Mathbabe wrote a very interesting two posts about math competitions and the harm they do. In summary, her argument gives three negatives to competitions:
- Math competitions discourage most participants because low scorers conclude they are not “good” at math.
- Those who do well at math competitions get an inaccurate picture of success, only to be stymied in (for example) grad school, which requires more sustained attention, gives less instant feedback, and does not always have a “right” answer.
- Competitions particularly discourage girls, who in general are not as “into” competitions and are more susceptible to feeling mathematically inferior.
Her posts are interesting and provocative (just see the comments thread!). Yet there are also numerous advantages to competition.
Perhaps the biggest advantage is that competitions are an incredibly scalable model: a single competition can involve tens of thousands of participants at low cost. Good competition problems and preparation books offer a solid curriculum for math team coaches across the country (even those who are not very good at math) and hence create mathematical communities where they could not otherwise be.
There are other advantages as well. Competition really does appeal to something primal in young boys, and so it can get them into doing math when other techniques won’t work. (Is this infrastructure why boys are still good at math when they fall behind in many other subjects?) In fact, being on a math team can help raise your social standing relative to just being a generic math geek—it provides a cover similar to being on a sports team, if somewhat less revered. Competition math also spurs the creation of lots of good math problems(*) that help students to reflect on deeper mathematics.
(*) Some argue that competition problems are not good problems, but that they are forced and arbitrary exercises that do not arise in nature. I see the point, but I still find them intellectually stimulating.
I believe that competition math is valuable up to a point. I think that intense studying for high-level performance is generally not very useful; it’s not learning real math (just competition strategies) and it exposes competition problems as arbitrary. But for someone who has not spent hours upon hours studying, competition problems are engaging and a good way to challenge yourself.
However, mathbabe’s arguments point to two very important changes that we should bring about in the pathways towards studying mathematics.
- We must create more alternate ways to become mathematically successful. Math circles are a start, but they still rely on a strong mathematician/educator in the area. Some specific, scalable ideas:
- Can we harness the internet to create better opportunities with video lectures and online communities? Art of Problem Solving does a great job at this (see, e.g., Alcumus) although still with competitions as the main focus.
- Instead of timed competitions, can we distribute deep problems for students to unravel over the course of a month, submit, and receive feedback without grading? The best work could be published, sufficient reward in itself.
- Can we develop deep mathematical curricula for the formation of “math clubs” that could be used out of the box by motivated students or teachers?
- Can we expand math summer programs and create more national opportunities for students? Successfully doing this also requires new selection mechanisms, as right now most summer program quizzes are competition-like problems.
- We must change the culture around math competitions. We should emphasize the joy of the problems over numerical success and treat competitions for what they are—fun, stimulating, but arbitrary collections of problems. Students should understand that competition math is an inaccurate reading of their chances at becoming a real mathematician. In particular, they should understand that studying hard and doing well at the top levels of competition math is like training hard to be a good sprinter—you’re great at running 100 meters, but it doesn’t help for the marathon. There is nothing wrong with that and the competitive spirit should be encouraged for those who want it. However, it should not be forced on those who don’t.
I feel that throwing out competitions is throwing out the baby with the bathwater, and mathbabe notes much the same towards the end of her second post. But building a new infrastructure for mathematical success is not easy, and will require some real initiative on our parts to make it happen.
Consider Learning 
If someone asked me to come up with a list of all of the bad things about math competitions, I’d probably end up with something resembling those three bullet points too. If I had to list the positives I think I’d get a longer list.
To add to some of your rebuttals, I think the problem solving aspect is completely missing from the discussion. AOPS’s articles section (http://www.artofproblemsolving.com/Resources/articles.php) has some nice stuff about this, such as http://www.artofproblemsolving.com/Resources/Files/problemsolving.pdf. Rusczyk said somewhere that when he faces heat from professional mathematicians about overemphasizing contests he points out that it’s not so much about math as about how solve seemingly intractable challenges with basic tools. AOPS’s goal is not really to make future mathematicians; they’re trying to make people who can be really successful at any number of things, whether in science or industry or whatever. In short, the original post seems to be written with the mindset that encouraging a math competitor to become a professional mathematician is the ultimate objective, which is hardly true.
As a follow-up on the above, she mentioned that competitions emphasize being able to think really fast. Even ignoring how that is not too applicable at the olympiad level, in many other jobs besides being a professional mathematician it’s a useful skill. In some of my summer jobs, being able to wizard out a complicated Excel sheet in not much more time than it takes to type in all the needed formulas has made me a pretty valuable employee. Even away from a computer, when I’m organizing something and a snafu comes up in the middle, keeping things going smoothly pretty much requires coming up with a solution on my feet. I’ve found I’m pretty good at this, and I think I know what to owe it to. I wouldn’t argue with the assertion that being able to think deeply is more important than being able to think quickly, but I do have a bone to pick with the statement that being able to think quickly is useless.
