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I’m going to push back on the advice to learn higher grade math rather than competition math, as I feel the author is ignoring an important skill that competiti

I’m going to push back on the advice to learn higher grade math rather than competition math, as I feel the author is ignoring an important skill that competition math helps develop. They allude it in passing: > A student can wrestle with a competition problem for long periods of time, and all the teacher needs to do is give a hint once in a while and check the student’s work once they claim to have solved the problem. Wrestling with a problem for long periods of time is not just a convenience for the teacher, it is a skill that will serve students well for decades to come. Sitting with a problem that you don’t know how to solve for hours, trying various approaches, failing and failing and trying again, is a life skill that learning calculus two years early won’t teach you. Many of the tactics used in competition problems are also useful in general quantitative situations: identifying symmetries, invariant quantities, properties that can only increase under perturbations.

alephnerd 2y ago
I did competition math in middle and high school, and the only reason I was able to build the base needed to do decently in the AMC, AIME, and CEMC was because I was introduced to various concepts in math much earlier than when American or Canadian curricula would introduce them. Competition math becomes a zero sum game when you are competing with students who have both built strong fundamentals AND then concentrated on technique and problem solving. You can't run if you can't walk. > failing and failing and trying again, is a life skill that learning calculus two years early won’t teach you But learning Calc for 2 years, and getting a 5 on the AP Calc BC exam means you can take 2 additional courses in college or graduate early. > Many of the tactics used in competition problems are also useful in general quantitative situations Agreed. But at the end of the day, the kids getting into AIME or USAMCO were already doing high school or even college level math by 9th grade
- CrazyStat 2y ago
  You don't have to "do decently" or worry about beating students who are already doing college level math, though. You can just do it for "fun" (and learning value). It may be a zero sum game if the outcome you're concerned with is beating other people, but that doesn't need to be the objective.
- bee_rider 2y ago
  I actually did learn how to run before I learned how to walk. It caused my parents all sorts of stress. I guess, though, there’s room to quibble about where controlled falling forward is really running. Anyway, it seems like a shame that there’s a problem solving strategy beyond fundamentals for competitive math. What makes the puzzles in the game different from the sort of typical math somebody in STEM might do?
  - zozbot234 2y ago
    Walking is just as much "controlled falling forward" as running is, it's just slower.
    - gowld 2y ago
      No it isn't. When walking, you can revert mid-step. It's not a dynamic movement, but running is, leaving you temporaily airborne.
qsort 2y ago
Conflict of interest since I was very much into competition math in high school, but I definitely agree that at the HS level it's just about the best thing you can do. It develops your mathematical maturity in ways that simply front-loading calculus or linear algebra won't. A LOT of competition alumni go on to become great academics or successful professionals. And just by the way: competition math is definitely "higher math" in a lot of cases. To be competitive at a decent level you have to know stuff like "real" algebra (groups, fields, etc., stuff like Burnside's lemma is pretty much table stakes), vectors, barycentric coordinates and so on for geometry problems, how to handle recursion for combinatorics, generating functions etc. It's by no means only silly tricks.
- gowld 2y ago
  By those parameters, "decent level" refers to the harder questions on USAMO/IMO... the top 100 in USA in a given grade level. Burnside's lemma is a silly trick, in the sense that it's easy to memorize without knowing why it's true.
  - qsort 2y ago
    > By those parameters, "decent level" refers to the harder questions on USAMO/IMO... the top 100 in USA in a given grade level. No, those are table stakes. To reach that level you have to know the theory and be good at the game. > Burnside's lemma is a silly trick, in the sense that it's easy to memorize without knowing why it's true. By that standard pretty much 100% of mathematical education is silly tricks unless you are literally getting a PhD in pure math. Calculus without real analysis? Silly trick! Statistics without measure theory? Silly trick! Learning how to code without an in-depth understanding of computability theory? Silly trick! Applying Burnside's lemma in a competition setting requires at a minimum a fairly sophisticated understanding of what is a group action and an orbit, which is definitely "real math". The broader point is that competition math isn't some kind of parallel universe where you learn "fake math" that doesn't exist in the real world. You learn the real thing -- perhaps less rigorously than in a university course, but the real thing nonetheless.
zozbot234 2y ago
