Math contests kind of suck
I’m going to annoy quite a few people with this post, but I’ve been thinking about this for a while and it comes down to this: I think math contests for kids kind of suck.
Here’s the short version of my argument.
Math contests discourage most people who take them, because most people don’t get close to winning, and in particular give those people the impression that because they lost a contest they don’t “have it” when it comes to math. At the same time, although they are encouraging for a few people, it’s not clear to me that the kind of encouragement they give those kids is healthy. Finally, they are bad for women.
Now I will argue this more thoroughly.
The way math contests are set up nowadays, they start in middle school, at the school level, and if a student does well at a given test they move on to a larger stage, perhaps at the state level, and they typically culminate in a national test, or sometimes even an international test (in the case of the IMO).
This system sets up nearly all the participating students for a feeling afterwards of having not been good enough. It encourages competition over collaboration, which is a huge problem in my opinion, but even worse, it tends to make young people feel like they aren’t smart enough to be mathematicians. It is in fact 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 (why else would Stuyvesant kids consistently beat other kids? Is it really possible that smart people somehow know to be born in New York?). The bottomline is that these extremely young, impressionable kids get early impressions that the contests are measuring their genetic abilities, and that they aren’t cutting it.
When I was in middle school, there were no math contests. I was lucky enough to have a great teacher in 7th grade, who let us nerds debate amongst ourselves for an entire class whether 0.999999… is equal to 1 or not. He put himself in the position of a mediator. It was a great moment for me, and made me realize how much creativity and originality could be involved in the process of making and understanding math.
When I got to high school, I was on 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 is really a testament to my internal love for mathematics, combined with finding this math camp that I’m teaching at now, that motivated me to become a mathematician. If I had not had that 7th 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! I discussed this with another mathematician today and he refined that argument as follows: some people are good at understanding concepts but can’t work out the details, and some people are good at working out details by rote but don’t understand the concepts- you can’t really be a good mathematician without both, and perhaps the contests select for the details people, but after all you need that aspect.
But I would go further: although I agree you can’t be a good mathematician without both, I don’t think the contests select for the details people. They actually select for people who do or don’t understand the concepts (probably do for the higher level tests) but who in any case are extremely fast at the 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 tests- the idea that you need to be original and creative really quickly is just a silly, arbitrary way to select for talent.
I guess if you could have math competitions that aren’t timed then I might start being okay with them. Especially if they were collaborative.
The reason I claim math contests are bad for math is that 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 do they expect 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.
One related complaint to this rant: it seems like there is way money out there for math contests for young people than there is for math enrichment programs like the program I’m working at now (I’m looking at you, NSF). Why is this? Probably a combination of the fact that’s it’s easier to organize, it seems quantitatively measurably “successful” because there’s a winner at the end, and maybe even because it makes the United States look good compared to other countries to have a winning IMO team- in other words, spin. Booo! How about throwing a little bit of money towards programs that sponsor a sense of collaborative, exploratory mathematics and which encourages women?
Before I get people too riled up, I will say this in favor of math contests: they do tend to expose kids to different kinds of math than is normally offered in their classrooms, which can be really great, and expansive, for kids that have drab math curriculums with drab teachers. Lots of kids first find out there’s math beyond quadratic equations by going to a math contest. That’s cool, but can’t we do it in a better way?
mathbabe 
Agreed on all counts. On the other hand, I was one of those kids who was reasonably good at competitions, wasn’t particularly discouraged when I got to the level where I stopped advancing, and overall got a lot out of the whole experience, so I feel slightly hypocritical saying that.
I do think that a lot of kids in my high school math club were eventually worn down by the competitions we went to, though, especially the girls. It just isn’t fun to not win, especially as a girl in an environment where the best competitors are almost without exception guys. As far as I know, not a single one of them is still doing math. In fact, the only female friend of mine from high school that I know is still doing math wasn’t in math club and didn’t go to any competitions.
Creative, engaging lesson plans would be a great alternative, but someone has to write them, popularize them, distribute them, and train teachers how to follow them properly…
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I pretty much agree. Maybe like Qiaochu, I was quite good at the local math contents in high school days, but it was geared towards HS level math skills and we didn’t do any special preparation or take it beyond the region. In college, I discovered the Putnam exam, did it maybe two years and could not get motivated about it.
I feel a bit bad taking anything away from the people who excelled at that, but it is a different kind of thing. You need a library of clever tricks, and I think it is essential to drill to do really well — it isn’t any direct measure of your understanding or talent.
I liked your posts about HCSSiM, which sounds really worthwhile. It reminds me of things I heard about Promys and also Ohio State from their strong proponents. (I never did anything with any of these and don’t know how they all compare, or what others I am missing.)
In terms of the real math skills you need, I would also put a lot of emphasis on having an somewhat adversarial approach to things, so you look for holes and flaws, you remember counterexamples and pathologies, and you really appreciate everything behind a fully formed idea and sold proof, including limitations and key hypotheses. These also serve very well outside of mathematics.
So let’s encourage the programs like you are doing now that inspire so many.
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I think you’re overlooking one key useful function of math contests: they cause schools to have math teams, which give the students interested in math a place to go to meet other like-minded kids. I’d be delighted to see their format changed to something more collaborative, more creative, and less oriented towards “extremely fast at the details,” certainly. But the existence of math team means there’s some social norming that might not exist otherwise.
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I have many fond memories of socialising at maths contest training camps, mentoring sessions, and meeting (and horsing around with) the teams from other countries at the international mathematical olympiad.
I have almost no memory of the actual competitions themselves, though. For me, the competitive aspect was actually not the most prominent part of these contests, and perhaps because of that I always enjoyed them.
But this was over two decades ago. It seems nowadays that maths competition has become as professionalised as more traditional sports, and I think this is actually something of a pity. Though I did visit a recent international olympiad and it still seemed the kids were having a blast at all the social activities outside of the actual competition itself…
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I’ll agree with Michael: I got social benefits out of math contests. But I’m pretty dubious about them as well; and it seems to me that a lot of the complaints you make about math contests seem to me that they could be leveled at traditional math teaching as well. (Selecting for a certain kind of math ability, encouraging competition over cooperation, leading people to feel that they’re not smart enough to be mathematicians or to do math at whatever level might be useful/enjoyable for them.)
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I had few opportunities to enter math contests and did poorly whenever I did. The Putnam was the most memorable example. Yet I managed to survive as a research mathematician. So I am biased towards the view you hold. But I’m curious about whether you think competitive sports in schools has a similar discouraging effect on kids who don’t make varsity.
Also, I recently witnessed the Mathcounts competition held at Courant this year in which my son participated. I don’t think I’ve ever seen a larger group of kids showing so much enthusiasm for doing math, even though most of them did not do exceptionally well. It really made me wonder if math competitions, done well, can be a lot more fun than I realized. I was also impressed by number of girls that participated and that were among the finalists.
So my views have evolved into a state of uncertain ambivalence.
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I largely agree with all you said, with two exceptions.
I am rather dubious about the sentiment that “I will say this in favor of math contests: they do tend to expose kids to different kinds of math than is normally offered in their classrooms.” Most math contests I was involved in in school went out of their way to make the questions reasonably approachable for a student whose only maths education is from the classroom. (The three prominent exceptions are the AIME, the USAMO, and the USAMTS which I will write about in the next paragraph.) For example, as far as I can remember, every question in the New Jersey Math League can be brute forced from first principles using “school maths”; but that way you’d only have enough time to do a third or half of all the problems. Don’t get me wrong, I do approve of training “problem solving skills”, but I don’t think problem solving is the same as “different kinds of mathematics”.
The other exception is that not all math contests suck. I have really fond memories of when I participated in the USA Mathematical Talent Search. For each round you get 4 or 5 problems (I think the format was slightly different in my days), and have an entire month to do them. The answers are supposed to written up in free-response format (so kinda like a proof, but not so rigorous). It is run on an honor system, through the mail, so you hardly ever find out how anybody (besides yourself) did on the contest (though now they do have a leaderboard posted on the internet, which invalidates what I am about to say). Of all the math contests I did, that one felt most like what I do for a living now. It is not you against other contestants. Your foe and your prey is the problem itself. And furthermore, the problems did introduce some new mathematics. I remember for one of the problems I almost “discovered” convolution.
Unlike some other commenters, I don’t feel like I got any social benefits from math competitions. There was only one other student who was as “into” math contests as I was, and we are both mathematicians now and remain good friends, and there were a couple more students in the “math club” who actually enjoyed mathematics. Most everyone else in it were there to pad their resumes. But that maybe because we never had a real “math team” per se. For the competitions where there was actually a team structure (science olympiad, physics team, quiz bowl), the social interaction did play a large role in my participation.
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hi Cathy – I remember you from UCB and the Noetherian Ring. I think your blog writing is wonderful. This essay rings very true to me… I took the AIME and its qualifier without really studying for it and I found the competition pretty stressful. I never felt attracted to Putnam style activities, partly out of awareness that there was a mini-industry preparing students for it that I never got recruited to.
I’m helping put on a Julia Robinson Mathematics Festival at SF State, which is one of those alternate kinds of math enrichments that is less individualistic and time-driven. It seems to me there may be space in our enrichment ecosystem for this range of enrichment, and I wasn’t aware that NSF and other funders were so heavily skewed towards the competitions.