Finally, having risen pretty far in a couple competitive areas myself, one of them even less useful than contest math (when am I going to need to know how to speedsolve a logic puzzle?), I believe there are at least a couple benefits to hitting the top through an intense amount of effort. For one thing, after having gotten that far, having been frustrated with some particularly stupid and contrived problems you’ve gone through (and know you have to know how to do), and having seen 95% of the ideas that come up, you’ll know damn well the arbitrariness of competition problems and that a different mindset is needed for later mathematical study. Another thing is that there’s some value in the experience of knowing how to master and improve at something, and once you have that experience it makes it easier to learn another, and maybe more useful, discipline. I’ve programmed as a hobby for a long time, and I’ve often spent time trying to get better in anticipation of using those skills in the job market. The perspective that getting as far as I did in contest math and logic puzzles gave me has made this process much smoother in several ways, most prominently in allowing me to trust my own judgment and make/learn from my own mistakes more rather than immediately googling some expert’s answer when I stumble across a sticky design issue. In all of this I have not even mentioned the discipline it takes to sit down and work hard to improve a weak spot, particularly if you don’t enjoy it. (Damn you Sudoku and Kakuro!)
USAMTS seems to fit the second bullet point quite well, except that it does have grading. There is about a month to do the problems, there is some (limited) feedback, and the clearest solutions are published as the solution set.
Dan, I totally agree with pretty much everything you said. There is one thing I would emend: I think that in a long-term contest, publishing the best student work may still discourage people from participating. If they’re new to mathematical thinking or mathematical writing, and their example is some beautiful paper that solves the problem and makes generalizations and conjectures, they might think, “There’s no way I could ever do that.”
Of course, it would take years to get to that level, and even people who don’t ever learn to write like that can still benefit a lot from math competitions.
On a similar note, there is a fact which I think needs to be emphasized much more to laypeople:
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Hard math problems take a lot of work, even for math whizzes. (Maybe less work than it would take you, but still a lot of work.) If a math whiz is able to do a problem without lots of scratchwork, it is not because math is effortless for him. It’s probably because he has thought about essentially the same problem before, and remembers how the solution went.
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Papers do not show this at all. A paper is the result of hours and hours of scratchwork, spread over at least many days, usually much longer. This is not the same kind of thing as the “worked examples” you find in grade school textbooks — those are mechanical. It is much harder to demonstrate a creative problem-solving process. But in my opinion it is important for people to know that problem solving always involves a lot of unpublished work and frustration, and that that is part of the relationship between a mathematician and his work.
Maybe people do already know this. Or maybe they would if they cared to consider the issue. But very often I get the “it’s in his genetics, it’s not in my genetics” cop-out, and that annoys me.
There is absolutely a lack of cultural knowledge about what doing mathematics is. People don’t realize that we aren’t just adding really big numbers together all the time, but that lack of knowledge extends to thinking that math problems are “easy” for some people or assuming that smart people do all problems quickly. If people really understood how long it takes to do a difficult problem and what it looks like, perhaps they’d be able to put their own struggles in context.
In the comments to Dan’s Google+ post about this blog post, David Cordeiro wondered why people who aren’t in the “top” seem to be more discouraged in mathematics than in sports. That seems to apply also to your comment about publishing the best work — does watching pro atheletes on TV discourage ordinary folks from playing sports? It’s just as true there that “it would take years to get to that level, and even people who don’t ever learn to [play] like that can still benefit a lot”. A pro athlete’s performance is also the result of hours and hours of practice which makes the result look effortless. Why is math different?
And a propos of showing people the unpublished work and frustration that goes into a math paper, recently some of my co-bloggers decided to write a paper on our blog (see http://golem.ph.utexas.edu/category/2011/06/categorytheoretic_characteriza_2.html for the conclusion, with links to the preceeding discussion). It was interesting to watch the back-and-forth process of figuring stuff out take place online, although of course I personally am already familiar with how mathematics is done. I don’t know how edifying it would have been to someone else, or to who else — perhaps a graduate student just trying to break into doing research. But maybe group work towards solving simpler, less specialized, problems could also be done in a public way. On the other hand, would anyone have the patience to read it all?
Ah, but there are other ways to communicate that doing math is a rewarding challenge for everyone! For example, most people don’t appreciate exactly what it is that makes performing surgery so hard, but popular culture has successfully transmitted that it is quite difficult and requires steady hands. Similarly, many people who don’t understand law very well still understand that it requires lots of research and grunt work. I wonder if the same could be transmitted to “the world at large” about mathematics, even if folks don’t understand the math itself.
Incidentally, the most edifying G+ comment I saw was the idea of earning “belts” in mathematics a la karate, implying continuous improvement and showing what the next steps look like. I think that’s an awesome idea.
I don’t quite follow; what part of my comment is the “Ah, but” a reply to? Also, probably I am being dense, but I don’t see what that sentence has to do with your examples — people know that surgery is hard, and that law is hard, but in my experience people also know that math is hard. I don’t understand how surgery and law have solved any problem that we are trying to solve for mathematics here.
The idea of “belts” is an interesting one. I’m not sure I’m convinced, though. For one thing, I’m suspicious of the tendency to assign simplistic gradations of skill or ability to *people*. It feels kind of elitist. I don’t have any experience with martial arts communities, though, so maybe it doesn’t work that way there?
The “Ah, but” referred to your comments about the online paper working out math in detail, especially “On the other hand, would anyone have the patience to read it all?”. I agree that any sort of leakage into popular perception of mathematics would be unlikely from an exercise like mathematicians showing their process in public. However, there are other ways to penetrate the public consciousness which, while less precise, still capture the essence of sustained thought on a problem.
I think the elitism of belts would depend on implementation, but you’re right that it’s a big danger. It’s not as if contests aren’t already fairly elitist in terms of measuring individual performance, though.