The SMART board had been unresponsive all morning and my students (being helpful problem solvers!) wanted to troubleshoot with me even though I'd already tried their ideas in the previous class. I appreciate that they wanted to help, especially because most of the tricks I've learned on the SMART board have come from students, but the class wasn't settling down and getting to work on the do now. By the time I finished checking homework only a couple kids had gotten started (usually this isn't such an issue, but we also usually start class with a quiz, a habit thrown off due to the blizzard- I hadn't see this class in 5 days!). Once I focused my attention on getting them to work, they were engaged in trying to come up with the shortest possible definitions.
They did a nice job with this task. Some students wanted to include properties in the definition, for example defining a rectangle as a shape with two pairs of congruent sides and four congruent angles. I got the chance to differentiate between definition and property, as well as preview the rest of the lesson- proving properties. By the time we finished this and going over the homework (which sparked some good debate over whether a rectangle is always/sometimes/never a rhombus) we were about half an hour into class. Normally this would have been great, but I neglected to consider the early release schedule due to PD when planning the rest of the lesson. Instead of 90 minute classes we only had 60, which I know is still a lot for many of you, but I only see them once more before vacation and wanted to test on quadrilaterals before all this knowledge escapes over break!
While I started distributing the proof cards, I bribed my students- any property they proved today I would provide on the test, the rest they would need to remember or figure out again on Friday. This received comments from both ends of the emotional spectrum: proof evoked "is that the given thing? I can't do that!" while the potential to get extra information on the test got students to perk up.
I showed them the chart below and reviewed how some quads are subsets of others, so anything we prove for a parallelogram will hold true for all the other shapes "underneath" it. Then I asked what property they wanted to start with. One student suggested "a diagonal divides a parallelogram into congruent triangles" which sounded good to me since I knew many of the proofs depended on use of congruent triangles. We started with the definition of a parallelogram (which they knew, but I didn't have a card for), then they stalled. I prompted a bit and we got consecutive angles are supplementary. This is a key property, so I ran over to the computer (since the SMART board was broken) to add this property to the slide- one thing proven! On the way back to the board I glanced at the clock- panic! Not only would we run out of time before they proved many properties on their own, at this rate we might run out of time before finishing this proof! As I stood at the board I was wondering what I could possibly do to get them through the rest of the steps without giving everything away. I suggested we draw in the diagonal, look at their cards, look at particular color cards (triangles and lines are different colors), draw the triangles separately... Lots of wait time in between as I desperately searched the eyes of my students to find one who had the spark of an idea I could pounce on. In the end I gave some bold hints, did a lot of pointing, and they figured it out. But by that point there were only 5 minutes left in the period. Definitely not enough time to send them off to do some other proofs on their own, especially after the defeated looks most of them had about a minute into this process. In the steps of that one proof we did prove three properties, so there was that redeeming fact.
As we started cleaning up I was feeling frustrated that I’d had to drag them through the process rather than the students being able to figure it out on their own; I didn’t think they had any investment in the problem. But then the student who had figured out the last steps yelled out “wait! Don’t erase mine yet- I want to post it on Instagram!” And then I stood up a bit straighter because if even one student was proud enough of the work she had done to want to share it with the world, it wasn’t a total failure of a lesson. (By the way, this was my One Good Thing for the day, if you haven't seen that blog yet you need to subscribe, it's good for the mind and the soul.)
I think my first mistake was letting them choose the example. I should have chosen a short but representative example ahead of time and structured this more carefully so that the proofs built up from previous properties. Then I could have emphasized how once they've proven something, they can use it as a reason in the next proof. If I'd set up a flow then I wouldn't even need to do an example as a whole class; just reminding them that the proof cards on lines are still useful would have been enough to get them going. I'm struggling with when it's better to let them come up with the questions (that worked great for discovering properties) and when it's better for me to provide some guidance. If we had unlimited time then it would have been fine for them to start a proof, realize they needed some other information, go do another proof first and jump around until finally they'd proven all the properties. But between the time crunch (2 classes between blizzard and vacation! Must wrap up unit now!) and the defeatist attitude they already take with proof, this class just isn't there yet. And I need to remember that's okay. Sure, in my ideal class we sit around and play with ideas, teasing apart details until they all have eureka moments and see math as a thing of wonder and beauty. But, the world isn't ideal, students have past histories with math, preconceived notions, are distracted by valentines day and vacation and snow. That's why they need a teacher- I have to find the right amount of support and independence that allows them to engage in productive struggle.
What I'm hoping to hear from you is that you've done this exact lesson and can hand me the perfected version and that you've also perfected the next lesson- proving these statements are biconditional. Barring that, what I really want to know is how you approach proof in geometry. I hated proof when I took high school geometry because they were fill in the blank two column format. I could never figure out where they were going, how that reason could possibly provide a helpful statement or what theorem number went with the statement. I've tried to make changes that I think help, but students still have strong emotional reactions to the word proof, so I'm not there yet.
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