Thanks for sharing your great writing!
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I was going to comment about the USA Mathematical Talent Search, and Willie Wong beat me to it. That was wonderful, as it was the only exposure I got to proofs (I don’t count the deadly boring “proofs” we did in geometry class) until college.
I agree with much you say — I can date my unfortunate tendency to shy away from academic math to taking the Putnam and doing terribly on it, after having had a history of doing well on math competitions. (That being said, there were other factors going on as well — my parents’ focus on external validation surely didn’t help — but that’s another story.)
Like other commenters, I also got quite a lot of social benefits from math competitions, so on the whole it was positive. The wonderful math coordinator for my state (who is now retired, so I don’t know if this still happens) used to take the top 30 or so in the state math contest and send them to the American Regions Math League competition in Pennsylvania. (Which itself has a collaborative aspect, which I really enjoyed and which was my first introduction to working with others on academic problems.) Since we aren’t that near to PA, this was like math camp and road trip rolled into one — I had an amazing time and met several people who later became good friends, including my best friend.
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Anecdotally, I attended the Math Olympiad Training session one summer, which is where I heard about the Hampshire College Summer Studies in Mathematics program, and immediately applied. HCSSiM definitely made a bigger positive impact on my life in mathematics, by far, but both were meaningful.
Part of your commentary was on the value of math contest vs. math enrichment, and I heavily side with the latter. My own opinion is that both are valuable. Think about how to make the biggest impact on the mathematical development of students. Go back in time even just a few decades, and contests were probably more cost-effective. But these days – I would argue that math circles, math festivals (like the Julia Robinson festival) and other similar activities deliver a whole lot more.
I especially like your suggestion about high level collaborative math activities. But for teachers to implement them, we need to have great lessons available, and great examples of collaborative pedagogy easily accessible. Most of us were trained primarily with lectures, and it’s difficult to move on from that.
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Cathy, I love your blog, and I also loved the talk you gave at the 2009 Math Prize for Girls.
The concerns about contests that you raise are very serious ones, and I constantly try to bear them in mind as I encourage math circle students that “There are many ways to measure success.”
The true value of a contest, as I always tell my students, are not the trophies and medals you might win that will eventually gather dust, but the “Aha!” experiences and the discoveries and–above all–the mathematical friends you made along the way to preparing for it.
Yes, we need *far more* than contests–we need more programs like Girls Angle, HCSSiM, Canada/USA MathCamp, PROMYS, and Ross, which are treasures. For some students, contests can open doors of awareness to programs like those. Indeed, at Math Prize for Girls 2011, we will have a panel of presenters from some of those programs.
At the moment, however, contests and local math circles to prepare for them can scale to create include far larger and more inclusive mathematical communities than those amazing summer programs currently can.
Our math circle has completely awesome students who put their heart and soul into applications into some of those wonderful summer programs. Some were accepted–but a number of absolutely terrific students were not accepted. A parent of a student who was not accepted at one of those programs wrote to tell me that “[The program] had an overwhelming number of applicants – they have had to turn a lot of students down. (They also said that this year a student would need to have a perfect Interesting Test to get in. [My daughter] feels bad as she could not do some of the problems.”
I think a major issue is making sure that students do not dismiss themselves as “not being good in math” because they do not solve all the problems on the Interesting Test or rank as highly as they might have hoped on some contest or other. There are so many contests these days that students can see they are a roller coaster, with many random ups and downs, so that scores and ranks can’t be taken as the be all and end all.
Indeed, I find it striking that you yourself wrote above “When I got to high school, I was on the math team, and although I wasn’t bad, I also wasn’t good – and I felt bad about that, consistently.” And yet, your Alice Schafer prize biography begins, “Catherine O’Neil’s serious interest in mathematics began in her first year of high school when she received the highest freshman score in a statewide mathematics competition.” If you dismiss your own high school math contest performance as not being “good,” then how should all the other ninth graders in the state feel about their performance?
I think the key thing is to create communities that encourage and support students in challenging and growing mathematically, that encourage students not to write themselves off as “not good at math,” because they did not outperform everyone else on a math contest.
As for competition vs. collaboration, as others have pointed out, many contests do have strong collaborative elements–team rounds, GUTS rounds, Power Rounds, etc. But beyond that, there are ways to turn even the individual contests into a collaborative enterprise. Another of my mantras to my math circle is that if all students in the circle work together to encourage and help one another at our weekly meetings, then all students have a shared sense of ownership in the accomplishments of every member of our circle.
We need to help young women–and young men–redefine what it means to be “good” at mathematics. We need to try to help young people avoid the trap of defining themselves by imperfect measures–whether it is contests, admissions processes, journal referee opinions, etc.
This beautiful essay (written by a young woman who was in the audience for the talk you gave at the Math Prize for Girls 2009) eloquently illustrates how a young person can rise above the point where she lets math contests define her:
http://www.artofproblemsolving.com/Forum/viewtopic.php?p=2239381#p2239381
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Hi Cathy,
I really like your new blog. Regarding math competitions, I agree that they have the drawbacks you mentioned, in addition to another major drawback: they have little to do with “real” (i.e. research) math. That said, high school math competitions played a crucial role in my development, and without them it is quite possible that I would not have become a mathematician, since I had other strong interests before my math interest was really awakened.
My life changed profoundly the day I walked in to the Stuyvesant high school math team. For the first time I realized that other kids like me existed! My modest successes in the competitions made me feel that I had a talent for mathematics. And from the math team I was chanelled into HCSSiM (which was one of the few noncompetitive math activities available at the time — I think I would have loved math circles) and to ultimately becoming a math major in college.
I agree that there are drawbacks to competition. Both winning and losing were harmful in different ways to my immature ego. But I really enjoyed the competitions. The math team met at 8:00 am every day. In retrospect it is really hard to imagine how one could motivate me, or any other high school student, to get up 45 or 90 minutes early each day. The motivation probably came not just from the excitement of competition, but also from the social, noncompetitive activity of preparing for the competitions together as a team.
When high school ended, I despaired that there would be no more math competitions, except for the Putnam. And since I was not competitive at a national level, I felt inferior to my college classmates. In my freshman year I did well enough on the Putnam to be invited to the “Putnam dinner”. At this dinner Andrew Gleason gave a little speech in which he pointed out that Putnam problems, which you solve in 30 minutes, have little to do with real math research problems, which might take 4 years to solve. Before this I don’t think I had realized how disconnected math competitions are from real math. That was my last Putnam dinner; I did worse and worse on each successive Putnam exam. I like to think that this is because I was getting better at real math.
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Further thoughts in response to: “One related complaint to this rant: it seems like there is way money out there for math contests for young people than there is for math enrichment programs like the program I’m working at now (I’m looking at you, NSF).”
As far as I can tell, NSF is not spending “way more money” on math contests for young people than on other non-competitive kinds of math enrichment programs.
When I look at the websites for major math competitions (e.g., MATHCOUNTS, AMC8/10/12/AIME/USAMO, ARML,USA Math Talent Search, HMMT,etc.), they all include sponsor lists and I did not see NSF listed as a sponsor of any of those contests.
However, NSF does sponsor the Julia Robinson Math Festivals and other noncompetitive math circle enrichment activities, which I agree are wonderful–and we need more of them!
But backing away from the NSF question to the broader question of why there is lots more funding (from other non-NSF sources) for math contests than for math enrichment programs such as HCSSiM. You wrote: “Probably a combination of the fact that’s it’s easier to organize, it seems quantitatively measurably ‘successful’ because there’s a winner at the end, and maybe even because it makes the United States look good compared to other countries to have a winning IMO team- in other words, spin. Booo!”
There is one other reason you don’t mention: scale. Contests are not just “quantitatively measurably ‘successful’ because there’s a winner at the end,” they are quantitatively successful in engaging a huge number of students–as well as their teachers on a scale that is hard to do in other ways.
And thanks to the Internet, it is no longer just the students at Stuyvesant who have a critical mass with whom to discuss interesting contest math problems at 8 in the morning. If you visit the Art of Problem Solving website, you can find students from all over the country–indeed all over the world–discussing interesting math competition problems, both those from old competitions and those of their own creation, any time of day or night.
Reading those on-line discussions, I am frequently struck by the generosity of spirit and the genuine helpfulness with which students and coaches from opposing teams collaborate with one another in discussing problems.
It is particularly noteworthy that the most successful math team in the country by far, the Lehigh Valley Math Team (National champions at ARML in 2005, 2009, 2010, 2011) has freely published their “playbooks” on the open Internet, freely available to their competition and the entire world.
It is worth noting that the playbooks were written by a remarkable father-daughter team, Ken and Maria Monks. Maria, who won the Morgan Prize for her undergraduate research earlier this year, was a member of Lehigh Valley’s first national championship team (in 2005) and she has continued to coach the team throughout college and now grad school in math. Maria was also a student leader of the Harvard-MIT Math Tournament during her time as an MIT undergrad as well as a coach of the US teams at the China Girls Math Olympiads. But Maria has also been extremly involved in other kinds of non-competitive math activities as well–she was a mentor in Ken Fan’s model Girls Angle program throughout her time at MIT, as well as and she was also a research mentor at the Duluth REU.
