Using spaced repetition systems to see through a piece of mathematics (2019)

https://cognitivemedium.com/srs-mathematics

by sebg • 3 years ago

160 74 3 years ago

cognitivemedium.com

nickcw2• 3 years ago

In case you are looking for the essay on the = sign, this is what Google Bard said:

The reference to the essay by Andrey Kolmogorov in the paragraph you provided is:

Kolmogorov, A. N. (1930). О понятии числа [On the concept of number]. Matematicheskiy sbornik, 38(3), 3-19.

This essay is in Russian, but an English translation is available in the book "Selected works of A. N. Kolmogorov, Volume I: Mathematics and mechanics" (1991).

lsedgwick• 3 years ago

Do you think it's true? I got my hands on the pdf see no trace of it in table of contents or text searches.

nickcw2• 3 years ago

A good question. I've found Bard to be quite good at this type of question in the past but maybe it has hallucinated the answer!

crawfordcomeaux• 3 years ago

Could it be the author isn't Kolmogorov, but Kronecker?

lexh• 3 years ago

  > Perhaps it is the article On Sense and Reference by Gottlob Frege?
  >
  > http://www.scu.edu.tw/philos/98class/Peng/05.pdf

from previous discussion https://news.ycombinator.com/item?id=18895613

dkural• 3 years ago

ChatGPT thinks it might be Paul Lockhart's essay, titled "A Mathematician's Lament".

raidicy3• 3 years ago

Slightly tangential but are there any anki decks/csvs of definitions, theorems, axioms, terms Etc that start from basic arithmetic and get more advanced?

I have an idea that being intimately familiar with mathematical operations and ideas, as like one would with vocabulary and grammatical in a language, would help one internalize math better.

xyzzyz5• 3 years ago

With math, the hard part is not to remember, but to understand it in the first place. Once you understand it, remembering is easy (because understanding is placing things within the whole context of your mind), but just being able to flawlessly recite the definition will not help you understand it at all.

I think a lot of people just cannot come to terms with the fact that mathematics is inherently difficult, and being good at it is not just a matter of memorization. For example, I often see people complain about mathematicians using arcane notation, as if it was the notation that was preventing them from understanding what’s going on, when in reality it is just lack of understanding of underlying concepts and experience with arguments being put forward.

voidhorse2• 3 years ago

Much of what I know beyond the basic mathematics taught in elementary school I've learned on my own, and accepting this basic difficulty and learning to go slowly has been an absolute boon. I've simply accepted the fact that I might need to spend almost two weeks on a single proof or even definition, continually returning to it or ingesting it in different ways until I am ready to move on. I'm able to read a typical piece of somewhat difficult literature (fiction or non-fiction) in about two weeks, so it was certainly an adjustment, but as soon as I got used to this I found that my actual understanding of the concepts had increased tenfold.

I think part of the problem is that mathematics is so precise that subsequent understanding is tightly coupled to a comprehension of prerequisites. In so-called "softer" disciplines, the concepts are less precisely delineated, and their relationships are fuzzier. If you don't quite understand some prerequisite concept, you can sort of muddle your way through even the later notions that depend on it. Not so for mathematics. If you fail to get a complete and precise idea of the basics you'll eventually face complete doom or confusion when working through the subsequent portions of a theory.

Almondsetat4• 3 years ago

Understanding is remembering, remembering is understanding.

I seriously don't get why people keep separating the two

hiAndrewQuinn3• 3 years ago

Once you understand it, remembering is easy (because understanding is placing things within the whole context of your mind), but just being able to flawlessly recite the definition will not help you understand it at all.

I found this less and less true as the math I did got harder, hitting a breaking point when I took abstract algebra. The course just didn't make sense at all until I pulled a perfect score on my final by whipping out the Anki decks and drilling the theorem proofs and homework problems from class on a schedule. I would go right back to that method if I ever returned to mathematics for my master's degree.

raidicy3• 3 years ago

I feel my comment is slightly misunderstood. I don't mean to say that strictly memorization is the path to mathematical understanding.

