Thing OneStructural stuff matters when you're learning Algebra. And a great way to see structure is by comparing different particulars for similarities. That's why it's great to do things like pushing kids around to teach the commutative property. But sometimes I find that students have a hard time connecting those abstractions to the problems that are sitting in front of them. So sometimes it strikes me as a good idea to connect things back to the problems that they're going to have to solve, almost all the way, and then just holding up at the last minute.
I like this because (1) it worked* and (2) I've got a good, quick conceptual story to tell kids when they're stuck that emphasizes structure. I just get to ()(x + 5) with the student, turn that to ()x + ()5, and then fill in the parentheses that we just drew with a little "x + 1".
*About 2/3 of the class could just do things like (x+3)(x^2 - 3x + 6) after our unit. We devoted no whole-period class time to this.
At the same time, I feel like I'm providing students with something that's basically a procedure. And I worry that I'm falling back into a game where I'm trying to provide students with a better procedure. But I think that if my students can make the connection between ()(x - 5) and the correct distribution, that's enough conceptual understanding for the moment. Maybe.
Here's another thing that basically worked. How do you get kids to avoid just crossing out random stuff in a fraction and calling that simplified? Chastise them for it? Urge them to evaluate the expressions all the time? Reps?
Here are a few observations:
- For Heaven's sake, don't use the word "canceling" in the classroom. Ever. The language that you use in the classroom matters, and the word "canceling" means something like removing, expelling, kicking out. And that's not what we're doing.
- So what are we doing? We're essentially factoring out a "1." But that doesn't have a fun name.
- So call it a "Special 1," and then ask kids to find it. It's challenging, and it's the fun part of simplifying anyway.
By the way, if you want any of these files, they're all on Google docs. Google Drive has made it ridiculously easy to share my curriculum with my students and anybody else who wants it.
What's the advantage of Google Drive over Dropbox or any other cloud service? With Google Drive I have a folder on my desktop that is synced with the web. Anything that I alter or add to my desktop folder gets automatically added to the web. Maybe other services have this feature, but I wasn't able to figure it out with Skydrive or Dropbox.
Here's my curriculum. Chill out. It's my second year. I write it hours* before I use it. The stuff in the second semester is better than the stuff in the first semester. And I haven't done much curriculum writing for Geometry yet.
There are things there that aren't my own, that I don't have the rights to redistribute. Sorry. I'll worry about that when you, my faithful readers, are more numerous.
Things Four through Seven
- Clinometers are a great project for Algebra 2. Easy to pull off. Dice are fun for probability. Post-its make great instant histograms. Google Maps is pretty good for getting distance from you to a thing of your choice. Astral Weeks is a great album; I'd take it over Moondance any day.
- Some things I've learned about making problem-solving work: I either need an answer key available for the kids, or I need to check in with each kid or group, or I need to devote a ton of time in whole-group to talking about the problems and developing a way to check the process. Those are my tools for giving feedback. The choice differs, depending on the day.
- Also, giving "Leveled Problems" is something I stole from Dan Meyer, and I like it. It helps kids see that there's a hierarchy of complexity to the problems, it gives us a better vocabulary for discussing issues ("I'm fine with Level 3, but...") and it's good for giving kids a sense of how much they're progressing as they master different sorts of problems. I do that whenever we take on something that's algebraically complex. When I remember.
I don't think that I really care about whether my kids know math. I mean, I do. But not in the way that some other math teachers care. Unless you can point me to some evidence, I don't buy the idea that there are Mathematical Habits of Mind that are transferable to non-mathematical contexts. I have a hard time saying that anything past Algebra 1 is really helpful on anything resembling a regular basis.* I agree with those of you who point out that math is like comic books, history, novels, music, or any of the other parts of life that are fun and amazing. At the same time, what about the parts of math that just aren't that interesting to me?
*"Dammit, doesn't anyone here know how to factor a fourth-degree polynomial!" I wish.
What motivates me is making sure that my kids have good feelings about learning. Kids should leave school with the firm belief that learning is something that makes life better. Not math. They can forget math, for all I care, I think. But I want them to leave school with a respect for real learning and all it entails. They should know, from experience, that deep conceptual understanding out-flanks the sort of flimsy procedural knowledge that hucksters try to sell on the cheap. They should know how to learn something new, and they should believe that there are times when doing just that can make their lives better.
That's it for now. As always, comments are open.