> Wrestling with a problem for long periods of time is not just a convenience for the teacher, it is a skill that will serve students well for decades to come. And one of the best ways of developing that skill is... learning higher-level math. This can also include 'competition math' topics of course, but they should be approached as self-contained subjects of their own, not just as a bundle of disconnected "tricks" to be applied solely in a competition- or puzzle-solving context.
- CrazyStat 2y ago
  Depending on how the course is set up, maybe. Most math courses are not set up to make students wrestle with problems for extended periods of time, even through University level. I took courses in topology and number theory in undergrad that were set up this way—the professor did almost no lecturing; we were given a series of results to prove and expected to wrestle with them ourselves (mostly alone as homework). Once you thought you had a proof you presented it to the class. But this is very atypical. Your typical calculus or differential equations or linear algebra course does not develop this skill.
conductr 2y ago
Good advice but not good general advice. This will benefit some but many more people will get frustrated and learn to dislike math.
- CrazyStat 2y ago
  This is not my experience. If they see the task as “solving the problem is success and anything else is failure,” like they might be used to from most school math classes, sure. If you set up the context properly my experience is that most kids enjoy working on hard math puzzles.
  - conductr 2y ago
    Perhaps on basic math with younger kids but I expect this will hit a wall at a certain level. Or, the audience of kids doing this is already a skewed/biased sample of kids that just love math (or it's parent driven)
- colonwqbang 2y ago
  It seems that OP assumes you are already targeting a career where maths will be useful to you. If so, I disagree. Everything in school becomes much more fun when you understand what you're doing.
insane_dreamer 2y ago
> Sitting with a problem that you don’t know how to solve for __hours__ my child is very good at math, able to grasp advanced concepts quickly, years ahead of his school curriculum, etc. there is __0%__ chance I could get him to do the above for __hours__
- CrazyStat 2y ago
  You have to work up to it, of course. I don't know how old your child is, but when I was in elementary school I also probably could not have focused on a problem for hours. When I was in middle school I had a long (45 minutes) and boring bus ride and would often spend the entire ride thinking about some problem I was trying to solve. By the time I was in university I could sit and think about a problem for 3 hours (the length of one session of the Putnam exam) without much trouble. And of course, you (usually) don't have to literally sit! I spent many lunch breaks in school pacing around thinking about problems.
  - insane_dreamer 2y ago
    Fair point. He's 8. One challenge is the amount of distractions today (internet, devices, etc.) seem to have greatly shortened children's capacity to 1) handle "boredom" in creative ways (like trying to solve a problem, as per your experience) and 2) have the sufficient attention span to see things through. (I used to spend hours as a teen doing math problems on my own because I enjoyed it; but this was pre-internet and my parents didn't even have a TV (out of principle). There was only books, math, and my bike. I feel like kids today are at a huge disadvantage.
bulubulu 2y ago
I did math competition when I was in high school and the prize I won opened the door to a top university for me. I would say wrestling with a problem with limited tools (no calculus, no advanced algebra etc), finding a solution as elegant as possible is a game that I enjoyed at that time and still enjoy to this date (I'm close to 30 now, still challenging myself with IMO problems from time to time). But that's it. A game is a game and whatever you do inside the game world, no matter how fancy it is, is not practical in the real world. I didn't feel any privilege compared to fellow students without that experience after entering the university. Getting into the "higher math" world was really painful for me actually. Seeing how some "modern" techniques (which has already existed for ~400 yrs, not literally modern at all) have solved the problem I struggled with in such an ultimate way made me feel overwhelmed. I felt like a track athlete, gloating over how fast I was running, realizing how modern transportation has revoluted - people no longer move long distances with human power. The quality a good driver needs is attention - on the road and the car simultaneously - not a pair of sporty legs. The transfer from high school math olympics towards "higher math" requires not only practically a major upgrade in knowledge and toolsets, but also some shift of thinking paradigm - the task is not looking for an elegant and ingenious shortcut to a particular problem, but a highway that is general and inspiring to a fully new field (like how equations opened the door to linear algebra and everything subsequent). I pushed myself to embrace the transition, but it didn't seem to work. I managed to pass the exams and obtained a higher degree, with some expertise in a particular field of application. Still, I always have the feeling that my mathematic understanding is like house built on sand and lack a solid foundation. There is some sort of chasm that I failed to break through... I agree with the part about mental skills, partially. Experience with math competition improved my concentration and persistence. I found piano practice more contributing in this means.

