This is a story of how I ended up with problem-solving at the (sometimes) center of my classroom. It happened, more or less, by accident, and now I'm trying to figure out what to do with it.
Here's the short version: I used to start by teachering* for about 10 minutes, then easing students into practice work with choices. That was going just OK. Now, though, I'm starting with exploratory problem sets and reacting with small-group instruction and explanations and activities that react to how folks are doing on the problems. That's going way better.
*Teachering is like plain old teaching, but with my teacher voice. You know teachering when you see it.
This past September I told kids that every day there would be a "Warm Up" assignment ready for them when they walked in. I encouraged them to work with partners, and I gave feedback on how well they were working during the opening assignment. At that stage, the "Warm Up" was for reinforcing or subtly extending old ideas, or taking a pass at prerequisites for the day's main lesson.*
*I'm teacher-uneducated, so my teaching education has really been scrapped together from experience, books and blogs. My slides originally were just ripping off Dan Meyer's. Now I find that my slides still look like his. It's Tahoma's fault.
Some kids started finishing the Warm Up early, and wanted to know what they were supposed to do. I started putting challenges and extensions into the opening problem set. The problem sets started getting longer and I started telling kids that I didn't expect them to finish everything. I started putting more interesting, weirder problems towards the end of the opening problem set.
*I remain pretty fond of those last few composition problems, especially the second to last one. I think that playfulness is an important quality in hooky problems.
Kids started complaining when I told them that we had to move on from the Warm Up. They wanted more time. So I started giving it to them. The Warm Up problem set started stretching out, and so I started putting more and more of the day's new content into there. It started becoming a nuisance to put it up on the projector -- I was running out of room on the slide -- and I still wasn't writing problems that were interesting to the quickest students.
I had a very, very clear problem in the classroom, and it was the classic one: kids all need different things. I didn't know how to handle it.*
Well, I knew about giving choices to students, but I didn't really know what choices to offer them other than something stupid and lame like "You can either do these 4 practice problems or these 4 practice problems." "Oh, really Mr. P? Really? Oh boy oh joy. I get to choose!" I shouldn't joke. It was fine -- it just wasn't working super well.
It was around this time that I started playing around with old PCMI problem sets. I started staying up really late to do math problems, and I started thinking about the difference between what I was providing students and what was keeping me up at night.
I thought about it over a weekend, and I came back wanting to rip off PCMI. So I tried to.
This has been going on for three weeks now, and here's my new cycle:
1. Students work on problems in small groups. Usually pairs.
2. I circulate, observe, and help.
3. I come back for whole group stuff. This is either a short explanation/demonstration (which is how I try to weasel out of the word "lecture"), an activity, a question, a game, whatever.
To contrast, here was my old cycle:
1. Warm Up
2. Relatively short whole-group instruction, with pair activities interspersed.
3. Activity, usually practice problems
4. Something else whole-group.
Take a guess: Which cycle is more efficient? Which cycle do students enjoy more? Which cycle gives me more face time with more students? Which forces students to learn how to justify themselves mathematically?*
*The fact that I can't be there to help all of them is a feature, not a flaw, in having problem solving sessions. They have to figure out the right answer, and I literally can't be there to help. I don't even have to be annoying about it.
I mean, there were a lot of problems with my old cycle. It's not like I thought that I had it all figured out. But I do feel like I understand a really important component of teaching that I didn't understand before. This whole process is more efficient and effective if my job is to react to the students, once we're in the classroom. It's the difference between me imposing information on uninterested bystanders, versus me lending a hand to people in the grip of a problem.
I still need to figure out how much support to offer, and I still need to figure out how to find and write really grabby, hooky problems. But, for now, I'm sold.
One last thought: I don't think that problem sets are right for every lesson, though I am leaning on them quite a bit at the moment. I do think that, fundamentally, it's better for me to be in reaction mode rather than performance mode. At the moment, problem sets are the way that I'm giving students a chance to encounter important or interesting things. I'd like to have a few more tools under my belt for getting students working hard on math other than problem sets.
Blog posts are supposed to end with a snappy line, but this one just peters out.