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How to learn mathematics: the asterisk method (geometry.org)
350 points by VitalyAnkh 88 days ago | hide | past | favorite | 76 comments

I did this during university, but made it much more efficient. I wrote a program that would work like this:

- while reading, instead of copying, the software would ask me to enter "facts" in form of questions with the most important piece of knowledge being the answer. I would type the question and the answer into the software - much faster than writing anyway

- after I have gone through all the material, I would start the Q&A part of the software, which would ask me all the questions in either random order or sequentially (it was an option).

- at first, it would only show the question and wait for a key press to show the answer. After the answer was shown, I could mark whether I knew the answer or not. If not, it would mark that question to be asked in the next loop. This is basically the same as the asterisk method.

- once I got through all the questions, it would go into next pass, asking only those questions that I didn't know the answer to. And then filter out the remaining ones, and loop again and again until the all answers are known.

- then I would restart the whole system with all the questions to check.

What I learned after using this for about 2 years, is that there's a short term memory problem. Often I would know some of the answers on the first pass, but a week later I might forget it.

I found a way that works much better: Even if you mark that you know the answer, it will come up once again in the next pass. If you mark that you know it twice in a row, only then it would be removed. For some reason, this made the knowledge stick much better.

You've implement some lite version of Spaced Repetition! There's a tool that does exactly that and it's called Anki

Yeah this is spaced repetition. In this case the purpose is to memorize facts. But the asterisk method asks if you understand a concept. The purpose of revisiting asterisks is not to aid memorization via spaced repetition but to check if the understanding has improved after reading additional chapters. This seems like an important distinction.

This. There's also Mnemosyne, SuperMemo, and others. More here: https://www.gwern.net/Spaced-repetition

I use Quizlet for exactly this (https://quizlet.com/). Works great. I paid for the "Plus" plan and am happy with it. (I can't recall if I needed the paid version or just wanted to support the project, but recall being pretty happy with the free version too).

Lots of great Flashcard apps out there. Definitely a great way to learn - especially things that fade quickly if not using them regularly (looking at you, Powershell ;-p ).

This site has many useful methods and lists its technicality in details. Thanks.

gwern is an amazing resource and has great articles on a range of topics, including productivity stuff; his posts on nicotine [1], modafinil [2], and melatonin [3] would be 3 highlights.

[1] https://www.gwern.net/Nicotine

[2] https://www.gwern.net/Modafinil

[3] https://www.gwern.net/Melatonin

This is a good method for memorizing facts about some material that you understand. The asterisk method is about getting a grasp of material that is hard to understand. Your method is typically what is needed in studying subject like (foreign) languages and history, while the asterisk method is for subject where the primary problem is to understand the material, such as mathematics and physics.

Have you ever tried learning a spoken language by doing this? Like another commenter mentioned, it sounds a lot like Anki Spaced Repetition. It's very cool that you just rediscovered it, though.

Duolingo certainly takes this method to an extreme. I’m not sure about spoken but I’m much better now at reading Spanish than I was this time last year

FluentForever is based on this method - the initial product was actually a set of Anki cards.

Intuitively, it does make sense. If you only ever encounter something once, it probably does not make sense to spend brain resources storing it.

I think I've reinvented trigonometry at least fifteen times in my life when I needed to animate something roughly circular, spherical or wave-like. I still have no freakin clue which thing is a sin, cos or tan. I failed precalc (and they still let me work on video games!) But I do remember that a hypotenuse is the square root of A^2 + B^2 and from that I can pretty much figure out the rest.

As someone who did trigonometry every week for over a decade, I can say that reinventing it often will really slow you down. There is merit to memorizing the identities (of course, you should understand them well enough to derive them if you need to).

I was also better at solving problems involving trigonometry than anyone I knew - including my professors.

You could, of course, argue that if you needed to use it every week, you would know all the identities merely by using them so much. Don't assume this is correct, though. One of the reasons I used them so much is because I had memorized them. My colleagues who used them as often as I did and who didn't memorize them did not, in fact, have them burnt into memory after so much usage.