alephnerd 2y ago

I did competition math in middle and high school, and the only reason I was able to build the base needed to do decently in the AMC, AIME, and CEMC was because I was introduced to various concepts in math much earlier than when American or Canadian curricula would introduce them. Competition math becomes a zero sum game when you are competing with students who have both built strong fundamentals AND then concentrated on technique and problem solving. You can't run if you can't walk. > failing and failing and trying again, is a life skill that learning calculus two years early won’t teach you But learning Calc for 2 years, and getting a 5 on the AP Calc BC exam means you can take 2 additional courses in college or graduate early. > Many of the tactics used in competition problems are also useful in general quantitative situations Agreed. But at the end of the day, the kids getting into AIME or USAMCO were already doing high school or even college level math by 9th grade

CrazyStat 2y ago
You don't have to "do decently" or worry about beating students who are already doing college level math, though. You can just do it for "fun" (and learning value). It may be a zero sum game if the outcome you're concerned with is beating other people, but that doesn't need to be the objective.
bee_rider 2y ago
I actually did learn how to run before I learned how to walk. It caused my parents all sorts of stress. I guess, though, there’s room to quibble about where controlled falling forward is really running. Anyway, it seems like a shame that there’s a problem solving strategy beyond fundamentals for competitive math. What makes the puzzles in the game different from the sort of typical math somebody in STEM might do?
- zozbot234 2y ago
  Walking is just as much "controlled falling forward" as running is, it's just slower.
  - gowld 2y ago
    No it isn't. When walking, you can revert mid-step. It's not a dynamic movement, but running is, leaving you temporaily airborne.

qsort 2y ago

Conflict of interest since I was very much into competition math in high school, but I definitely agree that at the HS level it's just about the best thing you can do. It develops your mathematical maturity in ways that simply front-loading calculus or linear algebra won't. A LOT of competition alumni go on to become great academics or successful professionals. And just by the way: competition math is definitely "higher math" in a lot of cases. To be competitive at a decent level you have to know stuff like "real" algebra (groups, fields, etc., stuff like Burnside's lemma is pretty much table stakes), vectors, barycentric coordinates and so on for geometry problems, how to handle recursion for combinatorics, generating functions etc. It's by no means only silly tricks.

gowld 2y ago
By those parameters, "decent level" refers to the harder questions on USAMO/IMO... the top 100 in USA in a given grade level. Burnside's lemma is a silly trick, in the sense that it's easy to memorize without knowing why it's true.
- qsort 2y ago
  > By those parameters, "decent level" refers to the harder questions on USAMO/IMO... the top 100 in USA in a given grade level. No, those are table stakes. To reach that level you have to know the theory and be good at the game. > Burnside's lemma is a silly trick, in the sense that it's easy to memorize without knowing why it's true. By that standard pretty much 100% of mathematical education is silly tricks unless you are literally getting a PhD in pure math. Calculus without real analysis? Silly trick! Statistics without measure theory? Silly trick! Learning how to code without an in-depth understanding of computability theory? Silly trick! Applying Burnside's lemma in a competition setting requires at a minimum a fairly sophisticated understanding of what is a group action and an orbit, which is definitely "real math". The broader point is that competition math isn't some kind of parallel universe where you learn "fake math" that doesn't exist in the real world. You learn the real thing -- perhaps less rigorously than in a university course, but the real thing nonetheless.

zozbot234 2y ago

> Wrestling with a problem for long periods of time is not just a convenience for the teacher, it is a skill that will serve students well for decades to come. And one of the best ways of developing that skill is... learning higher-level math. This can also include 'competition math' topics of course, but they should be approached as self-contained subjects of their own, not just as a bundle of disconnected "tricks" to be applied solely in a competition- or puzzle-solving context.

