In high school, a relatively good high
school (public, but my year MIT came
recruiting, and the year before three
guys went to Princeton and ran against
each other against some fourth SOB for
President of the freshman class), I took
four years of math. Since my brother was
in a private college, to save money I
did my freshman year at a not very good
state college. There they wouldn't let
me take calculus and, instead, forced me
into some 'college algebra' -- from my
four years in high school I already knew
99% of it. So, a girl told me when the
tests were, and I showed up only for those.
Teacher said I was "The best math student
I've ever had.". Well, maybe, but the
main reason was I'd long since learned
the material.
So, for calculus, I just got a good book
and dug in. So, yes, first I did the
chapters on analytic geometry, i.e.,
the conic sections -- hyperbolas,
parabolas, ellipses -- and then one with
calculus. For my next year, my brother
was out of the good, private college,
so I transferred in as a sophomore
and hopeful math major. So, I started
with their sophomore calculus. Did fine.
Wrote my honors paper in math and got
800 on my GRE Math knowledge test.
But, right, I 'skipped' freshman calculus,
never took it! Didn't get credit for
it, either.
In grad school, there was an advanced course
in linear algebra. I'd never had a course
in linear algebra but still told the faculty
that I didn't need that course. They said,
with a patronizing smile, "Take it anyway.".
Okay. I blew away all the other students
with by wide margins the highest scores on
homework, tests, and midterm and final exams.
Prof wrote, "Best performance in the class.
Knows this material cold.". Yup. I told
them so! Heck, I'd wanted to study optimal
control theory, not waste time with linear
algebra. How'd I do that? That is, essentially
'skip' a first, and, really, an advanced,
second course in linear algebra? Easy? I
already knew the material from independent
study of a stack of books, including some
of the best, especially Halmos, 'Finite
Dimensional Vector Spaces'. But in total
it was a big stack. How? Why? A long
list of various projects in school and in
my career in physics, signal processing,
applied math, numerical analysis, multivariate
statistics, and more along with some careful
independent study.
Lesson: Independent study can work fine.
Then can 'skip' about whatever you want,
at least in math.
Really, guys, what the heck do you think
a research prof or any researcher does?
Actually know all that stuff just from
sitting in classes in school? Heck no!
Instead they learn, from texts, papers,
seminars, etc. Actually, if need to
know something and can get a good text,
then usually can get good understanding
of just what need from that text for
less than 10% of the effort that would
be required if had to learn the text
well enough to get an A on all of it.
Net, independent study is crucial.
So, for calculus, I just got a good book and dug in. So, yes, first I did the chapters on analytic geometry, i.e., the conic sections -- hyperbolas, parabolas, ellipses -- and then one with calculus. For my next year, my brother was out of the good, private college, so I transferred in as a sophomore and hopeful math major. So, I started with their sophomore calculus. Did fine. Wrote my honors paper in math and got 800 on my GRE Math knowledge test.
But, right, I 'skipped' freshman calculus, never took it! Didn't get credit for it, either.
In grad school, there was an advanced course in linear algebra. I'd never had a course in linear algebra but still told the faculty that I didn't need that course. They said, with a patronizing smile, "Take it anyway.".
Okay. I blew away all the other students with by wide margins the highest scores on homework, tests, and midterm and final exams. Prof wrote, "Best performance in the class. Knows this material cold.". Yup. I told them so! Heck, I'd wanted to study optimal control theory, not waste time with linear algebra. How'd I do that? That is, essentially 'skip' a first, and, really, an advanced, second course in linear algebra? Easy? I already knew the material from independent study of a stack of books, including some of the best, especially Halmos, 'Finite Dimensional Vector Spaces'. But in total it was a big stack. How? Why? A long list of various projects in school and in my career in physics, signal processing, applied math, numerical analysis, multivariate statistics, and more along with some careful independent study.
Lesson: Independent study can work fine. Then can 'skip' about whatever you want, at least in math.
Really, guys, what the heck do you think a research prof or any researcher does? Actually know all that stuff just from sitting in classes in school? Heck no! Instead they learn, from texts, papers, seminars, etc. Actually, if need to know something and can get a good text, then usually can get good understanding of just what need from that text for less than 10% of the effort that would be required if had to learn the text well enough to get an A on all of it. Net, independent study is crucial.