On a recent visit to a small district in the Midwest, I got the chance to visit a third grade class that was working on division (3U5, Session 3.4). When I joined Nicole, she was in the middle of working on the following problem:

Gil loves toy cars. He saved enough money to buy 32 toy cars. How many 4-packs of toy cars did he get? (SAB p. 333)

Below the problem, Nicole had written

I asked her to tell me about her thinking. She said she wanted to start with something she knew, which I agreed was a smart strategy.

**Nicole**: I know that 4 x 5 = 20. Then I did 20+12=32. … And now I’m going to add 5 and 12, and…that’s 17.

*I wasn’t expecting that! Thinking about it now, it probably shouldn’t have surprised me. Combining partial answers is often part of a successful strategy. There’s also the challenge of keeping track of the different units the numbers in this problem represent: the number of groups, the number in each group, and the number of toys. Once you’ve abstracted that information to just numbers and equations, it can become challenging to hold on to what each number represents, and what part of the problem you’ve solved/have left to solve. That’s the work of Math Practice 2 in the elementary grades.*

**Me**: So 17 is your answer? (She nodded, hesitantly.) Let’s go back and see what the problem was asking. (I reread the problem aloud.) So, 17 4-packs would give you 32 toy cars? (I jotted the following in my notebook and showed it to her.)

17 4-packs –> 32 toy cars

**Nicole**: (frowning) That seems way too big.

*Nicole was happy to pause and re-think through the problem. She didn’t hesitate to suggest that her answer seemed unreasonable. I was seeing Math Practice 1 in action!*

**Me**: Hmm. I agree with you, 17 4-packs sounds like it would give you a lot more than 32 toy cars. You told me you wanted to start with something you knew, and you just knew 4×5=20. What does 4×5 represent in the story?

**Nicole**: The 4 is four cars in each pack. And the 5 is five packs. Five 4-packs is 20 cars.

**Me**: And then you did 20+12. What were you figuring out there?

**Nicole**: I need 12 more cars to have 32.

**Me**: Ok. And how do those 12 cars come?

**Nicole**: Oh it’s how many 4-packs make 12.

She thought a moment, then began writing.

*I was fascinated that, after recording her first step as 4×5=20, she now recorded 3×4=12. As she wrote 5+3, I had to pause myself. “She’s adding again? … Oh, right! Five 4-packs made 20, and three 4-packs makes 12. So (5 + 3) 4-packs makes eight 4-packs, or 32 toy cars.” As I worked to follow her thinking, I was again struck by the challenge of keeping track of all these different units. As an adult!*

We talked through the connection between her work and the story again. As she did, I added to my notes:

17 4-packs –> 32 toy cars

8 4-packs –> 32 toy cars

**Me**: Does that seem like it makes more sense?

She agreed it did, just as a colleague joined us. I explained to Keith that Nicole and I had just had a really interesting conversation about this problem, and I wondered if she could tell him about it. Nicole was happy to. Using both my notes and her work, she talked him through our process. As she finished, Keith asked: “And, so what is your answer to this problem?”

*As confident as I was in Nicole’s thinking, I held my breath. This question of what the problem was asking, and whether it was about groups, the number in each group, or the total number was slippery.*

**Nicole**: 8.

She added one final piece to her work.

*As Nicole went off to play *Multiplication Compare*, I was left thinking about the ideas 3rd graders need to make sense of and understand as they learn about the operations of multiplication and division. But, mostly, I was thinking about Nicole’s classroom. Nicole was a problem solver. She expected to explain her thinking and was comfortable doing so. She expected the problem, and her solution to it, to make sense. When she reflected on the reasonableness of 17 as the answer, she didn’t try to force that answer to make sense; she re-connected to the problem and re-thought it through. She did not seem phased, nor embarrassed, by an error, but persevered in finding a strategy that made sense. As a one-time visitor to this classroom, this speaks volumes about the work this teacher must have done to create the mathematical community of which I was momentarily a part. *

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There were so many ways this problem could have gone, but I really value your move when you asked–>

Me: So 17 is your answer? (She nodded, hesitantly.) Let’s go back and see what the problem was asking. (I reread the problem aloud.) So, 17 4-packs would give you 32 toy cars? (I jotted the following in my notebook and showed it to her.)

17 4-packs –> 32 toy cars

You took her back to the misstep and she found her way with a strategically placed question.

Thanks for the comment, Chris. It did feel like an important moment, and I think a lot of possible responses went through my head. Luckily, the one I chose this time had a good outcome!

I agree with you that mathematical community building is so vital to our students’ learning. Explaining her thinking, persevering and making sense of problems are obviously the normal, expected way that Nicole’s classroom works!

I agree with you Kathy! Every time I’m in a classroom like this one, I find myself wishing I’d been a fly on the wall at the beginning of the year, to see how that community-building began. And, how it developed over time. It can feel magical to people (e.g. But how did she _do_ that?), and so I’m always thinking about the hard work, the concrete examples, that contribute to creating such a mathematical community.

My district is in the process of piloting, and soon adopting, Investigations 3. We have been working on the “Ten Minute Math” problems that ask which one of the three estimates of the answer is reasonable. I was remarking to the students just last week how well they are able to describe the reasonableness of their choice or how another estimate is not reasonable. This comes only from having math conversations like the one you had above where we ask them about their thinking. While this type of classroom culture doesn’t happen overnight it is very doable with planning, patience and effort. It is also so much more fun for the students and I when they are asked to dig into their thinking. When it is done often enough we are creating true mathematicians who are able to “construct viable arguments” and then go on to “critique the reasoning of others.”

This is so nice to hear! I saw MP1 and MP2 in my work with Nicole, and now you’ve added the idea of MP3 as well 🙂 I think of the Routines and Ten-Minute Maths as a great place to develop a math community – regular practice with similar ideas build kids’ familiarity and confidence, encourage risk-taking, and provide opportunities for short discussions focused on their ideas. Thanks for writing, Nina!

It looks to me that Nicole made a mistake in her notation, but that mistake did not interfere with her correct solution of the problem. When she wrote 5 + 3 = 6, she was adding the packs of cars from the previous two steps of the problem. However, it seems that she kept the correct sum of 5 + 3 = 8 in her head when she moved to the equation 4 X 8 = 32 => 4(cars per pack) X 8(packs of cars) = 32 (cars).

As you have noted, in this kind of problem there are so many parts to keep track, and the symbolic representation of the process and solution is another level of abstraction and complication. My intuition would have been to point out the mistake. Would you?

What a good eye you have Don! You caught something no one had yet seen – which is that when we darkened Nicole’s student work to be more readable, we turned her 8 (which looked quite 6-like) into a 6! We will be redoing the images to reflect this. Had she, in fact, written 5+3=6, I definitely would have asked about it. Thank you for bringing this to our attention!