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!

Saturday, July 30, 2011

Summer Learning, PCMI Edition: Deeper Criteria

This is a series of posts that are reflections from the Park City Mathematics Institute Secondary School Teachers Program.

One afternoon we listened to a lecture/powerpoint by Douglas Corey of Brigham Young University about comparisons of effective teachers at home and abroad. Toward the end of his talk, he seemed to have partaken of a generous serving of the edureformer kool aid and came across as anti-teacher or at least teacher-concern-dismissive, which obviously turned many people off. However I took away some notes about his research that struck me as important.

Based on classroom-level comparisons between different countries from the TIMSS video study, researchers found
  • there is no single effective teaching method
  • all high-achieving countries teach quite differently
  • we can not judge a lesson's effectiveness by methods used, but rather
  • effective lessons have deeper criteria in common he called "instructional principles"
He asked us to predict the instructional features which must be present for students to learn with understanding. These were the guesses that my group brainstormed. If you want to play the home version of the game, take a moment to jot down what you think they are too.
  • teacher content knowledge
  • problem-solving
  • students have to be working and thinking
  • deeper explorations
  • students making connections
  • continuous assessment that informs instruction
  • deliberate metacognition is part of instruction
  • teacher believes all students can learn rigorous, conceptual mathematics
  • students need to spend time thinking about math outside of class
However researchers only found two:
  • "intellectual engagement" - the teacher has to get the kids thinking about a problem. Students have to struggle. "Struggle" means students expend effort to make sense of math, to figure something out that is not immediately apparent. It does not mean needless frustration.
  • "connection-making" - the focus of the teaching has to be on making connections. Connections don't come by accident but must be an explicit focus of planned instruction.
The struggle thing rang true for me. At some level I internalized that idea long ago. I'm still coming to terms with the connection-making point. The same concept was approached earlier by Gail Burrill with respect to the Common Core standards. She pointed out that in American classrooms, teachers can plan and ask connection-making questions and activities, but students mostly still end up doing procedures. A big question I am still grappling with is how to design and deliver instruction so that the students are doing connections. I have only vague notions about what that would even look like. I don't really know what to do with this yet beyond hang a sign on the bulletin board next to my desk at school that says "make the students do connections."

Friday, July 29, 2011

Summer Learning, PCMI Edition: Keep Learning

This is ('s a goal but I am distractable...ooh! shiny!) a series of posts that are reflections from the Park City Mathematics Institute Secondary School Teachers Program - the best professional learning out there, in my opinion, except it sounds like next year it's basically canceled or at least way diminished (thanks, dysfunctional government!) We wrote personal reflections on a regular basis, and I spent 20-30 minutes before bed every day writing down things I didn't want to forget. The reflections were supposed to be private, but you all know I'm an exhibitionist about my learning.Without further ado,

Lesson 1: The most important thing I can do to keep improving as a teacher is to keep placing myself in the position of learning new things. The discomfort, confusion, coping - I have to keep coming back to it and back to it. The goal is perceiving myself, even when facing thirty teenagers, as an "accomplished novice" instead of an "answer-filled expert." How do teachers move from expert to novice? That is hard to do and even hard to think about - a desirable pathway but a difficult one to find. But the "accomplished novice" attitude is one I respect in the best teachers I know. If I don't keep placing myself in the learner role, I forget, forget, forget. "Answer-filled expert" is the pattern of habits I fall into when I forget.

Saturday, July 2, 2011

Virtual Conference on Core Values: Mistakes are Made

To sum up the center of my classroom in a phrase: We Make Mistakes. Sometimes deliberately and sometimes not, but we celebrate both kinds.

Non-educators are maybe puzzled at this statement, but all the teachers in the audience have already started nodding along with me.

See, lots of people think that learning happens like this:
and that that sparkly rainbow is the maaaagical majesty that separates good teachers from bad.

But those who have spent some time, you know, actually responsible for other people's learning know that it really happens more like this:
We could just call it "Piaget for Dummies." Other authors have likened the confusion to the conflict in a story. That little star is a place a teacher's skill is really important, because that's the point where kids might feel stupid and check out and also probably hate you for making them feel that way.

So, I'm very deliberate about centering lessons around mistake-making in non-threatening ways. If all my lessons have one thing in common, that would be it. Here are two specific strategies.


The first one became a staple in all my classes late last year and is really just Cornell University's GoodQuestions project, by way of Helen Doerr at Syracuse University, who sums it up this way: "A good question is divisive."

Pose your question. Here is one I plan to use next year in Calculus:
Compel the students to register a choice. You could use clickers, poor-man clickers, or polleverywhere. But it is important that they commit to a choice. And it's best if they can't see what other people selected until everyone's vote is registered. Ideally the responses will be more-or-less evenly distributed, or at least there will be no clear winner. Which is awesome! Because even if it ends up that you're wrong, you can't feel too bad about it if a third of the class agreed with you.

The point is to instigate an argument that can be settled with mathematical justification. If you're not sure how to get the justification ball rolling, try "Would anyone care to defend choice A?" If you're desperate, pass out slips they can write their name on and hand in for participation credit whenever they present an argument. Whatever you do, for goodness sake, don't tell them the right answer. (Like, ever. Let them come to consensus. Learn how to ask helpful questions without giving away the store.) Unless for some reason you want to completely shut down discussion. And thinking. You know you've got a really good question when you don't have to pull teeth to get kids to talk because they are so compelled to explain their reasoning that it overcomes their fear of everybody looking at them.

The Tyranny of Randomness

My second strategy is, I imagine, common in most math classrooms, and that's getting kids to the board to present their work. When I get them at the beginning of the year, they are freaked out by this. Of course. Standing in front of a room of your peers, potentially exposing your ignorance, is super intimidating until you do it a few times and everybody is doing it and you realize it's not such a big deal.

"Will you check it first?"
   "We'll all check it together!" (This is not optional!)
"What if it's wrong?"
   "That would be awesome! I'm kind of hoping it's wrong actually!" (If anyone points and laughs, I'll kill them!)
"I didn't get an answer..."
   "We'd still like to see the progress you made!" (Get up there, kid!)

I've always done this but sometimes have used volunteers, and sometimes have played with ways to randomize selecting students. For example, use the random integer function on the TI and match it to a numbered list of names. I also tried the popsicle sticks, but I am not that organized and kept losing them. Neither of these methods was very satisfying, because they couldn't see the process and had to take my word for it being random. But random selection really helps, here - when any of them could be selected at any time, they become more likely to give a problem some honest effort. Also, if they see a machine pick, they don't blame/hate me for picking on them.

Then last year I found the best thing - SMART Notebook has a flash random word chooser in the gallery (if you can't find it you might need to update the Lesson Activity Toolkit) - so I just have a file saved with a different page for every class with the students' names. When it's time to select a student, I just fire it up and we can all see it ping around before it settles on a name. It's very suspenseful.

Whatever you do, you can't let them off the hook. When selected, they're going to stand in the front of the room and pick up a marker and write something. (Alternatively, going to put their work under the document camera, or whatever process you've worked out.) Or you might as well go back to asking for volunteers, and it will be the same three kids who are the only ones doing work and going to the board.  You'll probably have to be annoyingly insistent for a little while until they become convinced of the inevitability of the situation.

There is a free random word chooser here that seems to work pretty well, but I have not tried it in a class. (And you'll want to turn the sound down/off because, ugh.)

The moral of the story: confusion and mistakes are necessary for learning. So much so, in my opinion, that I center my classes around them! Thanks for attending! Check out the other presentations!