Last week, I facilitated a few PD sessions for elementary teachers in my district. There were some predictable — and wildly unpredictable! — tech hiccups. I stumbled over some words. Despite the fact that I’ve been facilitating PD via zoom for the last 3+ years, and sport battle scars to show for it, glitches persist.

But one thing that went well was sharing participant math thinking.

Sharing mathematical thinking builds community. It communicates that everyone is a mathematical thinker and learner, and that we can learn from one another. It supports students in the development of a positive math identity. It is also something that must be done deliberately, with attention to the discretionary spaces and all of the ways we either reinforce or disrupt patterns that “perpetuate injustice and oppression of marginalized groups.” (Deborah Ball)

I like to engage everyone in math as soon as I can in a professional learning session. In preparation of one session last week, I reviewed some lessons from the first unit in fifth grade, and decided that the Quick Image routine would fit nicely for the PD session with upper elementary teachers. The routine is quick to explain: learners need to figure out how many dots there are. The facilitator shows a dot image, and hides it quickly. Repeat. Then learners share their thinking.

What I love about deceptively simple routines like this people catch on quickly, which allows us to focus on the enactment. I used the five practices for orchestrating classroom discussion to support my planning and facilitation.

### Anticipating Mathematical Thinking

Here was the first image I wanted to use.

I had a sneaking suspicion that students who are engaging in this for the first time might need an entry point. It can be overwhelming to have 81 dots flash before you and then disappear! So I built the image onto a jamboard.

…and, for the sake of some organization, I put four copies of the image on each page, and labeled each one with the name of one of our K-8 schools.

After presenting the image, I asked teachers to find their school’s jamboard slide, and annotate the image to show how they determined the total number of dots.

### Monitoring, Selecting & Sequencing

The beautiful thing about Jamboard is that I was able to flip through the slides as participants worked so that I could select 2-3 work samples for discussion. I decided that I wanted to showcase one example that used additive reasoning, and another that highlighted multiplicative reasoning, and to elicit ways to connect the two.

After selecting two participants, I was able to send them private messages via zoom to make certain that they were comfortable sharing their thinking. Often, this isn’t a big deal, especially when the answers are correct. Later in the session, however, I wanted to share a participant’s inaccurate but deliciously mathematical thinking, and, because we’d never worked together, I felt like it was especially important to ask her for permission to share.

### Connecting

I showed the first work sample, and asked participants to share things that they noticed. I wanted this to be a hands down conversation, but, as it was our first meeting as a group, we didn’t have a strong culture built up yet. Instead, some participants used the ‘raise hand’ feature to signal. Others wrote things in the chat. When it became particularly silent, I called on someone who I thought might be game to answer.

**Christy’s Thinking**

- I think she figured out the dots by doing 6 and adding 1.
*(Where do you see the 6?*) In the array. It’s a 2 by 3 array. - I noticed she outlined the array to make it easier for us to see.
- There are lots of groups of 7. The 7 is boxed in.
- She counted the groups. 1, 2, 3, 4, 5…
- She wrote the equation 7 x 8 = 56.

**Melinda’s Thinking**

- She circled the 7. We don’t know how she figured out that there are 7.
*(The creator of the work shared her thinking.)* - I see that she wrote 7 x 4 and 28.
- There is another row with 4 groups of 7.
- 28 + 28 = 56.
*(I see the 28 + 28 in the dots. Do you see that in where she recorded equations?*) - I wrote 7 x 4 x 2 because I doubled the number of dots to figure out how many are in two rows.

From there, we engaged in the Quick Images routine. Having examined different ways of calculating the total number of dots made it easier for participants to chunk subsequent dot images.

### Revising Thinking

After showing this image, I asked participants to send me a private chat message (on zoom) with an equation that represents how they determined the total number of dots. Most people wrote 9×9=81. But then I saw a few others, including:

3 x 3 x 3 = 27

I sent a private message to the person who submitted that answer, and she agreed to let me share it with the entire group. She agreed.

**Elise’s Thinking**

Using zoom’s annotation feature, I elicited from participants where we could see the 3×3, and then the 3x3x3. It reminded people of the multiplicative work we had seen from Melinda in the first problem. We had a 3×3 array of dots, and the first row represented 3 copies of that 3×3 array, for 3x3x3.

That meant that the first row had 27 dots. So how many in all?

I cold called on a participant I had never met — we will call her Sarah — to revise the equation. We wanted 3 x 3 x 3 ________ = 81.

It took me 2 tries to accurately record Sarah’s thinking. She wanted to use nested parentheses. It started with **(3×3)** to show the grouping of the array. Then:

**((3×3) x 3)**

“Is it okay to have parentheses in there?” Sarah asked.

“Yes, that’s actually called *nested parentheses*.”

“And then there’s three rows of that… so… add a 3 onto the end?” Sarah continued. I recorded:

**((3×3) x 3) x 3**

“No, no,” Sarah again corrected me. “I was thinking, like, just put a 3 on the end.” So I tried again:

**((3×3) x 3)3**

I saw some funny expressions in the gallery of zoom faces. I assumed that most people preferred my original attempt. I asked for people to comment, but the room was pretty quiet.

“Let’s check it out. So we start with 3×3,” I said, “in that nested parentheses. That’s 9. Then 9 x 3, which is 27. Then we have 3 copies of all of that, which is the 3 at the end. Other people might have had other ways to record 3 x 3 x 3 x 3, but this is mathematical accurate!” I think I saw Sarah breathe a sigh of relief. I could see Elise nodding in recognition, too. She understood how to revise her work, and everyone else got to delve deeper into ideas about multiplication and expressions.

### Communicating Values

This year, grades 3, 4, and 5 in my district are piloting *Investigations*. The guiding principles for the curriculum are guiding principles for my practice as an educator, too. They are:

**Students have mathematical ideas**. The curriculum supports all students in developing and expanding those ideas.**Teachers are engaged in ongoing learning**about mathematics content, pedagogy, and student learning. The curriculum supports them in this learning.**Teachers collaborate with the students and curriculum materials to create the curriculum as enacted in the classroom**. The curriculum provides a clear, focused, and coherent mathematical agenda and supports teachers in implementing in a way that accommodates the needs of their particular students.

By engaging in math with my colleagues, I was able to reinforce and share my commitment to these values — values that I want to guide us in everything we do together this year.

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