> You proposed enrichment materials such as number theory puzzles and other recreational mathematics. That's already what the math olympiad contests consist of.

No, I did just the opposite: I said that
others proposed such materials;
I did not propose them but criticized such
proposals. Again, for a high school
student to be pushed into seeing
lots of tricky things in Pascal's
triangle, attacking number theory
exercises, doing recreational math
won't do the students much good in
later life -- so as in the title
of this thread, we won't expect to
see many such students successful in math
later in life. Instead, the efforts
there in middle and high school will
rarely come to anything.

Helping middle and high school students
do better in math than what is
in just the usual courses would, could,
and should be doable and done, but
the path is not recreational math, etc.
but, instead, as I outlined, just
proceeding with the main line of
math education -- rush through the
high school math of algebra and
geometry, likely also trig,
and then get a college calculus
book and dig in.

E.g., the high school math
teachers I had were eager to
say that we "were not ready for
calculus". Nonsense. It was
the teachers who didn't know
calculus.

I very much would
have rushed ahead had
I had some good guidance.
Eventually I knew enough to rush ahead
when I was
a college freshman: I'd had
four years of math, grades 9-12,
at a relatively good high school
but for my college freshman year
went to a state school that was
cheap and close enough for me to walk
there and back. They wouldn't let
me take calculus but pushed me into
a course beneath what I'd already done
in high school. So a girlfriend told
me when the tests were, and I showed
up for those. The teacher said I
was the best math student he'd ever had --
in no very real sense was I his student!

I asked to be permitted to start calculus
but was turned down. So I got a good
book and dug in. I did well (in high
school mostly I'd been teaching myself
from the books and sleeping in class anyway).
For my sophomore year I went to a good
college with an excellent college math
department, started on their sophomore
calculus, from the same text used at
Harvard, and did well.

I'm hot on this stuff about what
middle and high schools do to
good math students: My eighth grade
arithmetic teacher gave me a D and
took me aside one on one and fervently
urged me never to take anymore math.
In some significant ways,
my father was good at education
and just laughed.

That eighth grade teacher
didn't have a clue about what math was:
For the actual math, I'd done well
in her class. What I did poorly on was
multiplying two four digit numbers, and
the reasons were (1) my handwriting
was sloppy (common for boys), so sloppy
I didn't get the columns lined up and
then added wrong for the final answers,
(2) my clerical ability was just awful
(common for boys), (3) I understood the
ideas right away but never worked
on the clerical skills to
get correct answers, (4) my parents
never urged me to work to get right
answers (my father had some
high-end views of education
and just wanted me to learn,
especially out of interest,
and didn't care at all about grades,
and I did learn).

But in the ninth grade,
the teacher saw right away that
I was his best student, and he sent
me to the state math tournament.
In the summer after the 11th grade
I was sent to an NSF math and physics
program.

So, net, I was a good math student,
was very interested in math, was
eager to race ahead, but the school
did a poor job helping me.

There is a little more to the
story: When the SATs came back,
I was called out of study hall
to get my scores, from the same
teacher I'd had in the sixth grade.
Nice woman. Sweet. Ignorant!
She, along with most of the rest of
the teachers, thought I was
a dunce. So, she read me
the Verbal SAT score -- 538.
She said "Very good". BS!
It sucked! But she was sweet.
She may have thought that I would
get 250 or some such!

Then she looked at my Math SAT
and hesitated. She looked
worried. Confused.
She said
"There must be something wrong."
Then she said
the score
was 752 and
"That's uh, uh, very good.".
Darned right it was good.
But she was correct that there
"was something wrong" and had
been for 4, 6, 12 long,
wasteful, painful years.

I had had no idea what the SATs were
about and had made no effort to prepare.
But I finished the Math SAT early,
checked my answers, and still had
15 minutes. I'd very much like
to know what the CEEB thought I'd
missed. Maybe they scaled the
scores and didn't give anything much
higher.

My Math SAT was second out of a grad
class of about 180, which is about
right from the statistics, 2.5 sigma
above the mean of 500. The #1 guy
beat me by a few points. The best
student in the grad class was a little
behind me -- he went to MIT and
burned out his fuses in his freshman
year.

I was a good math student, had been eager
to rush ahead in math, physics, and chemistry,
but I'd had poor guidance. So, I'm hot
on grades 6-12 continuing to give
good math students bad advice. So,
here I say, mostly, for the actual
Math Olympiad competitions, waste of time.
To do well with math, just go down the
main road to a ugrad math major as I outlined.
Simple. And to heck with,
say, learning to play Nim in
Courant and Robbins, What is Mathematics,
about Pascal's triangle, etc.

Yes, I'd learned about Pascal's triangle.
So when I was in grad school and had
been pushed into a silly ugrad class
in probability, the prof went all
arrogant and emphasized that due to
the factorials it can take some
really long precision arithmetic
to calculate even relatively ordinary
binomial coefficients. BS: Just
work in from one side of Pascal's
triangle and never see a factorial.
Indeed, I raised my hand and stated
so in the lecture. I'd long since
written software do to just that.

Next he did a sloppy
job explaining the glory of the central
limit theorem, and I dropped the course.
But I was taking and grading in the
Neveu measure theory level course
at the same time, my first official
course in probability, and did well.
E.g., I was the only student who showed
that there are no countably infinite
sigma algebras.

The schools who want to send their
good math students to Math Olympiads
are likely also doing a poor job
helping their students. Then, as in
the title of this thread, in later
life we mostly won't see much
effect from such high school efforts.

Net, my hot button is that the
schools in grades 6-12 don't know
enough math to do well helping
their students eager to race ahead
in math.

> > Good grief: Lots of people in middle and high school get pushed into math competitions such as the Math Olympiad but don't go on to be math majors in college and grad school.

> This isn't true at all.

Of course it's true: E.g., you
gave some examples of how high school teachers
try to get some of their students into such
competitions, i.e., "push" the students.
That's lots of students, and
in college math is not a very popular
major. So, lots of middle and high
school students in math competitions
don't major in math in college, just
as I claimed.

For more, you mentioned some activities
at the college level; I was addressing the
high school level. Sure,
college students who do well on the
Putnam are likely college math
majors who are taking math very seriously.
Again, I was addressing high school.