Hello, reader! If you intend to post a link to this blog on Twitter, be aware that for utterly mysterious reasons, Twitter thinks this blog is spam, and will prevent you from linking to it. Here's a workaround: change the .com in the address to .ca. I call it the "Maple Leaf Loophole." And thanks for sharing!

Monday, August 26, 2013

Building Functions

How do you motivate vertex form for a quadratic? Do you just drop it on them? That's pretty much what I used to do in Algebra 1. Hey, kids, you want to model this u-shaped path, like you get when you toss a basketball. I'm just going to tell you to start with y = a(x - h)2 + k. Start messing with a, h, and k, and see what happens. Let's see what we can say about how they each affect the graph.

There's an awful lot of "building functions" in the common core, and an awful lot of modeling, and I think it's great. The whole F-BF header should be a playground. I'm just not clear on how you take a class there.

You can look at sequences of patterns easily enough that result in y = ax2, and I suppose patterns that result in y = (x - h)2 and y = x2 + k. Do those arise naturally anywhere? Or do you choose carefully something from Visual Patterns?

What else? "Fit a quadratic function to a photograph" seems to be a favorite of presenters at conferences who want to browbeat teachers for not making class real-world enough. But how do students develop those functions in the first place, in an authentic way? I feel a little awkward for asking, because I feel like I should already know this. But I also suspect that not that many people have a great answer.

Sunday, August 25, 2013

The Fault in Our Stars has some wrong math (spoilers)

"There are infinite numbers between 0 and 1. There's .1 and .12 and .112 and an infinite collection of others. Of course, there is a bigger infinite set of numbers between 0 and 2, or between 0 and a million. Some infinities are bigger than other infinities. A writer we used to like taught us that. There are days, many of them, when I resent the size of my unbounded set. I want more numbers than I'm likely to get, and God, I want more numbers for Augustus Waters than he got. But, Gus, my love, I cannot tell you how thankful I am for our little infinity. I wouldn't trade it for the world. You gave me a forever within the numbered days, and I'm grateful."

I love that this character used mathematics to express her love to her dying boyfriend. It still bugs me that the math is wrong. Gus gave her the same size infinity between 0 and 1 as she would have gotten between 0 and 2, or 0 and a million. I'm afraid this makes me a heartless asshole.

(The author acknowledges that the math is wrong, and claims it was intentional. I believe him. I don't know what to do with it.)

Saturday, August 24, 2013

A little bit of what I intend as thoughtful pushback

I went to Steve Leinwand's global math presentation this week. It was thought-provoking and worth the time. You should go watch it. I asked a question and got an answer that helped me grow.

So, Steve's all like, hey teachers, you need math smarts. Check! You need good tasks. Check! You need effective instructional practices. You suck at these!

These are his major recommendations, as I understand them. And they are very good. Everybody, listen and heed.

1. Students do more justification as a regular part of math class. Note: this doesn't mean "proof" as I clumsily tried to ask about during the session. Proof, of course, has a precise meaning in mathematics with a level of ironcladness that we're not talking about here. Rather, this is more informal, could be written but often verbal, justification. Which is just as well, as far as I'm concerned. The word "proof" scares K-12 teachers more than having class outside, chaperoning dances, and unannounced classroom observations combined.

Practically, this means students solving fewer problems, but being required to write about their process and reasoning.  Additionally, student responses ought to be normally followed up by a request for justification.

2. Ask students to estimate more often. Before solving or mindlessly applying a formula, students are explicitly asked to estimate a reasonable answer. Fine. Awesome. Takes zero time. If you're not already doing this, get with it and start.

3. Collaboration centered around classroom video. The recommendation is, everyone tapes him/herself regularly, and clips are randomly selected at staff or department meetings to watch and critique.

Review: I'd be fine with this. I'd be all, look at how amazing I am, bitches. Let me hear your critiques; I am eager to learn from them. I wish Steve Leinwand would come observe me, so he could see one teacher asking kids to explain the reasoning behind their answer (which is evidently rare in the classes he observes) and approve of me and everything I do, and maybe adopt me. 

95% of my colleagues would not have been fine with this.  And I'm pretty sure that where there are strong unions, you can't make them.  And where there aren't strong unions, life sucks anyway.

Case in point: central Virginia. I worked with some great teachers this week. Whip smart. Love their kids. Doing their wholehearted best to do the best thing. In a nutshell, good people. This year, the high school teachers will be teaching 6 classes a day instead of 5, with no change in salary. Computing...20% increase in workload, 0% increase in pay. 

Talking to more people locally, evidently this is a trend in Virginia. Where unions and contracts are not commonplace features of school employment as they were in New York. School boards can basically demand whatever fuckery, and rely on compliance, because, teachers are part of their communities and are committed to their work and don't want to do something else or move.  So they're taken advantage of. 

But then, to suggest they collaborate with nonexistent time in their work day is a bit laughable.

Don't kid yourself, it's a 20% increase, because it's 20% more children, when you teach children as opposed to teaching a subject. There are 8 periods in a day. So, 3/4 of their work day, they are executing carefully-as-possible planned interactions with kids. They have two other periods to spare. One is lunch. Humans require a meal in the middle of their day. It's civilized. The other is given over to a duty, like study hall or checking hall passes or whatever. They probably have one hour after classes end when they are expected to be available to help kids. Then, they need to leave and pick up their own kids from daycare, because they are humans who have lives. Our active-on-twitter friends take a ton of time to plan instruction outside their normal workday because they are math ed nerds. And they're awesome. But can we expect more than a small fraction of our colleagues to travel the same course? Not really.

When are these folks supposed to collaborate? When are they supposed to do the cognitively demanding work of figuring out how to get another person to understand something? When exactly? Leinwand, from what I've seen, brushes these concerns off. He says, teachers typically teach 5 out of 8 periods a day, and can find time to collaborate if they really want to. But at least where I live, these are real questions and not excuses. Act like teachers are kicking back and drinking coffee during luxurious planning periods, and you'll lose them.

Please note I'm not arguing with the premise. But we need to acknowledge the dearth of and fight for collaboration time. Which I don't believe will happen without funding that gives professional educators more time in their day not in front of the children. In places that do this right like Japan and Finland, teachers spend half as much time in front of students as American teachers - and they spend all that extra time: planning, collaborating, developing lessons, and generally kicking our asses. Policy matters. Funding matters. Yelling louder at classroom teachers isn't going to solve this problem.