Besides brushing up on my math, starting from simple arithmetic, In my spare time I also study Japanese. And one of the things that has helped me the most in my fluency and understanding has been the memorization of vocabulary and of grammar patterns and their usage.

Of course, I read materials at my own level and listen to material at my level and above and practice writing. However, I noticed the biggest boost in my comprehension after I memorize a large amount of words or really internalize grammar patterns. And I do this mainly through flashcards.

I have to spend much less mental energy to catch on to what is being expressed allowing me the ability to potentially comprehend more.

And analogously to my language studies, I would like to approach math in a similar fashion.

galaxyLogic4• 3 years ago

The difficulty in understanding a proof I think is partly because whoever wrote the proof omitted many "trivial" steps in the proof. But they were trivial to the person who wrote the proof, not to whoever is reading the proof.

  To understand the proof you have to accumulate  enough mathematical knowledge that the trivial, omitted, steps in it become trivial to you as well.

This has to do also something with the fact that to understand the proof it cannot be longer than what fits into your memory. That is why proofs omit the trivial steps because that way mathematicians can understand the proof both in terms of remembering its outline, but also understanding how each step in it follows from the previous ones.

BeetleB2• 3 years ago

The two are not independent - it is not a dichotomy.

When I did undergrad level engineering math, I pretty much never had to memorize. Just solve a lot of problems, and it's ingrained in your head. It helped that I utilize that material all over engineering and physics courses. Even now, over a decade since solving problems, I can still recall most of it and use it if needed.

Once I got to upper and grad level math, the approach of simply understanding and solving problems failed me, because you are usually not provided enough problems compared to earlier courses. It may have been good enough for the course to get an A, but not good enough to retain the material beyond the course. I've studied complex variables (Cauchy Goursat, residues, etc) at least 3 times. It's always a breeze, and it's always forgotten soon after. The same with statistics. With the latter, what did finally help me was spaced repetition. Even though it's been a few years since I last reviewed/relearned statistics, I can still read some material involving it and understand it. This is entirely because of those flashcards.

Understanding definitely has to precede memorizing. Insisting that memorizing is a sign of poor understanding, however, is parochial thinking. It's simply not true for the majority of folks.

brian_spiering• 3 years ago

My mathematics anki cards are here: https://ankiweb.net/shared/info/1715995729?cb=1697579387129

They are organized by category with a focus on fundamentals.

raidicy3• 3 years ago

Thank you. I use flashcards to study for Japanese and I'd like to use flashcards to study for math. Thanks again

waprin• 3 years ago

I am a strong believer in spaced repetition. My current primary focus is building a spaced repetition platform for learning poker solver solutions with the aim of winning money at live poker games ( www.livepokertheory.com ) . The motivation for the project is just to work on an indie hacker project that self-funds itself since I use my own tool to make money. My app is not based on Anki since , besides wanting to independently monetize my work , I also wanted to build a poker-centric UX for it.

Now, within the poker community there is an idea that “GTO robots” (players who memorize game-theory equilibrium solutions produced by solver software) are inferior to intuitive players who exploit their opponents mistakes rather than play an equilibrium strategy which is suboptimal vs a non optimal opponent. For example, a GTO strategy will bluff very aggressively which is not good against opponents that never or rarely fold. Especially GTO play is criticized in live games where there is other info such as opponent “tells” such as shaking hands when making bets.

But it’s a false dichotomy as I believe studying equilibrium gives you a strong baseline understanding of the mechanics and in live play you can still listen to your gut and make adjustments.

The OP talks about math and I’m talking about poker but I’ve seen this concept discussed in a few other places. Basically, some education systems way overindex on rote memorization. And just rote memorization is a really bad way to learn and understand since you miss high level reasoning, problem solving, and abstraction. But some rote memorization can be good as a foundation for abstraction. I’ve seen some studies suggest people who can do arithmetic and basic algebra more quickly can do higher level proofs more effectively. They are less “bogged down” with low level mechanics when doing higher level thinking. Musicians who want to creatively and improvisationally “jam out” might still drill scales to build that low level technique to unlock the high level creativity on top of it.

