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.