My wife was at the top of the class in
nearly every course she took in school,
K-Ph.D. I was at or near the top of
the class in material I liked, essentially
just math, physics, and chemistry. From
those two examples, here are some lessons
on how to make good grades and/or do well.

I will start with what my wife did: First,
care about making good grades; care a LOT.
Second,
actually do at least most of the reading
and homework. Take good notes in class and
study them. Work with other good students
in the class to share notes and thoughts on
the content of the course. It helps to have
a terrific memory, and for that it helps to
really care a LOT. It helps to cut out
nearly everything except course work and to
have the ability to do well with relatively
little sleep.

Then, with the basic
learning done, work with other good students
in the class to get a good view of what the
teacher likes to see on tests, term papers,
etc. I.e., figure out how to please the
teacher, i.e., 'read' the teacher. So,
before tests, work with those other
good students to write out likely test
questions and together work out
good answers. Then
on a test with, say, four questions, may
have written out good answers to two or
three of the questions the night before.
Tough to compete against that.

For the
other good students she worked with,
she had a room in the girls dorm
on the 'academic unit' with other
astoundingly bright girls, and they
shared draft test questions, etc.

Lucky
was not in a class with any of those
girls because would have come in at
best second. For making As in college,
those girls were fantastic.

Of course they were beyond belief in
the humanities courses, e.g., could
actually make some sense out of the
mush. But for a while my wife was in
pre-med so also made As in the
'filter' courses in organic chemistry
and comparative anatomy.

It also helps to have some just fantastic
academic talent: My wife wanted to take
a course in European history but did not
want to have to work to make a good grade
so just audited the course. The prof
asked audits also to take the tests, so
my wife did. At the end of the course,
a lecture course with 300 students, the
prof told my wife that she should have
taken the course for credit because she
made the highest score in the class.
And she didn't even seriously try.
Tough to compete against that!

For a term paper, she organized all the
information on index cards, arranged them,
then typed the paper.

She made high
school Valedictorian, in college was
'Summa Cum Laude', Woodrow Wilson,
PBK, and won two years of NSF graduate
fellowship in one award.

Then what I did: In K-8, the girls
were much better students than I was
in penmanship, spelling, writing,
memorizing poetry, artistic drawing,
working in groups, clerical accuracy,
etc., all as should be expected. Also
all the teachers were women and clearly
liked the girls much better than the
boys. So, I gave up on trying to
get good grades from the teachers
and just pursued what interested me.

I was so often treated with such
contempt by the teachers that
for all the rest of my time in
school I was unconsciously terrified
of criticism from the teachers. So,
if something didn't go just right
in a course, then I was terrified that
the situation was hopeless so gave
up. So, really, I could be comfortable
in a course only if the material was
pure math with a level of precision
about like that of Bourbaki so that
I could be iron clad sure that my
knowledge could not be questioned.
It was also good if I had carefully
read an excellent text before the
course and, thus, really already
knew the material. K-12
teachers: Quit hurting good
students.

Things made a big change in the
ninth grade in math: I liked the math
and really cared. I had no study
skills but began to develop some.
The teacher sent me to the state
math tournament; I was likely at the
top of the class. He realized that
mostly I was learning just from the
book and told me that for the tournament
I should learn the last two chapters
on trigonometry he was not going to
be able to cover in class, so the
weekend before the tournament I did.

That pattern went on: I really liked
math and learned mostly from the book.
I really made no attempt to get good
grades from the teacher, but I
understood the material so well I
got A or B from the teacher but
led the class or nearly so on
state standardized tests.

That pattern continued in college:

It worked out that I never took
freshman calculus! The college
I went to for my freshman year didn't
want me to start with calculus
but put me in some course beneath
what I'd already covered
in my high school (that had a
relatively good math sequence).
So, I showed up only for the
tests and otherwise got a good
calculus book and dug in.
For my sophomore year I went to
a much better school, with a quite
good math department, and just
started with their sophomore
calculus. Did fine.

In freshman physics, I really
liked the material and
led the
class, effortlessly.

It was nice:
Often I studied the physics in the
gorgeous
reading room of the library. There
some of the really pretty girls were wearing
some short, slightly full, plaid, heavy wool
'wrap around' skirts, each held together
with a huge, chrome diaper safety
pin! Still I got some physics done!
Got to really like physics to
learn under such circumstances!

