How fast?

A decent day, coming back from Thanksgiving. Opened each IMP3 class with this:

Thanksgiving Dinner.
Marcie's family went to her aunt's house for Thanksgiving, 120 mile away. They left home at 1:30 pm, arriving at 3:30 pm. After dinner, they drove back home, leaving at 9:00 pm and arriving home at 10:30 pm. Compare their speed each way. Which direction was faster, and why might that be?


Predictably, I got a lot of "coming back was faster, an hour and a half versus two hours going there" stuff. The thing that made it sing (in 2nd and 7th period, at least) was me taking notes on the board: Going there Coming back. 2 hours. 1 and a half hours. Then, I made a big production of it: "How is 1 and a half hours FASTER than 2 hours? Shouldn't the bigger number be faster?"

A good platform for re-teaching the idea of rate. Launched from there into mechanics of calculating rates, and then into the Big Idea: it's all about answering the question, "How fast?" Then into questions that start as "how fast does..."


I feel like I'm getting better at showmanship, which is good: if it's not interesting, they won't be interested. My passion for the subject needs to be a draw, to attract interest (think of busking your own show...) Interesting, how I started from the idea that I want to make it as little about myself as possible, and have gradually scaled back from that (also interesting, if dark, to consider how much of that is driven by this recognition: "Yeah, well, it's not like I'm actually succeeding at making it about THEM...")


Potentially good tape from today's Calculus lesson: put the limit-based defn of derivative on the board, said, "Sort out what it means. Explain it." I was worried, when one kid jumped straight to, "It's the slope formula", and relieved a few seconds later when he said, "oh, man, now I'm confusing myself..." From confusion, they can build their own clarity. It's when one kid gets it cold, and everyone else is lost, that I worry... Too easy for everyone else to just throw in the towel at that point...

VideoQuest

Just taped a discussion on the difference between "precision" and "accuracy", with my calc class. I haven't looked at the tape yet, but I fear it's not going to be usable for National Boards-- the situation felt a little contrived (possibly, because this was the first time I had a student, one who's planning to transfer out of the class, do the taping, and other students even said that that was off-putting)-- a big part of that was, I think, that my carefully-crafted set of "focus questions" fell a bit flat: the students said that they weren't sure what I was looking for from them (so, that's a launch problem: I should have focused more on the notion that EVERYTHING is pointing back to the difference between "accuracy" and "precision", and I should have made it clear that there's more than one right answer to each question, so long as you can justify it).

For future use, I'll definitely change my first question (which gives six similar measurements of the same object, and asks "what do you think the actually length of the object is?") to "which OF THESE measurements do you think is most accurate?" Possibly, I'll go back to putting the camera on a shelf (though, that has problems, too...)

A major problem is that I think my materials didn't do a good enough job of emphasizing that it's possible to be BOTH accurate AND precise: the consensus opinion quickly was that there's a sort of exclusive relationship, or at least that "accuracy" is a "rough estimate"... Late in the prompting questions, I asked "Is it possible for something to be both accurate and precise?" Maybe it'd be better to ask "What is an example of a measurement or set of measurements that is both accurate and precise?"

Still, there was a great moment when Sanchez was struggling with the English, and so I encouraged him to use Spanish (after some cajoling, it went much better, and I was able to follow along). I suspect that my mentor will say, "It'd be really great if you could use that..." We'll see how the 15 minutes around it looks.


Anyway, I might re-use this material with my IMP3 classes, too. Maybe even next week, if I can't think of a better time for it.

Latest piece in the philosophy

So, there's a thread on the AP-Calculus mailing list right now, about rationalizing the denominator (that is, if you have a fraction with, say, a square root in the bottom, you multiply the top and bottom both by that square root, which effective puts the root in the top instead of the bottom). A couple of people defend the practice, for reasons I couldn't much understand. A few others slam it, for reasons that I do very much understand. One person, who is maybe in the middle ground, pointed out that it's really a pretty neat hack, and so might be taught just because it's kinda cool (see also most of the "neat" series, Pascal's Triangle, etc).

Here's what I just wrote to the email list. I'm posting it here, because I think it sums up a piece of my current teaching philosophy. It's a long way from two years ago, when I was dying to find a science teaching job...


Taught well, rationalizing the denominator can make for an intriguing half hour for a student who is at the right place in his math-student career to spend that half hour contemplating it. Excellent teachers will help put that student in that right place, by routinely, consistently inspiring students to be interrogative, and to see the beauty in mathematical patterns.

Without those routine elements to drive the exercise, rationalizing the denominator is a dry and meaningless exercise, with no context or meaning for the student-- just another "math rule" that they do because they were told to.

As you point out, there are LOTS of beautiful mathematical patterns that a good teacher can rely on to create those intriguing half-hours. Rationalizing the denominator isn't one of the ones that I choose to rely on, usually. Sometimes it comes up on its own, and then we spend that half-hour, but it's never something I plan to teach, on grounds that an interrogative student, in the exceptionally unlikely event that she ever needs to know it, will figure it out for herself in that same half hour...

To me, THAT'S what math education is about: to develop core problem-solving skills, and enough of a sense of pattern to allow students to understand patterns that I DIDN'T teach them, for themselves.

Quarter 2: the new beginning

I took some the time during the last couple of days of the first quarter to take to each of my classes about policies and procedures, and I think that it was a very good idea. It gave me some better frameworks for day-to-day operations (homework policy, grading policies, etc), and it gave my students a chance to give their input to those frameworks, which not only made the ideas stronger (especially because I now have a better idea about what's important to them), but also gave them more of a sense of fairness.

Second quarter is off pretty well with the juniors: we're getting into rates, which is a favorite topic of mine, and I'm getting good buy-in. Calculus is a bit rougher: limits, which is not my favorite topic. Today, I got well behind on timing the new material, because I tried to "graft on" an exploration assignment to assess yesterday's topic. It sort of slowly fell apart from there. And calculus is going to be a bit rough for a little while, because this week and next are all lecture-and-practice, without much room for class discussion or interactive/manipulative assignments.

Possibly, what I ought to do with this two weeks is to use it to focus on individual seatwork: drill-and-kill until the class knows how to stay focused for more than 30 minutes (the last part of class today was ridiculous, with the off-topic chatter). It sounds a bit like punishment, and I want to be careful to avoid that idea: really, I think the class has just swung a bit too far towards a relaxed, collaborative atmosphere, and we need to balance back towards scholarship.

Some great ideas from last week's AP conference. The students elected two leaders today, who will take charge of programs and projects (hopefully to include family night and after-school study sessions, as well as creating a t-shirt for the class), and I gave them all my phone number yesterday. We have an email group now, too. So, some good stuff coming together, even if it's likely to be a rough couple of weeks.