*What do I need to know about student thinking today before I teach tomorrow?*

Towards the end of a math lesson, this refrain plays in my head.

______

In third grade, we recently started an *Investigations* unit about multiplication and division called “Cube Patterns, Arrays, and Multiples of 10.” The first investigation involves an exploration of the relationship between multiplication and division, using connecting cubes and thinking about multiples.

Here is a 3-Train.

The first green cube is cube #3. The second grade cube is cube #6. This makes sense because it’s a repeating pattern, iterating chunks of red-blue-green (R-B-G), and so every third cube will be green.

3 x 2 = 6

3 x 3 = 9

So the third green cube is at position #9.

Here’s a 4-Train.

Generating a list of the position of green cubes will generate a list of multiples of 4. Students started to link multiplication equations to the patterns that they noticed.

## What do we want to know about student thinking?

As classroom teacher Caitlin and I wrapped up Thursday’s lesson, we were curious to see what students might do if we inverted the situation entirely: instead of generating multiples by thinking about groups of cubes, what if we asked them to determine the number of groups given a set length?

**There is a 4-Train with 32 cubes. How many times does the R-Y-B-G pattern repeat?**

A student distributed a blank sheet of paper to each of her classmates, and students recorded their thinking. Caitlin and I planned to use this to figure out exactly what ideas to emphasize the following day, which happened to be the last day before a week of vacation.

## Why do we want to know about student thinking?

Towards the end of class, it’s time to think about tomorrow. For example:

- What understandings that surfaced today will students need to leverage tomorrow?
- What knowledge and experiences do student’s already have with tomorrow’s content?
- How can we center student ideas tomorrow using what we learn today?

In the lesson to follow, Caitlin and I wanted to generate more thinking around the relationship between multiplication and division, which is why we came up with the “how many times does the R-G-Y-B pattern repeat” question. It also felt important to keep in mind:

- Assessments capture a moment in time.
- Just because a student didn’t demonstrate a particular understanding, doesn’t mean that they haven’t engaged with that idea in the past.
- A student might have different strategies and different thinking about a similar problem that has different numbers, e.g. a 6-train with 54 cubes or a 7-train with 23 cubes.

We want to center student thinking. If we learn about what students already know about tomorrow’s content, we can start to plan for how we will amplify their ideas. Maybe we would see student thinking that we’d want to leverage in the launch the following day. Maybe we would change our launch in order to bring up certain ideas that might help students problem solve the following day.

## Watching Students at Work: Why Use a Diagram?

As we worked, we noticed that many students were drawing diagrams. Some of the students used the diagram to help them solve the problem, while others used it to justify their reasoning.

**Aria **drew the four boxes on the bottom layer. She continued to draw 4 boxes at a time, often individually, until she had drawn all 32 boxes.

She then labeled the layers to show how many groups of 4 there would be in the train with a length of 32.

**Lev** wrote the division equation 32 ÷ 8, and then drew the tape diagram train to represent how many groups of 4 are in 32.

Lev used his diagram to justify his thinking rather than to determine an answer.

**Lara** had also used a diagram to solve. Her diagram didn’t exactly match the cube train problem, which is a quotative division situation (examining ‘how many groups of a fixed size’). Instead, she wrote 32 ÷ 4, and then reimagined the problem as a partitive, or sharing, situation. She drew one dot in each circle before adding a second dot to each circle, etc. The total of 8 dots per circle means that 32 split up into 4 groups is 8.

## What can we learn from the student work?

Caitlin and I poured over the student work samples. I started to think about buckets that we could put the student work into: direct model by 1s, skip counting, thinking of division as missing factor multiplication, etc.

However, the more buckets we created, the less helpful it felt.

There are times when we might want to inventory student thinking, but this wasn’t one of them. Caitlin and I had no more than 10 minutes to talk, and then I likely wouldn’t see her until we were co-teaching the following day. We wanted something that would be immediately helpful for our lightning fast planning session. We decided to discuss exactly what we had observed: who had used the diagram to solve the problem, and who may have solved the problem another way and drawn a diagram to justify.

We also noticed that some students made linear models, like trains, while others started to form groups or even draw arrays that emphasizes the “4-ness” it all.

