Spaced Repetition for Mathematics (2021)(cronokirby.com)
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Spaced Repetition for Mathematics (2021)
https://cronokirby.com/posts/2021/02/spaced-repetition-for-mathematics/
9 comments
You are missing a key part in your argument.
Any bits of knowledge that have solidified such that they can be retrieved effortlessly, can be used as building blocks to construct or understand higher order knowledge.
Memorization may not (initially) help with understanding the particular thing you're trying to memorise per se, but dismissing it altogether as not being generally relevant for creating understanding is wrong.
Also, knowledge is bidirectional. A higher order concept you learned because you were able to use a more basic building block to reach that understanding, may later provide the insight that then allows you to get a better, revised understanding of the lower building block too, without compromising any of its dependents.
Any bits of knowledge that have solidified such that they can be retrieved effortlessly, can be used as building blocks to construct or understand higher order knowledge.
Memorization may not (initially) help with understanding the particular thing you're trying to memorise per se, but dismissing it altogether as not being generally relevant for creating understanding is wrong.
Also, knowledge is bidirectional. A higher order concept you learned because you were able to use a more basic building block to reach that understanding, may later provide the insight that then allows you to get a better, revised understanding of the lower building block too, without compromising any of its dependents.
I agree with the claim that problem solving is essential to learning mathematics. And I have had the exact same experience where I did well on math exams by ignoring most of the theory and proofs of main results, and focusing solely on examples and problem solving.
However, I think straight-up memorizing definitions and theorem statements is really useful for problem-solving/exam prep, just so they're at your fingertips. There's no way you're passing a real analysis exam if you can't regurgitate the epsilon-delta definition of a limit in your sleep.
What seems to occur (at least for me) is that you naturally memorize all of these things in a somewhat inefficient fashion by doing problems. If the concept gets used in enough problems, it slowly burrows its way into your memory - and this is a very durable kind of memory, as you point out. But I do think for things like graduate school qualifying exams you can "juice" the process by explicitly memorizing core material.
Probably it's not as useful for doing research, though.
However, I think straight-up memorizing definitions and theorem statements is really useful for problem-solving/exam prep, just so they're at your fingertips. There's no way you're passing a real analysis exam if you can't regurgitate the epsilon-delta definition of a limit in your sleep.
What seems to occur (at least for me) is that you naturally memorize all of these things in a somewhat inefficient fashion by doing problems. If the concept gets used in enough problems, it slowly burrows its way into your memory - and this is a very durable kind of memory, as you point out. But I do think for things like graduate school qualifying exams you can "juice" the process by explicitly memorizing core material.
Probably it's not as useful for doing research, though.
I never remember explicitly learning the epsilon-delta definition. By the time I had worked through enough problems, read enough books etc I just knew it.
I saw other students learning definitions etc off by heart and to me it seemed liked they were doing it because they hadn't really understood the material and it would get some them marks. It was probably the right exam strategy if you didn't deeply understand the material to optimise your marks and get a reasonable pass mark but I don't think it was the right way to really learn mathematics or get one of the top marks.
I did learn and test myself on the structure of some of the more complicated proofs in my finals I mostly revised by doing old papers though.
(Feel I should offer some credentials but also don't want to brag, but feel I am about as qualified as one can be to talk about doing extremely well in maths exams :-))
I saw other students learning definitions etc off by heart and to me it seemed liked they were doing it because they hadn't really understood the material and it would get some them marks. It was probably the right exam strategy if you didn't deeply understand the material to optimise your marks and get a reasonable pass mark but I don't think it was the right way to really learn mathematics or get one of the top marks.
I did learn and test myself on the structure of some of the more complicated proofs in my finals I mostly revised by doing old papers though.
(Feel I should offer some credentials but also don't want to brag, but feel I am about as qualified as one can be to talk about doing extremely well in maths exams :-))
I think it's a good tool for quick access to many useful tools in a toolkit. Yes, you need to know how to apply the tools, but quickly being able to manipulate a problem into several other forms makes lots of problems much easier.
What if your spaced repetition puts the questions in the form of problems to solve?
In fairness, graph theory/combinatorics is the poster child for clever problem solving techniques as far as areas of math go.
"In order to be successful as a mathematician, you must commit to memory all definitions and statements of theorems, for these are the tools by which you can construct valid mathematical arguments. Of course, memorization is not always fun, but sometimes it is just simply required." -Steven Roman, Abstract Algebra: A Comprehensive Introduction, Volume 1: Linear Algebra
“It is a profoundly erroneous truism, repeated by all copy-books and by eminent people when they are making speeches, that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilization advances by extending the number of important operations which we can perform without thinking about them. Operations of thought are like cavalry charges in a battle — they are strictly limited in number, they require fresh horses, and must only be made at decisive moments.”
- Alfred North Whitehead
- Alfred North Whitehead
Do think you're refuting the parent?
The "mochi cards" link leads to Anki. But I think this is the one the author is talking about: https://mochi.cards/
I can wholly recommend Mochi: it’s great looking but also quickly becoming feature competitive with Anki. It’ll be a long time before anything hits the long tail of Anki features, but I find the trade offs to be more than worth it.
What are the tradeoffs?
In addition to the other comment, it's also missing a lot of card types that Anki supports. However, I think this is made up for by the knowledgebase features, so you can use it to take notes stored with / linked to by your SRS.
Mochi is only available through a subscription model (or a neutered free version).