So, yes, I agree–most emphatically–that we need to deliver over and over and over again the message that Andrew Gleason gave to the Putnam dinner Michael attended. In essence, math contests are like spelling bees. They are fun and they can stimulate interest in math or language arts, but just because you ranked high or low on a contest does not mean you will necessarily be a great or terrible mathematician or writer. And we need to give that message early–long before they get to college.
I started giving that message to the middle-school kids I began coaching in the mid-1990s, and I am happy to say that I believe they took my message to heart: almost all of them had very positive math experiences throughout the rest of their education, regardless of how they did on the contests. Although only a few of them are doing what Michael terms “real math” (i.e., research), they are using mathematics in all kinds of ways–computer science, statistics, chemical engineering, physics, economics, etc.
Sorry this is so long (and a second post too!) But I appreciate your raising this provocative and cautionary note. (Even though it kind of hurts!) It is one that those of us involved in math contests need to bear in mind constantly.
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I’m a little too old to have ever been exposed to math competitions, but I do know a little bit of history in regard to the NSF. In the 1960s the NSF sponsored math and science summer programs at universities for high school students. I think there were four general areas of study: mathematics, physics, chemistry, and biology. Students were selected for the mathematics programs on the basis their responses to problem sets. I believe that all but travel expenses were paid for by the NSF. Even adjusting the almost $3000 cost for the HCSSIM program down to the 1960s inflation adjusted cost, my own parents probably could not or would not have have paid for this.
The first that I attended was a seven week program at Notre Dame. We had formal classes and exercises (generally proofs) to do back in the dorm. There was little collaboration among the students, there were no organized fun activities or field trips, and outside of class we were mostly left to our own devices. The most difficult course was (unmotivated) abstract linear algebra (based on the Hoffman and Kunze book). This was probably not the best choice for kids at this age. I did learn some Fortran and got to run some programs on the main frame (using punch cards!). I remember being yelled at for using too much CPU time. The campus experience was rather boring, but there was the weekly movie night at the student union where I was first exposed to European movies. I remember Fellini’s La Strada in particular.
The next summer I spent six weeks at Berkeley. This program was a lot more fun. We lived in the International House that summer. I think the main subjects that we studied were combinatorial math and topology, and these were more appropriately intuitive for our age group than the subjects at Notre Dame. The book we used for topology was First Concepts of Topology by Chinn and Steenrod. This was an excellent book for us. There was more social interaction among the students than there was at Notre Dame. They took us on a fascinating field trip at the nearby national lab run by Berkeley, and one of the instructors took us to a local movie theatre for more exposure to art house films. This was 1968, so we were generously exposed to the social and political movements of the time. I remember picking up the Berkeley Barb at Cody’s bookstore, the demonstrations in support of the French student strikers, and the ensuing riots that turned Telegraph Ave and the campus area into a little war zone. Several of the kids (not myself) in the program were arrested during the riots, although my little cliche did at one time flee when a molotov cocktail was tossed at a cop near to us. I’ve wondered what happened to my best friend there — Jack. I recently did some internet searching, but could not locate him.
I’m not sure what happened to these NSF programs after that. There must be more older people like me out there with memories of the early years. I wish we could hear from them.
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Just wanted to mention that there is financial aid available for HCSSiM.
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Cathy,
That’s wonderful that financial aid is available.
I just had two additional thoughts.
I think that the summer programs I attended were probably the first attempts at summer activities for high school students, and it’s likely that the people who ran the programs had no prior experience with them and weren’t certain how they should be run. We were experiments. These early NSF programs were probably a result of the cold war and the “space race.”
I did experience a let down when I finally got to college. There were no longer any special mathematical things to do during the summer (although I think this has changed since then). While my brother, in physics, could get interesting work in labs during the summer, I had nothing except dreary summer jobs to help pay my way through college.
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This is actually a response to #16:
I just graduated from undergrad, and most people I knew who were interested in math did REUs during the summer. These are NSF-funded opportunities for research (in math, and in other fields), and were definitely way more fun than taking math classes. One needs to apply, but many of them are not particularly competitive to get into (although some of them are very competitive).
Some people also work as counselors for the summer camps for high schoolers, i.e. MathCamp, etc.
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Yes, most of the other summer math enrichment programs also offer financial aid as well, which is great. There is also a wonderful new summer math enrichment program at Bard College for kids from high-poverty New York City schools, which is totally free. (http://spmps.weebly.com/index.html)
Richard’s comment reminded me that my husband also attended a couple of those NSF-funded summer programs back in the 1960s. (He also did math contests–the contests and those NSF-funded programs did co-exist in the 1960s, and I am guessing participation in contests was likely higher in many places then than it is now. Certainly, there has been a huge decline in the number of American schools offering math contests–and the number of American students takings them–since the mid 1990s. The decline in numbers of American students taking the AMC contests is masked by the huge number of students attending schools in China, Taiwan, Singapore, etc. I seriously doubt that students from my husband’s old high school in NJ have the opportunity to participate in math contests, let alone learn about summer math programs like HCSSiM.)
I asked my husband about the selection process for the two NSF-funded summer math enrichment programs he attended. He remembered nothing like the effort that students need to make now to apply to programs such as HCSSiM, Canada/USA MathCamp, PROMYS, etc. As he remembers it, he just filled in an application, noting his math grades, and got a recommendation from a teacher and that was pretty much it.
Now, I think that the admission process currently used by most summer programs (a bunch of outside-the-box problems that are inherently interesting to work on) are a much better way to select a group of students who will enjoy spending the summer doing that kind of mathematical exploration.
I tell the students that the tests are worth working on even if they don’t get into the programs–which is true–the problems are great ones and they can learn a lot from doing them, but it would be nice if the programs had a way to provide at least a little detailed feedback and encouragement to the students who tried to work on their problems but were unable to make enough progress to be admitted.
Canada/USA MathCamp had a very nice “math jam” last year discussing the problems on their qualifying quiz. It was open to all students, not just the ones that had been admitted to MathCamp. http://www.artofproblemsolving.com/School/mathjams.php?mj_id=283
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Mary,
I did note that in my case I had to submit problem sets. I think admissions criteria was probably determined by each individual hosting institution. I’m guessing that the kind of problems I had to work on were similar to what kids are now exposed to in math camp. There was probably also a letter of recommendation involved, but I don’t have certain memory of that.
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I went to a high school in the midwest that did not even have a math club, so that is probably at least part of the reason I had had no exposure to math contests. And because there was no math club and no math contests, I also had no prior exposure to the sort of problems that I had to work on for acceptance to the NSF programs.
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I understand that, at least these days, the people who run the admissions processes at summer programs these days do take into account the mathematical context/background from which a student comes. If a student comes from the middle of (mathematical) nowhere, with no experiences with math teams, math competitions, advanced math classes at local colleges, etc., their application will be read in a different light than those from students who apply from a place like the Bay Area, Great Boston, Stuyvesant, etc., where they have had lots of opportunities to be exposed to problem solving methods. Of course, these days, any student with access to the Internet has opportunities for lots of exposure to problem solving–but only if s/he knows to look in the first place. That is a big if. How do you look for something you don’t know exists?
I grew up in the 60s also. I now realize I could have applied to those NSF programs, but I did not even conceive of their existence, and no teacher bothered to tell me about them. Perhaps they did not know about them. Or perhaps it did not occur to them that I might have enjoyed such a thing. Or that anyone would enjoy such a thing.
Indeed, it might not have occurred to me that I might have enjoyed such a thing–but taking the AHSME contest might have given me a clue that there was something out there more exciting than the rather straightforward and unexciting stuff I was doing in math class. Unlike Cathy, I was not lucky enough to have a teacher of the sort that encouraged debate about 0.999…=1. (Indeed, for most of my K-12 education, I was in parochial schools that did not encourage debate of any sort. Class sizes were large (56) in those baby boom years.)
One of the (three) high schools I attended was a public school that did offer the old AHSME contest, but perhaps due to being a mid-year transfer student with no track record at the school at the time of the AHSME, I was not invited to take it. Perhaps if I had taken it, I might have ended up on a mailing list for one of those NSF summer programs.
On another note, Richard’s observation that there were lots of NSF programs for high school students in the 60s, but none for undergrads is an interesting one. Now there are close to 100 NSF-funded REU-type programs for undergrads in math all over the country:
http://www.ams.org/programs/students/undergrad/emp-reu, and, in addition to those open programs, many colleges run similar programs for their own students.
I wonder why the NSF priorities have shifted from funding programs for high school students to funding programs for college students.
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This is a reply to Mary and to Richard. I have read not once but several times what you two wrote. I am from Hungary, but live in the US. I went to school, college and graduate school in Hungary. We had a program (KoMal – an abbreviation for High School Mathematical Journal –referred here further the “Journal”) that started in the late 1800’s and it is a problem solving Journal for grades 5-12, but aspiring 3rd graders start it as well. KoMal publishes separate problems for grades 5-8 and 9-12. It was a monthly, the solutions of submitted problems were later published.
We had one month to think about the problems and write up a solution with reasoning. Points earned were accumulated through nine rounds. The best problem solvers were featured with their picture in the “Journal”.
