*Recently, some interesting discussions on math education and mathematical philosophy have taken place on, of all places, my Facebook wall. Since Facebook is a rather restrictive medium, I feel it’s time to widen the scope of my blog to include such topics.
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I am currently sitting in the main common area in our residence halls at the Math Olympiad Summer program (abbreviated MOP) in the middle of Lincoln, Nebraska. It is rather quiet, as the students are all taking their 5th “MOP test” in the last two weeks, and my fellow instructors and graders are either proctoring the tests or preparing more material to teach these brilliant kids.

It got me thinking about a topic that is often discussed among mathematicians and among those in math contest circles: How correlated is mathematical contest ability with mathematical research ability? Does one help or hinder the other? Is the social environment that math competitions create hostile to noncompetitive students, especially women?

## Math, Marathons, and Spelling Bees

The first two questions are somewhat easier to address, and many successful mathematicians have expressed their thoughts. Terrence Tao, who earned a gold medal at the International Math Olympiad at the age of 13, has written a blog post on the topic. He links to this post on LessWrong, which lists a number of quotes by great mathematicians regarding math competitions.

Given how many prominent mathematicians were successful at math contests themselves, I was surprised that the general consensus among these mathematicians seems to be that success in math contests hardly correlates at all with the ability to do the slower, deeper, more tedious work required for mathematical research. Some go on to claim that being immersed in “math contest culture” may actually *harm* one’s ability to produce novel mathematical ideas, since it encourages a kind of impatience in the mathematician. William Thurston even goes so far as to compare math contests with spelling bees:

These 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.

In an ideal world, I would agree. That is, if schools gave as good of a sense of what doing mathematics was as they give a sense of what writing is, then math Olympiads would not be as necessary in terms of encouraging students to pursue mathematics. As it is, though, math contests such as the AMC, which at least require some level of mathematical thinking, are the closest thing most American high school students can get to experiencing a glimpse of what mathematicians really do.

Additionally, although I am in no position to disagree with William Thurston, and my opinion may change as I dive deeper into the research world, I find further problems with the spelling bee analogy. Math contests, especially at the Olympiad level, require much more meaningful work than memorizing a dictionary and spelling some words correctly. At MOP, we teach the students modular arithmetic, generating functions, projective geometry, the probabilistic method… all things that a professional mathematician might spend a lifetime studying. And the students who really understand these subjects deeply are generally the ones who perform better on the tests we give them. So what is going on?

Let’s consider an alternative analogy: Math Olympiads are to mathematical research as a 5K road race is to a marathon. Both are challenging endeavors that take a lot of training and practice. The 5K requires more speed, strategy, raw strength, and head-to-head competing, while the marathon is more about patience and diligence.

But are they uncorrelated? Are 5K’s just the spelling bees of running, and don’t necessarily predict future marathon or ultramarathon success? This is where I disagree. The best marathoners in the world would leave most runners in the dust in a 5K, and the best 5K runners in the world can place highly in any marathon after only a few months of endurance training. Similarly, I have no doubt that most of the best mathematicians in the world would do just fine on the USAMO, and I have seen firsthand that the best Olympiad students can usually produce some good research after just a few months at a summer REU.

Of course people who have never run a 5K in high school can start running at the age of 25 and run a marathon; I am not disagreeing on that point. But I do disagree that math Olympiad training gives no significant mathematical advantage; it’s always going to be easier for that high school track star to go on to do marathons later in life. Whether they choose to do so is a different matter and depends on a lot of personal factors that are hard to quantify.

## Social Environment and Women

The third question – whether the social environment created by math contest culture is hostile to noncompetitive students and girls in particular – is a trickier one. I bring it up because I just read a blog post from last year by “MathBabe” Cathy O’Neil. A quote:

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.

As a woman who has excelled at math competitions and is now pursuing a Ph.D. in math, I find this comment both interesting and very hard to relate to. Math contests, which started for me in middle school, were always a joy to me, because I loved the mathematics so much. Yes, I practiced to get faster, but I mainly practiced because I was amazed at how you can use modular arithmetic to find the units digit of $7^{2002}$ *without calculating the whole number*, and how the area of a triangle was equal to the product of these mysterious quantities, the inradius and the semiperimeter. My father would have me prove the Pythagorean theorem, or derive the quadratic formula from scratch, and with each new understanding I appreciated mathematics even more.

I made friends through math teams and programs, and the community support spurred me on to continue to practice and study. Rather than it being a hostile environment, I found the social circle of math geeks to be much *more* welcoming than the bullying crowd of “popular” kids that dominated my high school. So wherever the negative social reinforcement is coming from for girls, I don’t think it can possibly be coming from math contests themselves.

One interesting thing about Cathy’s comment, though, is that perhaps boys’ brains do develop in a different way. Perhaps we should have different divisions for women in more of our math contests, just as there is always a womens’ division in any 5K race.

But even as is, I find that math Olympiad training is useful and encouraging for students in mathematics, women included. I would never have gone so far with mathematics if it hadn’t been for the math contests that helped me realize that math was more than just memorizing your multiplication tables. Or the dictionary.

