point to this card is that if I want to be able to walk through the concepts in the right order, with the understanding that I could expand the concepts into full formal detail if needed (since I understand the subconcepts of e.g. the Lebesgue measure, simple functions, measurable sets, etc). I have the prereqs, and I just need to remember how to put them together.
Then I just add philosophical concepts and clarifications at the end, in case I get confused later / want to just add notes.
(Here’s a recent version of my Anki/math/physics deck, so feel free to look through it)
I’m really surprised at how big your cards are! When I did anki regularly, I remember getting a big ugh-feeling from cards much smaller than yours, just because there were so many things that I had to consciously recapitulate. It was also fairly common that I missed some little detail and had to choose between starting the whole card over from scratch (which is a big time sink since the card takes so much time at every repeat) or accept that I might never remember that detail.
I’m super curious about your experience of e.g. encountering the function question. Do you try to generate both an example and a formalism, or just the formalism? Do you consciously recite a definition in words, or check some feeling of remembering what the definition is, or mumble something in your mind about how a function is a set of ordered pairs? Is the domain/range-definitions just there as a reminder when you read it, or do you aim to remember them every time? Do you reset or accept if you forget to mention a detail?
First, the Lebesgue integration card is one of my is one of my largest; I selected it to show Isusr how I might handle a non-decomposable question. I recommend that new Anki users be careful not to do this kind of thing: find a way to decompose, whenever possible!
I generally tolerate missing details that I would have worked out were I actually expanding the mental sketch into formal writing; for example, if I make an arithmetical error but perform the correct kinds of calculations, I count that. If I make a sign error that I wouldn’t make if I had been writing it out pencil and paper, I’ll usually count it. If I think momentum is a scalar quantity or something crazy, though, I fail that card.
I want to penalize the kind of error that I’m unlikely to fix with unaided reflection, or instinctive errors (like my gut sense for a concept just being wrong).
I guess I use Anki to recall the “felt sense” of a proof sketch, and I don’t sweat the details. Here’s another long card I have:
Believe it or not, I can decompose this proof into a sequence of ~five “urges”, each of which I can (but generally don’t) unpack into actual math if pressed.
That said, this card was a PITA to learn, even with my level of experience. It took a lot of ’fail’s to get right consistently. Again: if one is thinking about using Anki for the first time, do not do this kind of thing! It’s hard to pull off at first, knowing if your mistake was worth redoing; if you make yourself recite the whole card above, you’ll be reviewing it forever.
Imagining that card, I first imagine the motivation before superimposing the formal definition “on top” of it to capture the relevant part of the situation. I remember the phenomenon of apple exchange, and then overlay the function to describe what happens. Then I recite the formal definition.
I don’t make myself remember to label the “domain” or “range”, because that’s not fair game—it wasn’t on the prompt. Those should go on separate cards. (I’d only learn one of those at a time, for similarity reasons)
Interesting. I have used multiple different approaches with Anki. Last year I had to learn all pre-university math to enter uni, and I decided to use Anki which worked out pretty well. For a lot of the problems it’s about remembering what steps to take in general. So I put in practice problems so I could remember the steps. The numbers don’t matter, they could’ve been anything.
I also tend to put in definitions, rules, theorems, standard and harder derivatives and primitives.
Now I’m in uni and I used Anki for learning linear algebra, I mostly tended towards clozes.
For linear algebra I tended to put in all theorems and facts, tried to keep them short so they’re easier to remember and then do complementary exercises handed out by the course. Being an autodidact myself, I must say that if the course is done right it can make learning a lot easier. Sadly many of the courses weren’t as good as the linear algebra course.
I don’t have much to contribute to your comment except to acknowledge that it’s useful, quality information. An upvote didn’t feel sufficient.
(Why not simply define the integral of f as the LM of {(x,r)|x in Omega, r ⇐ f(x)}?)
The definition I gave is the standard one, but that’s an interesting question. I’m not sure I know enough measure theory to find a hole in this right away; presumably there is one, because this would be a much more succinct definition.
EDIT: TheMajor pointed me to this SE question; looks like this can work if you can prove that the aforementioned set is measurable in the product sigma-algebra. I like this definition a lot more.
Mathoverflow has discussion on it. In short:
This area definition is equivalent to the standard definition, although this was (to me) not immediately obvious.
Some statements (linearity of integrals, for example) are obvious from the one definition, while others (the Monotone Convergence Theorem) are obvious from the other definition. Unfortunately, proving that the two definitions are equivalent is pretty much the proof for these statements (assuming the other definition).
The general approach of “given a claim, test it on indicator functions, then simple functions, then all integrable positive functions, then all integrable functions, then (if desired) integrable complex functions” is called the standard machine of measure theory, so there is educational benefit to seeing it.
I haven’t thought it through, but I suspect the theorems you want to prove about integrals are probably easier to prove with the definition TurnTrout gave than with that one.
What process do you use to review cards? Do you look at a prompt until you can say exactly what is on the card? Or if not verbatim what tolerance do you have for missing details/small mistakes?
Not verbatim. I mark the card ‘correct’ if I conjured the concept in the right way. If I was supposed to do a physics calculation, I don’t pass it if I could recite the answer, I pass it if I went through the correct reasoning steps. “Calculate the gravitational force of a .1kg apple on a 70kg person, when they’re 1m apart” → ”.1 * 70 * G / (1^2) = ?” is marked correct.
How do you find doing problems/exercises from these textbooks when you have prepared using Anki? And are you finding that earlier material seems obvious when reread?
Sorry if this is all coming across as critical and /or doubtful. I’ve tried to use Anki for theory before and dismally failed; the success you claim is very exciting and I’m trying to understand where I was going wrong.
So far I think I have focused too much on creating cards that can be memorised exactly (formulae and what-not), rather than having general concept cards that are used to develop fluency and familiarity (and later understanding), which sounds like what you are doing.
I think I do find Anki’d subjects much easier when I go back to them, yes.
I think that’s probably it—focus on concepts. If you like, I’d be happy to take a look at a deck you make / otherwise give feedback!