Using spaced repetition systems to see through a piece of mathematics
cognitivemedium.com
cognitivemedium.com
(...)
(I found Kolmogorov’s essay in my University library as a teenager. I’ve unsuccessfully tried to track it down several times in the intervening years. If anyone can identify the essay, I’d appreciate it. I’ve put enough effort into tracking it down that I must admit I’ve sometimes wondered if I imagined the essay. If so, I have no idea where the above story comes from.)"
Can anyone find the essay mentioned above? Maybe a Russian speaker could track down the original (as presumably what Michael Nielsen is referencing here is an English translation).
http://www.math.harvard.edu/~mazur/preprints/when_is_one.pdf
https://www.turpion.org/php/reference.phtml?journal_id=rm&pa...
Perhaps "On the school definition of identity", Matematika v shkole 1966
* * *
Maybe a Russian speaker wants to go search? There doesn’t seem to be an English translation of most of Kolmogorov’s essays online.
https://www.google.com/search?tbm=bks&q=Математика+в+школе+К...
Polar is an SRS based document management platform and personal knowledge repository I've been working on.
It syncs with anki and allows you to create and read flashcards directly in the reader itself.
Supports text annotations, highlights, comments, keeps track of where you're reading in documents, supports tags, rich formatting of notes, etc.
It's also Open Source and supports cloud sync across your desktop machines.
Particularly happy about offline first!
Vocabify is a tool to help you remember the words you come across. It uses a basic implementation of spaced repetition to help you remember the words and phrases you add.
Having Polar sync with Anki is an interesting idea, thinking about it now, I might have been able to get further with my own tool by leveraging Anki as well. Food for thought.
Cool idea. I suggest improving the onboarding experience of your product dramatically with minimal effort by adding this sentence to your site. Like so: https://snag.gy/iqOlFC.jpg
The Vocabify "definition" is added for any new users, so you'll need a fresh browser instance to check it out. Thanks!
It seems he and I approach SRS very differently. I see it as a place to drop bits of information I want to remember, and once the info is in there, I no longer have to think about it, outsourcing the effort of remembering to the algorithm. He seems to use it as a tool to mentally digest a subject, constantly revising his understanding and readjusting his cards. That process isn't foreign to me, but I prefer to use something besides SRS for it. If I had to constantly revise my cards, that would seem to defeat the purpose of using SRS anyway.
Very interesting to get a different perspective on this though, so I'll have to think more about his approach. I'm also interesting in the creation details of his math-cards. Does he use LaTex? Maybe he goes into it in one of his earlier posts...
This has, incidentally, made many of my matrix-based Anki cards rather ugly. I'll probably update a bunch of answers with cut-and-paste from this essay.
Not to put too fine a point on it, but it's pretty bad I'm using raw LaTeX in this way - I'm sure it hurts my understanding! On the other hand, I've written so much LaTeX over the years that reading it seems really natural.
Even in something as basic as language learning, it would allow to commit idioms and common collocations (words "often used together") to memory once you know the original words - a nice "next step" after pure vocab.
The hard part might be the UI for finding and selecting the dependencies for a new item.
1. learn the concept
2. break it into atomic question/answer pairs
3. define some dependence relation for all these pairs (X is foundational to Y and Z but not A)
4. input them into your SRS of choice
Each of these steps require significant amounts of time and brainpower. Also it would be harder if you were still learning the concept since you'd continually need to go back and redefine relations as your understanding grew.
On the one hand all this would be a great way to understand a subject. On the other, one could argue that it's very inefficient, and it would be better to not bring SRS into it all all. Compare this to say a vocabulary card, which you make, enter into SRS, then you're done. So far this (the vocab thing I mean) is the only SRS use-case that makes sense for me.
