
How to read a mathematics textbook (2016) - Tomte
https://www.drmaciver.com/2016/05/how-to-read-a-mathematics-textbook/
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joker3
As one of my grad school professors used to say, the proper way to read a math
book is with pencil in hand. If you don't completely understand something,
take the time to work it out. Don't settle for good enough, either--dot all
the i's, cross all the t's, and get the details right.

The suggestion to take material out of order is a bit of a mixed bag. A
typical book will have a few core chapters, and some extra topics later on.
Most chapters will follow a similar outline. You can skip the optional
chapters and the optional sections within chapters, but if you skip chapter 2,
you probably won't be able to follow much of chapter 3.

One last comment: there are huge variations in how well different books are
suited for self-study. Rudin's "Principles of Mathematical Analysis" is a
collection of true statements with zero motivation or exposition, and it's
impossible to follow on your own. On the other end, something like Herstein's
"Topics in Algebra" clearly spells out the reason for everything and is a joy
to read. Most books are somewhere in between, so if a certain book isn't
working for you, try something else on the same topic.

~~~
aportnoy
> if a certain book isn't working for you, try something else on the same
> topic

I spent a year with Baby Rudin in college, what would you recommend as a
replacement fit for self study?

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joker3
I don't think there's an exact replacement. Strichartz seems like a very
readable book, but it's at a very slightly lower level and I think you'd
occasionally have to supplement with Rudin.

~~~
aportnoy
Thanks.

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rofo1
This is a very underrated comment:
[https://news.ycombinator.com/item?id=18055053](https://news.ycombinator.com/item?id=18055053)

I've came to this conclusion on my own years ago.

I wholeheartedly agree that there are books whose authors don't understand the
subject.

There is no other explanation for writing books so badly.

I keep thinking that if you know a subject well enough, you'll surely be able
to write a comprehensible book on it, be it maths or anything else. They say
that Feynman said something similar: "If you can't explain this to kids, you
don't understand it well enough". Not sure if I agree completely with that
cause advanced math. subjects typically require prior knowledge (just the way
it is, no way around it), but the _concepts_.. perhaps you _can_ explain them
to a certain degree. Some times, at least in maths, older literature is
written in such a better way, it's unbelievable. Modern literature is terse
and dry. This is the skeleton: -definitions -some axioms that we take for
granted -theorems based on the previous two -and (usually) the simplest
examples the authors can find

And then a list of exercises. Some of them extremely simple, some of them
postdoc. research (say, Engelking's topology exercises - if you can solve them
all, you can probably publish double digit amount of papers)

There's usually no explanation, no historic importance of certain results, no
motivation for what truly moved the mathematician to discover/invent that, and
so on.

It's like we don't get to the bottom of things, only to the surface..

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bmurray7jhu
I think the writer is understating the importance of the exercises. In
graduate level math textbooks, the exercises frequently serve as a truncated
exposition of related topics and concepts.

~~~
nanolith
This is true. Complete understanding rarely occurs until one can understand
how to wield this knowledge.

~~~
thomasfortes
My calculus teacher used to say that you learn calculus through your wrists,
when you read the theory you get a grasp of how to solve some problems, when
you learn to solve the problems is when you really can start to understand the
theory.

The same is valid for writing software, reading code is great, but you need to
write a lot to learn how to program.

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philhartmanonic
My thing is the symbols. Stopped studying math in college, and every time I've
tried to get back into it, seems like my options are either starting over from
4th grade or make sense of a formula that appears to be written in some form
of ancient hieroglyphics. I think I can grasp the concepts if I could
understand the syntax, but the best source I've found is the Wikipedia page on
math symbols, which is little more than a list (and quite a few have a good
half a dozen meanings).

It'd be cool if there was a good book focused on just understanding the
symbols in as plain english as the concepts allow, and teaching you what
context you need to look for when the symbols have multiple potential
meanings. Maybe there is a book like that (and if so, please let me know, as
that would be super exciting).

~~~
nerdponx
What's an example of a subject you struggled to learn? In a good introductory
calculus book you shouldn't see anything more outlandish than an integral.

~~~
kbwt
Not OP but you could write a whole book on "what exactly does dx (and its
variants) mean and why can you sometimes manipulate it algebraically?"

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mjfl
Step 1 to reading a mathematics textbook is finding 4 hours per day to work
through it. Most people cannot do that.

