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> One big problem with the plan, which a lot of people don't realize, is that your typical university math degree isn't that much more challenging than doing well in high school math.

That wasn't my experience. (I went to a top liberal arts college - it didn't give minors, but I took enough math courses that it would've been a minor at other schools.)

In my experience, post-calculus college-level math is qualitatively different from high-school math in focusing on proofs and abstractions over problem-solving, techniques & algorithms. I had an intuitive sense for high-school math. College math kicked my ass, because I was asked to prove everything rigorously, and the proofs just kept coming, dozens of them per homework assignment. That required a very different way of thinking than high-school math did: you needed to remember this large body of theorems and combine them in creative ways to derive new proofs, you couldn't just memorize recipes and techniques that will get you the answer.

I found that physics was more like the continuation of high-school math, except that then your courses had all this lab work and experiment design and fundamental physical principles. One of my friends (a math/physics double major) described it like this: "In physics vectors are objects that live in a vector space, while in math a vector space is a set of axioms from which the concept of a vector and rules for manipulating them naturally fall out."

Edit: hmm, just looked at the username. I know you, and I also know you're better at math than I am. I'm curious just what kind of math you did in high school and what you did in college where the college math was no harder than high school. Either your high school had some really great math courses or your college had some pretty shitty ones. Maybe it's just where we went to school.

If he was doing olympiads in high school, he was already doing proof-based math then.

I think Terry Tao describes the stages of mathematical maturity really well: https://terrytao.wordpress.com/career-advice/theres-more-to-...

Well, I should be more specific. I think once you get into the sort of number theory where you prove things about modular arithmetic, and the sort of analysis where you are talking about abstract spaces rather than n dimensions of real numbers, those courses are notably different from high school courses, and all the focus on proofs is intellectually stimulating to advanced kids.

The courses which are typically boring for advanced kids are calculus, linear algebra, differential equations, probability, statistics, combinatorics. Most of an undergrad math degree usually falls into this bucket but it really varies by university.

Also, there’s a big difference between colleges. Not because of the curriculum as much as the other students who will set the pace of the class. Often the same kid who could attend Ohio State at 14 could also attend Princeton at 18, and that makes a difference even if the course titles are the same.

Ah, I'd agree with that. Calculus/probability/statistics/combinatorics were usually freshman courses at Amherst, and you could often place out of them with APs. Linear algebra was taught proof-based; it and DiffEq were usually sophomore-level courses. The rest of the major was all proof-based.

>The courses which are typically boring for advanced kids are calculus, linear algebra, differential equations, probability, statistics, combinatorics. Most of an undergrad math degree usually falls into this bucket but it really varies by university.

That's just not true. That's the freshman/sophomore curriculum. And there are big differences between the diff eq, stats & linear algebra course which are which are 'proof' vs. those which are 'practical'.

I think once you get into the sort of number theory where you prove things about modular arithmetic

That’s one small part of the intro algebra course at my university, which all math and CS students take in the fall of first year. The first year calculus courses are considerably proof-based as well, and lots of first years end up failing despite being a high 90s student at their high school.

A more typical high school student who got 80s in their sleeves and calculus courses would fail everything and be asked to withdraw.

I’m not sure how else you’d describe a vector than something that lives in a vector space. The approach of a mathematician might be different but I do believe the idea of scaling and adding things (and then stuff like inner products) is a lot more important than axioms. And at least to me the rules for manipulating them fall out after a huge amount of technical work and details.

How would taking the AP exam in high school be appropriate in this case? That would be the equivalent of taking an AP exam in math 5 years after graduating EE in college.

People are going to have different opinions on whether or not it's worth a 13 year sentence of boredom in order to socialize with age peers, but I really question whether the default high school experience is, for most people, a good thing. Activities like sports and music seem like a vastly better return on effort for advanced kids, with more positive values in the milieu than the typical high school popularity contest.

Ultimately, the world is defined by a vast diversity in ages, backgrounds, and capabilities. If you want to do something in that world, and you're really young, you'll first have to get used to being the youngest person in the room.

Throw all of this argument out the window if the student actually enjoys school, of course.

Taking the AP exam in high school is just a tactic. A situation like:

Student: Man I don't want to take this boring mandatory calculus class.

Teacher: You have to.

Student: But I already know all this stuff!

Teacher: Mmmmm, well that's just like your opinion, man.

Student: What if I ace the AP exam?

