[graycat]

No, the OP is giving bad advice.

Reading good foundational text books carefully is darned good advice. But for solving every exercise before moving on, no, that's not a good idea. Instead, be willing to be happy solving some 90-99% of the exercises. For the rest, guess, with some evidence, that they are incorrectly stated, out of place, just too darned hard, or some such. If insist on solving 100%, then get on the Internet and look for solutions.

Next, if read some foundational text books, then in each subject also read several competing text books, perhaps just one mostly but also look at least a little at the others for views from 'a different angle' that can be a big help. Why? Because likely no text book is perfect and, instead, in some places is awkward, unclear, misleading, clumsy, etc. So, views from a 'different angle' can make it much easier to learn both better and faster.

His description of doing applications by just getting what really need and forgetting the rest can be done but is not so good. Instead, having a good foundation helps a lot. And, commonly for an application in an important field, there really is some good material in that field that should understand with the application. Else risk doing the application significantly less well than could have.

His description from Wiles is more or less okay for doing some research but, really, not for learning. And for research, more of a 'strategic' overview, i.e., with the 'lay of the land', would be good, i.e., for publishing not just one okay, likely isolated, paper but a series of better papers that yield a nice 'contribution'.

[nbouscal]

You misread the article. He is not recommending that students solve every exercise; he's recommending the exact opposite.

[graycat]

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.

I'm correct.

[nbouscal]

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.

[j2kun]

For the record, I'm saying two things:

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.

[graycat]

You wrote:

> 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.

[drivers99]

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.

[graycat]

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.

[GFK_of_xmaspast]

Come on, dogg, re-read the article and see what he's actually arguing.