How do you read a math book?

[/0]

Math books are quite different from other books such as novels, as they condense a lot of often complicated information into a small amount of space.
I'm just wondering, what are everyone's strategies for reading and learning from math books effectively?



I used to skim through math books really fast and do exercises to actually learn the material, that's how I self-studied Methods 1/2. But now with the more abstract stuff there are a lack of examples which show you step-by-step what to do, and the depth of the material has me doubling back every time I try to read like I used to.
I guess right now I just stare at theorems for a long time. The longer I stare the more I absorb.

[Cthulhu]

Sometimes with great difficulty. If there is a huge wall of text I can't focus a lot of the time and lose interest. For maths I've always been a learn by example kind of guy but I've been trying to change this recently by referring back to the text quite a bit.

This is strange because with walls of text in physics books I'm fine.

[TrueTears]

i like reading maths books from head to toe (books like stewarts etc) because the order helps alot. i dont normally like to skip around and read different chapters, although i only do it for problem solving books like art n craft, the chapters are not interrelated that much and you can skip around freely.

[zzdfa]

Don't spend too long trying to understand something the first time through, just do some exercises and read ahead and the understanding will come. You will feel like a fraud doing this (doing exercises and proving things without complete understanding of what is happening) but it works!

Read blogs and papers too. They often provide more insight and are much more interesting than textbooks. (It is alot easier to stay focused and motivated on a 3 page blog post than a 500 page textbook).

[humph]

I generally read maths books by osmosis - if I read a theorem/definition enough times over some long period of time, then eventually I'll be comfortable with it. I always borrow several different books on the same topic so that I can look through all of them in order to find the approach that makes the most sense to me. If I really have to learn a topic then I might sit down with one book and read for as long as I can until I get bored/my mind wanders, which takes at most half an hour... And I definitely never bother doing exercises or anything, though if it's some new technique/concept I've learned, I might look through the exercises section just to check that I've learnt enough to at least understand the questions.
I do try to learn lots from random areas though - I check various blogs daily, as well as reading math.nt on the arxiv, and reading (and occasionally responding) on mathoverflow.

[Cthulhu]

Which blogs do you read humph? I only read Terence Tao's from time to time and it is way over my head.

[humph]

I read Gowers's as well (http://gowers.wordpress.com/), and my favourite is Emmanuel Kowalski's (http://blogs.ethz.ch/kowalski/) because it's closest to my area. Gowers's blog has a long list of other blogs that I check occasionally but a lot of them aren't that interesting. In any case, I find these days that I spend a lot more time (and learn the most) on mathoverflow rather than via blogs.

[/0]

And I definitely never bother doing exercises or anything, though if it's some new technique/concept I've learned, I might look through the exercises section just to check that I've learnt enough to at least understand the questions.

So.. if you don't do any of the exercises how do you prepare for exam-type questions? Do you just understand the material so thoroughly that it's not really necessary?
I would like to do more exercises if I could, except... the lack of solutions manual to pretty much every math textbook is a huge demotivator. I reckon the author's excuse that you'll just cheat yourself by looking at the solutions is bullshit. The real reason is so that teachers can pull assignment and exam questions out of the book. It's backward >_>

[Cthulhu]

the lack of solutions manual to pretty much every math textbook is a huge demotivator.

This. 100xThis.

[humph]

And I definitely never bother doing exercises or anything, though if it's some new technique/concept I've learned, I might look through the exercises section just to check that I've learnt enough to at least understand the questions.

So.. if you don't do any of the exercises how do you prepare for exam-type questions? Do you just understand the material so thoroughly that it's not really necessary?
I would like to do more exercises if I could, except... the lack of solutions manual to pretty much every math textbook is a huge demotivator. I reckon the author's excuse that you'll just cheat yourself by looking at the solutions is bullshit. The real reason is so that teachers can pull assignment and exam questions out of the book. It's backward >_>

Well yeah, I pretty much back myself to do well on the exams based on my understanding. Also most of my exams these days have been take-home exams, which means I usually have enough time to brush up on my understanding in an area if I'm a little rusty. In any case, I tend to find with higher-level maths courses that exercises aren't particularly useful, as by this point you're not so much rote-learning techniques (e.g. techniques of integration, to solve various differential equations, and so on) but learning concepts, which you then have to apply when you're asked to prove something.

[TrueTears]

Don't spend too long trying to understand something the first time through, just do some exercises and read ahead and the understanding will come. You will feel like a fraud doing this (doing exercises and proving things without complete understanding of what is happening) but it works!

Read blogs and papers too. They often provide more insight and are much more interesting than textbooks. (It is alot easier to stay focused and motivated on a 3 page blog post than a 500 page textbook).

yeah i agree with this so much, sometimes i understand how to apply things before knowing how they work, but naturally the understanding comes after a period of applying it.

however in most cases i do attempt to understand the proof before application, sometimes it works, sometimes it doesnt

[Martoman]

I need to know where things come from. Once I do I like to play around with it, fiddle with what ifs and constraints.