And within learning and memorization, spaces repetition works really well. It matches something with our brain chemistry to review mostly new stuff but occasionally review old stuff that might be fading, it’s the right mix of novelty and preventing forgetting. Anki is heavily associated with flash cards for language learning Vocab and MCAT test prep so nice to see some others exploring it’s possibilities outside of that.

BenFranklin100• 3 years ago

“But some rote memorization can be good as a foundation for abstraction. I’ve seen some studies suggest people who can do arithmetic and basic algebra more quickly can do higher level proofs more effectively. They are less “bogged down” with low level mechanics when doing higher level thinking. Musicians who want to creatively and improvisationally “jam out” might still drill scales to build that low level technique to unlock the high level creativity on top of it.”

I personally find this to be true. I became significantly better at higher math and physics after I memorized and drilled on lower level concepts and algebraic/trigonometric manipulations.

spencerchubb3• 3 years ago

The chunking concept really resonates with me. I have been a speedcuber for years (solving Rubik's Cubes really fast) and I can remember Rubik's Cube solves without much effort. If you gave me the scramble of my personal best solve, I could reproduce the same solution.

In some sense it gives me hope that I can also reach the same level of proficiency in more productive areas.

Blahah2• 3 years ago

Can you explain for those of us who aren't speedcubers what the scramble is? Is it what it sounds like - a number of pseudo-random twists to undo the solve?

spencerchubb3• 3 years ago

To make an "official" scramble, first they create a random state, which basically means random permutation and random orientation. Then they find a sequence of twists to arrive at the random state.

On the contrary, applying random twists will not give you a truly random state. Some states are more likely than others to be achieved when applying random twists.

heyoni• 3 years ago

Have you tried using SRS for anything productive and failed or just haven't gotten into it yet?

spencerchubb3• 3 years ago

A while back I used Anki to learn a bunch of French vocab. At the time it was great and I learned a lot. I eventually stopped and I've probably forgotten most of it.

Blahah2• 3 years ago

This seems intuitively true but much more generally - not just for mathematics, and not just for spaced repetition. Our understanding of fundamental principles in any field is deepened by exposure to them generally in different contexts. For me (as a biologist/computer science/systems design person) the quintessential examples are evolution and entropy. Both are trivial in their essence, but infinitely rich in their implications.

voidhorse2• 3 years ago

Tangential, but I think people tend to underestimate just how important and deep mathematical fundamentals are in general. The relatively "simple" notions on which we build all other mathematical concepts, such as equality, relation, function, symmetry, associativity etc. actually have incredible depth to them. Too often, I think teachers and texts gloss over these notions instead of plumbing their philosophical depth and implications (statements like "the proof falls out automatically" or "the proof is obvious" or "the proof is a natural consequence" are a real disservice to learners). Both set theory and category theory are illustrations of just how far you can go with a few of these ideas (category theory, for instance, is basically entirely built up around the notion of a function and only two other ideas, identity and associativity—yet these three "simple" ideas alone are enough to construct a theory rich enough to model all the other branches of mathematics!)

Understanding mathematics is really understanding how to start approaching metamathematics. What does it mean to distinguish one class of things from another? What does it mean to say something has a property? That a property is reflected or preserved? These are the sorts of questions that are fruitful. It makes sense that the spaced repetition approach can help because it might necessarily force you to continue thinking about (and thus questioning) a concept you'd otherwise have taken for granted or not plumbed to sufficient depth. Too often, we fail to recognize how complicated "simple" things really are.

Affric• 3 years ago

Too often, I think teachers and texts gloss over these notions instead of plumbing their philosophical depth and implications

I have a few questions:

What level of teacher?

Are you a maths teacher/teaching academic?

My understanding of Maths education in the west is that they essentially spend the whole time attempting to get children to the point where they can do precalc and that any attempt at reform is eventually defeated by conservative engineers running a concerted political campaign to convince the voting public that set theory and category theory are a leftist conspiracy.

