10 ms·

How to learn mathematics: the asterisk method

350 points by VitalyAnkh 5y ago

blaeks 5y ago
"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).
- SPascareli13 5y ago
  I didn't really understood this "learn then understand" concept in the article, and also didn't understood the "backdoor", can you illuminate me?
  - jkhdigital 5y ago
    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.
steve918 5y ago
I’ve been doing this for 25+ years without knowing it had a name
spekcular 5y ago
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?)
- GuB-42 5y ago
  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.
  - mkl 5y ago
    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).
    - eru 5y ago
      Yes. 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!
  - spekcular 5y ago
    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.
high_byte 5y ago
I love it. and it also applies to all kinds of learnings. I even love how the captcha is just a link.
visarga 5y ago
Sounds like the 3-pile method for flashcards.
paulpauper 5y ago
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. http://www.geometry.org/human/?i=/tex/conc/mathlearn.html http://www.geometry.org/human/?i=/tex/conc/mathlearn.html
- jedimastert 5y ago
  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
- treszkai 5y ago
  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?
babuskov 5y ago
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.
- thevinter 5y ago
  You've implement some lite version of Spaced Repetition! There's a tool that does exactly that and it's called Anki
  - normac2 5y ago
    This. There's also Mnemosyne, SuperMemo, and others. More here: https://www.gwern.net/Spaced-repetition https://www.gwern.net/Spaced-repetition
    - Trex_Egg 5y ago
      This site has many useful methods and lists its technicality in details. Thanks.
bjarneh 5y ago
> If you read something which is difficult to understand, stop and think about it until you understand it clearly. Ahh, ok.
- pash 5y ago
  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.
  - bjarneh 5y ago
    > How one acquires mathematical maturity is the matter of some debate. The asterisk method just reminded me about this (how to draw an owl): https://www.reddit.com/r/funny/comments/eccj2/how_to_draw_an_owl/ https://www.reddit.com/r/funny/comments/eccj2/how_to_draw_an...
    - drivers99 5y ago
      If you stop at step 5 and ignore the rest, including step 10 which includes asking people who can help you understand.
      - cratermoon 5y ago
        Then why isn't that step 6, or step 5a? "Think about it until you understand it" is really not helpful advice.
xyz11 5y ago
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!
foxes 5y ago
And then you come across a baffling Russian paper that takes months to understand a single line.
tyilo 5y ago
I get a 410 Gone when visiting the page.
- sva_ 5y ago
  I'm not sure why it doesn't directly link to http://www.geometry.org/tex/conc/mathlearn.html http://www.geometry.org/tex/conc/mathlearn.html
  - plumeria 5y ago
    This other link doesn't work as well.
  - MayeulC 5y ago
    > (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...
kamaal 5y ago
>> 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.
hyperpallium2 5y ago
> 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.
lordnacho 5y ago
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.
- Rainymood 5y ago
  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-of-learning-competence/ https://staciechoice1010.wordpress.com/2014/08/15/illusions-...
j7ake 5y ago
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.
saasaccount2 5y ago
This approach seems to be quite innovative.
graycat 5y ago
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.
sureglymop 5y ago
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.
Cthulhu_ 5y ago
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.
jimmyvalmer 5y ago
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.
timwaagh 5y ago
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.
- pas 5y ago
  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...)
- cratermoon 5y ago
  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 https://doi.org/10.17763/haer.81.4.t2k0m13756113566
  - timwaagh 5y ago
    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.
    - cratermoon 5y ago
      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.
k__ 5y ago
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).
- zaik 5y ago
  > 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_correspondence https://en.m.wikipedia.org/wiki/Curry%E2%80%93Howard_corresp...
  - k__ 5y ago
    Now, how can I use this fact, to finally become a decent math user? :D
- randomswede 5y ago
  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.
  - k__ 5y ago
    Fair enough! Good points, thanks.
jkhdigital 5y ago
> The mind is on the path between the eyes and the hands. Killer quote.
treeman79 5y ago
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.
deleted 5y ago
[deleted]

350 points by VitalyAnkh 5y ago

"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).

SPascareli13 5y ago
I didn't really understood this "learn then understand" concept in the article, and also didn't understood the "backdoor", can you illuminate me?
- jkhdigital 5y ago
  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.

steve918 5y ago

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

spekcular 5y ago

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?)

GuB-42 5y ago
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.
- mkl 5y ago
  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).
  - eru 5y ago
    Yes. 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!
- spekcular 5y ago
  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.

high_byte 5y ago

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

visarga 5y ago

Sounds like the 3-pile method for flashcards.

paulpauper 5y ago

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. http://www.geometry.org/human/?i=/tex/conc/mathlearn.html http://www.geometry.org/human/?i=/tex/conc/mathlearn.html

jedimastert 5y ago
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
treszkai 5y ago
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?

babuskov 5y ago

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.

thevinter 5y ago
You've implement some lite version of Spaced Repetition! There's a tool that does exactly that and it's called Anki
- normac2 5y ago
  This. There's also Mnemosyne, SuperMemo, and others. More here: https://www.gwern.net/Spaced-repetition https://www.gwern.net/Spaced-repetition
  - Trex_Egg 5y ago
    This site has many useful methods and lists its technicality in details. Thanks.

bjarneh 5y ago

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

pash 5y ago
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.
- bjarneh 5y ago
  > How one acquires mathematical maturity is the matter of some debate. The asterisk method just reminded me about this (how to draw an owl): https://www.reddit.com/r/funny/comments/eccj2/how_to_draw_an_owl/ https://www.reddit.com/r/funny/comments/eccj2/how_to_draw_an...
  - drivers99 5y ago
    If you stop at step 5 and ignore the rest, including step 10 which includes asking people who can help you understand.
    - cratermoon 5y ago
      Then why isn't that step 6, or step 5a? "Think about it until you understand it" is really not helpful advice.

xyz11 5y ago

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!

foxes 5y ago

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

tyilo 5y ago

I get a 410 Gone when visiting the page.

sva_ 5y ago
I'm not sure why it doesn't directly link to http://www.geometry.org/tex/conc/mathlearn.html http://www.geometry.org/tex/conc/mathlearn.html
- plumeria 5y ago
  This other link doesn't work as well.
- MayeulC 5y ago
  > (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...

kamaal 5y ago

>> 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.

hyperpallium2 5y ago

> 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.

lordnacho 5y ago

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.

Rainymood 5y ago
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-of-learning-competence/ https://staciechoice1010.wordpress.com/2014/08/15/illusions-...

j7ake 5y ago

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.

saasaccount2 5y ago

This approach seems to be quite innovative.

graycat 5y ago

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.

sureglymop 5y ago

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.

Cthulhu_ 5y ago

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.

jimmyvalmer 5y ago

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.

timwaagh 5y ago

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.

pas 5y ago
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...)
cratermoon 5y ago
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 https://doi.org/10.17763/haer.81.4.t2k0m13756113566
- timwaagh 5y ago
  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.
  - cratermoon 5y ago
    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.

k__ 5y ago

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).

zaik 5y ago
> 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_correspondence https://en.m.wikipedia.org/wiki/Curry%E2%80%93Howard_corresp...
- k__ 5y ago
  Now, how can I use this fact, to finally become a decent math user? :D
randomswede 5y ago
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.
- k__ 5y ago
  Fair enough! Good points, thanks.

jkhdigital 5y ago

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

treeman79 5y ago

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.

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