Some of the more advanced physics
was badly taught
or from a poor book, and then my
grade fell to a B; I had no patience
with poor quality material. But in math
the books were much better, and
I did very well on both learning
the material and grades. I got
Honors in math and 800 on the GRE
test of math knowledge and got
sent to an NSF summer program
in axiomatic set theory, modern
analysis, and differential geometry.
The differential geometry was
lectures by a Harvard graduate
and student of A. Gleason. He
said that I needed only the
inverse and implicit function
theorems, but so far I'd not
seen either of those (I
later got them from Fleming's book).
I was
too intimidated to realize that
I could have hit the library for
an afternoon and evening and
walked out with good knowledge
of both theorems. The theorems
are just local non-linear versions
of the standard and fairly obvious
general solution of a system of
linear equations. There is a nice
proof using contractive mapping.
Alas, due to those two theorems,
I walked out of the class --
bummer, it was material I would
have liked to have learned, especially
for relativity theory.

In my career I continued a lot of
independent learning from some of the
best math texts, e.g., I took
a second pass through Rudin's
'Principles', went carefully through
Halmos's 'Finite Dimensional Vector
Spaces', Fleming's 'Functions of
Several Variables', the math
parts of von Neumann's 'Quantum
Mechanics', and much more, a big
stack more.

When I went for my Ph.D., what I
had learned before I entered was
nearly enough for the course work.
For the research, I brought my
own problem with me to graduate
school, had an intuitive solution
I'd worked out on an airplane
flight, got enough math in my
first year to turn my intuitive
solution into some solid applied
math, later wrote some corresponding
illustrative software, and that
was my Ph.D. research. For a
Master's, there was a question in
a course without an answer. I thought
for two weeks in the evenings and
saw a first solution and asked for
a 'reading course' to attack the problem.
When the course was approved, I gave
my first solution right away. Two
weeks later I had a much nicer solution,
wrote it up, and that was the end of
the 'reading course' and the last I
needed for a Master's. Later I published
the paper. When I published I did some
more library work and discovered that I'd
invented a theorem comparable with the
classic Whitney extension theorem, that is,
H. Whitney long at Harvard. I also
discovered that I'd solved a problem
stated in the famous Arrow, Hurwicz,
Uzawa paper in mathematical economics.
So, that work I did in that 'reading
course' was publishable.

Eventually
I concluded that working the more
challenging exercises in the best
pure math texts -- Rudin, Halmos,
Royden, Neveu, etc., is good training
for doing original research. Eventually I
discovered that if not afraid of
being whacked in the neck with a bad
grade by a prof in a course, then
usually can get what need for
a given research problem from
stacks of books and papers much
more quickly than the rate of
coverage in a course.

Eventually discovered a 'way'
to do research: Do a lot of
intuitive guessing; then test
the intuitive guesses with
some intuitive filters or
checking on special cases.

For a simple outline,
"Is A true? Okay, likely
if A is true, then B is true.
Is it believable that B is
true or would that be asking
too much? Naw, likely B is
false. So likely A is false.
So, check A on some simple
special cases. Okay, still
A seems false. So, try
C. Is C true? If C is
true, then likely D is true.
Okay, maybe D is true. Check
out C on some simple special
cases. C might be true! So,
maybe try to prove C is true.
Now to prove C is true, likely
the proof has to make essential
use of all the hypotheses we
have for C, so in looking for
a proof, be sure can make good
use of all the hypotheses.
Else are trying to prove something
stronger than C which is likely
not true. Now, if C is true,
just what, intuitively, is going
on?"

Can do a lot of this while
writing little or nothing.
When the intuitive work seems
good, then try to write out
some actual math with careful
derivations, any new definitions
and then theorems and proofs.
If the intuitive guessing goes well,
then have a good shot of seeing
how to do the proofs right away.

Mostly don't write longer than trivial
algebraic derivations
without having a fairly good idea
what is going on intuitively. That is,
mostly don't expect to get much just
from pushing symbols around.

In nearly all of K-12 and college, what
my wife did worked much better than what
I did. For graduate work, a Ph.D.,
publishable original research, and
applications in a career, what I did
worked much better.