My computer seems to be crashing in weird ways today so a bunch of my blog entry for today seems to have vanished. Sigh. I wonder if it picked up a virus from one of these Russian computers. I know they said not to drink the water; maybe the computer wasn’t listening. Anybody know what could make the computer screen go blank, and then start changing colors one after the other (grey, red, green, blue, or something like that)? Weird. Now it mostly just goes black. A loose battery connection? Hm.
Anyway, it’s all for the best, because about half of what was lost was me whining about being sleep deprived.
Today was great, again. We got a few hours in the afternoon for some sightseeing, and besides a brief driving tour of Moscow (yuck!) we got to stop in Red Square and walk there and into the Kremlin (which I didn’t realize has been open to tourists since 1955; I thought it would be a post-Soviet thing). Pretty amazing buildings, that’s for sure. Pictures maybe if I have time to upload them. I took a lot the last couple days, most of which are bad, but more of which have people in them. You might be curious to see what all these people look like! Still Joseph has the best pictures and we’ll try to get a CD from him tomorrow. We also saw the outside of Moscow state university, which is sure an important building to our friends here.
It’s also quite amazing to me how I’ve gone from pretty scared (coming into customs in Russia, for example, and hearing about all the pickpocketing that happens in StP, and the mafia, and all that; and even the police scared me at first when I arrived) to actually being quite comfortable here (and believing that the police for the most part are doing their job, not planning to find an excuse to fine tourists to raise money for the police department). Tomorrow night I want to get on the metro, which has a reputation of being very safe and efficient, and I want to walk across the most beautiful pedestrian bridge across the Moskva (Moscow River), which apparently is pretty new and was moved a couple km recently and then glassed in or something? I’ll try to get some pictures.
Another part that got wiped when my computer crashed was all about the entrance exams to school #57. Do you know school #57? It’s probably the best math school in the world. It’s not even a specialized math school any more; it has a good humanities program (which the Russians always translate as “humanitarian”, which gives us all a good laugh and yet we are all too embarrassed to correct them, so the laugh stays internal except here in this blog! Which, by the way, I am telling a few of our friends here about, so I suppose they’ll learn it this way.)
Anyway, this morning’s agenda was about 4 hours at school #57, which was simply amazing. I can only compare it to some of the best summer programs in the US. Even there it was maybe better. And it goes on year-round. I don’t think our specialized math schools come anywhere near this place.
First we went to a meeting where we learned about the school. The teachers are all charming, charismatic, passionate about their subject and about kids and seemingly about life in general, especially Sasha Shen (Chen?) who really stood out for his great English (including command of geek-speak phrases like “free as in beer”), his great energy, his honesty, and his thoughtfulness. I wish we had more time with him, or a chance to see him teach! Oh, just as one more example, on the handout he prepared for us he wrote about schools like his that “teaching is performed not only by professional teachers but also by amateurs (graduates of the school who have become students of math departments, professional mathematicians of different levels, etc.)” What a statement! Calling professional university mathematicians “amateur teachers” is essentially correct; they are paid for doing research, not for teaching, at least for the most part, besides which in general they have no training in how to teach. But still. What a statement. I love this guy. Oh yeah, and even better, he said about Russia’s move to national exams like the NCLB stuff we have in the US “A national exam is a good idea, because we can’t have teachers evaluating their own students and their own performance, but not to combine with college entrance exams and absolutely not to be multiple choice.” Good advice for us, I think.
Hm, other good tidbits from the morning: the music analogy is better for math than the sports analogy. Maybe I’ll write more about that later. Oh, and their principal, who I also was very impressed by (obviously intellectual, but not about to get in the way of teachers doing their jobs, and very friendly yet also careful and precise), told us about how important an integrated curriculum is, to prevent students from drawing artificial borders between subjects based on which teacher is talking to them, to make the world more understandable. He also talked about new “project” classes, sort of like service learning projects or science fair projects or research projects of other types, and described how each had at least two teachers involved: one for the technology, the “how to” do the project, and the other for the “what to” do. I wish we had time to learn more about that!
