Have you ever thought about how difficult it is to truly understand another person’s perspective? I think about it all the time, especially when I am working with students. As a facilitator of learning, it is my job to interpret what students are thinking. It seems so attainable when I type the words, but my experience tells me that the quest is actually quite elusive because interpreting is different from judging and thinking is often different than saying.

Last week, I showed this picture to a group of fourth graders and asked them, “what do you see and how do you see it?” I chose this image because I was hoping it would prompt students to use strategies and numbers that would encourage an exploration of the associative property.

I went into the lesson trying to capture two strategies that we could compare and connect during our next math class. I stumbled upon something else. As always, my students got me thinking more deeply. This time it was about units.

J was the second student to share his thinking. He doesn’t usually volunteer to talk. In fact, when he first came to this class, he hardly talked at all. It is often difficult to truly understand what J is thinking and I have to tread carefully when he shares because I don’t want him to shut down. As soon as I saw his hand up, I knew it was important honor his initiative. When he started to describe that he saw 9 by 12, I almost assumed that I knew what he meant. Fortunately, the little voice in my head told me to shut up and listen because my assumptions would have been wrong. Check it out:

Did you hear what he said? He said he saw 9 *by* 12 and his classmates and I assumed he saw the larger rectangle made up by the holes in the pretzels, but when he described what he saw in more detail, what he really saw was 9 *pretzels, each with *12 holes in them.

After J shared his strategy, several other students shared how they saw the pretzels. Here’s what they said:

- “I see 3 by 3.”
- “I see 9 three by fours.”
- “I see twelve holes in each pretzel and 9 pretzels.”
- “In each pretzel, I see three groups of four.”

One student said she saw nine 1 by 12 pretzels. She struggled to articulate it. First she said 12 times 1. Then, she said 1 times 12. Finally she settled on nine 1 by 12 pretzels. Then, a different student explained, “I see one group of itself. I see 1 times 12 in E’s pretzel.” Again, they are mucking around with this word “by” that they have heard and are now trying to make sense of. Is she wondering about the relationship between the individual pretzel and the larger picture of pretzels? I wish I had asked her to tell me more. After listening to her, I see something new: I *could* see one group of 12 nested inside a larger group of 12 if there were 4 pretzels in each of the three rows. I *could* see one group of itself.

I asked each of the students to show us how they saw what they saw. I had intentionally chosen this image because I was hoping students would use three factors to describe it. I asked one more time if anyone saw it a different way. One student said he saw three, three by twelves

Another student said he saw three, four by nines:

The last student said that he saw 88 holes. He went on to describe how he calculated 88. He said “I knew that five 12s is 60 and ten 12s is 120 so if I take away one of the 12s… oh wait. I think there are 108 holes.” I thought is was interesting that he was the only student to calculate the total number of holes.

We ended our number talk here. I did use two of the strategies to plan a number talk for a 4th grade learning lab the next day, but that is another blog post. Right now, one weeks later, as I sit and listen to the number talk, I still wonder so much. I wonder about the word “by”. I have always associated the word “by” with measuring area. If I am shopping for a rug that is 5 by 8, I am looking for the rug to be 5 feet long and 8 feet wide. J and E were thinking about the word “by” as a general way to describe a multiplication situation. We had some trouble interpreting what they were saying. We had to navigate from the math to the pretzels and then back to the math again.

I used to think Number Talks involved me bringing the numbers and my students operating with them, but now I realize that there is so much value in letting my students bring the numbers to the Number Talk. By starting with an image, I put the students in charge of using math words and symbols to describe what they see. Will they describe it as a relationship or a quantity? Or maybe a relationship between quantities? Starting with an image also lowers the floor and raising the ceiling of the task. I am not assigning an operation to be used. The students choose which operation will help them make sense of and describe what they see and, as they describe, I can listen and try to interpret what they understand.

I am not saying I will never use numbers in a number talk. I am just enjoying the opportunity to change my perspective, push my thinking, and be slightly uncomfortable with the math I think I know.

I used to think the properties of operations were all symbols, numbers, and rules, but they are so much more than that. I think, maybe, they are more like filters with which we interpret and explain our world. I have often heard people say that math isn’t “breakable”. At first, I thought they meant that Math was impenetrable, predictable, solid. I never really felt comfortable with that definition. Now, I wonder if they meant something different. I am realizing that math is so much more bendable than I ever knew.

Thank you Christine Newell and Number Talk Images for the pretzels.