[l_t]

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]

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/ 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]

> 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.

[l_t]

> 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.

This is a great point and example of the problem with a one-size-fits-all strategy. For some books, exercises are an essential part of comprehension. For others, not so much.

> 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.

My experience is that by taking excellent notes and asking why, you engage with the material to a similar degree, if not a greater degree, than by doing exercises. (Once again, depending on the book, as you mentioned.)

> Ignore the exercises if you want, but you almost certainly will end up knowing about the math, but not able to do it.

I would argue that's the point. Usually self-taught math is about self-growth. Getting new ideas, being exposed to new concepts, recognizing patterns. Being able to actually "do it" on-the-spot is beside the point (and is the quickest level of skill to evaporate once you stop focusing on that material, anyway.)

[jjgreen]

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 ...

[throwlaplace]

>Ignore the exercises if you want, but you almost certainly will end up knowing about the math, but not able to do it.

Isn't that literally exactly what I said?

> 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

[ColinWright]

The submission and this entire thread is about learning math. That, to me, implies learning to do, not learning about. Yes, you said:

> 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

There's ground in the middle, and this thread is about that. This thread is not about learning for tests and qualifications, nor is it about "being exposed", it's learning how to do the math.

And for that you need to do the exercises. You don't need to do all of them, you don't need to be completionist about it, but if you don't do the exercises, if you don't actually do the math then you won't actually be able to do the math.

Specifically, you said (quoting again):

> if you're ... just interested in learning ...

There's a difference between learning about and learning to do. If you meant just "learning about" then you are at odds with the entire thread. True, in that case you don't need to do the exercises, but I don't think that's what people are talking about here. I think people are talking about being able to do the math.

And if you meant "learning to do" then in my opinion you are wrong, and one needs to do a large slab of the exercises.

Otherwise it's fairy floss, and not steak.

My apologies if all this seems overkill, but there's a real danger of talking past each other and being in violent agreement, and I wanted to state explicitly and clearly what I mean, and why I thought you said something different.

[throwlaplace]

> you won't actually be able to do the math

but i'm not a mathematician. i don't need to be able to do math anymore than i need to be able to do history (while reading serious history books).

>And if you meant "learning to do" then in my opinion you are wrong, and one needs to do a large slab of the exercises.

no i didn't. that's precisely why i used the word "exposed".

>violent agreement

we don't agree but i'm not being violent. but my responses are short and yours are long.

i do not see the exercises as essential for anyone other than practicing mathematicians. i have read a great many serious math books (i just recently finished Tu's Manifolds book and am now reading Oksendal's SDEs). i read them without doing absolutely any exercises but following the rest of the guidelines in the post i responded to. the experience is gratifying because i learn about new objects and new ways of thinking about objects i've already learned about. that's absolutely the only thing that matters to me.

but let me ask you something

>That, to me, implies learning to do, not learning about.

here's a fantastic explanation of the topological proof of Abel-Ruffini

https://www.youtube.com/watch?v=zeRXVL6qPk4

would you say that I don't understand that proof if i haven't done any exercises related to it? and therefore would you say I didn't learn any math by having watched that video?

[ColinWright]

We agree that if you want actually to be able to do the math then you need to do the exercises.

Do we agree that if you don't do the exercises then you probably won't actually be able to do the math?

You are discussing learning about the math, and not eventually being able to do it, because you say that you don't care about becoming a mathematician, therefore you don't need to do the math. Fair enough.

But my reading is that that's not what this thread is about. This thread, and the original submission, is about learning how to do the math.

> i do not see the exercises as essential for anyone other than practicing mathematicians.

I think you're wrong. Knowing how to actually do the math has proven useful to many people for whom it is a tool in their craft/job/employment. Learning Linear Algebra properly, being able to actually do it rather than just talk about it, can be enormously useful in Machine Learning.

>> That, to me, implies learning to do, not learning about.

> here's a fantastic explanation of the topological proof of Abel-Ruffini ... would you say that I don't understand that proof if i haven't done any exercises related to it? and therefore would you say I didn't learn any math by having watched that video?

Understanding a single proof implies very little about one's ability to actually do the math. I've met many people who are math enthusiasts and who have watched hundreds of math videos. They say they understand all of what they've seen, and yet they are unable to do the simplest proofs, or the most elementary calculations.

My experience of people's abilities is that if they haven't done the exercises, they usually can't actually do the math.

But you complain about the length of my replies, so I'll stop. I think I've made my position clear, and I think I understand what you're saying, even if I don't agree with it.