So sorry about not keeping up. I knew I wouldn't be able to post every week, especially as the adoption started to kick into gear. We had more paperwork to take care of for a while, and we were getting ready to go out of town for Thanksgiving. But here's the update:
Things are getting a little crunchy now for the weakest students. About three of them earned a poor enough grade on a test that it finally started to sink in that maybe it wasn't a fluke. Only one is mature enough to be serious about improving. In fact, he has been working very hard all along, and students like him are the hardest on me: we get along, he works hard, he's been trying hard for weeks, and he still struggles. His parents have a hard time accepting the fact that he's not very good at this. He's really stuck between them and my class, and there's little I can do other than meet with him to go over his math in excruciating detail repeatedly, and hope that his still developing adolescent brain finally makes the necessary connections. The other two struggling students are more immature. The older one is starting to fall far enough behind that his lapses earlier in the year are starting to catch up with him. And the younger one is...well...just very young. She simply doesn't see math as anything but a series of algorithms, and she has no basis to judge when she might have made a mistake.
That's a more general problem that I think we teachers have to stress to our students: we all have common sense that tells us when something might be wrong, and it's okay—no, it's encouraged—to use that sense in a math class. Each year there are some students who say, "But I keep making careless errors! How can I fix that?" I tell them that we all make careless errors. I make them, too. But I rarely get through the whole problem without catching them and correcting them. Making careless errors comes from a lack of attention and 'reality check' on your problems. Students make them more often because they don't have as much ability to use that 'reality check'. For example, a student has to subtract: 18 - (-4) and he gets an answer of 14. If she's still working on the rule for subtracting negative numbers, that mistake is forgivable. But it's less forgivable if she has no way of realizing that they just can't be right. 18 - 4 is 14, so 18 - (-4) can't also be 14. But when you explain that to them, many kids understand for only as long as you're standing there explaining—that ability doesn't stick with them easily. Other examples are kids who reach for the calculator to multiply something like 24*5. If I'm working one-on-one with the student, I'll take the calculator away and ask, "What's 24 times 10?...Okay, then, what's 24 times 5?" Almost all can follow and come up with the right answer in their heads. Or they start stacking up 25 and 14 to multiply them. So I ask, "How much money is 14 quarters?" That one takes a little longer, but after about 10 seconds, most will cluster 3 groups of 4 quarters, and realize that it's $3.50. And then I repeat the original question "What's 25 times 14?" Their eyes light up with "350!"
It's like they have a part of their brain that can do common sense math, and they have a part that they use to do classroom math, and they're not letting those parts talk to each other. They've been convinced, I think, that classroom math is somehow so detached from real life, that real thinking about it can't possibly be useful. This is brought home in another way, too: I give them the classic fixed-point camping problem. On day one of a camping trip, you hike up the mountain trail, taking from 9am to 3pm to complete the hike. On the next day, you hike back down on the same trail, again taking from 9am to 3pm. But even if your schedules (speeds, rest times, etc.) on the different days are completely different, there has to be a point on the trail at which you arrive at exactly the same time on the first and second days. (If you don't believe it, imagine that on the second day, a ghost version of you goes up the mountain following your exact schedule from the day before. You have to meet your ghost at some point, and that's the required point.) The kids usually love this proof because it's so convincing and because it doesn't seem possible that it's true. But almost inevitably, one student will say, "I get it, but that's not really math." They don't see any calculations, pictures, or anything that reminds them of previous math problems. Of course I have to tell them that it this is a fundamental math problem, in the sense that it uses ideas at the very basis of math (continuity, in particular, comes to mind). If they don't think it's math, it's because they don't know yet what math is really about. It's a more advanced version of the 24*5 problem: students don't think that this very basic kind of thinking is truly mathematical. That's one reason that I think it's so important to spend occasional time in class working on non-standard problems; students need to see how to apply reason and common sense to problems in unexpected situations.
Okay, those are my teaching thoughts...I need to get going. But on the personal side: my wife and I are preparing for some big news next week. It's looking increasingly likely that we will get our baby's referral (picture, description, known history, etc.) towards the middle or end of the week (if it doesn't come then, it will come at the end of December). After the long wait so far, it's hard to imagine that it's true. But true it is. This baby is going to become much less theoretical very soon.