Make Training Non Zero-Sum

Some thoughts about some modern trends in mathematical olympiads that may be concerning.

I. The story of the barycentric coordinates

I worry about my geometry book. To explain why, let me tell you a story.

When I was in high school about six years ago, barycentric coordinates were nearly unknown as an olympiad technique. I only heard about it from whispers in the wind from friends who had heard of the technique and thought it might be usable. But at the time, there were nowhere where everything was written down explicitly. I had a handful of formulas online, a few helpful friends I can reach out to, and a couple example posts littered across some forums.

Seduced by the possibility of arcane power, I didn’t let this stop me. Over the spring of 2012, spring break settled in, and I spent that entire week developing the entire theory of barycentric coordinates from scratch. There were no proofs I could find online, so I had to personally reconstruct all of them. In addition, I set out to finding as many example problems as I could, but since no one had written barycentric solutions yet, I had to not only identify which problems like they might be good examples but also solve them myself to see if my guesses were correct. I even managed to prove a “new” theorem about perpendicular displacement vectors (which I did not get to name after myself).

I continued working all the way up through the summer, adding several new problems that came my way from MOP 2012. Finally, I posted a rough article with all my notes, examples, and proofs, which you can still find online. I still remember this as a sort of magnus opus from the first half of high school; it was an immensely rewarding learning experience.

Today, all this and much more can be yours for just $60, with any major credit or debit card.


Alas, my geometry book is just one example of ways in which the math contest scene is looking more and more like an industry. Over the years, more and more programs dedicated to training for competitions are springing up, and these programs can be quite costly. I myself run a training program now, which is even more expensive (in my defense, it’s one-on-one teaching, rather than a residential camp or group lesson).

It’s possible to imagine a situation in which the contest problems become more and more routine. In that world, math contests become an arms race. It becomes mandatory to have training in increasingly obscure techniques: everything from Popoviciu to Vieta jumping to rectangular circumhyperbolas. Students from less well-off families, or even countries without access to competition resources, become unable to compete, and are pushed to the bottom of the IMO scoreboard.

(Fortunately for me, I found out at the 2017 IMO that my geometry book actually helped level the international playing field, contrary to my initial expectations. It’s unfortunate that it’s not free, but it turned out that many students in other countries had until then found it nearly impossible to find suitable geometry materials. So now many more people have access to a reasonable geometry reference, rather than just the top countries with well-established training.)

II. Another dark future

The first approximation you might have now is that training is bad. But I think that’s the wrong conclusion, since, well, I have an entire previous post dedicated to explaining what I perceive as the benefits of the math contest experience. So I think the conclusion is not that training is intrinsically bad, but rather than training must be meaningful. That is, the students have to gain something from the experience that’s not just a +7 bonus on their next olympiad contest.

I think the message “training is bad” might be even more dangerous.

Imagine that the fashion swings the other way. The IMO jury become alarmed at the trend of train-able problems, and in response, the problems become designed specifically to antagonize trained students. The entire Geometry section of the IMO shortlist ceases to exist, because some Asian kid wrote this book that gives you too much of an advantage if you’ve read it, and besides who does geometry after high school anyways? The IMO 2014 used to be notable for having three combinatorics problems, but by 2040 the norm is to have four or five, because everyone knows combinatorics is harder to train for.

Gradually, the IMO is redesigned to become an IQ test.

The changes then begin to permeate down. The USAMO committee is overthrown, and USAMO 2050 features six linguistics questions “so that we can find out who can actually think”. Math contests as a whole become a system for identifying the best genetic talent, explicitly aimed at weeding out the students who have “just been trained”. It doesn’t matter how hard you’ve worked; we want “creativity”.

This might be great at identifying the best mathematicians each generation, but I think an IMO of this shape would be actively destructive towards the contestants and community as well. You thought math contests were bad because they’re discouraging to the kids who don’t win? What if they become redesigned to make sure that you can’t improve your score no matter how hard you work?

III. Now

What this means is that we have a balancing act to maintain. We do not want to eliminate the role of training entirely, because the whole point of math contests is to have a learning experience that lasts longer than the two-day contest every year. But at the same time, we need to ensure the training is interesting, that it is deep and teaches skills like the ones I described before.

Paying $60 to buy a 300-page PDF is not meaningful. But spending many hours to work through the problems in that PDF might be.

In many ways this is not a novel idea. If I am trying to teach a student, and I give them a problem which is too easy, they will not learn anything from it. Conversely, if I give them a problem which is too difficult, they will get discouraged and are unlikely to learn much from their trouble. The situation with olympiad training feels the same.

This applies to the way I think about my teaching as well. I am always upset when I hear (as I have) things like “X only did well on USAMO because of Evan Chen’s class”. If that is true, then all I am doing is taking money as input and changing the results of a zero-sum game as output, which is in my opinion rather pointless (and maybe unethical).

But I really think that’s not what’s happening. Maybe I’m a good teacher, but at the end of the day I am just a guide. If my students do well, or even if they don’t do well, it is because they spent many hours on the challenges that I designed, and have learned a lot from the whole experience. The credit for any success thus lies solely through the student’s effort. And that experience, I think, is certainly not zero-sum.

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3 thoughts on “Make Training Non Zero-Sum

  1. Another problem is age “restrictions”. I’ve never had a real opportunity to participate in math olympiads because I started too late. And when I say “late” I’m not saying like 20 or 25. When I first got interested in math olympiad problems, I was 15 or 16 and, surprisingly, it was already too late. If by the age of 15 (or even 14 or 13) you’ve never had any olympiad training, you are pretty much out of the game. The system makes the student believe it’s his or her fault. “If you didn’t find your passion earlier, screw you! We not only look for genetic talents, but we also look for people who discovered their passion by the age of 12”. Today the “max” is 12. By 2040, the “max” gonna be like 6 or 7. (well, it already is, depending on the country and competition)

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    • While this is certainly true to some extent, there certainly are cases of people who started in 10th grade or something who have done extremely well.

      However, I do agree with your point. The important thing about math competitions is not math but it’s the social component. In that regard it usually takes longer than ideal to become part of the community, which is all the worse.

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  2. I think you’re really underestimating the value of training and overestimating the value of training materials, especially expensive ones (in a group setting). IQ tests can be trained for: https://youtu.be/W3oUqKUx2o0. In the book Outliers, a study was done of students in some music school: nobody who put in 10k hours wasn’t world-class and nobody who put in 4k hours wasn’t just teacher material. Intelligence can be changed by definition since it’s basically the ability to solve problems or be creative. You can learn this per subject. Talent is very minisculely spread: Singapore places in the IMO highly but they are a tiny country: can you imagine Manhattan placing at the IMO with that population of talent is random.
    I am critical of expensive (anything over a 1000 dollars) camps, seminars, etc. that are in a group setting. I attended AMSP and found it a waste of money because I can get free good-quality HMMT problems with solutions. Aops books are reasonably priced IMO for concepts. All you need is discipline. In a group setting all you can do is teach the concepts, not provide feedback that’s one-on-one or focuses on weaknesses which’s what training is.

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