In my last post, I shared an experience about trying to explore the properties of operations with K-12 teachers and students in order to understand the progression of how these properties are taught and learned. Since then, so much exciting learning has happened – a lot of it has been mine.

One of the fourth grade teachers in our K-12 Professional Learning Group approached me to talk about her experience exploring some complicated expressions with her students. She said she showed her students this expression, (7×8) + (8×3) and asked them what they thought the solution was. You can read more about her experience on her blog, but essentially, this is how a few of her students approached the problem:

Chrissy was thrilled that her students were decomposing 7×8, but she was hoping they would see that you could recompose the factors into 8×10. When she came to me, she said, “I got the students to see how you could combine 7×8 + 8×3 to make 10×8 and we talked about how “seeing” 10×8 would be so helpful because they know 10×8 is 80.” Then, she shared something provocative.

She said, “one of the students, Emerson, said, ‘you can’t do that. You have to do what is in the parenthesis first.'”.

Wow.

“What do I do with that?” she asked me. “What should I do next?”

“Great question. I am not sure.” I wondered what some of my colleagues on Twitter would recommend as a next step.

Wow. We got so much instantaneous feedback. It was awesome. We were still left with a lot of questions:

- Is there such a thing as the “reverse distributive property”?
- Is factoring out the 8 an example of “using the distributive property” or is it something else?
- How do we get these students to shift their perspective about decomposing factors so they can see the potential of recomposing factors? How can we get them to connect decomposing and recomposing?
- How do we get them to see and use parentheses as a tool instead of a rule?
- Is it possible to help them figure out that this “regrouping” only works when the expressions in the parentheses share a common factor?
- Will they, can they, figure out why?

We asked our Twitter friends for some advice about how to pursue these questions, ** our** questions, and still honor

*‘s original disequilibrium.*

**Emerson**David Weese actually started a planning doc so that multiple people could contribute to the brainstorming. You can see it here.

Christopher (@Trianglemancsd) suggested using a picture of Eggs!

We decided to ask the students, “what are all the different ways we could figure out the total number of eggs in the picture? Don’t tell me the answer. Just tell me all the ways you *could* find the answer.” Chrissy and I didn’t have a ton of time to connect in person. We were connecting via Twitter, David’s planning doc, and short conversations in the hallway en route to and from the bathroom. This was all happening 24 hours before the last day prior to Thanksgiving vacation. I anticipated what the students might say. You can read about this in the planning doc that is linked above.

I emailed Chrissy and begged her to let me come in to math the day before Thanksgiving break to try this lesson with her kids. I told her I completely understood if she wanted to tell me to shut up and go on vacation already. She didn’t. She was equally as curious to see what her kids would do with this picture of eggs. We didn’t have a ton of time to hone our plans, but so we decided to jump in because we really wanted to revisit the topic before too much time had passed . Keep in mind, what you see and read below grew out of ** unrefined **plans. It is bumpy. Chrissy and I are unsure at times. I am sharing it with you so I can learn from the experience and so we can learn together.

We showed the students the picture of eggs and asked them to record ways that they could organize them to find out how many.

This is what they did:

Chrissy and I started a discussion by asking Chase to share his thinking. The expressions below were written *before* Emerson asks her question. You can’t see the video of Chase, but you can hear him. The black lines indicate what he was drawing on the smart board. The pause in the middle of the video is because the white boards kept falling off the ledge. Hence, my high pitched reminder of why I had suggested NOT putting the white boards on the ledge. #reallifeclassroom

Have a listen.

As I listen to him now, I wonder if I missed an opportunity. I got caught up in the fact that Emerson noticed one of Chase’s expressions didn’t match the image. I didn’t catch the fact that he says, “a*nd I would add those together.*” I wish I would have asked him what he meant by that.

At one silent point during the audio, Chase changed one of the expressions on his whiteboard from (2×4) to (2×3) and handed it back to me:

We only had 15 minutes before lunch.

“Emerson, it is because he is adding….”

**Let her finish, Sarah!**

It is so hard to listen to myself in this video. I want to jump through the recording and put my own hand over my mouth. I wish I hadn’t cut Emerson and her classmate off. The more I listen to and/or watch myself teach, the less I want to talk in a classroom. (This is a good thing!) I talk so much less than I did when I started as a math coach, and yet, I still think I talk too much.

At this point in the lesson, a few boys came in. They weren’t present at the beginning of the lesson so we tried to catch them up by explaining what we were doing.

Have a listen. This conversation is where we start to really wonder about the role of those parentheses.

As I listen to this clip, I wish I had prompted Gavin to tell me more about what he was thinking. I was so focussed on getting these students to discover what ** I **wanted them to discover that I missed a golden opportunity with Gavin. I wish I had asked Gavin, “Why do you want to take the two out of the second set of parentheses?” (Tracy Zager, you’ve got me thinking about asking better questions to encourage relational thinking.)

I wish I could rewind and pause. I was listening ** for** answers instead of

*my students. (Where did I read about this last weekend? On Twitter? I think it came out of the #CMCmath North conference? Was it Zak Champagne who said it? Whoever said it, it has really stuck with me.)*

**to**I wonder what would happen if I showed Gavin the video above, paused it, and asked him why he wanted to “take the two out”?

At this point, it was time for lunch. How many times have you been in the middle of some deep, messy thinking and the bell rings? I wish I could have spent the whole day with Gavin and Emerson. Interestingly, they hung around and chatted with me while everyone else got ready for lunch. We continued our conversation for a few minutes.

As I reflect on this lesson, I think I was rushing. I was so desperate to get Emerson and Gavin to figure IT out before the bell rang.

What was I thinking? The IT is huge! Gavin and Emerson are wrestling with some giant ideas about the distributive property. They are wondering about the limits of parenthesis. They are trying to figure out when to add partial products and when to multiply them. They are manipulating expressions to match a context so that the math makes sense. This kind of learning doesn’t happen in 15 minutes or less.

I was so worried about losing the opportunity to connect big ideas that I hurried right past several opportunities to connect big ideas.

It has taken me weeks, literally, to write this blog post. It takes me a loooooong time to process experiences because I have to understand * all* of “it”

*before*I can make sense of

*of “it”. See Pam Harris’s post about the three groups of people. I am a classic “c”.*

**parts**I am so grateful for taking the time to process this post. I used to think all of Gavin and Emerson’s thinking would disappear over Thanksgiving vacation. I thought I had to be the super hero math teacher lady who swooped in and helped them organize their thinking in a neat and tidy pre-lunch math chat.

Now, I realize that Emerson and Gavin are doing some serious thinking. I need to let them be the hero’s of their own math stories.(Thanks Dan Meyer for planting this seed.) It takes a long time to construct the understandings that they are wrestling with. They haven’t even started their multiplication and division unit this year.

Wow. I need to say that again.

They haven’t even started their multiplication and division unit this year.

I wonder what’s on the menu for that unit?

How about eggs?

Lots and lots of eggs.

Oh, watching videos of yourself is so awful 🙂 I need to do it … I know a teacher who used plastic eggs that you take apart to put stuff in for this kind of thing… including dividing by fractions.

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I had to look at this for a while and I’ve been teaching mathematics for 20 years. Just goes to show you how procedural my thinking has been for so long. In the last video…wouldn’t it be nice if they could rotate the 2×4 so that it could be 4 columns(same as the 3×4) by 2 rows and then stack it on top of the 3×4? Then they may see that the common dimension is multiplied by the sum of the other dimensions.

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