My experience of university mathematics was that set theory was assumed knowledge (fair, it has managed to stay in the curriculum), same with category theory (lol).

I would say that if it weren’t for the rise of the computer and the prevalence of relational databases in the late 20th C they would have managed to cut all non-precalc out entirely.

galaxyLogic4• 3 years ago

Many people lack basic "numerical literacy". They cannot understand different magnitudes, percentages, probability and most importantly LOGIC.

If you only understand 0, 1, 2 and MANY, you can be led astray with all kinds of politics, and advertising.

Gerrymandering, I think we should teach people how that works, and leave Category Theory to those who are truly interested in math for math's sake.

hyh10485762• 3 years ago

If you want to see how some very simple notations greatly simplifies some math, check out J. H. Conway's proof of Morley's theorem.

Background: Morley's theorem is a non-trivial theorem in planar euclidean geometry stated in 1899 (first proof appeared 15 years later). The proofs are not easy. One can use complicated trignometry identities to prove it. Even the "simple" proofs are sometimes quite involved.

Conway introduced some notation and almost trivialized it. The notation he introduced was just a* := a + 60 where a is the degree of an angle. No one would believe this notation can do anything good, but with them (and some other insight) Conway can explain the proof in just a few sentences! (One might think anyone who understand that the interior angles of a triangle will always have a sum of 180° can come up with this simple proof, but that just didn't happen for 100 years until Conway revealed it.)

See page 3-6 here: http://thewe.net/math/conway.pdf

AlexErrant2• 3 years ago

Michael Nielsen (with Andy Matuschak) also wrote about how Hindu-Arabic numerals enabled more powerful kinds of thinking (compared with Roman numerals.)

https://numinous.productions/ttft/#how-to-invent-hindu-arabi...

...the Hindu-Arabic numerals aren’t just an extraordinary piece of design. They’re also an extraordinary mathematical insight. They involve many non-obvious ideas, if all you know is Roman numerals. Perhaps most remarkably, the meaning of a numeral actually changes, depending on its position within a number. Also remarkable, consider that when we add the numbers 72 and 83 we at some point will likely use 2+3=5; similarly, when we add 27 and 38 we will also use 2+3=5, despite the fact that the meaning of 2 and 3 in the second sum is completely different than in the first sum. In modern user interface terms, the numerals have the same affordances, despite their meaning being very different in the two cases. We take this for granted, but this similarity in behavior is a consequence of deep facts about the number system: commutativity, associativity, and distributivity

hyh10485762• 3 years ago

Yep. Imagine doing multiplication in Roman numerals :)

ChrisKnott• 3 years ago

It's a wonderful proof, thanks for sharing, but the notation is really just syntactic sugar, no? It would have only been slightly clumsier to have "+60" everywhere. The real insight seems more to have done the construction "backwards", starting with the equilateral.

raincole• 3 years ago

Syntactic sugar is powerful.

There was a time the China accepted calculus, but rejected Leibniz notations.

It looks this like: https://i.imgur.com/foUem6w.png

Edit: Technically this is still Leibniz notation.

bmacho• 3 years ago

I like to keep all of my proofs very very very simple. 3 big steps, where each big steps should be intuitively true. Sure sometimes you should internalize the properties of your objects before you go on (big and useful theorems are just properties of the objects), but that's it.

I believe every natural problem has such a simple proof, that's why I use BFS when I try to solve a problem instead of DFS.

vjerancrnjak• 3 years ago

There's also many assumptions in the initial m^2 != 2n^2 proof that many would have a hard time instantly grasping. Problem is stated in three different ways, takes quite a bit of knowledge to instantly see they are the same problem.

The proof stops at 2n-m and m-n, subtly, because that is another smaller case of can two squares fit in a big square. If you do the expansion of 2(m-n)^2 and (2n-m)^2 to see if there's a fit, you see there isn't but the proof does not need to deal with that and to me the reason is very subtle, wordy.

Graph theory proofs feel very similar.

