I had planned to spend my first time in Room 12 observing the classroom teacher and getting to know her students. Instead, the teacher texted me the day before to let me know that she had come down with pneumonia. I immediately offered to teach the class.

…not that I had any ideas for a lesson. First Grade. I didn’t know the students. The classroom teacher, Natalie, would probably be out all week.

Natalie is a force to be reckoned with. She’s high energy, organized, and has a clear and thoughtful vision for instruction. Her mind is trained on intended learning outcomes. Even her little transitions and classroom energizers are imbued with playful connections to content; she asks students to review backwards sequencing while jumping up and down, or ‘sneaky e’ vowel changes while dancing and posing. Because she works so carefully to connect ideas, walking into her classroom was intimidating. I didn’t want to ‘waste’ a day with a significant detour. How could I push students forward while also spending some time to learn more about them as mathematicians?

I knew it would be impossible for the lesson I chose to fit perfectly into what the class had been working on, but I wanted to give it my best shot. I pulled out an old book from TERC Investigations, 1st Edition — Grade 1, “Number Games and Story Problems.” One of the first investigations is called “Dot Addition.” I assume it’s still in the current version of the curriculum, because it’s pretty brilliant. The task has an effortless genius to it: it’s open and accessible for all the learners in first grade, and the lesson plan includes questions designed to push the thinking of any first grader in the class, even ones who are learning how to do long division in their outside math class.

Students are given sets of dot cards — 2s, 3s, 4s, and 5s — and asked to make target numbers on their game boards.

Students can approach this any number of ways. Some might use familiar combinations. Others may choose some cards, and then count on, or adjust their work, until they reach the target.

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Most use a combination of strategies.

I knew the task had the potential to be well suited to anyone in the class… but what was I hoping to uncover about these learners? And what story did I want to tell?

I decided that I could accomplish a lot of what I wanted with a conversation driven by the 5 Practices for Orchestrating Classroom Discussion: anticipating, monitoring, selecting, sequencing, and connecting. (I’ve written about the five practices in other posts, listed here.) While reading the lesson, I couldn’t help but start anticipating student thinking.

I decided that I would probably want to tell the story of using a knowledge of doubles to form new numbers. First graders are often more comfortable with doubles facts, and I want to build up a culture of making connections. Of course, I reserved the right to change my mind mid-math block if I saw noticed more urgent in student work and thinking.

Anticipating

To create a target number using the dot card, students might…

  • Count every dot and remove cards that go over the target, or trade for small numbers.
  • Start with a large quantity card, like 5, and work on counting on to hit the target. (“5+5+5=15… and 1 more to get 16.“)
  • Use their knowledge of doubles to hit specific targets. (“8 is 4+4.“)
  • Apply their knowledge of making a 10 to hit specific targets. (“12 is 10+2. 5+5=10, so 5+5+2=12.”)
  • Decompose numbers into chunks, e.g. replace the 5s in 5 + 5 with numbers that make 5, like 1 + 4 + 1 + 4.

Monitoring

During the lesson, I monitored students at work. I sat down next to students, conferring with them about their thinking. I asked a lot of questions. There’s only so much I can infer by looking at the student’s work.

Selecting

I chose four students: Daysha, Jackson, Priya, and Xavier. Three of them are students of color. This was not an accident. About 50% of the students in this class are students of color, and I wanted to position them as capable mathematicians to learn from… from Day 1.

Daysha:
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Yes, this is Daysha from the “Assigning Competence” blog post! I chose her to share on the first day. Something about her attitude in the classroom — the way that she seemed to raise her head with confidence while also scanning the room, eyes wide — made me feel like she needed to share. I was interested in her use of doubles to make the first 12. 5 + 5 = 10, and then 2 more to make 12.

Jackson:
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Jackson finished quickly. In fact, he completed three sheets. He wasn’t even using the dot cards themselves while I conferred with him. He was thinking a lot about equal groups, and I asked him questions about how he visualized this in his head. (He said that he “just saw the numbers coming together.”) It was interesting to watch him tackle a prime number later, but for the sake of our class discussion, this would do nicely.

Priya:
niketa
Priya liked using doubles. I was interested in how this changed for how she made 20. “I couldn’t make it 10 + 10, so I kept adding 2s, because I can skip count.” (Note that she actually made 21, and not 20. I asked if she was okay sharing anyway. She said yes.)

Xavier:
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Xavier used more than just doubles — he talked about skip counting to make 8. “2, 4, 6, 8.” I was fascinated with how he came up with 5+5 and then 4+4+2. “Next would be 3+3+…3? No. 4.”

I did not choose Daniel’s work. He had an fascinating way of making 15 — he made 4+4+4+4, and then, upon realizing it was too large by 1, he “traded for a 3” to make it 4+4+3+4. This is interesting, but I didn’t feel like it fit into the math story I wanted to tell.

Sequencing & Connecting

So what is the math story I wanted to tell? I wanted to focus on how we can build off our understanding of doubles to make new equations. I also wanted to show that we can count more and more efficiently, eventually writing equations for our work. Thus, I decided I would show Jackson’s equations last.

The question became: how does this work show how we can take what we know about doubles and build a new idea?

So my order became:

  1. Priya, focusing on her use of repeated 2s to make 20
  2. Xavier, transitioning from how he used repeated 2s to make 8 (like Priya) to examining how he made 10 from 5+5 and then 4+4+2.
  3. Daysha, examining how she used 5+2+5 to make 12.
  4. Jackson, looking at how he recorded an equivalent equation (5+5+2) and how he was able to apply his understanding of doubles to make this without counting dots.

 

 

The Discussion

Quickly, I transferred the photos I took while conferring with students to my computer (using AirDrop) and threw them in a blank google slide document. You can view the actual document I created here. Because I was working quickly, I didn’t have time to edit the photos or do anything fancy — just copy and paste. I’ve left the doc as is for you, although I removed one students name (to preserve their privacy with the pseudonym. All student names that I use on this blog are pseudonyms).

So I had a story I wanted to tell. But did the students want to tell the same story?

We started with Priya’s work. Students instantly noticed the doubles patterns.

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Then we examined Xavier’s work. I expected that students would see his work the way Xavier had. He knew that 5+5=10, and so he went down to the next number — 4 — and noticed that he needed 2 more after doubling it. 4+4+2. Then 3+3+4, and so on.

But, instead, Xavier’s classmates started to notice how the 2 in 4+4+2 could be redistributed, split across the 4s to make it again 5+5. They were proving why it worked form a different angle.

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…which meant that they saw Daysha’s work differently. Students could picture the 2 being split across the 5s, so that 5+2+5 became 5+1+1+5, which students recomposed as 6+6. Of course 6+6 is 12 — they know their doubles! Students were digging even deeper into the idea of decomposition and composition than I had envisioned.

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We closed with Jackson’s equations without dots. Many students were able to compose and decompose numbers fluidly during the discussion.

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Through this activity, students were able to play around with numbers and notice patterns. Through conferences with students, I was able to nudge their thinking further, to think about adjusting numbers and decomposing them. Then, through our class discussion, we were able to synthesize learning collectively. Students were building meaning together about what it means to break down (decompose) and then compose numbers. Maybe not everyone took away the exact same message from the closing discussion, but the way that I planned for it — using those 5 practices — meant a greater chance of students developing parallel ideas.