> 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 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.