Teacher: OK, then I would believe you.

In my experience it's pretty rare for all of high school to be boring. You probably haven't read the particular works they're discussing in English class, you probably haven't read about the particular time period they're discussing in history. And even if you have, there's plenty more you can learn about those things. The most common area to be bored by is math, because you can pick that all up yourself relatively easily.

I suppose it depends on when this all kicks in. In my case, in kindergarden I was tested into 11th grade; in response I was placed into 1st grade. Which was fine, honestly, I spent a lot of time drawing and so forth, but things got really unpleasant and pointless in 7th grade, so I dropped out, and dropped into college. I'm reading a lot of people complaining about the damage to one's social prospects, and surprisingly few calling out what I experienced: school was minimally useful, the social environment was pretty toxic, and in my case, the teacher was tyrannical.

University was so much better! For highly motivated and gifted students, early university can be a great option. It is important to place it on a larger arc of one's own journey of learning, development, and self discovery, however, that must continue for as long as one keeps growing.

As an important aside, girls and boys have approximately equal interests in science math and technology earlier on; for girls this changes pretty severely in middle school and high school. I'm sure there a very large social component to this, parents would do well to inoculate against it by finding alternative environments, and university is certainly one.

> One big problem with the plan, which a lot of people don't realize, is that your typical university math degree isn't that much more challenging than doing well in high school math.

I think this is very US-specific. Here in Europe there's definitely a huge difference. You enroll in a pure math-only 5 year degree, and are talking about topology in your third semester already. That might be a specific thing only my analysis prof does, but quite a lot of time is devoted to neatly delineating different levels of structure you're equipping a space with, proving things with as less structure as possible or at least discussing it. You are expected to juggle tricks on a totally different level, which takes a lot more qualitatively different practice than in HS.

When I talk to people who take math classes in the US it's usually not on that level of engagement until the graduate courses. Maybe it depends on their university, in smaller European countries you often have just like one main math program that is then high-intensity.

I think the keyword is "typical." Which is probably true that the math is pretty basic at most universities and in most programs.

But you can go to schools like the University of Toronto, which is a strong math school and has very challenging programs like ASSPE1165 (https://fas.calendar.utoronto.ca/mathematics-specialist-scie...)

> It's better to wait and go to a top-tier school rather than to start college super early at a medium-tier school.

I can actually agree with you from experience. I was homeschooled from age 13, but my parents didn’t know enough math to home school me in it, so my mother got me into a dual-enrollment program with a community college for math and CS. Overall I did ok, but calculus really threw me for a loop, and I ended up scraping through calc I & II with C’s. I often wonder if I had taken those classes when I was 20 rather than 15 if I would have done any better. My own son took the test (at his math teachers urging) to get into high school algebra while he was still in middle school. He didn’t pass the test, so he had to take plain-old middle-school algebra, and he was actually a little bummed out about it. I told him that when you’re 30, the only thing that really matters (if anything) is where you went to school, which depends a LOT on your test scores… not how old you were when you took the tests. He knew some of the kids who “made the cut”, and they struggled with high school algebra, got B’s and C’s, and one actually dropped out of high school last year while my son is still a straight A student.

>> and they struggled with high school algebra, got B’s and C’s, and one actually dropped out of high school last year while my son is still a straight A student.

I keep hearing this when the school wants to put kids on the advanced track. All it does is give them more work and risk lowering their GPA, which is far more important for admissions. One district allegedly give a 0.5 bump just for taking the bait to make up for that. It's all a game.

That’s definitely not just an allegedly thing, my private high school gave a GPA bonus of 0.5 for honors classes and 1.0 for AP classes to avoid this issue. And I’m certain it wasn’t something unusual or hidden, many other schools must have done it also.

Goodhart’s law and all that - if unadjusted GPA were the sole number to target, it would make way more sense to take the easiest classes possible and never challenge yourself.

Does where you went to school really still matter when you are 30?

It depends on where you went to school.

Well, it may be that it goes like this: When you’re 30, where you’ve worked matters. Where you’ve worked depends (a lot) on where you went to school. Where you went to school depends (a lot) on what grades you got. What grades you got depends (somewhat) on how old you were when you took the tests.

Does where you worked depend that much on where you went to school, though?

Sorry if I am not convinced. In my country most schools are considered somewhat equal. Maybe in the US it is very different.