OK, so I do have to rewrite at least a little of the school #57 entrance exams. Most notably, they are primarily oral: students (a few hundred of them, at first) come in and sit ten or so to a room with 3 teachers (including TAs from Moscow State University). They get a set of problems and work on them individually, and when they think they have a solution, they raise their hand and then explain it to one of the teachers. If it’s satisfactory, they get the point (no partial credit here!) and if not, they get a hint maybe or just sent back to the drawing board for that problem (maybe the hints are a form of partial credit). They get at most three tries per problem. To end up at the school, they go through SIX rounds like this in the course of a month, during which students mostly self-select out of the pool. Finally the best 50 or so go to summer camp, and the best 20 or so are the new entering class to the math program at the specialized school #57.
We observed a superb class on plane geometry: superb because of the students, and the teacher, and the particular topic. The students knew a lot of math: the lesson opened with a review of what they knew about the nine-point circle. The teacher called on people, no hand raising, and the students stood and answered or came to the board and drew, as appropriate. What stood out here was the wait time: yes, the teacher would call on someone who perhaps didn’t know, but the teacher would wait … and wait … and wait … compared to a US classroom (mean wait time under 5 seconds) it was pretty amazing to watch. Then they went on to do my favorite, the transformational geometry of the dilation of magnitude -1/2 centered at the centroid (they use different vocab here, but it’s the same). In calling on students to help explain that idea, the teacher wouldn’t let students say that BH = 2OJ because B maps to J and H maps to O; instead, they had to carefully say that the segment BH maps to OJ so OJ is half as long as BH. In other words, reminding them that just because all the common transformations have the property that if they map the endpoints to the endpoints they also map the segment to the segment, you shouldn’t assume that the property always holds (e.g. inversion in a circle and other nonlinear transformations like that.)
Anyway, then I was surprised to hear what I would have thought of as a more American question from a student: “Why do we need to prove these things again? We already knew these facts.” But then the teacher did a great explanation, about how before we needed separate proofs for these many separate facts about nine-point circles and Euler lines and stuff, but now this one method gets you all these powerful facts. But what really stood out was that at the end of the explanation, the teacher looked the student right in the eye, and said “Have I answered?” Wow. I mean, you get that occasionally in the US, but for the most part you’d get “Do you understand?” or just nothing and a move on to the next topic. But what a huge difference between “Do you understand” and “Have I answered”! I’m going to start using that phrase all the time. Maybe with a “Have I answered your question?” instead – I think that English was this guy’s second language. But it’s the spirit that counts.
Then they went on to the famous (and in my opinion, with my weak geometry skills, very hard) problem of Fagnano: find the triangle inscribed in an acute triangle with minimum perimeter. The teacher did some great stuff, like asking why do we ask about the min instead of the max? Nice problem-posing kinds of stuff, as well as solving. He also emphasized that it was only for an acute triangle, though he left it hanging (for the students to solve as homework, is what that means) about what goes wrong when it’s not acute. Then he asked them to guess the answer, and essentially immediately one kid (14 years old. About 18 kids in the class, since I don’t think I mentioned that earlier) raised her hand and guessed. Correctly. Now, if there’s an American 14 year old who can do that, I would be willing to bet that they’ll be on the US international Olympiad team for two reasons: one, they have a great intuition, and two, there are just so darn few American kids who even know the word for the answer, much less the whole set of theorems about the answer that these kids know. (Yeah, maybe I should give the answer here, but I’m sure you can find it on cut-the-knot.org if you want to, or plenty of other places on the web, not to mention in Coxeter’s book.)
So the teacher was apparently pleased but not surprised by this, and he went on to ask why that guess seemed good. He got a bunch of unsatisfactory answers, and explained clearly why those answers showed “fuzzy thinking”, to use my old history teacher’s term. He reminded the students of some properties they knew of this conjectured answer triangle, and then related them brilliantly to the key properties of light travel in physics class: equal angles on reflection, and shortest path as well. This was some fantastic insight into the problem, and a great clue to the answer as well if you know it. It’s unusual to see such good connections between disciplines in the US, too.
So then he let them work on the problem for 10-15 minutes before class was over. When students asked him about their answers, he’d say things like “Not clear” which means to think more, write about their thinking, and ask again. Or “OK, write it.”, which means the idea of the solution is there, but they need to fill in some details. Or “I don’t know that method, show me” which I think translates to “That looks like a dead end to me, but I don’t want to squash your curiosity, so please try it and maybe we’ll both learn something.” I’m sure that last possibility, that the solution the teacher is skeptical about works out, happens quite often with kids like these.
