5 ms·

This is so difficult. I've been doing it off and on for twenty years and not made much of a dent in things. The hardest part I think is understanding and meas

This is so difficult. I've been doing it off and on for twenty years and not made much of a dent in things. The hardest part I think is understanding and measuring your progress. In school you've got exams and classmates to compare against, profs to talk to. Alone it's much harder. "Do I understand this well enough?" "Did I do the problems right?" (Especially with proof problems, how do you know you're right?). "I can work through some problems one by one, but it feels like something fundamental I'm missing. Am I, or is this chapter really just about some tools?" Then it's way too easy to say well I'm never actually going to use any of this so why am I doing it ... and take a few months off and come back forgetting what you'd learned.

polyphonicist 7y ago
Shared a similar thought here: https://news.ycombinator.com/item?id=22401420 https://news.ycombinator.com/item?id=22401420 Yes, it is really important to learn math with study-mates. Just like in code, we do reviews, in math too, we need someone else who can review our proofs. It is even easier to make an error in a proof and believe that something is proven when it isn't. A study-mate helps to prevent us from fooling ourselves.
l_t 7y ago
I've tried a few things recently that help with that: 1. Don't do exercises unless you want to. Completionism is a trap. 2. Take notes. Rewrite things in your own words. Imagine you're writing a guide for your past self. 3. Ask questions. Anytime you write something down, pause and ask yourself. Why is this true? How can we be sure? What does it imply? How could this idea be useful? 4. Cross-reference. Don't read linearly. Instead, have multiple textbooks, and "dig deep" into concepts. If you learn about something new (say, linear combinations) -- look them up in two textbooks. Watch a video about them. Read the Wikipedia page. _Then_ write down in your notes what a linear combination is. Anyway, everyone's different of course, but these practices have been helping me get re-invigorated with self-learning math. Hope they help someone else out there. I welcome any feedback! (edit: formatting)
- throwlaplace 7y ago
  this is excellent execellt advice. seriously anyone interested in learning math, chancing on this comment, should write it down. i wish i could upvote many more times. i have a bachelors in pure math and am 10 years out. i have time and again revisited things and didn't make good substantive progress until i came to these same exact conclusions. especially the part about skipping the exercises. if you're not trying to write a dissertation or pass a qual (and you're just interested in learning and being exposed) then you don't need to do them. a lot of exercises are a hazing ritual or imagined by the author to be a dose of bitter medicine (i'm looking at you electrodynamics by jd jackson) since they mistakenly believe all readers are formal students. the most important exercise is to mull over and consider what you're reading/learning. naturally dovetails in to asking question: what happens if i remove a hypothesis from a theorem, what happens if i add one, is there an analogy to another object/group/measure/etc, etc. also read multiple books (http://libgen.is/ http://libgen.is/ is your very very good friend and generous friend). a lot of math authors (no matter how esteemed they are) are terrible writers or make mistakes (look up errata for previous editions of your favorite book). the only thing i'd add is to learn to use LaTeX to take notes - it is much easier and faster and neater.
  - ColinWright 7y ago
    > this is excellent execellt advice ... especially the part about skipping the exercises. if you're not trying to write a dissertation or pass a qual (and you're just interested in learning and being exposed) then you don't need to do them I think this is deeply mistaken. In a well-chosen book, such as the ones in the submitted article, doing the exercises is not to test your memorisation, it's to develop your understanding. Math is not a spectator sport. Reading about math is fine, but it will not take root and develop unless you engage with it, and the exercises are the way to do that. Ignore the exercises if you want, but you almost certainly will end up knowing about the math, but not able to do it.
    - jjgreen 7y ago
      This, 100 times. Mathematical understanding can only be obtained by doing, fighting with the concepts, causing that pain that you get behind the eyes. Just reading the text will give you a surface knowledge, maybe enough to impress at interviews or parties, but nothing more ...

polyphonicist 7y ago

Shared a similar thought here: https://news.ycombinator.com/item?id=22401420 https://news.ycombinator.com/item?id=22401420 Yes, it is really important to learn math with study-mates. Just like in code, we do reviews, in math too, we need someone else who can review our proofs. It is even easier to make an error in a proof and believe that something is proven when it isn't. A study-mate helps to prevent us from fooling ourselves.

l_t 7y ago

I've tried a few things recently that help with that: 1. Don't do exercises unless you want to. Completionism is a trap. 2. Take notes. Rewrite things in your own words. Imagine you're writing a guide for your past self. 3. Ask questions. Anytime you write something down, pause and ask yourself. Why is this true? How can we be sure? What does it imply? How could this idea be useful? 4. Cross-reference. Don't read linearly. Instead, have multiple textbooks, and "dig deep" into concepts. If you learn about something new (say, linear combinations) -- look them up in two textbooks. Watch a video about them. Read the Wikipedia page. _Then_ write down in your notes what a linear combination is. Anyway, everyone's different of course, but these practices have been helping me get re-invigorated with self-learning math. Hope they help someone else out there. I welcome any feedback! (edit: formatting)

throwlaplace 7y ago
this is excellent execellt advice. seriously anyone interested in learning math, chancing on this comment, should write it down. i wish i could upvote many more times. i have a bachelors in pure math and am 10 years out. i have time and again revisited things and didn't make good substantive progress until i came to these same exact conclusions. especially the part about skipping the exercises. if you're not trying to write a dissertation or pass a qual (and you're just interested in learning and being exposed) then you don't need to do them. a lot of exercises are a hazing ritual or imagined by the author to be a dose of bitter medicine (i'm looking at you electrodynamics by jd jackson) since they mistakenly believe all readers are formal students. the most important exercise is to mull over and consider what you're reading/learning. naturally dovetails in to asking question: what happens if i remove a hypothesis from a theorem, what happens if i add one, is there an analogy to another object/group/measure/etc, etc. also read multiple books (http://libgen.is/ http://libgen.is/ is your very very good friend and generous friend). a lot of math authors (no matter how esteemed they are) are terrible writers or make mistakes (look up errata for previous editions of your favorite book). the only thing i'd add is to learn to use LaTeX to take notes - it is much easier and faster and neater.
- ColinWright 7y ago
  > this is excellent execellt advice ... especially the part about skipping the exercises. if you're not trying to write a dissertation or pass a qual (and you're just interested in learning and being exposed) then you don't need to do them I think this is deeply mistaken. In a well-chosen book, such as the ones in the submitted article, doing the exercises is not to test your memorisation, it's to develop your understanding. Math is not a spectator sport. Reading about math is fine, but it will not take root and develop unless you engage with it, and the exercises are the way to do that. Ignore the exercises if you want, but you almost certainly will end up knowing about the math, but not able to do it.
  - jjgreen 7y ago
    This, 100 times. Mathematical understanding can only be obtained by doing, fighting with the concepts, causing that pain that you get behind the eyes. Just reading the text will give you a surface knowledge, maybe enough to impress at interviews or parties, but nothing more ...