**Felix** started out drawing a model of the train, with every 4th square labeled “green.” It took only three iterations for Felix to notice the pattern. “Oh, so it’s going to be 4 x __ = 32!” he announced, as he recorded the missing factor equation on paper. He looked up towards the ceiling, tapped his fingers a few times, and recorded *4 x 8 = 32*.

“How did you get the 8?” I asked him.

“That’s what you need to do times 4 to get 32,” Felix announced.

“I noticed you tapping your fingers,” I told him. “Did you skip count? Were you keeping track there?”

Felix shrugged. “No, I just knew it after a little bit.”

**Qiang** drew boxes of 4. He had drawn 8 boxes. “How did you know when to stop counting boxes?” I asked him.

“When I had 8 boxes,” Qiang replied.

“How did you know to stop at 8?” I probed.

“Because that’s how many to make,” Qiang answered, circuitously. Still, I gathered that he actually hadn’t actually skip counted to figure out how many to draw. His diagram justified his answer, and thus is emphasized the 4-ness o the repeating pattern.

**Ellie** had drawn an array not unlike Aria’s towers with layers of 4, but she had drawn hers differently. While Aria had added on each block individually, Ellie had drawn a rectangle then partitioned it into a 4×8 array. She then labeled the rows of 4 with more rectangles (and labels of “4”) and showed skip counting to prove it, as well. I didn’t get a chance to check in with her until later, but she had also used the missing factor strategy that Felix had used to solve the problem. How many 4s are in 32? Well, 4 x ___ = 32.

#### The Biggest Takeaway

While we observed many different approaches, *every single student* showed equal groups in some way. With this as a foundation, we can make stronger connections between multiplication and division in the lessons to come.

## The Next Day

We wanted to center and build upon student thinking.

Most students had made the connection between the cube train work and multiplication during the lesson, when we were figuring out when the 6th green cube would appear in a 4 cube train, or the 7th green would appear, etc. (4 x 6 = 24, 4 x 7 = 28) However, we wondered whether they were all making connections to multiplication when we inverted the problem to make it a division context. Maybe it would help the following day if we made certain to show some models like Ellie’s that showed arrays, or Qiang’s that emphasized the iterations of 4.

We launched the lesson by showing Ellie’s work.

It doesn’t look like a train. How does it still show the problem?

What multiplication equation might we connect to this?

Felix had written 4 x ___ = 32 first. How does that relate the the problem?

Where do you see it in Ellie’s diagram?

Lev wrote 32 ÷ 4 = 8. How does that relate to the problem?

Where do you see it in Ellie’s diagram?

From there, students worked on an activity in their *Investigations* workbook called “How Are Multiplication and Division Related?” Students investigated how they could use cube trains to show multiplication and division. We were making the connections that most students made *implicitly* the day before **explicit**.

## So What *Do* We Need to Know For Tomorrow?

There is no singular answer to that question. For this particular lesson:

- It was helpful to know that every student saw the 4-train with 32 cubes problem as an opportunity to leverage what they know about
**equal size groups**. - It was helpful to observe which students used the visual model to solve, and which used the visual model to justify.
- This did not mean that we dissuaded students from using diagrams to problem solve. It helped us determined how explicit we needed to make the connections between the diagrams, multiplication, and division.

- It was helpful to remember that we did not
*need*to classify and catalogue all of the student work.- There are days when formative assessment IS more evaluative, and there are days when we DO classify and catalogue student work.

- It was helpful to position ourselves in a listening stance; we were learning from students, and not just assessing “got it/didn’t get it.”
- It was helpful to know what we most wanted students to learn the following day — e.g. lesson objectives — so that we could identify student thinking or work that would help us build those ideas.

Most importantly: it was helpful to remind ourselves that teaching is **fluid** and **dynamic**. It’s essential to **listen to students**. It is also important to develop our own content knowledge as teachers, because Caitlin and I were only able to determine how to nudge students the next day because of our own experiences exploring the connection between multiplication and division, and different conceptions of division (as partitive/sharing, or quotative/fixed groups).

Which is all to say: I do not have a singular blueprint for an answer, but it’s always helpful to ask, “*What do I need to know about student thinking today before I teach tomorrow?*“

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