I spent two days last week at a conference led by our county ESC entitled Focusing on the Mathematical Practices of the Common Core Grades 9 - 12. Honestly, a lot of the reason I try to go to different seminars is just to get out of the building and shake up my schedule a bit. It's hard to prepare for 2 days of a substitute, but changing the routine is sometimes worth it.
I was surprised by how much I enjoyed my two days. Instead of focusing on what's going where in our classes, we talked a lot about how to focus our instruction. There are 8 standards for Mathematical Practice that I feel like I know inside and out.
1. Make sense of problems and persevere in solving them. Perseverance was a big issue that we talked about. If you have students who will keep trying when they feel like they don't know how to do a problem, then I salute you. Most of mine won't. I get so tired of seeing blanks after kids have "done" their homework... or big ?s. (Know what I mean?)
2. Reason abstractly and quantitatively. Important here was the ability to decontextualize (make a problem more abstract) and contextualize (apply the numbers at hand). Tough for a lot of kids.
3. Construct viable arguments and critique the reasoning of others. Although it doesn't have to be a formal/written down process, I've started having the kids "check" the work of their classmates (they just accuse me of being lazy and making them do it). It's amazing how much they learn by going through a problem to try and find something wrong.
4. Model with mathematics. The kicker to us on this practice was the last line: "They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served it's purpose." What?! Check your work? Make sure it makes sense? And if it doesn't make sense, try to fix it?! Blasphemy. And you'd think I was torturing students in asking them to do that. Darin Hausberger, (@dhausberger) had a quote that I liked: "You may have an answer, but is it a solution?"
5. Use appropriate tools strategically. Don't automatically reach for the calculator (as we were reminded, paper and pencil are tools too!). But if there's a calculation that you can't do, go for it. Or if there's something online that you need to use, do that too.
6. Attend to precision. I originally saw this as more of a "watch your rounding" type deal. But it turns out that the premise is that students need to make sure to label axes, units, and use the equal sign "consistently and appropriately".
7. Look for and make use of structure. My big take-away on this one was the use of scaffolding. In Algebra 1 we do a lot of solving equations in the beginning of the year. I preach "show your steps" so that they get in the habit of knowing what they're doing to solve. Now, though, we're solving quadratics by factoring, and showing how to solve 2x - 1 = 0 is something they should be able to do without all of the steps. Those who are able to do that do. Those who need a little extra time are still writing it down.
8. Look for and express regularity in repeated reasoning. Once you understand something it's ok to use patterns and shortcuts. But I wouldn't suggest teaching that from the beginning. (#7 and #8 seemed very similar in nature)
So that's my 2 days in a nutshell. We talked a lot about giving "rich" problems (ala Dan Meyer, who was mentioned quite often), giving guidance but not answers, and the general idea of "Be Less Helpful".
It was a good 2 days; I brought away a lot.
HCESC (who led the program) has set up a blog and will be updating with different resources. Check it out.
(They probably won't be pretty colors like mine, but you can deal with that.)