The closest similar to this in the US is the USAMTS, initiated by George Berzsenyi (a Hungarian mathematician) 24 years ago. However, USAMTS is for high school only and it has three rounds. Berzsenyi is retired and the USAMTS is taken over by the National Security Agency.
I started to work on those problems in 4th grade (because a teacher suggested it to me). For the first glance, I could not understand the questions(never mind solving them). However, one month is a long time, I kept thinking about the problems, looked at the previous months problems (these were featured with solutions) and finally I solved some (some others I believed, that I have solved). The solutions had to have detailed reasoning (like a proof on the level of a child). In a few months I got the taste of challenging problem solving. I was waiting for the “Journal”. It was social, I started to learn the names or faces of other problem solvers, some of them I met in summer programs – in many ways this was very social for those who like problem solving. Often I envied that some of them could solve a problem I could not solve, the more their exemplary solution was published in the “Journal” with their name. Sometimes I even missed a day of school to finish the write up of the solution, other times, when I was older I was finishing the last solution write up in the post office before the mail was picked up.
Looking back, this was better than any summer program, because it gave us continuous involvement (and maybe improvement) in problem solving. The “convenience” of “living with the problems for one month” allowed us to participate in athletics competitions (instead of math team competition) and in chess tournaments.
I wish, some of you would share this view with me and together we could start the nine month long problem solving programs.
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This sounds pretty good to me. I have never been good at math competition problems and survived as a student and mathematician, only because I was allowed ample time to ponder questions in math.
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Deane,
The moth long time frame for preparing solutions naturally improves your problem solving skills. The practice assists the problem solver in sorting out and disclosing some possibilities, etc. On this way, intended or unintended the “one month deadline correspondence program” is good training for timed tests. I could see this back when I participated; with a 20-25% error rate, the top problem solvers of the correspondence program were the top problem solvers of the national competition.
The correspondence problem solving serves students to teach themselves for problem solving. Without competition, I have never seen any student in elementary or high school reading even the best mathematics book and solving the problems one after the other, page after page, chapter after chapter..
If, however, they have the excitement of competition, the students may read part of the same books and narrow down the pages or actual examples that helps them in the direction of one given problem.
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I would also like to point out that the disparities between the proportions of boys and girls in mathematics is exasperated in math competitions, especially the farther along in the competition you get. I personally was a girl who competed in math contests regularly throughout my youth, successfully, although never actually enjoying them. Yet math contests still remain where I’ve gotten the most sexism in the math community. My first exposure to the math community at all, when I was a sixth grader, was a sexist proctor telling me that since I was a girl I was unable to be the captain for my team. Although it improved slightly, I bore through many math competitions with no desire to do them, feeling it was a good day if I scared the boys enough to avoid most sexist comments. And math competitions would have permanently scared me away from mathematics if it weren’t for HCSSiM, which introduced me to the joys of mathematics. And the collaborative, creative process of math, which is what I truly love and enjoy.
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Cathy, the article Explaining the Gender Gap: the Role of Competition by Muriel Niederle and Lise Vesterlund (https://docs.google.com/viewer?url=http%3A%2F%2Fwww.stanford.edu%2F~niederle%2FNV.JEP.pdf) provides some experimental and empirical evidence that might be of interest to you.
Their data supports the hypothesis that competition systems tend to increase the performance of boys and to decrease the performance of girls, compared to assessments conducted in non-competitive contexts. Figure 1 and footnote 2 illustrate that point particularly well. The entire article is pretty interesting, especially their discussion of the differences between the performance of women in single-sex contests vs. mixed sex contests.
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[In a mathematical examination of high standards and traditions] the examiner is not allowed to content himself with testing the competence and the knowledge of the candidate; his instructions are to provide a test of more than that, of initiative, imagination, and even of some sort of originality. And as there is only one test of originality in mathematics, namely the accomplishment of original work, and as it is useless to ask a youth of twenty-two to perform original research under examination conditions, the examination necessarily degenerates into a kind of game, and instruction for it into initiation into a series of stunts and tricks.
— Godfrey Harold Hardy
These [mathematics] contests are a bit like spelling bees. There is some connection between good spelling and good writing, but the winner of the state spelling bee does not necessarily have the talent to become a good writer, and some fine writers are not good spellers. If there was a popular confusion between good spelling and good writing, many potential writers would be unnecessarily discouraged.
— William Paul Thurston
Professional mathematics is not a sport. In particular, whether one is “better” than one’s peers is not really the right thing to focus on; the more important thing is to ensure that one can do good mathematics in one’s chosen research area.
— Terence Tao
What is important is to deeply understand things and their relations to each other. This is where intelligence lies. The fact of being quick or slow isn’t really relevant. Naturally, it’s helpful to be quick, like it is to have a good memory. But it’s neither necessary nor sufficient for intellectual success.
— Laurent Schwartz
There was little about me that identified the kind of bright student who wins at prestigious competitions or assimilates, almost by sleight of hand, the most forbidding subjects.
— Alexander Grothendieck
The most profound contributions to mathematics are often made by tortoises rather than hares.
— William Timothy Gowers
I tend to be slower than most mathematicians to understand an argument.
— Stephen Smale
It’s true that I’m not good at solving problems. For example, I would never be good in the Math Olympiad. There, speed counts and I am certainly not a speedy worker. That’s one pleasant thing in mathematics: it doesn’t matter how long it takes if the end result is a good theorem. Speed is an advantage, but it is not essential.
— John Tate
Let me stress that creating new mathematics is a quite different occupation from solving problems in a contest. Why is this? Because you don’t know for sure what you are trying to prove or indeed whether it is true.
— Sir Andrew Wiles
Problem-solving should never be practiced for its own sake; and particularly tricky problems must be excluded altogether.
— André Weil
In one way I think my early involvement with Mathematical Olympiads and suchlike has damaged my mathematical taste: I still have this feeling that short, elegant problems should have short, elegant solutions, and that one should be able to do mathematics without absorbing a lot of difficult stuff first. Unfortunately, this is all false.
— Gareth McCaughan
I was never interested in artificial puzzles.
— Goro Shimura
I was never fascinated by puzzles or intellectual games.
— Robert Phelan Langlands
…because people in general, and teenagers especially, tend to feel sad and anxious when they are alone and thus try to avoid solitude as much as possible, it is very hard for them to tolerate the long hours alone that serious study in science and mathematics requires. Gifted young persons might give up not because they lack the cognitive capacity to process the relevant information but because they cannot stand working alone.
The introduction of competitive math tournaments to a certain extent mitigates the loneliness of this pursuit, but at a price: The cut-throat pressure of math teams tends to alienate those young men and women who prefer a more supportive environment. It has been our impression that many promising young mathematicians – especially many young women – become disengaged from math not because they are unable to keep up with the cognitive challenges but because they cannot bear the supercharged atmosphere of the math clubs, where the topic of conversation always includes each member’s latest standing on the team. There are a great deal of envy, barely veiled hostility, and jockeying for power and sympathy in these conversations, and some people just don’t wish to put up with the stress involved. Unfortunately, when a student drops out of the field of mathematics in high school, he or she is likely also to drop out of the domain altogether.
— Mihaly Csikszentmihaly, Kevin Rathunde, and Samuel Whalen
(from “Talented Teenagers: The Roots of Success and Failure”)
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I suspect that all of us here are missing the forest for the trees.
You say is that the quote-on-quote “losers” of math competitions supposedly feel bad about not doing well, thus turning them away from math, etc. Then, you go on to say that there were people “absolutely dominant in math contests in their youth who really seemed to suffer later on from that, especially in grad school”. So you say that misery happens both ways, and I completely agree that it’s possible. I don’t agree, however, that the culprit is the competitions.
I think that the common thread between these two groups of people is not that they both took math competitions, but that they are both American students. This is an extremely complex topic, and I’m going to completely inappropriately boil it down by saying American students have fragile self esteem due to the way we’re brought up through K-12 system. American students don’t experience failure in their youth, or are shielded from it, so that the first time they experience failure (which tends to be during the math competitions, or for those who don’t fail during the competition process, grad school), they crumble and don’t know what to do. I speak from experience: I went to IChO 2007 and I experienced the exact same thing as you, down to the way I tried to “deal” with my failure by completely overanalyzing it.
The answer is as simple as that: we set up a no-failure zone in K-12, and the first time that students fail in the “real world” or some simulation thereof, they get turned off, because clearly if they failed at it, they’re doing the wrong thing.
See more at http://moderndescartes.blogspot.com/2011/05/deciding-what-to-do-with-rest-of-your.html
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I’d like to add a point to the many great ones made here already. 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. 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.
It is a certainty that math competitions have had a positive effect on the mathematical education of many many students (including numerous career mathematicians), but we have no idea how these positive effects balance against the negative ones. It’s especially hard to weigh this when the likelihood is that the negative effects are highly concentrated on girls.
Contrary to Thurston’s quote above, I do not believe that competition math is as divorced from research math as spelling bees are from writing novels. Rather, competition math is just one facet of the big world known as mathematics, which includes not only research in pure math, but also service courses for engineers, etc. (It is fitting that Mathcounts is sponsored by engineers, not mathematicians.) It’s not that competitions are bad. It’s just that we need more outlets for mathematical enrichment.