I think that the degree to which Olympiad training is useful in research depends a lot on the style of mathematical research being done. I can imagine that in some of the more theory-heavy fields that it would be a lot less useful than in say graph theory. The more problem-solvey fields of mathematics have research problems that look a lot more like Olympiad type problems, with the main difference being that contest problems are guaranteed to have a nice solution while research problems aren’t. On the other hand I have known important research problems that could reasonably have been a B7 or B8 on the Putnam.

That’s a good point. I thought about that; people of the Erdos/extremal graph theory school of thought are definitely very Olympiad-y in the research that they do. However, what’s striking is that the best Olympiad students don’t all go on to do those kinds of topics; while people like Po-Shen Loh are making a name for themselves in graph theory, we also have people like Kiran Kedlaya and Bjorn Poonen going into much deeper fields.

Dear Maria,

How do you measure “depth” of a mathematical field?

It often boils down to measuring the height of the wall of definitions, which prevents many from enjoying mathematics on the other side of the wall.

Be well,

Alexander

Math competitions are certainly training, but they’re not trying to train kids to do research mathematics. Research math has this odd “home-field advantage” when it comes to the pros at math competitions, but if we realize that many math competition beasts go into fields like engineering or finance, we can see why the priorities of competitions might not align with those of professional mathematicians.

Just look at some of the sponsors of Mathcounts and the USAMO: Akamai, DE Shaw, Jane Street, Raytheon, the National Society of Professional Engineers. None of them primarily do pure mathematics. True, the AMS, NSF and Pi Mu Epsilon also make the list, as do some educators like AoPS (clearly a symbiotic relationship there) but the pure math donors are drowned out by those who want to train and recruit future employees to do something besides pure math research, especially for Mathcounts. Competitions, like anything else, tend to follow the money to some extent, and while the competitions continue to do a good job of training and distinguishing strong future employees, they’ll continue to exist through sponsorship by these companies.

Hi Maria,

I have a big problem with this debate, which is that its perpetuation implies there is an existential debt owed to math research by the math contest community. I suppose this is because the people who run math contests are mostly mathematicians, but it just makes their egotism more explicit. Contest-style problems are (if they are good) interesting and beautiful in their own right. Training for and participating in math contests heightens logic and reasoning skills, it provides an arena of success for those who might be shy, modest, or socially awkward, and it helps compensate for deficient math teachers. And while those who excel at the highest level in contests often go into math research, purely by the numbers I can say that 100 times (I made that up) as many students participate in math contests than will ever do math research. It is astoundingly selfish of mathematicians (who do so) to endorse changing the system with the goal of getting 1% more research mathematicians out, while ignoring the interests of the other 98%. However, I guess I shouldn’t be surprised, since in the math research community research is all that matters, while teaching ability counts for almost nothing and public outreach for even less. If you want to talk about the IMO or the Putnam instead of high-school contests, the numbers are not so lopsided but the argument is the same.

The problems with the math contest system are all in deficiently serving students, not in inaccurately representing math research, which is not its purpose. It is fine to caution an undergrad interested in pursuing math that research math is not the same as contest math (though actually a summer research project serves that purpose much better), but trying to correct this misapprehension in the general public just makes math seem more inaccessible. Contest math is inaccessible to over 90% of people as it is, and then we’re supposed to tell those that get it that even that isn’t real math? While we’re at it, let’s tell Vi Hart to stop making videos because they don’t accurately represent math research.

That’s a really good take on it, Stephen – thanks for your comment. I think that is indeed one thing that bothers me about some research mathematicians’ downplay of the value of math competitions. It doesn’t take into account the other benefits – like teamwork skills, learning interesting “classical” mathematics, developing general problem solving skills – that math competitions provide.

I guess we could take the 5K analogy a bit further, then – doing a few 5K’s is worthwhile for your overall health regardless of whether you want to eventually run a marathon.

On the other hand, even though the purpose of math contests is not to give students training in mathematical research, I do think it still does provide a taste the kind of thinking that is required. The more well-designed problems on math contests, unlike the usual homework assignments you encounter in the classroom, are usually such that you don’t immediately have any idea how to approach it at first. So you play with it, you try some special cases, you take a few attacks at it that fail, and eventually either something randomly works or you gain a better understanding of the system and it falls apart. This is very similar to the research process, just on a smaller scale.

I’m going to add a comment to my own post in response to an email I received after posting. It has been pointed out to me that my claim above that math contests, such as the widespread AMC, are the only kind of mathematical enrichment that some students have access to, may not be accurate. There are new math circles popping up in many locations in America now, and countless summer programs that teach more in-depth mathematics. I am not sure to what extent these programs are available in rural or economically disadvantaged areas, but they are another great way for students to learn mathematics, in addition to the countless free math contests that are available, and the growing online community at Art of Problem Solving.

Hard unsolved problems are just hard Olympiads.

Odd perfect numbers, FLT,etc.

VC

UH