Using proofs in math is something I'd never considered until recently. But I've started delving into this idea so that I can expand my teaching pedagogy. I know I must seem like I stalk Fawn Nguyen @fawnpnguyen because I'm always following her lead and writing how her ideas have worked in my class, but once again she was doing something last spring that made lots of sense for my students at this particular point in time.

Here students must use logical arguments to convince other people that their ideas are true. During this activity, I witnessed students having mathematical conversations maybe more than in any other activity I've had in class so far. I also thought they were very focused....the undefineness of "right" and "wrong" answers suits the middle school mentality.

I gave them a large sheet of equations...and asked them to cut each one out and discuss it with their partner. We worked in pairs and triads for this activity. I thought that would foster more conversation than working in 4 person groups. I have to say that this worked OK...but not everyone likes to work this way. It became a source of disagreement because some students wanted to cut out everything. Inevitably that leads to some equations being "lost" between the two days....so I made sure to display the original worksheet on the SmartBoard so everyone could double check their poster.

There was definitely lots of head-scratching going on throughout the two days they had to prepare. I had shown them a few ways to test their equation--trying a positive number, a negative number, a fraction and possibly even a really large number. If they thought of a different classification of a number, I encouraged them to use that too! Learning how to test something in a mathematical sense is new. It's definitely new to me from a teacher's perspective....I've done it but never thought to formally include it in my teaching bag 'o tricks. That's surprising because I've been working on writing to learn and writing in math class for years. I don't know maybe I was just ready or this was an activity where it was finally accessible to me. Either way it's a good one and scalable to all sorts of children....age it up or down.

Big chart paper was the way I went so they could spread out...lay out all their equations and have plenty of room for adding their tests and rationales.

One of the things they learn early on is that a single counterexample can change the course of thinking. It only takes one example that disproves what is being said. Somehow that really appeals to the nature of middle schoolers. My job was to identify this strategy as "offering a counter-example" and rein it into something that is educational. I'm grateful to the work that has been done at the Field-tested Learning Assessment Guide(FLAG) at the University of Wisconsin-Madison. They've published some amazing papers on how to approach these teaching tasks and how to break apart the pedagogy that is needed for someone like me to try it out. In their resources section are a series of papers by Malcolm Swan and Jim Ridgway that lays out the teacher's tasks in short order. They call these Convincing and Proving Mathematical Tasks. I highly recommend reading through these if you are considering this activity because they have insights that will help you with guiding and facilitating your students.

Sort of tying up all sorts of disparate pieces of thinking around the room, if you will. By teaching them the vocabulary of example and counterexample, I think it gives boundaries to thinking. It also helps students formalize strategies for thinking about how to approach understanding what all these equations mean. They aren't just a jumble of letters and numbers on the page.

Student thinking becomes visable, but what I really like, is that this isn't about prettifying their poster. It's a working poster, not a pretty product. Does that make sense? So they are done in pencil, show lots of revisions and shows thinking. It's not meant to be a formal final project. You could do that, but it wasn't my objective for this lesson.

My job at the end of the two days was to act as a scribe and facilitator of the conversation. I picked questions where I thought everyone would have something to contribute....but I used my antedotal notes from the work time to pick the presenters. For those that struggled with this task, I knew I would ask them to present their thinking on easier problems. While I knew that everyone had some successes, I reserved asking the more abstract thinkers in my classes for the harder problems.

I also believe this will be a project that we can come back to again and again. They will be able to use this idea of proof. The whole trick is to make sure that students are able to identify examples and counterexamples. Can they discuss them with each other, the class and even with adults? Do they know how to test it with specific cases that they can turn around and generalize so they can formulate a proof? Can they use the results they found during their tests as evidence to offer in their writing and/or discussions?

You've turned this into such a lovely lesson, Marsha. I like the use of chart papers. The writing to explain is amazing!

"I also believe this will be a project that we can come back to again and again." Yes!! This structure is so adaptable to any level of mathematics. Please keep on stalking me; it's mutual! Thank you so much!

Posted by: Fawn Nguyen | October 24, 2012 at 12:34 PM

Nicely represented post. I too believe in using what you refer as " working posters", i normally hung them from the wall, a trick i learned from an excellent faculty from British council, it has a deep impact on students when they are surrounded by their own and their mates ideology about a common topic which improve their understanding about it.

Posted by: Swagat Das | November 02, 2012 at 11:43 PM

Mathematical proving and convincing truly gave me headache when I'm still in college. It's amazing though you use charts and wide space so that your students is at ease while solving.

Posted by: A Plus Tutoring | November 14, 2012 at 05:54 AM