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

> 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.