In the 'conditions for theorems' section, I wonder whether it would be easier (and just as effective) to put the whole theorem (including conditions) in at once, and then use cloze deletion to get tested on different parts.
I've been using Readwise.io for years to revisit my Kindle highlights and I find spaced repetition helps remember high-level ideas from books I've read in the past.
A specific book I've highlighted is Aurelien Geron's Hands-On Machine Learning Book. It's a great way to revisit machine learning concepts I want to interiorize and I had been thinking of turning those into flash cards. I may try Mochi Cards for that.
I's also note that spaced repetition in any form requires great dedication. In the case of Readwise, it's somewhere from one to four minutes to read about five daily quotes.
A specific book I've highlighted is Aurelien Geron's Hands-On Machine Learning Book. It's a great way to revisit machine learning concepts I want to interiorize and I had been thinking of turning those into flash cards. I may try Mochi Cards for that.
I's also note that spaced repetition in any form requires great dedication. In the case of Readwise, it's somewhere from one to four minutes to read about five daily quotes.
I played piano competitively in college and when my teacher revealed a version of this trick for memorization it seemed like a cheat code at the time. Playing a short difficult motif over and over again is needed sometimes to get it right, and the key was to lock in the “correct” movements and not retain anything else. So the idea was after you played it once correctly you would take a moment and let the section recede from short term memory and hopefully into longer term storage and then start practicing again.
This is good. It's the first decent post I've seen about this instead of someone either giving advice so vague as to be useless, or sneeringly dismissing the value of memorization in mathematics. Actionable advice.
I tried SRS a while ago while working my way through a math textbook, but I got bored by the process. I might give it another go with the tips in here.
I tried SRS a while ago while working my way through a math textbook, but I got bored by the process. I might give it another go with the tips in here.
While I can certainly see improving recall being helpful, for me, knowing when to apply something is the powertool I need. eg. I know what commuting is, I know how to do it . . . what I really need to know is when to commute.
Can you learn how to play the piano with Spaced Repetition? No. The same applies to any number of things that we should practice doing until it becomes automatic.
SR has its uses but it is not a panacea, nor it is as interesting as rolling up our sleeves to solve the hell out of a problem and in the process, learn a ton.
SR has its uses but it is not a panacea, nor it is as interesting as rolling up our sleeves to solve the hell out of a problem and in the process, learn a ton.
> nor it is as interesting as rolling up our sleeves to solve the hell out of a problem and in the process, learn a ton.
Only to forget it after not using it for a few years.
I tried using SR when reading a textbook on FP from a mathematical standpoint (theorems/proofs). I needed the knowledge for a job, and used it occasionally in that job for less than 2 years. I left the job almost 3 years ago, but it's only because of SR that I still remember a fair amount of it.
Indeed, even during that job, I would have gaps of months at a time before I would utilize the FP math - and the nice thing about SR was that I wouldn't have to consult the book(s).
I can safely say that without SR I wouldn't recall even half of what I know now when it comes to FP.
SR vs problem sets is a false dichotomy.
Only to forget it after not using it for a few years.
I tried using SR when reading a textbook on FP from a mathematical standpoint (theorems/proofs). I needed the knowledge for a job, and used it occasionally in that job for less than 2 years. I left the job almost 3 years ago, but it's only because of SR that I still remember a fair amount of it.
Indeed, even during that job, I would have gaps of months at a time before I would utilize the FP math - and the nice thing about SR was that I wouldn't have to consult the book(s).
I can safely say that without SR I wouldn't recall even half of what I know now when it comes to FP.
SR vs problem sets is a false dichotomy.
>Can you learn how to play the piano with Spaced Repetition? No.
care to back up this claim?
care to back up this claim?
But if you want to get dexterity in some field, i.e. being able to actual solve problems, I think it would not work so well. I think there is a difference in how the brain stores and recalls information in this case. If you learn a definition by heart or learn a proof technique by heart, you would be able to recall it perfectly when asked directly for it. However, I suspect that when you would be actually solving a problem and you would need to recall this information your brain would not be able to make the connection.
That is why in mathematics one is usually required to solve a lot of problems and why studying the theory by heart does not help much. When you do a lot of textbook problems you brain starts making connections between the chucks of proof techniques and the chunks of definitions and the chunks of whatever features your textbook problems have. Those connections are the most important part of learning mathematics and you would not get them by simply learning facts.
Basically, to say it in another way, you would learn the theory, but you would not learn the problem solving associated with this theory.
Source: I studied mathematics, the wrong way for many years.
EDIT: The most bitter experience I had studying mathematics. It was the first year of my graduate studies and I took an undergrad class in Graph Theory as my introduction to discrete mathematics. Since I was now a graduate student I decided to approach the class in the graduate student way. That means I focused really hard on learning the theory and learned all the proofs and all the proof techniques that we went through. And that was very fun because the proofs in Graph Theory tend to be very elegant. And I fell in love with the field. And then came the exam and I was feeling really good about all of this, because for maybe the first time in my life I had learned 105% of the theory required for the exam. That was the worst grade I got in my whole mathematics studying career. The problems on the exam were much simpler than any example or theorem encountered during class, but I just could not make the connections between the proof techniques that I had memorized and what I was looking at on my exam paper. I retook the exam three years later (it had no influence on my grade at this point), with very little preparation, but the preparation was 100% in solving exam type problems. I could maybe recall 60% of the theory and proof techniques. I got top grade.