Yesterday we worked on this pattern.
n + n + 5 or (n - 2) + (n - 2) + 9
Today we had to decide whether or not these two rules were equivalent. We had a brief discussion about the different ways students could make their argument: numerically, visually, symbolically or verbally. I asked each student to choose a method they preferred and spend a few minutes constructing an argument. The plan was to then have them pass their journal around the group and have their partners help them make their arguments more convincing.
As I circled around the classroom, I noticed the work of a particular student who doesn't yet have the confidence I believe will eventually show up. I stopped and asked him about his work.
Me: So, tell me about what you have going on here?
Me: What type of argument are you trying to make here?
Me: Ok, so what numbers are you choosing?
Student: I chose 55.
Me: Does it work for both rules?
Me: Now that I'm sitting here with you and hear you explain, I can totally understand what you're trying to say.
Me: Let me ask you something: Do you think that if you ripped this page out of your journal and left it for me to read after class, I'd be able to understand your argument?
Student: No, I don't think so.
Me: Can you treat this as a rough draft and try to convince me as if I wasn't here?
Me: Ok, great. I'll come back and check in a bit.
After a second pass around the class, I come back to this:
I asked if I could have his permission to take a picture of both and show it to the class. We'd keep it secret if he wanted, I assured him. When I projected the first iteration, other students tried to explain his thinking. When I showed the work of the "second student", we all agreed it was much easier to follow the thinking. Then I said, "This is the same kid."
Class: "Wait, WHAT?!
The coolest part of this was that when I wouldn't say the name of the student, many of his classmates said, "It's obvious Mr. Cox. Look at him."
He was beaming.