Turneyboy• 3 years ago

A similar thing also happens in the modern formulation of the Stoke's theorem. Once you set up the machinery of differential forms it becomes almost a trviality which is just amazing.

redbar0n2• 3 years ago

Kolmogorov discussed this in loving detail, and made many beautiful points along the way, e.g., that the invention of the equals sign helped make possible notions such as equations (and algebraic manipulations of equations).

And then programmers came along and mangled the mathematical notion of the equals sign with the imperative notion of assignment. And here we are. Using = for assignment when we ought to have been using : instead, if we were to rely on people’s intuitions from reading and writing.

Which of course is difficult now that : has been co-opted by type systems.

galaxyLogic4• 3 years ago

In Smalltalk '=' means equality, and ":=' means assignment.

redbar0n2• 3 years ago

They realized so many things!

mattdeboard• 3 years ago

For people who don't want to follow the author down the unnecessarily loquacious dive into matrices etc., but like me are interested in techniques for self-learning, here is a wiki article on what spaced repetition is https://en.wikipedia.org/wiki/Spaced_repetition

He said he uses this application to do spaced-repetition learning https://apps.ankiweb.net/

xrd• 3 years ago

I'm curious if anyone here has experimented with Anki in collaboration?

As the author of this post notes, and I concur, others might not come up with his questions. I read the discussion he presented and thought to myself, I doubt I would have created the set of cards he came up with because my matrix memory is weak.

I've had great experiences with pair programming and amplifying my learning software development. Feels like this could work with anything and Anki.

AlexErrant2• 3 years ago

Not really. There are options for sharing cards on Anki https://www.reddit.com/r/Anki/comments/14j2jfy/deck_sharing_... but their collaboration features are limited.

I myself am building an Anki clone https://github.com/AlexErrant/Pentive with collaboration built in as a first class citizen, though its far from primetime. Currently stewing on how to get the SR algorithm, FSRS, to compile to wasm.

g_log• 3 years ago

Michael Nielsen and Andy Matuschak wrote Quantum Country (https://quantum.country/), an introduction to quantum mechanics, with integrated spaced repetition questions. Andy develops Orbit (https://withorbit.com/), allowing other authors to do the same.

I agree with Michael that a lot of the value comes from formulating your own questions based on the source material, but I'm also curious to experiment with author provided questions.

ohduran• 3 years ago

In my humble experience, you gain a LOT from writing the questions yourself, I point that I'm not sure if the author is stressing enough.

There's also the problem with cards that are difficult to memorize, which is something that you don't know in advance, and that's a good thing! It shows you were you should pay more attention, write cards in different forms, etc.

enugu• 3 years ago

Here's a different proof of the identity just thinking of M, M* as operators (not as matrices). So, the invariant way of 'diagonalization by a unitary matrix' means that we have to find an orthonormal basis of eigenvectors. We can always find one eigenvalue λ and the complex numbers. The strategy is to choose one eigenvector v and say that its complement subspace is also preserved by the action of M. This complement subspace, being of lower dimension, already has an orthonormal basis by induction and this basis extended with v will be the required basis. This strategy can be done in a straightforward way if we assume M is hermitian M*=M (or symmetric matrix for real numbers). <v,w>=0 ⇒ <v,Mw>=<M*v,w>=<Mv,w>=0. So, M preserves the complement and the matrix is diagonlizable. This is the Spectral Theorem.

For a normal matrix, we have to choose the vector v which is simultaneously an eigenvector for M and M*. Then the proof still works as <v,w>=0 implies <v, Mw> = <M*v,w> = λ'<v,w>=0 where λ' is the eigenvalue of v with respect to M*.

We can find such a shared eigevector v as M, M* commute implies that M* will preserve the eigenspaces of M.

dan-g• 3 years ago

I highly recommend Andy Matuschak’s guide on how to create good prompts for anyone looking to get serious about spaced repetition: https://andymatuschak.org/prompts/

Andy and Michael Nielsen have collaborated in the past on things like quantum country, as others have noted.

Using spaced repetition systems to see through a piece of mathematics (2019)

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