Reading Math Textbooks
Katie88265
have to read a book until you have a good understanding of it. Is once pretty
much always necessary? Can you always complete all of the exercises (do you
know it's right?)?
I used to think that I was pretty good at math but it has gotten to the point
where I can't read it once and have a good understanding of what I just read.
At times I get to exercises and don't see how to do them or go about it the
wrong way when a problem should actually be really easy. I'm still struggling
with proofs at times. I would guess if you're really talented at something, it
would all just come really easily. Am I wrong?
Will Self
"Katie88265" <katie...@aol.com> wrote in message
news:20030624214805...@mb-m05.aol.com...
It doesn't come easily. Hilbert said that genius is 9 parts (out
of 10) "industry" (meaning hard work). And Edison put it more
amusingly, genius is 1 percent inspiration and 99 percent
perspiration.
It's not so much that you need to read a text over and over. It's
that you need to slow down (way down) and understand each sentence
before going on to the next. That's an ideal, of course, but much
of the time that is going to be what works. I learned to read
that way with Herstein's Topics in Algebra. It was a whole new
world to me.
You can also do as I did as an undergraduate trying to read
Rudin's Principles, draw little pictures of H-Bombs in the
margins. Or a friend of mine in graduate school -- throw the
book at the wall.
Most good texts have exercises that range from pretty accessible
to quite challenging. In some texts, you wouldn't be out of line
to spend days on a single exercise.
Chergarj
This can't be genuine questioning. Math is not natural human language. The
student must reread and reexamine groups of low level information and facts
which he would not ordinarily apply. The student will usually reread some
parts or some pages as much as 10 times and reexamine some certain exercise
problems to decide what the first few steps may be for the exercise problems.
Very few people are so adequately designed as to be able to just read the book
once and so quickly know how to do all the problems.
If that question set was truly your genuine inquiry, then you just entered an
area of Mathematics much more rigorous than what you have been accustomed to.
Stay in the study! This is powerful stuff. Learn as much math as you can
handle.
G C
David C. Ullrich
Heh-heh. I'm scheduled to teach the intro to epsilon and
delta class next semester - mind if I quote this as one of
the things to try doing different if things don't seem to
be working?
>Or a friend of mine in graduate school -- throw the
>book at the wall.
>
>Most good texts have exercises that range from pretty accessible
>to quite challenging. In some texts, you wouldn't be out of line
>to spend days on a single exercise.
>
************************
David C. Ullrich
Randy Poe
> This might be a dumb question but I was wondering how many times most of you
> have to read a book until you have a good understanding of it. Is once pretty
> much always necessary?
It's not usually a question of number of times for me, but how
much effort I invest when I do read it. If I really want to
get a good understanding, I need to have a tablet of paper
next to me, and be willing to stop every time the author
says "it is simple to show that..." and actually spend the
effort (it goes without saying that it is almost never
"simple") to verify what the author is saying.
In other words, I can't absorb math by reading it. I have to
be writing it, alongside the author.
> Can you always complete all of the exercises (do you
> know it's right?)?
No. And I rarely take the time. Good exercises can be
an entire evening's effort apiece. I just don't have that
kind of time.
I usually don't read technical books in order like a novel
anyway, but skip to the chapter I am most interested in,
the one that made me buy/borrow the book. If Chapter 9
refers to something important in Chapter 2 that I need
to understand, then I'll go back and work though Chapter 2.
> I used to think that I was pretty good at math but it has gotten to the point
> where I can't read it once and have a good understanding of what I just read.
> At times I get to exercises and don't see how to do them or go about it the
> wrong way when a problem should actually be really easy. I'm still struggling
> with proofs at times.
Struggling with proofs is a good way of learning your own
"aha" process, something that you will need to know about
yourself if you have a problem-solving career. Generally
mine involves scribbling various futile attempts for hours,
then finally giving up and walking away. When the "aha"
moment comes, it is almost always on the way back to work,
especially when driving or walking from the parking lot,
but sometimes when lying in bed in the morning too.
After the "aha" moment when the structure of the solution
becomes crystal clear, it may still take pages of calculation
and hours of writing to work out the details.
All of it: the head-pounding (which might be spread over
days or even weeks), the walking away, the pondering while
driving, is necessary. Somehow a critical mass of head-pounding
is needed to fuel whatever the intuitive engine needs to
finish the job.
> I would guess if you're really talented at something, it
> would all just come really easily. Am I wrong?
Yes, dead wrong. Concert pianists spend many hours every
day practicing. If it all came easily, they wouldn't need
to practice.
- Randy
Michael N. Christoff
I would have to say yes, you are wrong. I often have to read a text many
times through especially if there are a lot of details and variables and
definitions. After each reading, you begin to get more familliar with the
details until you could pretty much rewrite the proof yourself. Once you've
fit all the details in your head (all at the same time! :) ) then you can
really start to make sense of what's going on.