I had trouble with that, always going back to the "SohCahToa" mnemonic. Nowadays, I always use the unit circle. Cos is abs[cissa] (horizontal), Sin is ordinate (vertical). Rotate the unit circle (or your drawing) to align as needed, scale with the radius: https://en.wikipedia.org/wiki/Unit_circle

You can derive that back from the Pythagorean theorem, but only if you have the sine and cosine definition in mind (SohCahToa).

> I still have no freakin clue which thing is a sin, cos or tan.

Atleast two methods I still recall:

1. Look up the SOH CAH TOA method.

2. Knowing that Sin(0) = 0 and Cos(0) = 1 can give a hint which one to use. You may know an equation should have a Sin or Cos but unsure which one so you pick the one that gives the expected results at 0 degrees. Works great for engineering type of problems.

First covid vaccine dose provokes a short-term response; second dose provokes a long-term immunity response.

Parallel adaptation of probability estimation?

Not long enough to require a third or fourth booster per Israel and some other countries, so not a good analogy unless we're all bad COVID students.

To the best of my knowledge the first COVID vaccine dose immune response is not short-term, except in the sense that it is weaker and therefore will fade below an effective level in a smaller amount of time. But it fades at the same rate as the second dose. The second dose just "raises" the response level higher than the first dose got it.

Couldn't find my original source, but here's another:

> The second shot has powerful beneficial effects that far exceed those of the first shot,” Pulendran said. “It stimulated [...], a terrific T-cell response that was absent after the first shot alone, and a strikingly enhanced innate immune response.


Can I recommend Mochi (https://www.mochi.cards/)? This is the tool I use to learn.

(I am not affiliated in any way apart from being a happy user)

This is not bad advice, especially for people new to proof-based mathematics. (I've noticed more mathematically experienced people do something like the linked method intuitively, without writing things down.)

But, it's only half the story. After you learn the definitions and theorems, you have to learn how to apply them to do computations and solve problems. This means working at least a few "easy" problems to learn how to crank through rote computations, and a few harder ones to learn how to think through novel applications.

If you can't solve problems with the material you've supposedly learned, you haven't actually learned it. (Otherwise, what did reading all that stuff really accomplish? You picked up some cool vocabulary words?)

> This is not bad advice

I was going to say this article was a bad advice.

I second the solving problems approach. It's pretty mainstream opinion really, at least among math/phys students. You can try to follow the article and feel you "understand" the topic. And still unable to solve any of the homework problems, let alone exams.

Reading is necessary but insufficient, but you can at least make it efficient -- that's what the article is covering I think.

I second this

One thing that is hard is that not all areas of mathematics have obvious easy problems to work through or rote computations, although you should clearly do that if those exist. I do think problem-solving and 'getting your hands dirty' is important, but you are also unlikely to manage to reinvent hundreds of years of mathematics by yourself so there is value in reading too. You do also need to learn the vocabulary that other mathematicians use.

When reading textbooks, I would often try and prove theorems myself before looking at the proof - and when I looked at it because I was stuck I would just try and see what the next insight or step and go from there.

I suppose I am thinking somewhat of reading maths papers here which don't come with a nice set of exercises, which probably isn't what we are talking about - but mathematics does move gradually in that direction from areas where the routine computations are laid out to areas where you have to work out for yourself how to make the abstract seem less abstract and where there are assumptions that you will fill in lots of details yourself.

It seems like proof-based maths and applied maths are different fields. The latter is more closely related to engineering.

I am the kind of person who likes to solve more practical problems but it is more of a hindrance in advanced maths, where you manipulate concepts that are too abstract for practical use, like infinities.

Especially in abstract mathematics, you must practice computations with basic examples (e.g. computing some homotopy groups of spheres). If you cannot do this, you haven't actually grasped the mathematical content of the reading. The abstractions exist precisely because they are concise, powerful ways to deal with various examples.

I agree with you, but I think you might misunderstand what GuB-42 means by 'practical'?

Computing some homotopy groups of spheres is a good simple exercise for the right kind of abstract math. But probably not 'practical' by GuB-42's standards?

It is a little hard for me to interpret their comment. I took it as meaning that the learning strategies should differ for pure and applied mathematics, hence my response.

