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发表于 2013-5-24 11:26
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Dr. O’Neil’s essay on math contests, followed with an alternative perspective by Dr. Paul Zeitz......
Dr. Cathy O’Neil:
I think math contests for kids kind of suck.
The way the math contest system is currently designed, nearly all the participating students end up with a feeling of not having been good enough. It encourages competition over collaboration, but even worse, it tends to make young people feel like they aren’t smart enough to be mathematicians.
When I was in middle school, there were no math contests. I was lucky enough to have a great teacher in seventh grade who really let the class collaborate and discover mathematics. When I got to high school, I joined the math team, and although I wasn’t bad, I also wasn’t good — and I felt bad about that, consistently. In fact there were definitely moments when I doubted my chances at becoming a mathematician. It’s really a testament to my internal love for mathematics that motivated me to become a mathematician. If I had not had that seventh-grade teacher, and if I had had earlier experiences being so-so at math contests, it’s possible I would have been turned off of math altogether.
Perhaps you are thinking, well of course there’s a selection process for math contests, because they select for people who are good at math! But I don’t think so. Actually, I think they select for people who may or may not understand the underlying concepts but who in any case are extremely fast at working out details.
I have never been particularly fast at working out the details of something from the conceptual understanding — for example, it takes me a long time to solve a 7x7x7 Rubik’s cube — but it turns out the Rubik’s cube doesn’t mind. And in fact mathematics in real life isn’t a timed test — the idea that you need to be original and creative really quickly is just a silly, arbitrary way to select for talent.
It’s well documented that people seem to think that one is either born good at math or not, in spite of the fact that there’s ample evidence that practicing math competition-type problems makes you good at them. Women are particularly susceptible to feelings that they aren’t good enough or talented enough to do things, and of course they are susceptible to negative girls-in-math stereotypes to begin with. It’s not really a mystery to me, considering this, that fewer girls than boys win these contests — they don’t practice them as much, partly because they aren’t expected by others, nor by themselves, to be good at them.
It’s even possible that boys’ brains develop differently which makes them faster at certain things earlier — I don’t know and I don’t care, because I don’t think that the speed issue is correlated to later deep thought or mathematical creativity.
Finally, I don’t necessarily think that winning math contests is even all that good for the winners either. In spite of the fact that many of my favorite people are mathematicians who were excellent at contests, I also know quite a few people who were absolutely dominant in math contests in their youth who really seemed to suffer later on from that, especially in grad school. From my armchair psychologist’s perspective, I think it’s because they got addicted to the rush of doing math really fast and really well, and winning all these prizes, and when they get to grad school and realize how hard math really is, they can’t stand it.
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Dr. Paul Zeitz:
I’d like to offer a not-too-full-throated defense of math contests. I’ve been involved with them for most of my life, since seventh grade. I was rather successful at them as a contestant, and later, as a high school teacher and then as a college professor. I’ve created contests, coached teams and edited problems from regional events to the International Mathematics Olympiad. I’ve thus experienced the contest world from a number of perspectives.
Dr. O’Neil makes a number of very good points, all of which I agree with:
1. Contest math is not “real math”; even “slow” events like the International Mathematics Olympiad have time limits of 90 minutes per problem. Real math is not done in such tiny flashes of time.
2. “Losing” a math contest (which could mean, for some, not getting first place) can harm one’s confidence, sometimes irrevocably.
3. Even winning contests may not be the best thing for a growing young mathematician, as it potentially reinforces the wrong types of thinking and may not prepare a person for the entirely different world of research.
Nor have I recently come to these conclusions. Indeed, I was aware of them as a kid, and can say honestly that in the 40 or so years I’ve been involved with them, I have had very few non-ambivalent thoughts about contests.
Nevertheless, I am not opposed to them. Here’s why: Contests are not just problems, and winning, and losing. There is also participation, and most important, culture. Some kids like competition. Some don’t. This is true for math as well as sports. Just as sports has a continuum from noncompetition (say, a friendly game of catch) to casual competition (a pick-up ballgame) to mania (take your pick), there are many levels of math competition.
And like sports, math competition can — under proper conditions — benefit many (but not all) kids. A friendly contest creates a peer group and an intensified intellectual atmosphere. Many kids who participate in contests enjoy the camaraderie more than anything else. Team practices and events (such as the American Regions Mathematics League) bring kids together who often have no peers in their entire schools. And the intensity and immediacy of the desire to succeed can, when nurtured properly, promote a really great mathematical work ethic: the knowledge that the ONLY way to do math is by thinking for long periods, again and again and again.
My first experience of high-level competition was on the Stuyvesant High School math team in the 1970s. I got up an hour early to take the subway into school for “zeroth period.” A couple dozen of us would discuss problem after problem, arguing about solutions. Often we digressed into “real math.” It was no different in intensity and scope from seminars I attended as a graduate student, really. The actual math team events that we competed in consisted of short (5-minute) affairs, and we never confused them with the real stuff, which was our collaborative practices. It was not much different from, say, being on a swim team, where the actual events last seconds, and few team members “matter,” but the real growth comes from learning to swim, getting in shape, being on a team, being in the water.
So, in my opinion, contests — mostly — don’t suck. They are not for everyone, but if teachers and parents are sensible and sensitive, they can, for many kids, enhance the mathematical experience. But it never hurts to make some points explicit. I was a co-founder of the Bay Area Mathematical Olympiad in 1999 and each year, I help to give out prizes at our awards ceremony. I always warmly congratulate the winners, but I never forget to end with a brief and heartfelt speech, in which I remind the contestants that there is really only ONE purpose of BAMO: to get them to spend four hours thinking about math, and, after the contest is over, to keep thinking.
Like most things, moderation is the key. “Winning is Everything” is corrosive to young mathematicians and ballplayers alike. Contrast this with “Play Hard, Play Smart, Play Together.” The first promotes alienation and bruised egos. The second prepares us well to take part in the real business of mathematics, “the most social of the sciences,” according to Stanford professor (and former contest superstar) Ravi Vakil..