Oh, I also picked up two more kids’ email addresses for Ms McKee; if you’re reading this, and I haven’t sent them to you, please remind me to do so. Oh, and speaking of Russian History, one of the mathematicians I’m working with here is Tanya Kutuzova (I can spell it right in Cyrillic, and I think this is good transliteration). All the Russian-history-knowledgeable people immediately ask her “Any relation to THE Kutuzova?” and she replies “Yes, I am his grand-grand-(waves-hands-a-bit)-granddaughter.” And she points out that her name, like his, is not just Kutuzova: it’s actually Goleneshcheva-Kutuzova.
I also left my card with the principal of the school who promised to help arrange for more pen pals by passing it on to an English teacher in the US part of the English department (they also have a UK English branch: of course they do lots of culture and history and geography along with the language, so this makes some sense. Flaurie, are you reading this part? Can you imagine a separate program in Central American Spanish or something like that?)
Finally we participated in a “specialized math class” which are the extra classes for the math-focused people on topics beyond the standard curriculum. And they mean BEYOND. These were 8th graders. It’s a problem-solving curriculum, so the students are working individually on problems, and when they have answers they call over a teacher to explain the answer to. I got to listen to (in the teacher role, though I didn’t do a good enough job to satisfy our hosts, who look for much more attention to detail than I would require) a pretty detailed explanation of the Cantor set from one boy (usually a college junior real analysis course kind of topic, though here they stuck mostly to the elementary aspects of it at this point, like proving certain numbers were or were not in it, which reminds me that calculators are banned and this kid could prove that 19/27 < sqrt(2)/2 < 20/27 in his head, and then justify it on paper for me when I asked him to explain how he did it.). I also got a great lesson on elementary number theory (like in a college junior math major kind of course) from another kid (proving that Z mod p adjoin sqrt(-1) is a field iff p is 3 mod 4, if I remember correctly). And yet another kid was doing some elementary vector space stuff, again at the college sophomore kind of level. Yeah. 8th graders. Right. Born in 1992 or 1993, at least the kids I talked to.s
The teachers in this class also impressed me very much. They were able to give hints to some kids without giving away too much, and meanwhile with other kids they would probe for very careful details in the key steps of the proof. The kids knew (at 8th grade age!) not just how to prove these very advanced things, but also had good taste in how much to put down on paper to show the idea, and good understanding of the skipped steps so they could fill them in easily when asked. Absolutely amazing. At about that age, I might have been learning about 1/3 that much math in summer camp, but when I learned about rigorous proofs I certainly didn’t have the taste these kids have about how to show my work. That came later by far. Extremely impressive.
Then we did our sightseeing, and then returned to MCCME to see their math circles for 7th and 8th graders. That was a refreshing dose of normalcy: much more like what I see in American math circles, and problems that I (with my math circle experience) could do without thinking because they were all similar to problems that I had done back home (most of which, of course, from Russian books in translation). The kids seemed much more normal, and, well, human too. I don’t need to write too much about this because it’s familiar to me, so I don’t have so many thoughts to organize, although this was an important piece of the day if only to be reminded of what more normal stuff looks like after seeing all these amazingly exceptional schools. Still I’m sure I’ll remember enough about what it was like without having so many notes or blogs, because the experience was familiar to me. Besides it’s bedtime.
Oh, but nonetheless I do have to say a few things. One, we got to see Konstantinov again! He told us a few words about circles (with extreme paraphrasing here, since he spoke in Russian which was then translated and then I scribbled notes on that and now I’m just making this up from my notes):
The main point of circles is to show kids that math is not what they do in school.
We aim to widen the gap between school math and what they understand as real mathematics. (Talking about 7th grade in ordinary schools here)
The circles are built on discussion of problems, but the teachers are not to lead the students to the teacher’s solution. Rather, help the student see the strengths and weaknesses in their approach; if at all possible, help them follow their approach to its end; and when you reach that end, let them decide to start a new technique. In other words, build as much as possible on their ideas.
There was sometehing else I wanted to say, but now I’ve forgotten (computer crashed again, though this time I didn’t lose any work. I wonder what’s up with it?). It’s hard to believe that in lees than 24 hours I’ll be on a plane back to the US,. I’ve barely scratched the surface here, but then again I’ve also had the privilege of visiting many of the best schools in the two most famous mathematical cities in the country and perhaps the world, and meeting many of the best mathematicians too.
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