Maybe another way to say this is: pure math is just a kind of applied math where the applications are resolving theoretical problems. Essentially all of the big mathematical programs/fields/whatever were created to solve or understand some Big Central Theoretical Problem(s), and prove their worth by continuing to be useful in solving other problems. And generally these problems can be understood in terms of concrete examples.

Even Grothendieck, perhaps the canonical example of a "theory builder," had resolving the Weil conjectures firmly in mind while writing his famous texts (and then got annoyed at Deligne for doing it the "wrong way"; see https://webusers.imj-prg.fr/~leila.schneps/grothendieckcircl...).

That's an explanation that only makes sense from the perspective of pure math. Many of us engineers are wholly incapable of caring about the resolution of theoretical problems. Learning how to take a real world problem and map it into the domain of theory is a special skill that is not required for pure math, because it has an intrinsically messy interface into the real world.

I suspect that a pure mathematician would look scornfully at that as a waste of time, as the whole point is the math doesn't depend on the particular instance. An applied mathematician / engineer on the other hand sees no point in the mathematics unless it is manifested in the form of physical problems.

I don't think mathematicians look scornfully upon people doing the mapping between applied and theoretical domains. On the contrary, I think there's a lot of respect (and perhaps some envy). It's an entirely different set of skills, and a lot of great math has been inspired by applications, which would never have been possible if pure mathematicians worked in isolation. (The respect/envy comes from the fact that most mathematicians don't have those skills but recognize their value.)

I'm thinking in particular of a lot of stuff in mathematical physics and PDEs.

> I don't think mathematicians look scornfully upon people doing the mapping between applied and theoretical domains. On the contrary, I think there's a lot of respect (and perhaps some envy).

Proper abstract mathematicians view applications not with scorn, but as a challenge to come up with even more abstract math. :)

As an applied mathematician, I disagree. Proofs are often useful in applied maths, and most applied maths is much more closely related to pure maths than to engineering. Infinities show up in practical applied maths, but there are more abstract things that don't (yet).


Though just like libraries in programming mean that you don't have to worry about how to implement a hash table when using Python's dicts, advances in math mean that you don't have to worry about infinities and infinitesimals when you are constructing a bridge using differential equations.

That's progress!

Unrelated, but I am curious about the splash page and redirect. Is this good enough to thwart spiders without having to use Google's captcha? Making pages un-crawlable is a major topic of interest. I read that it's very hard to do nowadays.


I presume it's good enough until more sites pick up this approach and crawlers counteract it.

Why avoid spiders, to prevent bots stealing your posts? This also prevents search engines crawling them, which is a major downside, no?

I think it just stores a cookie, then checks for a cookie before redirecting. I actually have no idea if google's crawler stores cookies or not, there would be good design arguments either way

>> 3. Start copying the relevant part of the book or the lecture notes to a (paper) note-pad by hand.

When I started learning to code. I remember people would buy the famous books and do an intellectual dive into them.

As for me I bought 'SAM's Learn C in 21 Days.' Then for the next one month like a dumb bot I would just read the code and type it out on my desktop. Religiously. Like wake up everyday spend significant part of the day just to read and write code into the editor and run it. Most of the time I had a hazy understand of what I was doing.

But surprisingly this approach worked better than the intellectual fantasising exercise(Most of my friends quit after a while). I've used this technique of using dumbest possible method to study anything useful in everything I've touched since. Including workout and fitness.

Over the years I've also wondered why it works. One reason is when you just wake up everyday and do this ritual, you are basically spending time with the subject. Add a month or two to this journey it's now a habit. Even though your understanding is relatively poor, you haven't quit at a point most people quit. You are comfortable doing work in the subject and therefore you are already scoring wins everyday. This not only builds confidence, it will eventually take you to higher levels.

Turns out the hardest part of learning and doing anything is sticking with a subject for long. Once you have practice the hard parts become easy.

Students should not assume that all mathematicians try to learn all the material thoroughly.

Instead, in practice, even among good mathematicians, there is a fairly wide range of how carefully they study and how well they learn some material.

So, it's possible and common (1) to get mostly just an overview, and even the overview can be at various levels of thoroughness, (2) try to get the main ideas of the most important points, (3) think about the material mostly just intuitively to build good intuitive models that can be the basis of more in learning, applications, research, (4) deliberately go over the material more than once with only the later passes quite thorough. In short there is more than one way to slice an onion.