CrazyStat 2y ago
Depending on how the course is set up, maybe. Most math courses are not set up to make students wrestle with problems for extended periods of time, even through University level. I took courses in topology and number theory in undergrad that were set up this way—the professor did almost no lecturing; we were given a series of results to prove and expected to wrestle with them ourselves (mostly alone as homework). Once you thought you had a proof you presented it to the class. But this is very atypical. Your typical calculus or differential equations or linear algebra course does not develop this skill.

conductr 2y ago

Good advice but not good general advice. This will benefit some but many more people will get frustrated and learn to dislike math.

CrazyStat 2y ago
This is not my experience. If they see the task as “solving the problem is success and anything else is failure,” like they might be used to from most school math classes, sure. If you set up the context properly my experience is that most kids enjoy working on hard math puzzles.
- conductr 2y ago
  Perhaps on basic math with younger kids but I expect this will hit a wall at a certain level. Or, the audience of kids doing this is already a skewed/biased sample of kids that just love math (or it's parent driven)
colonwqbang 2y ago
It seems that OP assumes you are already targeting a career where maths will be useful to you. If so, I disagree. Everything in school becomes much more fun when you understand what you're doing.

insane_dreamer 2y ago

> Sitting with a problem that you don’t know how to solve for __hours__ my child is very good at math, able to grasp advanced concepts quickly, years ahead of his school curriculum, etc. there is __0%__ chance I could get him to do the above for __hours__

CrazyStat 2y ago
You have to work up to it, of course. I don't know how old your child is, but when I was in elementary school I also probably could not have focused on a problem for hours. When I was in middle school I had a long (45 minutes) and boring bus ride and would often spend the entire ride thinking about some problem I was trying to solve. By the time I was in university I could sit and think about a problem for 3 hours (the length of one session of the Putnam exam) without much trouble. And of course, you (usually) don't have to literally sit! I spent many lunch breaks in school pacing around thinking about problems.
- insane_dreamer 2y ago
  Fair point. He's 8. One challenge is the amount of distractions today (internet, devices, etc.) seem to have greatly shortened children's capacity to 1) handle "boredom" in creative ways (like trying to solve a problem, as per your experience) and 2) have the sufficient attention span to see things through. (I used to spend hours as a teen doing math problems on my own because I enjoyed it; but this was pre-internet and my parents didn't even have a TV (out of principle). There was only books, math, and my bike. I feel like kids today are at a huge disadvantage.

bulubulu 2y ago

I did math competition when I was in high school and the prize I won opened the door to a top university for me. I would say wrestling with a problem with limited tools (no calculus, no advanced algebra etc), finding a solution as elegant as possible is a game that I enjoyed at that time and still enjoy to this date (I'm close to 30 now, still challenging myself with IMO problems from time to time). But that's it. A game is a game and whatever you do inside the game world, no matter how fancy it is, is not practical in the real world. I didn't feel any privilege compared to fellow students without that experience after entering the university. Getting into the "higher math" world was really painful for me actually. Seeing how some "modern" techniques (which has already existed for ~400 yrs, not literally modern at all) have solved the problem I struggled with in such an ultimate way made me feel overwhelmed. I felt like a track athlete, gloating over how fast I was running, realizing how modern transportation has revoluted - people no longer move long distances with human power. The quality a good driver needs is attention - on the road and the car simultaneously - not a pair of sporty legs. The transfer from high school math olympics towards "higher math" requires not only practically a major upgrade in knowledge and toolsets, but also some shift of thinking paradigm - the task is not looking for an elegant and ingenious shortcut to a particular problem, but a highway that is general and inspiring to a fully new field (like how equations opened the door to linear algebra and everything subsequent). I pushed myself to embrace the transition, but it didn't seem to work. I managed to pass the exams and obtained a higher degree, with some expertise in a particular field of application. Still, I always have the feeling that my mathematic understanding is like house built on sand and lack a solid foundation. There is some sort of chasm that I failed to break through... I agree with the part about mental skills, partially. Experience with math competition improved my concentration and persistence. I found piano practice more contributing in this means.