...continued.
One of the best known math schools in the world, indeed.
In the bay area (roughly the same population as Petersburg) we feel happy to get a total of about 250 students taking the Bay Area Math Olympiad. Here they have 2500 SIXTH GRADERS taking the 6th grade "district olympiad" from which the top ones are selected for the city olympiad. They have similar but slightly lower numbers in the higher grades as well. Wow.
Kids are invited at around that age -- 10 -- to be part of the math circles program, which means doing math from roughly 4-6 twice a week. When we finally left, it was getting close to 8pm, so I guess there is a second session with some other kids that goes from 6-8 because there were lots of parents waiting in the lobby. The program follows the students, so the circle leader you get at age 10 is the same one you will work with until you graduate the circle at around age 16. People really get to know each other in detail. There are huge batches of volunteer teaching assistants from the local university, so students get a ton of individual attention, working through some really hard problems.
We watched some 8th graders. In their "regular math class", which is really rather special, because the students are the 25 or so who are in this math circle at the best school in the world, but still, an algebra class, they did the following:
Problems were written on the board, which they had to solve purely mentally. A few minutes after the problem was written, the students would be given a few seconds to write down their answer in their blue book. Then after the last problem, the teachers collected the blue book, went over the solution methods for these problems in detail (great detail! And they are very careful and thorough presenters, so much so that with my essentially nonexistent knowledge of Russian, I could still follow the explanations perfectly. The Russians do use our same numerals, and use letters like x for variables instead of the Cyrillic alphabet, so that helps. But there are few American math teachers who are so careful and thorough and attentive to detail when solving problems of this difficulty.), um, where was I? Oh, yes, then they wrote the homework on the board, and that was it for the half-hour or so of lesson that we watched.
Some examples of those problems: If it is true that (a^2 - 1) * x >= 3a - 1 for all x, what value(s) of a are possible? Yes, they are expected to do that entirely in their head. In eighth grade, age 13. Or, another example: What is an inequality whose solution set will be (-inf, -2) union [3, inf)? Or for another, solve 2/3 x - 1/6 < (x+5)/12. Well, that's more straightforward. Or how about solving |2x-1| > 3x? Yes, in your head. Take your time; they'll give you two or three minutes.
I have pages and pages of notes about Russian pedagogy for these kids, and how they implement their problem-based curriculum at this phenomenal school (Perelman's school -- and we met Perelman's math teacher, too). But no time to type them in right now.
Instead, how about a couple nice samples of problems they did for their circle? These would have been assigned Saturday, and mostly solved by today by a good fraction of these 8th graders.
Of the 8 problems, here are my favorites:
2. Stas draws diagonals of unit squares in an 8x8 grid of squares (chessboard). Nikita sees to it that they have no common points (including ends). What is the maximum number of diagonals that Stas can draw?
[I love that one because of the names, and also because it's easy to draw, and fairly easy to find the best answer, but fairly tricky to prove!]
3. Let ABCD be a convex quadrilateral and M, N the midpoints of sides AD, BC respectively. Points A,B,M,N are concyclic. Line AB is tangent to the circumcircle of triangle MBC. Prove that it is also tangent to the circumcircle of triangle AND.
[Hey, if you took geometry from me last year, you should know enough to be able to solve it -- but that doesn't make it easy!]
8. Asterisks are arranged in some squares of an n by n grid. In each vertical, horizontal, and diagonal (even diagonals of only one square count), the number of asterisks in it is known. For which n is it possible to determine where the asterisks are?
13 years old. Wow.
And the same "shortage" kind of issues came up. They talked about teachers in general being paid not only low, like we complain about in the US, but insufficient, by which they mean you end up homeless and/or hungry. That's their big education reform push: find enough money to pay teachers enough to eat.
They do have photocopiers and printers and very old computers, but when they have a handout, they print it with the minimum possible amount of whitespace so they can fit three or so copies of the problem set on one page, and then they (with scissors, not even a paper cutter!) cut them apart into individual copies for each student. So that shows what kind of fraction of their budget photocopying is going to be: making 8 copies instead of 24 for the class is a huge savings.
Oh yeah, one or two other facts: at this math school, something like 22% of the students are girls, and they are working on it some. Almost all the instructors are male. Almost all the TAs from university, too. So they have a ways to go. It may have something to do with teaching style. One comment, approximately quoted, from that discussion on why they are NOT allowed to work together -- well, the why is that in the long run it ends up with one person doing all the work while the rest of the group fades away. But then they added "We must work to suppress their urges to work together, especially for girls." Interesting.
Some other comments really hit home, too: "The government thinks the purpose of math olympiads is to help create resumes for people applying to college." I get that fear all too often myself, not that the state thinks so, but that students or their parents think so.
There's also some very interesting stuff about the emphasis on oral presentations of solutions, but I'm on my way to the train station. I hope to write more tomorrow from Moscow!
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