Oh yeah. Where you were/who you knew at 20/22 has huge, cascading effects on where you end up at 30, 35 years old. I won't get in to details here but a connection I made working at a coffee shop at 22 can be clearly traced to where I am now at 35. There is a lot of truth in "it's not what you know, it's who you know", particularly on the other side of 30.

Your story seems to indicate that it is not necessarily the school, though.

These days it seems like a reasonable alternative for a gifted high school stduent to take online math classes. So they can still socialize with their peers but take some advantage of their advanced ability.

I know online classes don't have a great record, but this is probably one of the instances where it makes the most sense. The student is presumably motivated, and if they're not, then it's not a big deal. They're already ahead and can do something else at their regular school.

I was in gifted programs until high school, whereupon I went to a magnet school that had lots of opportunities for advanced math people (you could take calculus freshman year and not run out of math courses). The schools (both private and public) broke up the students into effectively the "fast" math group and the "slow" math group. And very often, there was the "me" group, where I was working at my own pace and quickly burning through the math books.

As you say, students who have that aptitude for math tend to have a sufficient motivation to do the self-study, especially if the material can be presented as some sort of challenge (how fast can you get through this textbook?).

>My recommendation to kids who are great at math and thinking about taking college classes is, don't. If the math classes at your high school are too easy, ask your teachers if you can skip straight to the AP exam

I wonder if this sets the teacher up for reprimand though via 'no child left behind' type stuff. I wonder what the repercussions would be if a student complained that someone else was getting more attention (advanced work in this case) which received attention from the administration.

My fiance is a high school math teacher and just a few weeks ago she had a student claim she was bullying him, which got administration involved with a parent-teacher-administrator-student conference. When the student was asked how he bullies her "she makes me stay awake and do my work in class" which got her off and got her an apology from administration but she was legitimately nervous about it prior to the meeting because that's the state of public education now.

Anecdotally, she's doing her best just to get her students capable of doing the assigned math as they will just pop out their phones when she's not looking (or when they think she isn't) and use some app on their phones to tell them the answers (PhotoMath I think) so they aren't even getting the foundations they need to continue to advanced maths. At her current school someone wanting advance placement would be a pretty rare exception. She has multiple students that will write profanities as answers, will draw cartoons as answers, will write things along the lines of "I don't know I want to die" "this doesn't matter" etc the past few years. Seeing the 9 year old in the OP article seems miraculous to us both given the state of things at some schools.

Most people that attend IMO are prodigies but generally, sociable. I met many of them and they just love math.

> Most people that attend IMO are prodigies

Can we all agree to never use that word? Actual prodigies are measure zero in the general population. Terry Tao is a prodigy. The rest of us are just normal schmucks.

Prodigy suggests that talent comes naturally to and is dismissive of the massive amount of work that goes into preparing for something like the IMO.

Talent does come naturally to them. Instead of arguing about labels, it's better to think about talent quantitatively. If the USA's IMO contestants were generally slackers, they'd still be comfortably in the top 100 in their birth year (in the entire country) in the AMC/AIME combined score, just by showing up when the teacher told them to. (Well, back in the early 2000's. Maybe competition is stiffer nowadays.) And the gulf in talent between them and the worst kids in that top 100 list is large.

Why do you have to think about talent at all? Who is them??

I didn't end up doing IMO due to weird school situations, but I'd won some prestigious math competitions (two other winners I befriended one year went on to become IMO gold medalists) and was able to solve some IMO problems in middle school on my own. At the time I felt I was just different from others in terms of mathematical ability.

My general feeling now is that talent as perceived by others is mostly a function of hard work. Looking back, the level of persistence I was able to apply to mathematical problems at such an early age without any real supervision was incredible. I was almost entirely self-taught from age 7-10 and without being on a strict curriculum, this included some college level math. I think this was a function of a lot of things that don't have much to do with natural ability.

At least a few IMO gold medalists have younger brothers who are good enough to be involved in math activities in high school but are not remotely close to the same level. They are in fact different in terms of mathematical ability. You would be more aware of the value of natural ability if you had put work in, but then couldn't come close to winning prestigious math competitions.

My own brother was exactly that way (good enough to be one of the best in school and hold his own at an elite university, but not close to good enough for any kind of actual competition) but my understanding is that he was only ever trying, never doing.

There's a huge difference between learning something because some external pressure dictates that you learn it and truly immersing yourself in something and just letting go. Math isn't that exciting for most people, so the vast majority of people never cross over to the latter. So the extremely small percentage of people that are able to cross the chasm and deal with math natively can stand out.

