Bad, bad Polymath for not posting last week. But...last weekend was our fall break, so these last two weeks have only comprised 8 school days, and I was very busy all week and weekend, so I beg your indulgence and forgiveness.

Three things happened this past week that were very pleasant:

- A student who has been working pretty hard, but getting D's* on his tests got an A– on his recent test. Of course the grade itself made me happy, but the best part was the expression on his face as he turned it in. He didn't need me to grade it to know he had done well. He was almost able to keep a straight face, but he cracked a little smile and said "that went a lot better." After he left the room I skimmed his test, and it was clear he was right.
- The principal was talking with some students (she didn't say which) about how things were going with them, and one of them mentioned having me for math, adding "I love him". I know that my classes are basically going pretty well this year, but it's nice to know that at least one student likes coming to class every day. It's also nice to know that my principal is the kind of person who will tell me things like this. Although I did fail to come up with the obvious reply: "Oh, I guess my well-placed bribe worked perfectly, then!"
- It's only once every year or two that I'm just about to get annoyed with a kid for doing something disruptive, but then it turns out to be so funny that I can't be annoyed. Like when I was in a high school choir, and we were rehearsing a requiem. The words were "
*et de profundo*" (or 'out of the depths'), and we all cracked up, and the conductor was mad at us until he saw what we were laughing at: a large, sweaty guy walking behind the conductor across the stage wheeling a Porta-Potty on a dolly behind him. He had to admit that was funny. So anyway, I was telling my algebra students my well-rehearsed opinion about classic rate problems (like "a train leaves Chicago going 60 miles per hour..."). These are the kinds of problems that so many adults cite as evidence of how stupid high school math is, because they're so useless. To the surprise of my students, I actually agree with that. So I tell them that I'll give them a real problem that's not stupid: "Design a new kind of airplane that won't crash and kill everyone on board." They become silent. "Too hard a problem?" They nod. "Well let's start with something easier...a train leaves Chicago going 60 miles per hour...." That usually gets a laugh and a recognition that you have to start with easy mathematical modeling problems before you try the hard ones. And I proceed with an example. And about halfway through the example, one of my quieter and more talented students starts folding up a piece of paper. I let it go for a minute, but it soon becomes clear he's folding a paper airplane. I get a little bit annoyed. "Are you folding a*paper airplane*?" And he comes back with "I'm designing a new kind of plane that won't crash and kill people."

I also concluded that a *huge* difference between 7th graders who "get" math and ones that don't is their grasp of how division works. Using divisibility tests (not in this kind of detail, though I teach the ones for 7 and 11), finding prime factorizations, and understanding the details of remainders (that is, the basics of modular arithmetic) are completely unnatural for some students, and those intuitions are extremely hard to teach. A student who can fly through the long division problem for 37,574÷7 and correctly get 5367-R5 (remainder of 5), can still not tell me very quickly what 5367 times 7 is afterwards. When asked to find prime factorizations of 24 and 48 on the same page, it doesn't occur to them to use the factorization for 24 and just add one more two. It's a revelation to them when they finally understand that if a number is divisible by 15 then it *must* be divisible by 3 as well. I can only conclude that this is a more abstract idea than it seems at first, and/or that grade school teachers don't really spend time on what division really means, but just on the long division algorithm.

And this is the paragraph where I'm supposed to wrap it up with a nice observation on how all this applies to the world at large. But it's not coming this week, so I'll just leave it at that.

*I am aware that plurals don't require an apostrophe, but *B+s* just looks so wrong compared to *B+'s* that I just have to use one when pluralizing letter grades. So sue me.