Just as there is unlimited variety in math, we need more variety in math education. Summer programs and math circles are steps in the right direction, but one problem is that they do not scale as well as math competitions. Any effort in math enrichment must always grapple first and foremost with the following problem: Most students do not have direct access to *any* adults who have a good understanding of mathematics. This suggests that the internet is a good way forward. Unfortunately, the largest and most successful online mathematical community for kids is AoPS, which is focused on — you guessed it — math competitions.
P.S. Mary beat me to the point about the NSF. She also makes a great observation about your supposed “failures” in math competitions. There are many mathematicians who claim that they were “never any good” at math competitions, but it’s rarely clear what this means since they are often comparing their own performance with those of other mathematicians.
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I’m going to completely disagree with you!
I loved math contests. For one thing, I always liked tests in general, regardless of my ability in the subject being tested. This was due to my laziness. In math, as in every other subject, the exams were the place where in an hour or so I could make up for months of not having done much of my homework — a fantastic opportunity! For another thing, I often did well on the math tests.
I was not competitive. At the MOP, a fellow student from a high school neighboring mine said that his goal for the last year, his motivation for long hours of hard studying, had been to do better than me. This explicitly competitive attitude was so foreign and weird to me that I still remember it decades later. I mean, if you don’t like the math problems themselves, then why bother? After all, math contest success doesn’t take you anywhere except more math contests. It’s worse than sports. Math contests don’t get you into college. There are zero careers in math contests, even for the top people.
I never lost a math contest. Nobody loses a math contest. Sometimes I got a horrible score (yes, it’s true), but I never thought of that as losing. It’s not losing.
If you just view contests in terms of how far up the ladder you can get, you will be disappointed, for sure. This puts math contests in pretty much the same category as everything else in life, if you ask me.
I enjoyed my math contest success, but success was not a goal for me. When I crashed the NYCML one year (I’m not from NYC), and after each round of problems they had the people who had gotten everything right so far stand up (a dramatically dwindling number of people), I didn’t mind when they came around and told me to stop standing up even if I had gotten everything right. I wasn’t interested in recognition. I was interested in those tasty mathematical treats that get collected into contest problems. And anyway, the whole reason I was there in the first place was because of math friends. That’s right, real friends that would never have existed without math contests.
Our high school had a great math team, very social and working together. Everyone at the school was on the same team. That was where I found people like me. I remember Sam Greitzer’s Arbelos magazine grumpily threatening to cease publication because nobody ever sent in material, and three of us (one girl, two boys, did it matter?) stayed up late at one of our houses collecting and writing up enough interesting material for a whole issue. (We sent it to him, but it got sent back, because he had died in the meantime.)
HCSSiM was a fantastic math place. Its non-competitiveness (even the volleyball games there had people rotate from one team to the other, so you couldn’t tell who won) was so natural to me that I didn’t even notice it until an incident at the very end where the director was upset with the administration because the administration had promised a cake to the dorm floor that left its floor the cleanest, and the director strongly objected to this use of competition as an incentive.
So, “Math team” (focused on contests) vs. “math club” (focused on topics), does it matter? I would say no, there’s not much of a difference. Ours was a “team”, nominally focused on contests (mostly school vs. school, not individual). Did we explore open-ended problems where nobody knew the solution if any? Absolutely. Why? Because we were people who were interested in things like that. You don’t have to be a research mathematician to want to explore mathematics — those urges are instinctual! I don’t work on the 3x+1 problem to outdo my neighbor, I work on it because I can’t help it!
I am grateful to the existence of the math team for bringing like-minded souls together, and I am grateful to the math contests for making it easy for math teams to form.
So, what use are these skills now in research? Marginal. The main use of the problem solving skills, outside (as I am now) of an explicitly problem solving field, is that when well-defined problems do come up, I can typically immediately identify them as either easily solvable or intractable (or on rare occasions, in the oh so tasty thin margin in between). Not an enormous help.
What I never learned, in all those years, was how to pose my own problems. Contest problems (of a medium to high level) are in fact very difficult to come up with, and that skill is never taught or passed down in any way. I could solve problems, but always somebody else’s problems. When I went to grad school, I specifically aimed to learn the skill of posing my own problems, problems that I would find challenging but solvable. I wish this sort of thing were more actively taught. It would have been much more helpful.
Anyway, math contests are cool, because the problems on them are cool. Just like books by Coxeter are cool, because whatever you find in them will be cool. Just like Winkler’s puzzle books are cool. Those books don’t help me with life or my research. I just like them for what they are. I am glad there are other people like me in this world. I would be sad if math contests disappear.
Now I am older. My kids went to a math contest. I happen to know that their scores were right in the middle of the distribution. But I don’t tell them that, because who cares anyway. They enjoyed the math contest. They want to do another one. They are happy. I am happy.
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At ARML, we make a real effort to write beautiful problems and put them on the contest, so that many students will get to see them. In my capacity as problem writer, then, I see contests as one way to get interesting mathematics in front of kids. But they shouldn’t be the only way, and if I could line up all the math teachers in the US who either skip the interesting sections (because they’re too hard) or reduce everything to algorithms that they teach in bite-size chunks….well, I guess there wouldn’t be a lot of math teachers to go around, but the average math class would do a lot more actual mathematics. I’ve also started a math circle and pushed dozens of kids to apply to summer math programs that increased their love of math. And I tell my own students that “pyrotechnic problem solving” is just one kind of math skill among many, just like crossword puzzle solving is just one kind of English skill among many. So I agree with the other commentators that there should be more pathways into math.
That said, I have a seven-year-old boy who has discovered he loves chess tournaments. (Those of you who have seen me play chess may wonder, like I often do, where he gets this from.) If he wins a trophy, he comes back elated. If he doesn’t win a trophy, we talk about the fact that the way to win is to play more chess (read his chess books, etc.) between contests. But being able to compete has fueled a passion in him that, without competition, wouldn’t necessarily have grown.
One last thought. It’s true that boys outnumber girls in research mathematics. It’s also true that grad-school-and-beyond is currently the ONLY area where boys outnumber or outperform girls. At my (highly selective) public high school, girls outnumber boys in admissions, and when we kept track of the top 20 GPAs, they were something like 70% girls. Girls outnumber boys in high school graduations, college enrollments and college graduations. So in an educational context in which it is often harder for boys to succeed than girls, I’m not sure that it’s a disaster if one avenue AMONG MANY into interesting mathematics is a better way for boys to do math.
Thanks for doing this blog, Cathy–it’s awesome!
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Cathy, if I told you that a statistically significant proportion of Fields medalists qualified for the IMO (in their respective countries) and excelled at it, would that convince you that doing well in high school math competitions (like IMO) is a reasonably strong early predictor of future success as a mathematician ? If you don’t believe me, just check it out for yourself, it will surprise you just how many of them demonstrated their potential by qualifying for, and participating in the IMO. In fact, out of the 4 people awarded Fields medals in 2010, 3 participated in the IMO. And 50% of the past 16 Fields medalists (1998-2010) were IMO participants.
I’ve noticed this trend for other math awards at different levels e.g. the Morgan prize for undergraduates, the Clay Research Fellowship for freshly-minted PhDs, etc.
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Fact: Lots of Fields medalists qualified for IMO. Does this mean that IMO is a “reasonably strong predictor of future success as a mathematician?” Of course not. That is a logical fallacy. AT BEST, it suggests that if you are unable to qualify for the IMO, then you are (statistically speaking) somewhat *less likely* to be a Fields medalist, or win a Clay or whatever. This hardly seems like an interesting fact to me.
I don’t think it affects Cathy’s main point, which is that some students take this a step too far and start thinking things like, “If I can’t achieve X on competition Y, then I’ll never cut it as a mathematician,” for unrealistically large values of X and Y. Whereas the statement is surely true only for sufficiently small values of X and Y. (Consequently, I think that the argument that unfolds below here is mostly tangential to the original post.)
I would actually agree that there is some weak correlation between math competition success and a future career in mathematics. But the message that we need to get out there is that the correlation between *pure enjoyment of math* and a future career in mathematics is actually much higher.
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It’s interesting, though, that relatively few American Fields medalists have had much (if anything) to do with the Putnam examination. Milnor and Mumford were Putnam fellows twice, and Quillen once, but that’s it — none of the other American Fields medalists have been Putnam fellows. Not Paul Cohen, not Steve Smale, not John Thompson, not Charles Fefferman, not Bill Thurston, not Michael Freedman, not Edward Witten, not Curt McMullen.
The moral of the story would appear to be: don’t bother with contests after a certain point. So why not give them up sooner, rather than later, and start putting all the effort they demand into what really matters?
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Laurens you’re definitely right about the Putnam thing, I noticed that too. However, the other people you mentioned were not Putnam Fellows but still did well on it. If you read Joe Gallian’s article about Putnam history, he does mention that Paul Cohen ranked in the Top 10, and John Thompson received Honorable Mention on the Putnam. Fefferman was probably too young to become a Putnam fellow (he started college when he was 12), and Edward Witten majored in History and Linguistics in college, so taking a math exam was probably the last thing on his mind at that time. Who knows how well he’d have done if he took it ?
