Why compete?

My friend Cathy has written extensively about why math contests suck, and there's a lot to what she says.  Many--I'd even concede "most"--contests encourage high-speed, single-step problem-solving rather than thoughtful analysis, or the kind of synthetic work that leads to new ideas and big theorems.  They also discourage kids who don't have the particular skill- and mind- sets that lead to success on math contests:

• "Pyrotechnic" problem-solving ability.
• Near-perfect recall for theorems and situations that come up often.
• Ability to calculate by hand, flawlessly (sadly, because of the huge inequities and arms races that can result, many math contests--including my own ARML--don't let participants use calculators).
• Willingness to push forward with a single solution rather than considering all options carefully.

The real misfortune is when kids lacking in these mind/skill-sets "learn", not just that they're not good at math contests, but that they're "not good at math," period. 

So is there a benefit to competition?

I would suggest three:

• The opportunity for talented math students to learn that they have a lot to learn.  Even our strongest students wind up doing not-so-well on contests once in a while, and it's a reminder that math isn't always easy--and won't be.
• The opportunity for talented math students to learn that hard work can pay off.  For many strong math students, the experience of math class is that what they need to learn to do well comes naturally--so they don't learn to connect effort with success in the context of mathematics.  Our arch-rival Whitney Young is currently the second-ranked high school math team in the state, not because their kids are smarter than everyone else's, but because they work extremely hard--an hour every day, several hours virtually every Saturday.  (Our team mostly feels like we've reached an optimal point on the effort-reward curve:  we work pretty hard, practicing 3-4 hours per week, and make it in the top ten of the state; our students have more time and freedom to do other activities.  But I digress...)  They know it; their students know it; and what they've learned is more useful than any theorem or formula.
• The opportunity for students to learn to lick their wounds.  My son plays competitive chess, and one of the hardest parts of chess tournaments is their duration: you play four or five games in a day, and if you lose one, you still have two or three ahead of you.  I've seen Jonah learn to dust himself off after a loss and go back in swinging (okay, that's a metaphor).  And I've seen the same thing in math team: students screw up a contest, and instead of saying "We're dumb" or "the questions were dumb," students can be taught to go back, study the contest, and practice hard for the next one.  That's another powerful lesson.

I wish there were more other ways for kids to learn these lessons--by doing authentic mathematics research at an age-appropriate level, for example (New York and Chicago now have math fairs that do for mathematics what science fair do for science), or just by having that scrape-your-knuckles-and-try-again experience in math classes.  But the lessons I cite are important ones, and important ones to learn about math.