The book How Not to Be Wrong says that math can enrich your life. It doesn't matter whether you like the math drills from school. Those are for math like running is for football: useful, but not quite the fun part. Unless you turn pro. In that case, running exercises are necessary, rather than merely useful.
I agree with this message wholeheartedly. But I learned something from Kahneman's Thinking Fast and Slow: If you find yourself completely agreeing, then you're probably not paying enough attention. It's easy to do so, because attention is a scarce resource. People instinctively tend to conserve it.
For a while I was worried that I agree too much with Ellenberg. (He is the author of How Not to Be Wrong.) But then I got to the section about genius, and I stopped worrying.
This section is aligned with the main message. It says that it's stupid to give up math because someone else is better than you. That's not the problem. With that I agree. But then, Ellenberg starts saying why you should go on. One reason is that more brains will make math advance faster. That's not the problem either. But then he says things like this:
It can be hard for me to make this case, because I was one of those prodigious kids myself. […] [I] won a neckful of medals in math contests. […] That group of young stars produced many excellent mathematicians. […] But most of the mathematicians I work with now weren't ace mathletes at thirteen; they developed their abilities and talents on a different time-scale. Should they have given up in middle school?
Well, … this comes after a few sections warning of the dangers of ignoring base rates. Having a neckful of medals in math contests is a very rare event almost by definition. Given this, the statement that ‘most mathematicians don't have a neckful of medals’ contains very close to 0 bits of information.
Let's put it differently. Suppose a country has about 150000 students in a certain school year. Out of those 10000 participate in math contests. Out of those 30 do really well, and get some medals at the national level. (Incidentally, I think these figures are in the right ballpark for Romania.) Now, out of these 150000 students, 20 become professional mathematicians. Most professional mathematicians (say, 15) come from those without medals. The rest (5 in this case) come from those with medals. And yet, having a medal gives you chance of 17% of becoming a mathematician, while not having a medal gives you a chance of 0.01% of becoming one. That is, not having a medal decreases your chances more than 1600 times. (Again incidentally, this reasoning mirrors an argument from Ellenberg's book.)
So, yes, most mathematicians were not mathlets. But, no, this is not such great news for your math future as it may sound.
It is easy to misinterpret what I said above, so I'll linger on the point. Suppose you are a student and you don't have a ‘neckful of medals’. Do I think you should not try to become a mathematician? I most certainly do not think that. Do I think being a mathematician is a perfectly valid career choice for you? Yes, I most certainly do think that. But, the reason why I think so is not that most mathematicians weren't mathletes. You see, I can agree with the conclusion but disagree with the argument.
Why, then, should you not care about medals? I'll tell you some of my reasons.
My family encouraged me to take part in the Romanian Math Olympiad. But, I never did well. And I was never worried about it. In fact, I thought that being a mathlete would be rather boring. The reason is the following conversation about the problems of one contest. Me: ‘So how would you solve problem 1?’ Other: ‘Oh, you use Theorem X.’ Me: ‘Ah, I didn't know Theorem X. How do you prove it?’ Other: ‘I don't know. But it doesn't matter. You can just use it.’ Me: ‘What about problem 2?’ Other: ‘For that one you use Theorem Y.’ Me: ‘Hmm. I don't know that one either. Can you tell me why this theorem is true?’ Other: ‘I don't remember now. But it's in the textbook.’ After going through a few more problems, I decided that I could do a lot better in these contests if I knew a bunch of theorems, even if I don't know why they're OK. I thought that knowing theorems without understanding why they hold is no fun. And, more often than not, it seemed like the theorem was more interesting than the problem. It felt like someone knew the theorem and said ‘I'll make a problem out of it by applying some make-up’. Then, the contestants' job was to remove the make-up, an recognize the theorem underneath. This simply doesn't sound like fun.
I should disclose that this memory is so vague that I'm not sure I actually had that conversation. Whether I had it or not, it captures my point of view from high-school.
I no longer hold that view; not entirely. It's true: one can do well in these contests by memorizing theorems. But, one can do really well only by knowing many theorems and techniques. And one simply can't remember so many things unless one also understands them. Another thing I realized in the meantime is that contests cover a tiny corner of math.
So, here's why you need not worry about contests. Your goal should be to have fun. Understanding is the supreme kind of fun. Figuring out on your own something that you didn't know feels awesome. (I sometimes joke that it's better than an orgasm.) This feeling I describe doesn't come often, and certainly not after solving easy problems. The harder the problem, the better the feeling. But, it can be difficult to keep going. Here is were contests come in: they are a great motivational tool. But, only for people with a certain psyche. Contests aren't good in themselves. Contests are good because they make you prepare for them. The work you do while preparing for them is the valuable part. And also the part where you'll ultimately have most fun.
I couldn't see the value of contests because I was not preparing for them. I just showed up. Of course they weren't fun!
The important thing is to keep punching hard problems. Whether you use a contest as an excuse to do it is a secondary concern. (Obviously, the problems need to be within your reach; otherwise, you can't punch them.)
Going back to the book, I concede that my objection amounts to nitpicking. Which makes me worry again that I didn't pay enough attention. I plan to do a second reading with a more critical eye.
Here's a link: Ellenberg, How Not to Be Wrong.