The Putnam exam should actually be a slightly stronger predictor of future success in mathematics, because doing well on it requires a very deep and profound understanding of the fundamentals of college math, and reflects that a student has attained an unusual level of mathematical maturity for an undergraduate. I know for a fact that many grad schools are willing to overlook lack of research experience if an undergraduate applicant had a solid performance on the Putnam exam.
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I think it’s pretty striking, though, that Cohen and Thompson participated in the Putnam long, long ago — it was a very different test back then, and participation in it (and in contest exams generally) was a very different thing, too. There was no MOSP, there was no AOPS, there was no systematic “cramming” or “jamming” of any kind. No one thought of working their way systematically through the hundreds and hundreds of (highly artificial) problems in a book like Engel’s, because no such book existed. Now there’s a whole subculture connected with these contest exams, and winning them is perceived to matter a great deal, which is a relatively new phenomenon. And yet it’s hard to see any great progress deriving from this “advance.” Quillen was the last American Fields medalist to have anything to do with the Putnam. And that was back in 1959!
I also think it’s strange to be wrestling over this point, when the matter was analyzed quite clearly and persuasively by G. H. Hardy way back in the twenties of the previous century.
Hardy was passionately and publicly committed to the complete abolition of the Tripos exam, precisely because his analysis convinced him that it had undermined generations of English mathematicians, indeed, Hardy insisted that the importance attached to the Tripos by young English mathematicians largely accounted for their negligibility in comparison with their French and German counterparts.
Looking over the list of Putnam fellows, I can’t help noticing that the names that subsequently became celebrated correspond overwhelmingly to people who seem to have taken the exam exactly once.
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While I agree with your general point, I think your last paragraph is a little unfair. You didn’t normalize to account for the fact that there are statistically speaking more people who are fellows once than twice, three times, or four times. Among those who fellowed three or more times, Bjorn Poonen, Ravi Vakil, Noam Elkies, and Kiran Kedlaya are all very prominent researchers in their fields. Several more recent names on this list are strong mathematicians I have small personal connections to and I expect good things from them in the future.
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Perhaps you’re right, and I certainly didn’t mean to seem harsh. I too hope that the persons you mention will do grand and wonderful things, and I greatly respect their achievements to date.
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I think it’s pretty clear that individualist, time-pressure competitions do alienate a substantial number of very intelligent people who have the capacity to be good mathematicians. I think there is agreement that there is room for wonderful alternative enrichment.
However, I’m surprised by the preponderance of comments saying that these competitions have *nothing* to do with high level mathematics. I’ll admit that my own limited experience of these competitions is that they overemphasis a kind of tricky combinatorics and facility with certain inequalities and problem solving tricks. But most of the people I knew who were successful at the competitions did indeed go on to high level mathematical careers. Angela suggests there is evidence that this correlation holds in general.
I wonder how much of a factor Gladwell’s “January effect” is. E.g. many more pro hockey players are born in the early part of the year, supposedly because the age cutoff for team participation is Jan 1, so kids with earlier birthdays are older and more successful than the general pool and their early success leads to internal and external encouragement which leads to more success, etc. Similarly, early success in exams may lead to the same virtuous circle (at the expense of those who don’t succeed at the exams).
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Laurens I don’t know if you’ve seen or tried Putnam exam questions, but I can assure you that no amount of “cramming and jamming” will allow a student to do well on it. It is pretty common knowledge that the questions are intended to test creativity and out-of-the-box thinking, so one can’t actually “prepare” for an exam like this. And if you’re not somewhat gifted at math, practicing old putnam problems will NOT make you significantly better at them.
Also, Qiaochu already made my point by giving examples of prominent mathematicians who were Putnam fellows multiple times.
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“Laurens I don’t know if you’ve seen or tried Putnam exam questions, but I can assure you that no amount of “cramming and jamming” will allow a student to do well on it. It is pretty common knowledge that the questions are intended to test creativity and out-of-the-box thinking, so one can’t actually “prepare” for an exam like this. And if you’re not somewhat gifted at math, practicing old putnam problems will NOT make you significantly better at them. ”
I was a Putnam fellow twice and I’m here to tell you that you can most certainly prepare for the Putnam, and that practicing on old problems really does help improve your score. Ravi Vakil, a four-time Putnam Fellow, used to run pretty hard-core Putnam prep sessions at Stanford (http://math.stanford.edu/~vakil/putnam06/) so I’m guessing he’d agree.
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Well Jordan, I guess I stand corrected then. But I still think that the effect of practice on people who aren’t particularly gifted is negligible at best.
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Angela, I have indeed seen many, many Putnam exam questions. I’ve also read Bruce Resnick’s illuminating essay
Click to access putnam.pdf
about writing such questions. And the simple truth is that any mathematics problem that is guaranteed to be soluble within a half-hour is a contrivance — a puzzle devised by one clever person to entertain another.
As Reznick notes, the options available to the writer of such a problem are tightly constrained, and once you’ve studied closely several hundred of these things, it gets significantly easier to guess how the author of any particular one was led to formulate it. (Also see W. W. Sawyer’s superb PRELUDE TO MATHEMATICS for a memorable comment on this point, in a brief essay entitled “Reconstructing an Examiner.”)
I think you may want to ask yourself why Hardy disagreed so strongly with the proposition that any contest exam can serve as a measure of “creativity.” Certainly no serious Tripos student back in the day would have accepted your principle that no cramming and jamming are of no use. On the contrary, an entire industry of specialized tutors grew up around the exam, and every successful examinee relied heavily on their assistance.
It’s an interesting suggestion that “if you’re not somewhat gifted at math,” you can’t hope to do well on the Putnam. Have you ever studied the career of Arthur Rubin?
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The Reznick essay you linked to was very interesting, I’m surprised I hadn’t come across this earlier. The essay actually proves my point because the committee rejects questions that “sound familiar” or “must have appeared somewhere else”. These people are professors who teach from a variety of textbooks and are familiar with past Putnam questions, so they’d be able to spot a “familiar” question from a mile away.
I know about Arthur Rubin, but Rubin is just one example, it’d be wrong to generalize based on his particular case. I have no idea why he lost interest in academia, but I doubt it was due to inability to carry out original mathematics research. I’m afraid his particular case does not constitute strong evidence against the predictive power of putnam performance.
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You persist in ignoring Hardy’s argument. Why?
Have you read Andrew Wiles’s address to the IMO participants in 2000?
Do you dismiss his judgment too?
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Here is a rough and ready definition of a genius: somebody who can do easily, and at a young age, something that almost nobody else can do except after years of practice, if at all. […] Every year or two a mathematics undergraduate arrives at Cambridge who regularly manages to solve in a few minutes problems that take most people, including those who are supposed to be teaching them, several hours or more. When faced with such a person, all one can do is to stand back and admire.
And yet these extraordinary people are not always the most successful research mathematicians. If you want to solve a problem that other professional mathematicians have tried and failed to solve before you, then, of the many qualities you will need, genius as I have defined it is neither necessary nor sufficient.
— William Timothy Gowers
A very quick thinker may be able to pick up new ideas rapidly, to find snappy rejoinders to any question, and to complete tests and examinations in a remarkably short amount of time, but these attributes may in fact lead to excessive frustration when such a student encounters a genuine research problem for the first time – one that requires months of patient and systematic effort, starting with existing literature and model problems, identifying and then investigating promising avenues of attack, and so forth. In athletics, the best sprinters can often be lousy marathon runners, and the same is largely true in mathematics.
— Terence Tao
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Laurens it’s not that I’m ignoring Hardy’s argument, it’s that I just don’t see how the Tripos can be compared to the Putnam exam.The tripos has hundreds of questions and is taken over the period of a week or two by seniors, it’s really just a test of knowledge acquired over one’s undergraduate career, more than anything else. I wouldn’t even regard it as remotely close to the putnam exam.
Anyway, the comments from people here have been pretty insightful and I will definitely concede that doing well on the putnam exam is not an absolute guarantee of future success (arthur rubin perhaps being the most glaring example of this). But I agree with Dan above that there is some correlation. And as he has pointed out, this argument is no longer even pertinent to Cathy’s original point anyway.
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I agree with Dan, too (as does Bill Thurston.) And it’s clear that Hardy (if you’ve read his essay) disagreed strongly with your characterization of the Tripos (indeed, this is evident even from the short excerpt I quoted from him above.) Hardy explicitly insisted that it was not at all “just a test of knowledge acquired over one’s undergraduate career.” Rather, the Tripos purported to be just what you insist the Putnam is: a test of “out of the box,” “creative” thinking. Which is why I regret not learning your reaction to Hardy’s critique.
I agree, also, that the comments here include many penetrating observations. Besides Dan L.’s, I particularly value Matt’s remark that
“What I never learned, in all those years, was how to pose my own problems. Contest problems (of a medium to high level) are in fact very difficult to come up with, and that skill is never taught or passed down in any way. I could solve problems, but always somebody else’s problems. When I went to grad school, I specifically aimed to learn the skill of posing my own problems, problems that I would find challenging but solvable. I wish this sort of thing were more actively taught. It would have been much more helpful.”
To the practicing research mathematician, problem-posing and problem-selection are extraordinarily subtle, interesting arts, but they don’t lend themselves to exam treatment.
