Towards the end of class a few weeks ago, we proved (using truth tables) that the original statement and its contrapositive are logically equivalent.

I sent a Quick Poll to assess student understanding and was fairly pleased with the results. (Students had only learned which was which for conditional/converse/inverse/contrapositive/biconditional statements the day before.)

We began to think about the implications of the original statement and its contrapositive having the same truth value in terms of proof, in particular, indirect proof. And then the bell rang.

When we started class the next day, I sent a Poll specifically pertaining to indirect proof, wondering whether a) what we had learned last class stuck and b) whether they were able to transfer the statement-contrapositive-same-truth-value into how to begin an indirect proof.

Here’s what I got.

What would you have done next?

I didn’t show the results of the poll to the students. Instead I gave them a different question from my stash, with more information.

We talked about the responses, and then they tried an open response question.

We talked about those responses, and then I sent back the first question.

I’ve recently read Embedding Formative Assessment by Dylan Wiliam. Wiliam, along with countless others, suggests planning ahead a sequence of questions for a learning episode along with the instructional moves you’ll make based on the feedback you get from the students. What will you do next if very few of the students get it correct? What will you do next if half of the students get it correct? What will you do next if all of the students get it correct?

If most of my students had gotten the first question about indirect proof correct, I wouldn’t have sent the question with more information about indirect proof. I would have gone straight to the open response question.

I have the luxury of teaching the same classes and planning with the same teachers from year to year. Our stash of questions to ask during the lesson has grown based on responses from students from year to year.

And so the journey continues … teaming together to decide what questions to ask and what instructional adjustments to make, based on the feedback we get from our students.