There's actually an addon to help out with that - one I'm personally really satisfied with.
https://ankiweb.net/shared/info/1199027445
Long story short: Take notes in a LaTeX document (or take notes somewhere else and then put them in a LaTeX document, this will be your ground truth). If you format your document based on the addon's guidelines, you can automatically "compile" your document to Anki flash cards. That way if you have to change stuff, just change the document and recompile.
His recent forays into spaced repetition have been an interesting glimpse into his thought process. I am a bit of a spaced rep fanatic, but use it to internalize more general "Mental Models" (a la Farnam Street), useful tidbits e.g. how to horizontally align using flexbox, or surface-level concepts like product management frameworks.
I love the effort Michael puts into grappling with ideas to strengthen his intuition. This is something I'd like to do more of, but feel that few of the ideas in my day-to-day are sufficiently complex (I don't have any use for linear algebra proofs, for example).
Anyways, that's my word salad on Michael Nielsen and spaced repetition.
I’ll typically make a few cards at first trying to test different aspects of the MM (general concept, example applications, given definition identify the MM, quiz any internal associations I have with it like if I learned it in a specific context, etc.) then I can delete ones that aren’t useful and/or add others later.
Let’s use the mental model of inversion as an example. I found this one in a Farnam Street blog post about Charlie Munger. Here are some cards I might create:
Front - what is the principle of inversion?
Back - when you want to try to maximize something, instead try minimizing its converse. Or vice-versa.
Front - what mental model might help if you’ve unsuccessfully tried implementing programs to increase innovation in your company?
Back - inversion. Rather than thinking of ways to increase innovation, can you instead think of things that are decreasing innovation and eliminate those?
Front - what’s another way to think about reducing time spent on work tasks?
Back - invert the problem. Try to increase time saved on nonessentials (e.g. laundry service, meal prep, outsourcing)
Front - what’s it called when you work backwards through a problem you’ve already tried to work through forwards?
Back - inversion
Front - what would Charlie Munger ask you if you came to him with a tough optimization problem?
Back - “have you tried inverting it?”
This is a bit contrived, but I hope this gives you a sense of how I think about creating cards for mental models.
Are we talking about math for social studies majors, engineers, physicists, economists or mathematicians?
For example, there are books like [0] for engineers which serve as a boot camp of sorts. There are no theorems, proofs or deep math in it. There are many different kinds of engineers, so books like this don't include everything an engineer needs to know. For instance, there's no automata or group theory in [0]. It has a part at the beginning called Foundation Topics which could serve as "all the high school math one needs". In fact, this whole book could probably serve as "all the math an advanced high school student needs".
There are books like [1] for physicists. They are meant to introduce physics majors to a wide array of math topics in a relatively pain-free way. This book is much more rigorous and contains a LOT more material than [0]. Basically, a theoretical minimum for a physics major.
Economics majors also get somewhat rigorous math load where measure theory features a lot more prominently than it does in other majors (except mathematics proper).
List of subjects for math majors vary from place to place, but you'll have much easier time down the road if you master the (rigorous) rudiments of linear algebra (vector spaces), group theory, number theory and real analysis. Of course, knowing more math (say topology, complex analysis, category theory, combinatorics etc) is always good.
Before you get started with math for mathematicians, you'll want to learn their jive. A good intro is [2]. It's free and a really nice book. Another really nice book with non-existent pre-reqs is [3].
A pitfall that awaits a lot of people new to math is a concept of "multivariable calculus". This concept is a mess and means everything to everyone. Oftentimes it means surface level discussion of concepts in scalar fields (functions from R^n to R) and a little bit of talk about differential geometry of curves and surfaces (functions from reals and planes to R^n). The treatment is often not rigorous and n is limited to 2 and 3. After this laughable bullshit, one is thought ready to jump straight into the rigorous analysis of manifolds, granted they know a bit of real analysis. This is like jumping from 3rd grade straight to 9th grade. Along the way the most important thing missing is the rigorous treatment of vectors fields (say, at the level of Rudin). Some nice books here include [4]. Since diff geo (in particular, that of curves and surfaces) is its own thing entirely and there are a lot of really nice books for that like [5]. Also, note there's a nice new intro to "manifolds and stuff" [6] which is like what's calculus is to analysis.