Most of all, though, I agree with Cathy’s original premise, which I sought to indicate by quoting Thurston (and Csikszentmihaly et. al.): competitions repel many very talented people. And the population does not consist exclusively of shrinking violets who “can’t cut it” in the hard-scrabble world of professional mathematical research. When Ramanujan was invited to Cambridge by Hardy, one of his explicitly stated anxieties was that he might be required to sit for an entrance exam!
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Just to be a nerd, let me interject that this is not totally irrelevant to my point, and in fact it’s actually potentially supporting my point. Let’s assume for the sake of argument that the effect I claim happens to an extreme extent; that in fact anyone who didn’t do well on a test was told to leave mathematics, and they did so. They would we be surprised that the Fields medalists were also people who did well on tests? Of course not, because they were the only ones who became mathematicians.
In other words, in the total absence of contests, we may have had a different list of Fields medalists because way more different kinds of mathematicians would be prospering.
What I actually think is that some people are just good no matter how you measure them- under any metric they are really just superlative. But I still think it’s a focus on people I don’t really care about, not because they don’t do great math (they do) but because they would probably do great math in any environment because they are so good. I care more about people who do great in some environments (i.e. research math) but not in others (i.e. contest math).
In another post I may tackle the issue that the Fields medal, being awarded only to those under forty, is inherently biased against women.
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Actually, I sort of wanted to point this out (that if we select for people who are good at competitions, we shouldn’t be surprised that so many great mathematicians were great at competitions), but I refrained for the very reason you point out (that most Fields medalist-types would probably be Fields medalist-types no matter what selection criteria we use to weed people out).
Michael’s point below was also on my mind. There is a lot of competition in the research math world. It’s just a very different kind of competition.
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I think we shouldn’t forget that there is a lot of competition involved in real mathematics (i.e. mathematics outside of high school competitions and the Putnam) as well. For example, if you want to become a math professor (I’m not saying that this is the highest goal in mathematics, it is just what I am familiar with), you have to be prepared to deal with the following:
Competition to get into a good college.
Competition to get admitted to good summer enrichment programs like HCSSiM, REU’s, etc.
Competition to get into a good graduate school.
Competition to get the kind of job you want.
Competition to prove a particular result first. (One can try to avoid this last one, which I particularly dislike, by carefully choosing what to work on, or by teaming up with people working on similar things, but it is often there.)
A key part of recommendation letters for grad school, academic jobs, and promotions is direct comparison between the person under review and other people in a similar area and career stage. So at least until you have a permanent job, you are constantly engaged in a competition, even if it is less overt than say the Putnam.
All of this can be quite stressful and discouraging, maybe to some people more than others. By comparison, math competitions are innocent and fun, since nothing is really at stake except for fragile egos. (My main complaint about most math competitions is not that they are competitive, but rather that they emphasize the wrong skills for real math, either pure or applied.) Maybe we need to better prepare kids to deal with the competition which is so prevalent in real life.
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Michael, you literally took the thoughts out of my mind. I was going to point out this exact thing, that mathematics research is more competitive than one would think. In most areas of mathematical research, more than one person is working towards the same goal, and whoever reaches it first gets all the credit, most of the time. Of course, people tend to collaborate rather than compete but even groups of collaborators have other groups of collaborators competing with them.
Here’s a quote from Cathy’s post which I disagree with:
In fact, there is a timed element to “real life” mathematics research. One is constantly under some time constraint at every step in one’s career, whether it’s trying to publish enough original research as a PhD student (time constraint: 4-5 years) to be able to get good positions afterwards, or trying to publish enough original research in the short period before which you have to apply for tenure, there is always a time constraint and always some competition. You’re constantly having to compete against (and worry about) people who are pumping out quality papers at a faster rate than you are, and if you are unable to, as you put it, “be original and creative really quickly“, you might not progress through your career at a reasonable enough pace, or you might not be able to compete effectively for the precious few openings at the top universities, most of which go to the super-productive types who may be about the same age as you but have published far more quality research than you have.
Perhaps being exposed to the rigors of competition at an early stage is not such a bad thing. If you can’t handle failure in high school math competitions (which is at best a minor failure that has rather trivial consequences), how would you deal with bigger failures down the line, like being denied tenure, etc. If you’re the type who tends to quit because of failure, mathematics may not be for you, because even the best mathematicians fail some times.
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This is one of the longest comment threads I’ve seen on the internet since someone asked “should I show my child Star Wars in chronological order (IV-V-VI-I-II-II) or narrative order (I-II-III-IV-V-VI?” http://kottke.org/07/09/star-wars-viewing-order
Love to all nerds!
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You shouldn’t have to apologise for this legitimate viewpoint.
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It’s unfortunate that that vast majority of the brightest U.S. secondary school students do not have access to any type of local math enrichment. The most popular AMC contests reach less than 20% of schools. Only a small percentage of those schools have an active math team. Active, non-competitive math circles are even rarer, generally limited to the largest cities.
The potential math talent living in areas without enrichment often sail through school math at the top of their classes with minimal effort. These students are far worse off than those damaged by their exposure to competitive math. The brightest students lacking enrichment have difficulty understanding the value of effort, have rarely experienced failure at math-related academics, and lack a social network of peers and coaches to help identify future opportunities.
How are these students going to develop the mathematical expertise of typical math team members who’ve invested hundreds of extracurricular hours into secondary school practices or top members who may invest thousands of hours? They’ll have to work a lot harder and a lot longer! That can be daunting if you’ve never had to work at math before. Will they be resilient when they first confront academic or professional failure? Probably not as much as those who’ve learned to handle it on the math teams.
I’ve posted more on the value of math teams and contests at Spirited Dreams. The contests are not really about identifying future math researchers or those who can excel in math-related careers. Like sports and the arts, they are mostly about character development and social support. Mary in post #11 alludes to the need to help math students develop a healthy identity. Math teams can develop strong character traits that will serve members well in many endeavors.
While teams and contests are not without risks, we should compare these to the risks faced by those faced by the majority of top math students who lack any enrichment. There have been many efforts to develop alternative math enrichment strategies but the contests have created an unmatched level of student engagement. In my view, the teams and contests are well worth supporting with my time and talents.
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holy shit even the great terence tao commented on this…..
i’m always really curious on how somethings get more attention than other things on the internet and how random people(i mean “random famous people”) find people’s blog.
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I agree with a lot of what you say. My brother was the state mathcounts champ, went on to be on the US Olympiad team, and went on to MIT (took a class by you) and is now a post-doc in Math. Two years his junior, I always participated in the same competitions, but never came close to the success he acquired. I was the high school valedictorian but was convinced I sucked badly at math. It was a huge insecurity for me. In some sort of exercise of masochism, I decided to major in math in college, to “conquer my demon” so to speak. Its been useful. I’m a physician now, but have a math degree and feel more confident with numbers than my colleagues in medicine. I turned the math-contest induced inferiority complex into a positive,but it could easily have gone the other way …
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“I was always deeply uncertain about my own intellectual capacity; I thought I was unintelligent. And it is true that I was, and still am, rather slow. I need time to seize things because I always need to understand them fully. Even when I was the first to answer the teacher’s questions, I knew it was because they happened to be questions to which I already knew the answer. But if a new question arose, usually students who weren’t as good as I was answered before me. Towards the end of the eleventh grade, I secretly thought of myself as stupid. I worried about this for a long time. Not only did I believe I was stupid, but I couldn’t understand the contradiction between this stupidity and my good grades. I never talked about this to anyone, but I always felt convinced that my imposture would someday be revealed: the whole world and myself would finally see that what looked like intelligence was really just an illusion. If this ever happened, apparently no one noticed it, and I’m still just as slow. (…)
At the end of the eleventh grade, I took the measure of the situation, and came to the conclusion that rapidity doesn’t have a precise relation to intelligence. What is important is to deeply understand things and their relations to each other. This is where intelligence lies. The fact of being quick or slow isn’t really relevant. Naturally, it’s helpful to be quick, like it is to have a good memory. But it’s neither necessary nor sufficient for intellectual success.”
— Laurent Schwartz
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When I was in high school I enjoyed math competitions and also math projects. What I loved about the projects was that I could spend days on a single problem (much like I do now as a mathematician). The trouble with the projects was that every time I did a beautiful project, someone would say I had help from my parents. That made me feel lousy. My success on math competitions was never questioned because there was no opportunity for me to cheat. As for collaborating, I never liked collaborating or doing homework with other students. I liked being alone with the mathematics. I liked thinking a very long time about a single problem. I believe that math contests with fewer longer more difficult problems, similar to the AIME, are the most interesting. I agree that having elimination rounds does discourage students. When I was in high school only students who reached a certain level on the AHSME were allowed to take the AIME. There was no reason for the elimination round because they tested different skills and besides, the AIME was also a single answer for each question. Sadly, a math contest which includes graded proofs, is expensive to grade. Luckily at least NYS has Regents High School Exams that include proofs and are given to everyone. Striving for a perfect score on a Regents Exam is another way to prove one’s abilities in mathematics and missing by a few points is less discouraging than being eliminated.
