First things first: a little celebration is in order–Mara slept through the night 4 nights this week! Huzzah!! This is the first time she hasn’t waken up in the middle of the night, for multiple nights in a row, since school began. Fingers crossed that we are moving towards a new “normal”.
Perhaps it was (in part) because of the extra hours of sleep, but this week–despite being a full week after a short one–felt good. I had my students fill out midterm course evaluations (anonymously) on Monday. It took me a few days to get around to reading them. I always am a bit nervous about doing this, so I waited for a time when I knew I could decompress afterwards if that was needed. Turns out it wasn’t.
While the comments were by no means universally complimentary (something I appreciate, because that’s not actually helpful!), the students were forthright and constructive in their feedback. It also seemed clear to me that some of the items I’ve been working on (e.g. providing more variety of tasks) has been noticed and well-received by my classes. So that was good. And I got some usable suggestions that I can try to incorporate moving forward. Also good.
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One of the biggest changes that I see in myself in this year’s version of our proofs unit compared to previous years is that I am doing a good job of shifting the onus of responsibility for checking students’ work from myself as “arbiter-of-accuracy” onto the students themselves. I’ve done several gallery walks where they write comments to one another on the whiteboards about what is effective and what is confusing about their group’s proof. These followed a more structured proof “writing workshop” where they commented (on google docs this year!) on one another’s work and read the feedback from their peers. I’ve done the writing workshop in previous years, but then drifted away from building upon that. This feels so much better!
We did some proof puzzles–I wrote up statements and reasons for some problems and had students arrange them in order–and I got to model what I look for when checking a proof.
First, I check to see: do you have the givens?
Then, I check: did you end with the thing you were trying to prove?
Finally, I check the middle: is each statement supported by the ones that come before it?
This has taken place over the past three-ish weeks, but this past week it really felt like things came together. I had students work on the lovely TIMMS problem below and had different students put up their solution method on the board.
I carefully selected students who used different methods (add a parallel line, add a perpendicular line, extend the lines) so that students could see a variety of ways to solve the problem. [I overheard one student say “I think that’s how we were supposed to do it.” which made me shake my head…your way worked too!!!] Then I gave them additional numbers to work, using a method that was different than their own, until they noticed a pattern and a conjecture emerged.
I wrote the conjecture on the board and asked the class, “What do we do with conjectures?”
Class: “Try to prove them!”
And so they did. Afterwards, they exchanged papers with someone in front/behind them (so as to avoid exchanging with someone they may have worked with) and took 1 minute of silent reading time to look over their partner’s work then they talked it over. The room was abuzz with great conversation! They were helping one another hone their arguments and make things clearer.
It felt good.
The Space Between the Numbers 