I agree with what you're saying in general. I don't know whether you really quantitatively disagree with me. I think without any deep immersion like you describe, the IMO team members would still have done well enough on the AMC/AIME to qualify for the USAMO. Do you disagree with that? (Again, my difficulty scale is based off the early 2000's.)

I mean, my basis for that estimate is that I know a bunch of people without such immersion who got that far when I was a kid, who were on the regular school education track, didn't self-educate outside that, and didn't have much external pressure either.

Edit: But I don't mean to imply these people absolutely never thought about math or numbers outside of the classroom, because it's interesting, and they most certainly did, just no traversal through a curriculum, or deep immersion.

Things exist on a spectrum and I don't think we're really on the same page on what it means to be immersed - it's about how your subconscious mind decides to prioritize things, not about any kind of curriculum. I spent a lot of time in my life cleaning and organizing stuff, but I'm not good at it and not getting better at it either. I'm pretty sure this isn't because some lack of natural ability prevents me from being good - for whatever reason, this is just something that I don't find interesting. This is how most people approach math - as a chore to be done and forgotten. Even regular good-at-math people (which most people at elite schools would qualify) that aren't remotely close to competing at the likes of IMO and Putnam are qualitatively different in this regard compared to average students because they are broadly speaking more immersed in their studies. For most people, it's not merely that they don't think about math outside of the classroom - their mind is also too busy thinking about anything but math while doing math.

Certainly. But that sort of subconscious prioritization isn't the same thing as putting in the work, as measured in reams of scratch paper.

But when it comes to learning, it's the only kind of hard work that matters.

The flaw with this model is that it doesn't really explain the world, because if you compared most kids who do well on math contests to the average high school honors student, you'd have to argue that their performance in almost every subject besides Band and PhysEd is the result of mental prioritization. Also, observing that an underperformer just didn't mentally prioritize it the same is basically mind-reading; it isn't a refutable observation. We can see the weakness of the argument by using it to explain the effect of age on performance, without finding direct evidence to refute it. But the actual cause is that kids get smarter as they get older. It also fails to explain why kids rank so differently at contests like 24, versus later contests that test a mostly different sort of ability. That kids who do well on math contests like thinking about math can't be taken as a factor in more than a banal sense.

And imagine trying to make the same argument for basketball. Kids with low field goal percentages just don't want it enough?

As someone who attended the IMO, I can tell you it varies widely. For some people, they screw around a lot, attend math camp but spend a lot of time playing games, dick around doing normal teenager things, get in trouble for stupid teenager things, don't study too hard, and nevertheless do well enough to attend the IMO. And then some people are responsible, study a lot, work hard, keep their heads down, and also attend the IMO.

I'm not sure if your advice is for kids good at math who want to eventually do something else, or kids that actually want to be mathematicians? I always thought being a mathematician was something special in that there is ample historical evidence of very young prodigies in that field who made big contributions.

Or are you saying top tier schools for math are so much better, that aspiring mathematicians should wait if at all possible to go to a top tier school? What if he's admitted at a top tier school at 9?

I was a pretty mediocre student in high school (as in, I didn’t get into Duke, people like you would consider me a failure etc), what would you say was the biggest difference between the precocious crowd you describe and everyone else? Do you think it’d just inherent?

I did community college in high school so I’d say it’s better to eliminate the “boring” general ed classes first but that’s just my experience.

The key takeaway to me here is that if you are gifted you should go to a top-tier school, not necessarily wait.

The top-tier schools usually don't accept super-young students, though. What usually happens is, a second-tier school will accept a super-young student because they see no downside. The top-tier schools will tell you, just wait and apply again when you are older.

I don't know specifically about the Eindhoven University of Technology but it seems like roughly the 100th best school in Europe, which in the US would be like going to Saint Louis University. A fine school I'm sure, but if you can graduate there when you're 9 then maybe you should just instead go to Harvard when you're 18.

Lots of people do the equivalent of this, early college classes and then they go to a top tier school for undergrad. I think Reid Barton did this.

There are caveats, I know someone who finished the CS curriculum in high school, when he got to Berkeley the only credits he had left for his CS degree were non-CS, which bored him, so he just dropped out and started a startup. Seems fine now, as far as that goes, aside from the exodus from the bay area that the housing situation has forced on many of us...