Here is what did me the most good: First get an overview, i.e., what is the material really about? Second understand the details, say, after reading a definition, theorem, or proof, be able to write it down. Third, look back and get a relatively succinct, intuitive overview, model, that keeps all or nearly all the important content.

Uh, of the five Ph.D. qualifying exams, I got the best in the class on four of them. For my research, (a) for a paper I published and (b) for my dissertation, I did all the work with essentially no faculty direction. For the research, sure, needed to understand enough low level details of some material, but the real key was intuitive models that led to, permitted guessing, the original math with theorems and proofs.

The issue with this is it requires a fairly mature metacognition from the student. We've all fooled ourselves into thinking we understand something, only to be shown a question we can't answer later on. With some maturity one gets into the practice of asking the right illuminating questions, but it's a painful journey at times. It's also made harder of there's a lot of time pressure, eg if you are studying a bunch of courses at the same time.

This is the so-called "Illusion of competence/learning" [1] and I have become fascinated with this topic (and learning in general). The only way out of this trap is to self-test.

[1] https://staciechoice1010.wordpress.com/2014/08/15/illusions-...

I noticed a great overlap in learning math and programming.

Programming was easier for me to learn and now helps me to understand math a bit better.

Coding a real project is the equivalent to math's proofs, I think.

The book "Badass - Making Users Awesome" says, learning something just requires two steps. Perceptual exposure (of hundreds of correct examples) and deliberate practice.

I think, math falls short in the first step, and I don't know why, but somehow mathematicians often see much part of syntax/grammar as a given, and use different ways to describe the same thing (sqrt and power of 1/2, for example).

> Coding a real project is the equivalent to math's proofs, I think.

Surprisingly, this turns out to be true: https://en.m.wikipedia.org/wiki/Curry%E2%80%93Howard_corresp...

Now, how can I use this fact, to finally become a decent math user? :D

But, then does programming. In C, there's at least three (arguably more) ways of doing a loop.

You can use for (...) { ... }, you can use while (...) { ... }, and you can use do { .... } while ( ... ).

Of course, which one you pick depends on context. But, then, which one of sqrt or "raised to 1/2" you choose also depends on what's more convenient at the time.

Fair enough!

Good points, thanks.

Learning to learn is such an important skill that a lot of people just miss. Speaking for myself and I think my brother who was unfortunately labeled 'gifted' at some point, learning came automatically - I just absorbed information - until it didn't. During primary and secondary school, I never had to 'knuckle down' and actually study.

I’ve been doing this for 25+ years without knowing it had a name

> Connections can only be made between ideas when they are in the mind to be connected.

This is probably true. I always used to try to understand to avoid the bother of rote learning... but the truth is, I probably examined it so intensively while trying to understand, that I rote learned without realizing it.

I love it. and it also applies to all kinds of learnings. I even love how the captcha is just a link.

"First learn. Then understand. Insight requires ideas to be uploaded to the mind first."

Yes, our mind has a 'backdoor' which is imho a slow write/read/breathe manifold we all can summon and it's free (both as a freedom, and as a beer).

I didn't really understood this "learn then understand" concept in the article, and also didn't understood the "backdoor", can you illuminate me?

For me, the experience of learning how to read/write Japanese revealed this insight. In particular, I utilized a technique called RTK (for Remembering the Kanji, a book which describes the technique) in which each character is assigned an English word or short phrase that approximates its meaning, and then the student invents a colorful story that relates the key word to the sequence of strokes used in writing the character. Then, the information is committed to permanent memory through spaced-repetition study methods (including lots of actual writing of the characters).

This technique is often criticized for teaching only the writing and rough meaning of characters, while ignoring actual usage and pronunciation. However, for me it was an absolute revelation that allowed me to finally break through and eventually achieve something close to fluency in the language (passing the highest level of proficiency exam, the JLPT N1).