One thing that I have learned is that things make a lot more sense when you
put them in some sort of context. Try and get the gist of what it is you're
trying to solve. Sometime author's will under-explain something because
they believe that what they say, although not understandable in isolation,
makes sense in the context of the overall goal, or statements made earlier
on in the proof. Sometimes it takes a few readings for that context to
materialize in your brain. Once you get it, you have the 'Ohhhhhhhhhhh!
Now I know what s/he meant by that!' moment.
As a side note, I recently tried to do a theoretical computer science proof
that an oracle C exists such that NP^C != coNP^C (what that means exactly is
irrelevent). I finally solved it, but only after two false starts. Each
time I got a little bit closer. ie: the answer certainly didn't jump off
the page for me. (Unfortunately noone is willing to check my proof :( But
the point is, don't give up!
I also agree with what others have written on this.
l8r, Mike N. Christoff
Chan-Ho Suh
> This might be a dumb question but I was wondering how many times most of you
> have to read a book until you have a good understanding of it. Is once pretty
> much always necessary? Can you always complete all of the exercises (do you
> know it's right?)?
Let me make a distinction between two kinds of understanding a math text.
The first is in being able to follow, reconstruct, etc. a proof (proofs
make up the bulk of most math texts). The second lies in rewiring one's
brain so that it's obvious why the theorem is true, and indeed, there is
no way the theorem could be false in the context of all your knowledge.
Once one reaches this stage, the text's proof may in fact appear tedious,
redundant, etc.
Contrary to impressions you may have garnered from your math courses, the
first kind of understanding is not the primary goal for mathematicians,
but the second kind definitely is.
The first is usually easier to come by. In response to your query, I
should say that personally I usually need only reread a proof several
times before being able to follow the logic. Some advice: when you read,
dissect every sentence; math text tends to be concise, saying only the
bare minimum needed to impart the information.
So you see, reading a proof even once, depending on the "newness" of the
concepts involved and the attention to certain technical details needed,
can take a very long time, stretching over days, weeks, sometimes even
months. Of course, in the extreme cases, one usually has an outline or
something that the author or someone else was kind enough to provide, so
one doesn't really read the proof linearly, but in sort of patches.
Sometimes you have to be smart enough to realize when *not* to read!
Especially by the time you get to the stage where you are reading research papers, many
times you don't really need to read the author's proof, but by working at
procuring the deeper, second kind of understanding, you can work out your
own proof, using knowledge that you are more comfortable with than perhaps
what the author is comfortable with. So when you are trying to understand
your math text, it pays to stop reading, and just try and piece things
together on your own. You may very well save yourself some reading time,
always a good thing in my book.
In the end, understanding mathematics is a tough business, and can't be
measured by something like how many times one has read something. In my
experience, if I understand something fairly well, I may have great
difficulty reading a proof or exposition of it. Primarily because the
writer's own personal understanding differs greatly from mine. Perhaps
this points out what we are trying to do as mathematicians: understanding
something means making it yours. But once it becomes a deeply rooted part
of your own experience, it may become even harder to relay this experience
to others.
Stan Brown
>I used to think that I was pretty good at math but it has gotten to the point
>where I can't read it once and have a good understanding of what I just read.
It's not that you've grown stupid, it's that you are now tackling
harder topics. So it's natural to have to work harder to master
them.
Unless the material is very far below your level, it's impossible to
read any technical book the way you would read a novel. You need
pencil in hand, and to work through all the examples on your own.
Half an hour per page is a reasonable target for many students if
they are taking material that is a reasonable level of challenge for
them.
You might like to read some hints I wrote for my students:
http://www.acad.sunytccc.edu/instruct/sbrown/math/read.htm
--
Stan Brown, Oak Road Systems, Cortland County, New York, USA
http://OakRoadSystems.com/
"Walrus meat as a diet is less repulsive than seal."
-- Harry de Windt, /From Paris to New York by Land/ (1904)
Katie88265
pretty easy for us to do well in, say, a history class. I am a math major and
he is a computer science major. It seemed easier than our major classes. I
just wanted to know if that was a 'sign' that I just wasn't that good at math.
I haven't gotten too far in terms of classes. I took my first number theory
course this past year. It was my first major exposure to proofs. I had always
been able to sort of read through the book prior to that and get a good enough
understanding to do well on exams. That was just my experience. Number theory
was much more difficult. Sometimes my eyes would glaze over when I was reading
the proofs in the textbook.
Thanks for all of the responses. The webpage was helpful.
Karl M. Bunday
news:MPG.1965220bd...@news.odyssey.net, replying to the original
poster,
> You might like to read some hints I wrote for my students:
> http://www.acad.sunytccc.edu/instruct/sbrown/math/read.htm
Those tips and the linked pages you wrote are very helpful. I will share
them with some local parents whose children start the UMTYMP program at the
U of MN
http://www.math.umn.edu/itcep/umtymp/
this September. Thanks for putting such useful advice on the Web.
