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Monday, August 1, 2011

Summer Learning, PCMI Edition: Odds and Ends

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

This is a catch-all for things I want to remember and post that aren't big enough for their own post.

In a 5-minute short, Cal Armstrong presented his use of Livescribe smart pens. I had a little "holy cow" moment during his presentation, because I've long dreamed of kids' recording their problem-solving process, but there's only one smartboard in the room, and writing with a mouse is hard. Enter the Livescribe pen which records both your writing as you write, and audio along with it. And they are only like $100 a pop. I could ask kids to record a livescribe of them solving a problem as their reassessment, or record a tutoring session of them teaching it to someone else. We could put them on blackboard and build up a little library of these, or upload them to voicethread for feedback.

Google Forms for Recording Small-Group Discussion
I am pretty good at incorporating small-group or partner discussion, but I don't often have an efficient way for groups to share their thinking. One technique I noticed frequently deployed at PCMI was to give groups a link to a google form, so that each group could send in a summary of their discussion or response to a prompt. We aren't a 1:1 school, but it would be sufficient for each group to have one laptop for this purpose, and I'm pretty sure I could secure 5-6 laptops to keep in my room. Then again, I am supposed to have a TI-navigator system next year, so maybe I could just use it for this purpose.

Other Kinds of Tasks
Do you ever get stuck in a problem-writing rut? I do. Throughout, I was keeping track of all the tasks I saw that were something other than "find the missing value:"
  • write an equivalent expression
  • give an example
  • show that two expressions are equivalent
  • interpret expressions/equations in writing
  • interpret a graph in writing
Metacognition: See How I Think
We spent a few days talking about what is metacognition, and ways for students to "do" metacognition. We participated in an exercise that I think could be adapted for students to use. In a group of three, students take on three roles: problem solver, listener, notetaker. The listener is NOT HELPING solve the problem, just asking the problem-solver to clarify their process and state it out loud. Meanwhile, the notetaker is writing down any evidence of metacognition or "thinking about thinking" that she hears. I think this could be very beneficial in helping students see how the same thought processes (making use of structure, considering extreme cases, organizing data, etc) cut across mathematical content, but I wonder at designing it in such a way that they can see the point. I need to spend some more time thinking about this.

The Vampire Animations
I worked on a lesson as part of our working group, and I don't think I'm supposed to disclose all the inner-workings of the lesson because it may be reviewed for publication as part of a larger project, but I do want to share this super-fun simulation we made. If you can use it, steal away.

Here is a "question" video of an infection spreading up to 64 victims:

And here is an "answer" video up to 512 victims:

Arts and Crafts
Finally, what is camp without crafts?


  1. Live Scribe sounds great! Pencasts would be an incredible way to share math back and forth outside of class, and/or across the internet. I'd love to have students create presentations that we can post in share, praise, and critique.

  2. vampire animation was THE highlight of the project presentations... just need some teeth on those dots

  3. "In a group of three, students take on three roles: problem solver, listener, notetaker. The listener is NOT HELPING solve the problem, just asking the problem-solver to clarify their process and state it out loud. Meanwhile, the notetaker is writing down any evidence of metacognition or "thinking about thinking" that she hears."

    stupid management tricks. do what you're told *because* you've been told and never ever expect any better reason. run like hell i beg of you.

  4. Love the vampire animations! I'm sad, though, that when the uninfected population ran out, more potential victims magically appeared. I suspect that even the least sophisticated students, watching the "question" video, would realize that the vampires were about to run out of people to bite. This could be a great chance to give a lot of students the pleasure of making a non-trivial prediction and then seeing it come true, but instead the prediction gets trashed because someone arbitrarily changed the rules mid-game.

    I've heard anecdotally (and, to some extent, seen first-hand) that many math students are unwilling to draw on their intuition and "common sense" when solving problems, and I can't help thinking that situations like this are partly to blame. By sacrificing verisimilitude for the sake of clean, perfect results, we teach students to leave their intuition at the classroom door, and their problem-solving is the poorer for it.

    (To add insult to injury, the magical fresh blood infusion also demolishes a perfect opportunity to talk about predicting when and how a mathematical model might break down, which is one of the most important parts of modeling. If secondary school isn't too early to do infection modeling, it isn't too early to do infection modeling right!)

  5. Fair enough but it all depends on how it's used, doesn't it? The animation is a small part of a lesson that I spent three weeks writing, so, take it for what it is.

  6. Owen, I see your point, but...

    If taking on these roles helps students to see more clearly what mathematical thinking is, aren't they useful? The roles I talked about in my Complex Instruction post are different, but the same issue comes up, right?

    I'm thinking my students are scared by the lack of structure when I hand them a real problem. I'm hoping that giving them this structure to hang onto will help them find more courage.

    If you (or a kid like you) were in Kate's class, you'd maybe ask why? And she'd maybe explain that for lots of kids, this will help them see how to do mathematical thinking. (You'd already have been doin' it, and wouldn't need this exercise for that purpose. But maybe it would help you to communicate mathematical ideas better?)

  7. The disease propagation is handled nicely by this exercise. (Scroll to: Activity I: Dice and Disease) Student's randomly interact, and roll dice to see if they've had a 'risky encounter'(Uncovered sneeze, say). The simulation generates a logistics curve. I used about 60 students, coin flips instead of dice, and had them run it for 5 actual days, as they went about their normal school day.

  8. Unknown: The way you did that activity sounds really cool! When I was in primary or secondary school, I'm sure I would have been a lot more engaged in a simulation run by living my life for a week than I would have been in a simulation run by wandering around a classroom for twenty minutes. Somehow, that one little change makes the difference between a pointless busy-work exercise and a life-size science experiment with clear and interesting implications.

    How did you have your students track the number of infections over time? Did infected students know they were sick? Did the infections/time curve actually look logistic? (Sixty is a pretty small sample, and when you run the simulation "in real life," the shape of the students' social network affects the spread of the disease---although I suspect that the set of social graphs that give you logistic growth is larger than I would have guessed.)

  9. Aaron, the simulation definitely feels more realistic spread over several days. Each had a roster with all 60 names on it, with a columns for each weekday. In this matrix, they recorded the actual coin flips. They had to have one interaction per 'day' with each of the other 59 (so variations in social network were not an issue), though often they did all five flips in a single session. They didn't know who was sick; I followed the original authors' suggestion in randomly choosing a single original carrier, and keeping it secret. (They DID know if they'd had an 'unsafe interaction, which has lessons outside of math). I then collected the data, and tabulated the results. (Warning: This took about 15 man-hours to cross-check all the interactions; next time, I'll find a way to automate that). I also let them replay the results, to track the spread of the disease. There was much good-natured finger-pointing. I then randomly selected a second starter person. Results played out the same: logistics curve.


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