Before I forget, most intro to stats books are written for science majors and are entirely inadequate for math majors, but there are elementary intro to stats books for math majors like [7].
Originally, I wanted to write a more fleshed out and huge comment, but I am running out of time.
Good Luck.
[0] Engineering Mathematics by Stroud/Booth
https://www.amazon.com/Engineering-Mathematics-K-Stroud/dp/0...
https://www.amazon.com/Advanced-Engineering-Mathematics-Kenn...
[1] A Course in Modern Mathematical Physics: Groups, Hilbert Space and Differential Geometry by Peter Szekers
https://www.amazon.com/Course-Modern-Mathematical-Physics-Di...
[2] Book of Proof by Richard Hammack
https://www.people.vcu.edu/~rhammack/BookOfProof/
[3] Linear Algebra: Step by Step by Kuldeep Singh
https://www.amazon.com/Linear-Algebra-Step-Kuldeep-Singh/dp/...
[4] An Introduction to Analysis by Wade
https://www.amazon.com/Introduction-Analysis-4th-William-Wad...
[5] Differential Geometry of Curves and Surfaces by Tapp
https://www.amazon.com/Differential-Geometry-Surfaces-Underg...
[6] A Visual Introduction to Differential Forms and Calculus on Manifolds by Fortney
https://www.amazon.com/Visual-Introduction-Differential-Calc...
[7] Statistics for Mathematicians: A Rigorous First Course by Panaretos
https://www.amazon.com/Statistics-Mathematicians-Rigorous-Te...
It reminds me of the situation with books of transcribed jazz solos, which at first seem like gold. But they're actually worse than useless, as you miss out on everything you learn from transcribing solos yourself.
> Why would you want to miss this step?
Because I don't really have much time for every step. I don't need it to be fun, I need it to be efficient (requiring as little time and effort as possible). In fact I had always wanted an app of a sort that would feed me math in concise pre-digested twitterish pieces.
Of course I can't expect somebody to do this for me but if somebody has done and shared I'd like to make use of that. I am certainly going to share if I find time to make the cards.
In your 2 articles you tell about using it to understand and retain memory of an Alphago paper and to do the same for the theorem about orthogonal diagonalizability of normal matrices.
I wonder, how do you organize anki cards and in which order do you study them? Personally I put all cards in one deck and anki shows them in kinda random order, so I might get a question about convex optimization, then a question on numerical linear algebra, then a question on some dance moves. I take it you do it in a different way? Because you create many small cards for a topic I think you would spend too much time context switching if you did random order like me. For this reason I make larger cards, e.g. "Prove that for any linear operator on a finite dimensional complex vector space there is a basis such that the operator has upper triangular matrix in it"; and when preparing for an exam in institute I might even create a card "Definition, existence, uniqueness, and computational complexity of SVD". Also I think creating small cards might be bad for chunking, i.e. you won't get large chunks of all the related knowledge about a theorem and instead you will have small chunks - a chunk per card.
Another question is how do you add all this information to anki? Anki obviously sucks as an exploratory medium. I often find that even clearing up my paper notes, taking photos of them, cutting them, and putting them into anki takes a lot of time; typing it up in LaTeX is even longer. Any tips or insights here?
I've only just discovered Kolmogorov.. not being particularly mathematically talented myself I'm happy to find his work conceptually fascinating from an amateur's perspective, I hope the essay is found.
(Some of my earlier work: http://j.mp/aleahmad-thesis )
It is actually more difficult to prove. [2, 1; 0, 1] is diagonalisable (via [1, -sqrt(2)/2; 0, sqrt(2)/2]) but not normal.
On another note I also enjoy your essays on SRS and it encouraged me to start learning Japanese using it.