--
Karl M. Bunday "Christ has set us free." Galatians 5:1
Learn in Freedom (TM) http://learninfreedom.org/
kmbunday AT earthlink DOT net (preferred email address)
Kent Paul Dolan
> I'm still struggling with proofs at times.
> I would guess if you're really talented at something, it
> would all just come really easily. Am I wrong?
You've gotten lots of good answers, let me add a different
one.
I've found excellent computer programmers, and excellent
mathematicians, tend to turn their studies into language
exercises, and to create little neologisms or canned phrases
that let them somewhat "automate" thinking about a proof.
The obvious example is "given any epsilon, there exists a
delta"; just that much lets you get through lots of
beginning calculus. Letting each "aha" continue percolating
until it produces its own catch phrase that captures it for
your use "forever after" can add internal wiring to your
thinking that leads you more often than not down the quick
path to the "aha" for the next proof. Listen to any
undergraduate trying to cope with all the well known ways to
solve integral equations to hear examples.
xanthian, who learned this _listening_ to former co-worker
Terry Lambert write software; once I realized that the
muttering he was doing was an essential part of why he wrote
code all of better, faster, more accurately, and with less
pain than I did, I started listening a _lot_ more
attentively, trying to learn to imitate his success.
[Another part, of course, of why that was happening is that
he's much smarter than I am, including smart enough to
invent his own concise language as he goes along.]
And also:
Due to the way textbooks are written, if you are having
trouble with a proof, a good first step is to review the
immediately preceding material, backwards in chunks page by
page, section by section, chapter by chapter.
Very likely, since good textbook authors try to build on
stuff just learned, some part of the immediately preceding
stuff is needed but not yet in your active memeset. If you
re-read it, more than likely it will not only become clearer
than on the prior pass, but also right then and there
suddenly voice its applicablity to the problem on which you
are currently stuck.
Stan Brown
<u3RKa.14618$C83.1...@newsread1.prod.itd.earthlink.net> in
sci.math, Karl M. Bunday <kmbu...@earthlink.de.net.de> wrote:
>"Stan Brown" <the_sta...@fastmail.fm> wrote
>> http://www.acad.sunytccc.edu/instruct/sbrown/math/read.htm
>
>Those tips and the linked pages you wrote are very helpful.
Thanks very much. Any suggestions for improvements will of course be
very welcome.
I was inspired by an old article in /Primus/ by Keith & Cimperman
called "The Hidden Script". With their permission, I've put part of
their article on the Web as "Math Students' FAQ" at
http://www.acad.sunytccc.edu/instruct/sbrown/math/faq.htm
Shmuel (Seymour J.) Metz
at 01:48 AM, katie...@aol.com (Katie88265) said:
>This might be a dumb question but I was wondering how many times most
>of you have to read a book until you have a good understanding of
>it.
There is way too much variability to give a numerical answer. The
amount of time to read a book depends on the book, the level of the
reader, the temprament of the reader and other factors.
In my case, I could read, e.g., "Finite Dimensional Vector Spaces"
like a novel, but when reading a book on Algebraic Topology I would
slow to a crawl whenever combinatorics were involved.
>Can you always complete all of the exercises
Absolutely not - some texts have exercises of various degrees of
difficulty. Most text book will have notes for the exercises that are
too hard for the average student.
In general, I look for exercises that are interesting in some way.
They may be neither very easy or very hard, but they catch my eye. I
also do any exercises that the author labels as essential. Most text
books have enough exercises to allow the instructor to pick and
choose; the student is not normally expected to do all of them. When
in doubt, read the author's notes.
>At times I get to exercises and don't see how to do them or go about
>it the wrong way when a problem should actually be really easy.
That could mean that you don't understand the material, or that the
exercise is harder than it looks. It could also mean that you need to
take a break, relax and then read the exercise with a fresh eye.
>I'm still struggling with proofs at times.
That's not necessarily bad, depending on what percentage you have
problems with. If none of the exercises is challenging then the book
is at too low a level for you. Maybe the material really is beyond
you, but don't assume that to be the case just because you're
struggling. You might also try reading a different text book and see
whether it is any clearer to you.
>I would guess if you're really talented at something, it would all
>just come really easily. Am I wrong?
If by "really talented" you mean like Gauss, then certainly. For most
of us, we have to work to understand everything.
>Am I wrong?
Perhaps too hard on yourself. I suggest talking to your faculty about
the material that is giving you trouble; it could well be material
that gives everybody trouble. Have you talked to your instructors or
their graduate assistants about your frustration?
A word of warning; just because other students find one book easier
than another doesn't mean that you will. You need to judge what works
for you.
--
Shmuel (Seymour J.) Metz, SysProg and JOAT
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