A well placed Notice and Wonder routine can make all the difference when you are trying to elicit and use evidence of student thinking. Take a look at the picture below. What do you notice? What do you wonder?

We asked a group of fourth and fifth grade students these questions and here is what they said:

This Notice and Wonder routine was inspired by many recent conversations about interpreting remainders. As far as I understand it, the term “remainder” is a convention. There isn’t really anything “math-y” about it, yet it pops up in very “math-y” places. It is in the Common Core Progressions document:

Paul Lockhart says, “The general problem then becomes how to efficiently determine the division of the total, as well as the number of leftovers, if, any. Incidentally, the number of leftovers is usually called the *remainder* (from Latin remanere “to stay back”). He also says, “The thing about verbs is that whenever we have one, we always seems to get two. If I lock the door, then at some point I will need to unlock it. Tie a piece of string, and sooner or later someone will want to untie it. Actions that can be done almost always need to be undone. And this is especially true in mathematics, where symmetry is so highly prized and where the imaginary nature of the place allows us the freedom to reverse our actions so easily.”

There are situations where we will have to deal with remainders. We will have to interpret them. We often teach kids they have three choices when dealing with a remainder:

- Use it as a decimal or a fraction.
- Ignore it.
- Round it.

The problem is: how do we record it the remainder, especially when we don’t have a context? In the past, we have used the letter “r” to denote a remainder. For example, we might write 13 / 4 = 3 R1. This method of recording can lead to problems. 13/4 is not equivalent to 3 R1 because “r” isn’t an operation. Ideally, we would love for students to create an equivalent expression as a way to represent the remainder. Any of the following would work, if we were talking about money or cookies – something “soft” as Lockhart would say.

- 13/4 = 3.25
- 13/4 = 3 1/4

But how do we get students to write equations about remainders that can’t be split into smaller pieces. There are contexts where we have to round a remainder or possibly ignore it, but how do we represent them with equations? We could write this:

- 13 = 4 x 3+1

This works, but how do we create a context that lends itself to writing this equation in the context of division? How do we create a need for this equation?

One day, last week, I was planning with a colleague and telling her about my quest for a division context that prompted students to naturally think about writing the remainder as an expression of multiplication and addition.

She said, “Well, you need to think of a situation where there are pairs – like shoes.”

At this point, I remembered a picture I had taken a few weeks earlier. It was a picture of a big pile of footwear on my mudroom floor. We decided to show it to her class of 4th and 5th grade students that afternoon. The picture shows all pairs and then there is just one lonely sandal. We hoped this picture would elicit equations that might prompt a discussion about recording remainders with mathematically accurate expressions.

Then, we asked them to write as many equations as they could think of to represent this pile of footwear. Here are some of the equations they came up with:

We asked the students to explain how the equations related to the picture. This is where things started to get messy, in a really meaningful way.

Someone wondered, “Where is 4 x 2 + 9=17? Where did the 9 come from?”

“It is the boots.”

“No, it is the shoes.”

“It doesn’t matter.”

“Yes it does! It can’t be the boots!”

A really important argument ensued about what the numbers in the equation represented. One student finally convinced everyone that the 9 had to represent the shoes because there are 4 pair of boots in the picture. The shoes are the category that have the extra- the remainder. We have to deal with the remainder in the context of the shoes. We had a great conversation about what to call these units that we were dealing with. Do boots count as a sub category of shoes? Can we call the total shoes or do we have to call it footwear?

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