About a month ago, I saw this in Twitter:

At first, I thought, “how would any of those problems be easy to do in my head?” Then, I saw an entry point. I thought about multiplication as “groups” and I realized that I didn’t actually have to do a lot of complicated computing. I could combine the groups to make friendlier numbers. I thought of 23 x 37 – 13 x 37 as 23 groups of 37 minus 13 groups of 37. That is equal to 10 groups of 37 which is 370. All of a sudden, these problems seemed much more accessible. I was able to solve the others in a similar way, except one. I decided to reach out to @nomad_penguin, who originally posted the problems, to see if she could help me out.

Below, you can read our conversation, from left to right.

I almost edited out the part where I mixed up the problems, but I left it in because I think it is really important. Communicating math thinking is really hard. I talk fast and think slow. My mind is forever chasing my mouth and, in the case of Twitter, my fingers. I think it is important to share this because I need to remind myself that mistakes are incredibly valuable. In this case, my mistake was not slowing down to think about what, exactly, I wanted to share with Aimee. Thinking about what I want to share will help me think about what I currently understand and what I am still confused about. If I can’t clarify my thinking than I won’t be able to verify it.

I want to say thank you to Aimee. I am so grateful that she supported my struggle. She didn’t tell me the answer, but she also allowed me the space to take a risk. I felt okay telling her that I was having trouble.

I decided I was going to share this problem set with the K-12 professional learning community that I am a part of. We had some really interesting conversations. The elementary teachers used strategies that were similar to mine. Many of them thought of “groups”. Robyn, one of the high school teachers in the group rewrote the problems as factored versions of themselves. I *think* her work looked something like this: (Robyn, feel free to correct me if I am butchering your ideas.)

23 x 37 – 13 x 37 = 37 (23-13)

37(23-13) = 37(10)

37(10) = 370

I asked Robyn, “what do you call that?”. I can’t remember exactly what she said, but I am pretty sure she mentioned “factoring” and “the reverse distributive property.”

At this point, we got into a pretty lively argument about what, exactly, the distributive property IS. Is it all of the equations listed above? Is it just some of the equations listed? Can you “use” the distributive property without knowing that is what you are doing? Is “using” the distributive property the same as “understanding” it?

We decided we wanted to learn more about the progression of the distributive property, and other properties of operations. We decided to try this activity with students at different grade levels. We realized that some of the teachers in our group would not be able to try this activity as it is presented above because it would be beyond their student’s reach. So, we brainstormed other problems that would still present opportunities to use other properties of of operations. Here are some options that we came up with:

We all agreed to try one of these problem sets with our students and share what we found out via our blogs. Our guiding question:

- What do our students understand about the properties of operations?

When we left the meeting, I was still wondering about a lot of things. I continued to let my questions simmer. A couple of weeks after this meeting, I read one of the teacher’s blog posts. She tried the problems with her students and it was a really frustrating experience. “Oh no!”, I thought. “What have I done?”

As I read her post, I wondered if I should have structured this professional learning experience a little differently. I don’t think I had taken enough time to ground our conversations in a context. When we discussed how we approached the expressions, we didn’t have an image to anchor our understanding. As much as I love Number Talks, I think I need to be more intentional about grounding some number conversations in an image so we can really connect the numbers and symbols to a representation. We need the image to explore the structure of our number system more deeply.

I decided I was going to think about a way to incorporate a number talk image into our next PLC meeting. How could I use an image to get us thinking about rearranging expressions to show equivalence?

Super interesting work you’re doing here as a group!!

I had some thoughts about what I think might be going on – things I notice. I also had some questions about what happened and where you might go next – things I wonder.

I noticed that you and the other teachers began with an assumption that you could make sense of the expressions mentally – and that perhaps there was a ‘trick’ (or, better, a strategy/shortcut) that would keep you from getting bogged down in calculations. I noticed this led you all to considering the distributive property as a useful tool.

—> I wonder how the posing of the question led you there: “not easy to work out in your head” frames the task as though we CAN do most of them, so the ‘figuring out’ is now shifted to how, rather than which.

I noticed that you developed grade level – appropriate written expressions that would embody the same property as students thought about these.

—> I wonder whether students might need similar guiding directions, like, “Raul said he can solve these in his head quickly, though one is more challenging for him than the rest. How do you think he’s figuring out the answers in his head? Which one do you think he might be having the most trouble with? Explain your thinking.”

—>I also wonder whether it might behoove students to read the above question BEFORE being shown the expressions. Have them make sense of what’s happening so that when the expressions show up, students who typically see numbers as things to DO don’t get overwhelmed.

I noticed that in the teacher’s blog you referenced, the array model was used as the primary representation of multiplication.

—>I wondered: what work have students done with arrays so far? Do they think of arrays as something to take apart? Have they built arrays themselves? Have they considered how area, multiplication, repeated addition, and scaling of groups is represented with arrays before?

Sometimes I think students are better positioned to DECONSTRUCT something (e.g., think about how this array’s dimensions could be broken up into tens and ones) if they’ve had opportunities to CONSTRUCT them beforehand.

So, here’s what I was thinking: What if students in that classroom had started, for an introductory number talk, with a rather open-ended discussion: How could you arrange 24 (or 240, or 48, or something else with lots of factors, or something that is close to a 10, or something with easy tens/ones distinctions like 25) plants in a rectangular garden? (or tiles on the floor, or something). The goal being, have students think about how an amount could be representative of a whole group to be sliced and diced into different arrangements.

I am super fascinated by this framing of ‘which is easiest’ and how it might prompt students who consider themselves less competent to consider ‘hard problems’ as the outliers rather than the rule. I posted the question on my door for the week (my office is one of the first in my hallway at SU) and I hope to see some colleagues questioning similar things…. I’ll report back! And I hope you follow up on this post!

Charlotte

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also, i’ve got to stop making such long comments. sorry. my thoughts get away from me!! I’ll curb them next time i promise….

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