The problem with math competitions.
I like math competitions. They were probably my favorite part of high school (yes, I realize how pathetic this sounds). But I believe that we need to question their overall value to society. Math contests do seem to achieve some noble purposes, such as to (1) foster an interest in math, (2) identify students who have mathematical aptitude, or even (3) give hopeless math nerds meaning to their lives. However, it is unknown how much collateral damage is inflicted in the process.
When most people talk about math competitions, they think about the mathematical elite, but the effects that they have on everyone, from future engineers to future sociologists, are at least as important. One example of possible collateral damage is that while contests might cause some to become more interested in math, they could conceivably cause others to become *less* interested in math. Some students might be turned off by the competitive aspects of mathematics, while others may mistakenly believe that they "aren't smart enough" to "do math" because they don't win competitions.
(Most of the collateral damage I speak of is probably inflicted on girls, and if this is true, then it would be a powerful argument for completely restructuring the world of competitive mathematics. I ignored this issue only because it's too big to deal with in this space. By ignoring the elephant in the room, I think I've written a rambling pointless rant, but whatever...)
Also, there is this idea in the high school math community that your mathematical intelligence (aka your worth as a human being) is equal to your performance on the AMC. It goes something like this:
IMO team = super-geniuses
MOP participants = geniuses
USAMO participants = very smart
didn't qualify for AIME = moron
(For you old-timers, the USAMO is now administered to a much larger pool than before.) Of course, this is not really AMC's fault; it's really a sociological phenomenon. The AMC, the "Tiffany" or "Rolls Royce" of math competitions, seduces us with its well-written and challenging questions, inspiring these bizarre ideas. It's strange how many people believe that the AMC tests measure something that one might call "raw mathematical talent" despite the fact that *these same people* observe that their own performance gets better with age, experience, and practice. For many students in the know, there is a giant mathematical arms race to study as much "competition math" as possible, but to what end? (See http://www.artofproblemsolving.com.) Reasonable answers: (1) to win trophies, (2) to make other people think you're a genius, and (3) to have something to put on college applications. Admittedly, these are powerful and understandable incentives, but shouldn't there be better things for bright young kids to do with their time? (Okay, I'll answer my own rhetorical question: For most kids, the answer is no, and I guess that's the real travesty.) Math contest jingoists would claim that contests are invaluable because they improve problem solving skills. This is tautologically true, but no one can know how well these math contest skills transfer to other problems, and consequently we cannot know the actual value of this skill set relative to the value of doing some other educational activity. Opinions on this issue probably break down something like this: Adults who enjoyed math contests and succeeded in them are more likely to think that these skills are valuable, while successful adults who did not have such experiences are more likely to see this skills as unimportant. (It's only human nature. The same goes for other questions such as, "Does learning music make you better at math, or vice versa," or, "Do video game skills help cognitive development, or are they totally useless?" Anyone who thinks highly of themselves will approve of most aspects of the way they were brought up. It is a difficult thing to lay aside one's prejudices.)
Another problem with AMC, which is actually the AMC's fault, is its misplaced emphasis on the IMO. Now, I have nothing against the IMO. But should anyone really care if the US wins the IMO? I say no; what we care about is the state of mathematics education in this country, for everyone from our best students on down. Doing well on the IMO doesn't mean that your country is doing a good job educating its best students; it means that you're doing a good job identifying and training them for the purpose of the IMO.
My gut feeling is that math competitions are a good thing, but perhaps in need of some reforms. I just wish people would take this stuff less seriously, and that our youth would stop and smell the mathematical roses.