No, I'm correct: He set up an extreme straw man
to knock it down. I clearly agreed that his
extreme straw man is foolish. There is a common
reason students fall for his straw man: They are
concerned that if there is an exercise they can't work
they are missing something important. My advice was,
instead, for a very diligent student, to solve 90-99%
of the exercises and just let go of the last few as
illposed, stated in error, out of place, use the Internet,
etc.
To do just the "opposite" of his straw man is not good --
for solid foundational material, Halmos, Rudin, Royden,
etc., the exercises are darned important. Right the
Rudin exercises where have to consider uncountability
are not so good. The Royden exercises on upper and lower
semi-continuity are a lot of work for a little curiosity
but likely won't see again. The Fleming exercise on
every bounded linear functional on a intersection of
finitely many closed half spaces achieves a maximum
value is mis places. Etc. The abstract algebra book
I had had an exercise where the student had to reinvent
Sylow's theorem; a student wrote the author and got
back a letter that the purpose of the exercise was
to see if a student could reinvent Sylow's theorem --
bummer, misplaced exercise.
This is a good example of how being correct is completely irrelevant if you can't communicate it well. That said, I still maintain that you're thoroughly misunderstanding the position the OP was arguing for.
1. I hear about lots of people trying to do every exercise.
2. I think this is bad if it causes them to quit out of frustration.
3. I provide some tips on how to deal with frustration, and to know that you're in good company.
> I still maintain that you're thoroughly
misunderstanding the position the OP was arguing
for.
Okay, let's see. The OP wrote:
> The second approach is to try to understand
everything so thoroughly as to become a part of it.
In technical terms, they try to grok mathematics.
For example, I often hear of people going through
some foundational (and truly good) mathematics
textbook forcing themselves to solve every exercise
and prove every claim “left for the reader” before
moving on.
> This is again commendable, but it often results in
insurmountable frustrations and quitting before the
best part of the subject. And for all one’s desire
to grok mathematics, mathematicians don’t work like
this! The truth is that mathematicians are
chronically lost and confused. It’s our natural
state of being, and I mean that in a good way.
So he has "forcing themselves to solve every
exercise and prove every claim 'left for the reader'
before moving on.".
So, clearly OP and I agree that this "forcing" is bad.
So, here with "forcing" and "every exercise" the OP
was mentioning an extreme case, yes, one that too
many students fall for, but one that both OP and I
agree is bad.
My view is that OP is inserting this extreme case to
have a 'straw man' to knock down so that he can
propose something else.
For his straw man "forcing" case, a better response
would be that it can be okay for a good, diligent
student with high standards to work 90-99% of the
exercises as I wrote. Leave out a few exercises in
case some exercises were stated in error, are out of
place, that is, given before material needed for a
solution, are just too darned difficult, etc. But
OP didn't mention this approach.
Instead OP went on with "The truth is that
mathematicians are chronically lost and confused".
Here, no: A student working carefully through good
material is mostly not lost or confused, certainly
not chronically. OP wants to set up and then knock
down his straw man to propose that students should
feel "chronically lost and confused" which, for
students working carefully through good material, is
just not true and "bad advice".
In research? Sure, lost and confused might be one
description: That is, once understand, i.e., find
the light switch in the sense of Wiles, then move on
to more where are lost and confused again.
I don't think it's an extreme straw man, because I saw notebooks of my math teacher in high school where he decided to take classes and did exactly that, just to make sure he knew it really well. Just the other day I was wondering if I should try to do the same thing in order to get good at math.
It is extreme; too many people do do it, but I was
warning them that they shouldn't do it. Why? It's
a super good way to get totally stuck-o and give up,
all for no good reason at all. And if work on such a
tough exercise full time for two weeks, as I did a few
times as a ugrad or in some of my later independent study,
then that's mostly a waste of time.
Although too many students do this, for the OP it was a
straw man to have something to knock down so that they
could say something else. Not good. The exercises are
one of the best aids to a good student trying to learn;
but, can do well working 1/3rd of them, or half, or the
more difficult 1/3rd, or 90-99%, but on that last 1%
can spend more time than on the first 99% and just
shouldn't do that. Don't worry: In any decently well
written book, if get 90% of the exercises, then have
done well. If really insist on the last 1%, cover the rest of the
book and then come back with the additional understanding
and maybe some crucial results didn't have the first time
through. If really want to know the material well, then
get 2-3 competitive books and work through those also.
To do just the "opposite" of his straw man is not good -- for solid foundational material, Halmos, Rudin, Royden, etc., the exercises are darned important. Right the Rudin exercises where have to consider uncountability are not so good. The Royden exercises on upper and lower semi-continuity are a lot of work for a little curiosity but likely won't see again. The Fleming exercise on every bounded linear functional on a intersection of finitely many closed half spaces achieves a maximum value is mis places. Etc. The abstract algebra book I had had an exercise where the student had to reinvent Sylow's theorem; a student wrote the author and got back a letter that the purpose of the exercise was to see if a student could reinvent Sylow's theorem -- bummer, misplaced exercise.
I'm correct.