Later, I began to comprehend how the method itself--which is kind of abstract and involves quite a bit of raw repetition--achieved its amazing results by building a rock-solid scaffolding in my brain upon which I could hang all of my Japanese language knowledge. If learning is the process of building new knowledge structures in the brain, then understanding is the process of linking existing structures into tightly integrated patterns which can support higher-level reasoning. The stronger the base structures, the deeper the understanding that one can develop on top of them.

As for the "backdoor", I believe the original comment was referring to the way that the brain seems to naturally (and sometimes effortlessly) work to strengthen the connections between knowledge structures that are sufficiently "exercised". So, one can use conscious efforts to simply reinforce the raw knowledge as much as possible, and then trust in the "backdoor" to reveal the important insights and connections as they are uncovered.

Really great method! I think the hardest thing for me is to look back at the notes I have written and study the asterisks. I haven't been able to develop good habits to "re-read" already written notes.

This is great advice and it is definitely how i learn best too.

But what do I do if i don't have the time necessary to do this always?

At my university i have 4 graded assignments i have to hand in every week and they always consume a lot of time. Between working on those there is barely any time to even study, except on the weekend, which by this method might be enough to study two subjects max. I feel like this works but not for the pace of a college degree.

Copying stuff by hand or otherwise focuses you on writing not reading. Personally I felt experimentation with this method set me back in my studies.

It's probably an aid for attentive repetition. If you can do it faster without writing, great. Many people can't.

(Basically it's a consequence of ADHD. While it's a novelty it's easy to focus, connect it to other things, grasp the rough outline of the concepts, mash them together, be creative, talk about it, barf up ideas about it ... but when it comes to memorizing a proof glyph by glyph reading does not help, because it just makes me feel like reading something that I completely understand and remember the 1000th time. Of course it turns out without actually writing it down from memory it's hard to judge how well I could write it down from memory based on what I feel while reading it...)

There's quite a bit of research linking writing with reading and understanding. Writing to Read: A Meta-Analysis of the Impact of Writing and Writing Instruction on Reading https://doi.org/10.17763/haer.81.4.t2k0m13756113566

I'm aware of this kind of research existing. I guess my personal experience does not count as a scientific work. However it's not study on math and math is not normal reading. It's anything but. You need to understand every symbol, place it in context and do all of the proofs in your head so you can get some feel for it. If you are copying you are not doing this. I'm a terrible hand writer. I have a math degree.

Oh, but everything I know about learning mathematics tells me that doing problems, on paper, is key. That means writing them out. I'm sorry you think you're not a good hand writer, may I recommend Write Now: The Getty‑Dubay Program for Handwriting? You're probably not bad at it, but could use a little refinement. It really helped me improve and speed up my handwriting.

For me it’s been 20 years since college.

I tried taking a trigonometry couse. Something I aced in college. Some parts were easy. But I kept running into references that I could not remember.

Eventually I just had to back to pre-algebra. Most of it is trivial but I do keep finding things I had completely forgotten.

And then you come across a baffling Russian paper that takes months to understand a single line.

Understanding maths is a marathon. It's generally huh? at first, then holistic synthesis over many years. This depth-first, beating-one's-head approach is definitely not the way.

> If you read something which is difficult to understand, stop and think about it until you understand it clearly.

Ahh, ok.

This is the essence of the method described in the link, and the ability to pull it off is called "mathematical maturity". How one acquires mathematical maturity is the matter of some debate.

> How one acquires mathematical maturity is the matter of some debate.

The asterisk method just reminded me about this (how to draw an owl):


If you stop at step 5 and ignore the rest, including step 10 which includes asking people who can help you understand.

Then why isn't that step 6, or step 5a? "Think about it until you understand it" is really not helpful advice.

> The mind is on the path between the eyes and the hands.

Killer quote.

Sounds like the 3-pile method for flashcards.

I get a 410 Gone when visiting the page.

I'm not sure why it doesn't directly link to


> (You must permit one cookie.)

What a throwback to the early days of Internet explorer when it asked you if you wanted to allow a cookie for every cookie the website asked you. I think that was available later as an option, but it became unbearable to browse with that activated later on...

This other link doesn't work as well.

I love that there's a focus on improving learning methods in general and how it can apply to anything. Even captchas are utilized in a way that shows their potential when they're given less of a burden!

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