As an example, my students are in the middle of Unit 2 in the Illustrative Mathematics/Open Resources Grade 7 curriculum. We have been doing a lot of problems where the students multiply a whole number times a fraction, and they have been doing pretty well with it. Some use mental math, and some want to write out the algorithm and cancel (which drives me nuts, but it works for them). To push all of the students to think a little more deeply, I thought about asking an OpenMiddle-style question:
The first round did not push the students much out of their comfort zone. I got repetitions of problems we had been doing already: 2/3 · 5 = 10/3 (I didn’t specify whole numbers in the boxes. But later on I was glad I didn’t). There were other students who answered with whole numbers, but they were pretty comfortable with those already.
Next Day: Round 2 (10 minutes)
I wonder what they will do with this:
With a little bit of constraint, the answers revealed some additional insights on their part. I got things like (I apologize for not taking the time to put all the fractions in a better format):
- 1/1 · 6 = 6 , and general rumbling there are an infinite number of those.
- And lots of ½ · 12 = 6, 1/3 · 18 = 6, ¼ · 24 = 6, and again more seeming consensus that we could keep going with those.
- One student offered 1/(1/2) · 3 = 6. That was pretty cool. I wish I could remember how he got there… I think it was continuing a pattern he saw, because he asked if that was possible. I don’t think he did the division, but I wonder…?
I know exactly what to do for the next round.
Next Day: Round 3 (10 minutes)
I wonder what they will do with this:
What did I notice? What do I wonder?
First of all, I notice that the order of the steps was reversed. I started out wondering, running an experiment, and then noticing. Of course, it is not really reversed… notice/wonder/notice/wonder is really a cycle of inquiry; I guess I was just spending time on the other side of the circle. But it reminds me… as I have begun to I ask my students to notice and wonder more often in class, I need to continue the cycle around, and help them to formulate actions inspired by “I wonder” for another round of “I notice.”
While writing this, I noticed that my understanding of the math behind this problem and the levels of understanding that my students might achieve evolved and changed tremendously.
I noticed that I can’t tell you exactly who understood what, or who progressed. But I did see energetic, engaged conversation, and there were a variety of strategies shared. I don’t think I would spend more time assessing individual students, but I think it would be worth pushing their understanding further with more of these questions, and also worth the time to shift to a different modality working the same concepts, like a number talk looking at:
2/3 · 9 =
2/5 · 15 =
2/7 · 21 =
2/7 · 5 =
2/14 · 10 =
I noticed that I had just been taking answers from the students, and I never stopped to ask how any of the solutions were related. In using resources like Visual Patterns or Which One Doesn’t Belong? or OpenMiddle questions, you develop a patter that can help highlight the deeper connections. And it takes many, many uses of those formats to develop those ideas. I will definitely be adding “How are the solutions related?” or “Are there any patterns you see in the different solutions?” or something like that to my OpenMiddle patter.
As I mentioned before, sometimes I wonder what concept I am actually working to develop and sometimes that changes. In this case, I wonder if the more general idea is important enough to spend more time on. I am not sure. But I don’t think that’s a problem. I am taking a skill or concept that the students are familiar with, and making some new connections, stretching it a little bit. I added time and day notations so that you realize this is just a warm up, and we do lots of other things in a class period. What gets learned will play out differently for different students. And I think that is perfect. What they all will get is more time practicing mathematical thinking. And that is my number one goal every day.