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threatofrain
I think even an hour a day is enough to concentrate on an author's discussion.

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your-nanny
I nearly failed calculus until I took a couple weeks off and read the damn
book. By read I meant: start from the beginning, skip nothing. Skip no
exercise, skip no example, skip nothing and only continue when you've
understood the chapter's material. I got a hundred on the final. Just school,
but do wish I had that kind of time.

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atpk
I'm curious if any of you full-time tech worker folks have ever read a
significant portion of a math textbook and done the exercises just for
fun/fulfillment. I've tried starting Concrete Mathematics myself a few times
during multi-day breaks from work, and it's tough to keep it up when work
starts back up again.

It's frustrating to me that it seems like this window during which you can get
to some of this knowledge (college) closes and the to lack of a teacher, the
structure provided by a course, and sufficient time seem to be unavailable.

~~~
closed
Last year I read a book on combinatorics [1], and did a fair number of
exercises each chapter. Normally, worked on it for an hour in the morning, and
then 4 - 8 hours a week spread over nights / weekends.

Looking back, it was pretty taxing--I think if I'd been meeting with a tutor
once a week it would have been a lot easier!

[1]: Basic Techniques of Combinatorial Theory, by Daniel Cohen

~~~
bordercases
How long did it take for you to finish the book?

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annadane
Is it also reasonable to include "Read good textbooks"? Some books are just
bad because the author is bad at explaining...

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bangonkeyboard
Or because the author does not actually know the subject:
[https://twitter.com/TheRealDrMcCoy/status/104379662417314201...](https://twitter.com/TheRealDrMcCoy/status/1043796624173142018)

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dragon96
The key to reading a math textbook is learning to be generative. In other
words, it is not enough to be able to _verify_ a proof, but to be able to
_construct_ the proof yourself. One of the most common failure modes is
reading a textbook and nodding along as the author presents proof after proof.
Most theorems, proofs, and definitions will sound similar to each other or
give a tautological "why is this even a theorem" vibe [1], and reading by
verifying is how to fall into this trap. (Cue a metaphor about NP-complete
algorithms being able to verify a proof but (probably) not able to construct a
proof in polynomial time.)

This is why, like other comments have noted, doing exercises is so important.
However, this is not the only way to train "generativity". I read textbooks
very slowly by reading theorems and constructing the proof myself. (Hard mode:
don't read the theorems and try to guess the next theorems and lemmas.) I like
this approach because you can get feedback afterwards by reading the solution,
while most textbooks don't have solutions for their exercises.

It sounds like OP's strategy is another approach for being generative that
I've yet to try myself. I like his approach because it seems to be more
effective at filtering out a lot of the noise from linear reading and,
instead, focusing only on the results and definitions that end up being used
later. But no matter the technique, it seems that the common theme is to spend
more time staring at your scratch paper than at the book.

[1] [https://xkcd.com/2042/](https://xkcd.com/2042/)

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zwaps
Maybe its me being dumb, but I don't get the description of what the author
actually does. I guess he picks some random theorem and then has two lists?

~~~
bzalasky
Here's the gist of it as I read it... the current list is a topic you want to
learn, and as you're working on it, everything that you don't understand is
added to the current list. As you check off the subtopics (and their
subtopics), you'll have the requisite knowledge to understand the topic you
wanted to learn in the first place. The pending list is stuff that will become
the first item at the top of your next current list.

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bordercases
For an even more demanding programme: www.topology.org/tex/conc/mathlearn.html

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sevensor
Looks interesting, but I'm not clear on the context. It seems like the
author's program is very ambitious -- why has he undertaken it?

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j2kun
A bit of shameless advertising: for those software engineers interested in
learning mathematics, I am _very_ close to publishing a book called "A
Programmer's Introduction to Mathematics." I hope it will accommodate readers,
specifically programmers, in a way that most mathematics books usually do not.
After reading it, you should know how to read a mathematics textbook, and have
a baseline of background knowledge to make the rest of mathematics more
accessible.

Sign up here: [https://jeremykun.com/2016/04/25/book-mailing-
list/](https://jeremykun.com/2016/04/25/book-mailing-list/)

Publication ETA is end of this year.

~~~
joker3
Is there a table of contents anywhere?

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j2kun
I should post one soon... good suggestion