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A) Math competitions are not these vicious, competitive things where participants try to take each other down and can claim to be ‘smarter’ than others just because they placed higher scores. I also think you’re underestimating participants’ intelligence. When I didn’t win a math competition, I knew it wasn’t because I was bad at math, but rather because other participants had either had access to more rigorous training or because plain and simple I hadn’t worked hard enough. And it was a motivation to WORK HARDER for my next competitions. And even when I didn’t win, I always enjoyed the intellectual challenges that math problems (including timed problems) presented.
B) By your logic, any form or competition is dangerous then. There will always be kids who think they’re not good enough at soccer, not good enough at swimming, or spelling bees, or gymnastics, or what have you. Competitions do not just reward natural skill, they reward EFFORT. We can’t compromise rewarding hard workers in order to make every participant happy. That doesn’t encourage intellectual challenging at all. Also, children’s self-esteem is only as fragile as we make it out to be. We might unfortunately be able to make kids feel bad when they don’t win, but we also have the power to make them feel good about themselves and their effort regardless of the outcome. What needs to change is the dichotomy between “good” at math and “bad” at math, which I do not endorse (and which you coincidentally also use quite a bit in this article).
C) Math competitions aren’t bad for women. They way you describe girls as ‘more susceptible to feeling bad about not being good enough’ and possibly ‘developing’ in a way that makes them slower than boys at math is what’s bad for women. I’m a girl and I was one of the best math competitors in my country in my time, and I didn’t let anyone or anything tell me that I wasn’t good enough or smart enough, ESPECIALLY not people who doubted my emotional or intellectual capacities because of my XX chromosomes. Math competitions aren’t bad for women. You should have heard the applause at the 2009 IMO when the United Arab Emirates team went on stage and the attendees realized that every single UAE competitor was a girl.
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When I stumbled on this blog a few weeks ago, I felt uncomfortable with the ideas expressed here. I have supported Math Contests for years, running them, coaching kids to prepare for them, and running a summer camp to support kids interested in math. I just read the article by Ellen Veomett, “An IMMERSE-Style Course Brings a Research Experience to Students and Faculty”, in the October 2012 Notices of the AMS. This article is on a different topic, but a couple of things it says about the IMMERSE-Style course I think are equally applicable to math contests.
“The greatest benefit for the students is the experience of the course. The students have a chance to struggle with topics that are on the edge of their mathematical understanding.” “The experience of a slow struggle eventually ending in a deep understanding is extremely empowering. The more chances that students have to encounter great reward for great struggle, the more students will be willing to undergo that great struggle in the future.” I would say this is true of math contests as well. What many have commented on as a negative about contests (too many students leave a contest feeling not good enough at math) could be one of their biggest benefits–at some point we all need that experience of struggle and perseverance and realizing later that we could have done better if we could have eliminated some simple mistakes or just looked at our problems from a slightly different viewpoint.
Later in the article the author states “I wanted the students to begin feeling comfortable with feeling uncomfortable. Reading a research paper is often frustrating and confusing, and those uncomfortable feelings will often discourage someone who is not used to experiencing them.” Again, many comments on this blog attribute contests to discouraging students in mathematics. There will be frustrating times in mathematics and those feelings aren’t comfortable. Does that mean they should be avoided entirely? I don’t think so.
So perhaps math contests aren’t the problem, but the expectation of what they are measuring and what they are accomplishing for students. I know that I try to advise teachers bringing students to contest to prepare them mentally for the contest by saying, “Contests are tough. You probably won’t get all the problems right. Getting half right would be a great result. Enjoy the challenge and see what you can do with the problems you’ll see at the contest.”
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Well said, Kevin Hopkins. There are too many times people are afraid of a challenge. Learning to face a challenge and succeed is an essential skill. For many students, the first time they face a challenging difficult assignment is in the first year of college. That can be a very difficult time to adjust. Grades and a scholarship can be on the line and there are often no parents around to provide moral support. Learning to face a challenge in the relatively low stress setting of an extracurricular club with parents at home to provide encouragement is ideal.
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Math is about research; Olympiads are a test. A gold medal at the Olympiads proves one’s ability in doing tests, not doing research. An important thing in research is to find new problems to solve, but Olympiads only train students to work on other people’s problems.
— Shing-Tung Yau
I am always unhappy when I see competition in mathematics.
— Viscount Pierre Deligne
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I agree that math contests are about doing tests not research. I see them as an opportunity to compete not an opportunity to discover. I don’t view them as a method of training for future mathematicians and scientists but as a sport that can be enjoyed by a far larger community. My own high school math team was highly competitive. Two of us became mathematicians: Jeremy Kahn and I. The rest went into other fields ranging from politics to medicine. I would venture to say that the mathematics contests and the training before them was as much preparation for those fields as it was for mathematics, if not more. Certainly high speed mathematical thinking is a necessary ability when responding to the intense emergency situations that can arise outside academia.
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Forged By Adversity
Guy Edson, ASCA Staff
I was following a school bus the other day when it stopped to pick up two middle school aged boys. Because of the framed glass emergency door in the back of the bus I could watch the two boys playfully tussling with each other as they made their way to the very last seat of the bus. Finally the bus began to move again – it seemed to take forever just to pick up two boys. I then thought back to my childhood days and riding the school bus. As soon as I crossed that white line on the floor at the front of the bus the doors closed and the bus sped off to the next stop, adding the dimensions of speed and bumps and movement to the normal tussling. It took balance and strength to make your way to the seat. Quite frankly, I sometimes didn’t make it to a seat without being deliberately shoved, or through my own clumsiness, stumbling and nearly falling. It was a challenge – but I never considered it as such. To me, it was “normal.”
Our school bus practices now are far more safe and part of a widespread effort to insure the safety and comfort of our children.
Who can be against that?
At the risk of getting some “what are you thinking” emails and maybe even a few cancelations, I am going to go out on a limb and suggest that our society’s collective efforts in protecting our children have, in subtle ways, removed “opportunities for falling down on the bus” and other failures. Failure is simply not allowed. Adversity is to be minimized.
Consequently, a healthy attitude toward failure and adversity is often undeveloped. A few years ago I was hired as the dryland training coach for a local high school. During a heavy weight lifting cycle I explained the concept of lifting to failure. (Lifting to failure, by the way, is an accepted and common practice in weight training. Widely written about and researched, it is the de facto method for improving strength.) Failure, as a concept, was so foreign to these high schoolers that they didn’t get it. Even when I demonstrated it, they still didn’t get it. Have we painted failure so darkly that no one gets the importance of it anymore?
I am happy to report that some do still get it. Recently we interviewed a young person for an open position and when asked what she was really good at she replied that she was very good at failure. She explained that it was through failure that she learned how to succeed. How refreshing it was to hear that! (And yes, she was a former national level swimmer.)
Last week I attended a lecture by a former Navy SEAL who, after over 30 years of service, is now part of the SEAL training team. He explained the SEAL Ethos and what stood out to me was the phrase, “Forged by Adversity.”
“Forged by Adversity” is at the heart of what we do with our upper-level, older age group swimmers and all advanced senior swimmers. Adversity, however, is not something normal people deliberately seek. Most avoid it. All good coaches find that it is one of the greatest tools for shaping swimmers not only into great swimmers, but into future grownups with one of the best of all the life skills.
Adversity provides the opportunity to build determination, build confidence, build mental strength, give perspective, and to build physical toughness. Are these not qualities we want in all our children?
Arnold Schwarzenegger said, “Strength does not come from winning. Your struggles develop your strengths. When you go through hardships and decide not to surrender, that is strength.”
And in swimming practice, adversity comes from sets, or possibly whole workouts, deliberately designed by the coach to make the athlete fail. The coach does that by creating a set where a combination of the distance, the intensity, and a low rest interval make it difficult if not impossible to make. There are many strategies and methods for doing this that go beyond the scope of this newsletter AND these strategies include a progression for how much adversity is presented at what ages, but the bottom line is this: Swimmers get better through a workout environment that offers the opportunity for failure.
And so, Parent, what is your role in all of this? I hope you refrain from seeking to protect your child from the adversity and opportunity for failure at swimming practice. To do so is to deny your child the opportunity for building the qualities described above. Instead, consider your role as the encourager. Encourage your swimmer to persevere, to break through, to come back the next day determined to work harder against the adversity placed there by the coach. Then enjoy and celebrate the moment when your child does break through. (And they will!)
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Despite being a mediocre math student I really liked math competitions–and I’m not even good at math any more, having pursued humanities in university. They gave me something to look forward to and work towards, and our teachers treated it like a fun extra instead of some kind of cutthroat competition. Most importantly, I remember my teachers being excited to see me there.
“It’s not how you win or lose it’s how you play the game” is a gross cliche, but I think it applies. Learning that I was part of a supportive environment where I could enjoy a healthy math-off was better for my development than avoiding competition altogether.
Most importantly, I think it’s kind of patronizing to assume that girls can’t handle competition because it will destroy their self esteem. Competition exists in life and in class, and there will always be someone out there who’s better than you at something. Shouldn’t we instead put the onus on our educators and family members to nurture girls’ talent, avoid implying that their female-ness makes them bad at certain activities, teach girls how to enjoy competition without taking it too seriously?
Your collaborative math enrichment project sounds cool though, and nerdy high school me would definitely sign up for it!
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