For one hour of my school day, I work with an amazing group of Seniors in a class I have called "Mathematical Thinking" (more about that in a later post, perhaps). Mostly, the class is a mixture of problem-based units and other miscellaneous open-ended puzzles, problems, and mathematical games.

Yesterday, we worked on a puzzle/game called Cartesian Chase. I played a few games against students to demonstrate the rules (we confined ourselves to a 3x7 rectangle) and then just let them play for while. Then, I had them stop and record anything they were noticing in terms of a strategy that seemed to be working. Then, they switched and played with new partners for a while longer. I stopped them again after a few games and had them record updated strategies. We ended with a class a few "undefeated" people playing each other. It quickly became apparent that there was a winning strategy at play.

In the process of all this, here is what I noticed:
- nearly ALL of the students were engaged and playing for the whole time
- students were having fun with each other
- we had a few early conjectures in place about what strategy might be best
- students uncovered structure in the problem, used it to win every time, & were able to clearly explain it
- after the game was "solved," a few students were curious: "what if we added another column?" and "what happens with other board sizes?"

I work hard to bring this same spirit of playfulness to other lessons. I work hard to make every day feel like a puzzle in our class. For some reason, I can never quite bridge that gap in the way I would like. I think I get pretty close most days, but for some reason "how many burger combinations are possible?" still feels more like a math problem and less like a puzzle to students. Maybe it has to do with our intent as teachers? Do we place too much emphasis on students "knowing" something specific by the end of the lesson? Could we set up the task better (slower?) so that it emerges as a puzzle? I have a lot of questions, but I do know that I value what students are learning about themselves as mathematicians and thinkers from a lesson just as much, if not more than, I value students knowing some piece of the thing we call "mathematics."
 


Comments

09/08/2012 12:48pm

I dunno if you would care for this, but some fun watching was found here...

http://www.youtube.com/watch?v=uNXwvBsiKP8

Reminds me of this.

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09/11/2012 11:50am

This sounds like a great answer/opposite to Dan Meyer's "5 Signs You're Doing Math Wrong" in your classroom (Lack of Engagement, etc). Nice to have a problem-solving class (I do too) where you can really spread your wings and explore this idea deeply. I always struggle with the balance of convergence and divergence in a class that involves a lot of play. Traditional lessons always converge so neatly and in the expected ways. With more play, there is more divergence. How and when to bring the class back together to share, debate and discuss is a real art form.

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09/12/2012 7:57pm

Hi Ana...
I totally get your point about convergence versus divergence. You wrote, "traditional lessons always converge so neatly and in the expected ways." I wonder about this often. Mostly, I wonder if we set them up in a way so that we are fairly certain they will converge in this way? I'm assuming you mean "traditional" lessons as ones in which students are building an understanding of a specific topic (standard). I wonder if the "non-traditional" lessons, the ones that are more likely to result in divergence, end up moving in that direction because we (the teacher) aren't dictating the direction as much? Can we let go because we don't feel the lesson HAS to end at some specific destination? And, further, does this change the mathematical experience for students? Does it let them follow their intuition, curiosity, and individual ways of thinking? Honestly, I have no idea about any of this....just interesting questions, I think. Thanks for your thoughts.

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