It’s my hope that there are other elementary teachers out there that might find this analysis useful if they want to use this compelling and fun lesson by Graham Fletcher in their classrooms to engage their students in exploring addition strategies with regrouping (2.NBT.5, 2.NBT.6, 2.NBT.9). This engaging lesson is very open in the middle. Students have a wide variety of addition strategies they can use including concrete models (base-10 blocks, place value discs, etc) and abstract strategies (arrow method, decomposing, bar method, etc).
Please feel free to plagiarize and make it your own as you see fit. I’d love to hear from you if you use it so we can get better together. (Special thanks to Graham Fletcher for providing such an amazing 3-Act Lesson!)
Before reading further, it may be useful to watch the lesson resources on Graham’s website. Go ahead and check them out here. I’ll wait.
And…welcome back! Let’s dive in.
Some quick background:
I conducted a lesson inquiry using Bright Idea with a team of 2nd grade teachers. Both lessons were unique experiences with their own twists and turns and choices. The lesson pathway below shows the main flow of the learning experience and took about 60 minutes. Further analysis and takeaways can be found at the end of this post, but I’ve tried to embed a lot of our thinking within the flow of the lesson.
This lesson was taught to 2 classrooms of 2nd grade students. Students had learned several abstract addition methods to build on their use of concrete manipulatives. We used this lesson as a way to formatively assess student conceptual understanding of addition and also their ability to persevere, communicate reasoning, and use appropriate tools strategically. We provided each table group with ample concrete manipulatives of a variety of types (straws and bundles, base-10 blocks, place value disks, etc). We were curious to see how many used the manipulatives versus how many went straight to an abstract procedure. What was their level of fluency and comfort with different abstract methods? What were their preferences telling us about how they were making meaning of place value and addition?
Our objectives (as teachers):
- We wanted all students hooked, engaged, motivated, positive, energetic….and to persevere in solving a problem.
- Specifically, we were curious about: What strategies will students use to make sense of a problem and solve it? What will my students do when I’m not explicitly telling them what to do?
- Bigger questions we were wondering about: How do my students talk about math? How do I navigate openness in a lesson in a way that is structured enough to promote student learning for all?
Learning objectives (for students):
- You will use strategies to solve an addition problem and communicate your reasoning with each other.
The objective was posted clearly on the board and referenced throughout the lesson at transition points.
Into (Act 1):
Hello everyone. Today, we’re going to start today by watching a short video. We’re really curious what you notice and wonder about it be we’re interested in how you talk about your (mathematical) thinking with each other.
We played the short Act 1 video showing Graham filling up a container with Skittles. After the video, we did a little pair/share before launching in to a Notice-Wonder exploration.
The Notice-Wonder strategy is a very effective discussion strategy to get all students talking (because everyone can describe something they saw and ask an interesting question regardless of their math ability). It also lowers the floor into a math task because it’s a low-risk strategy that allows student discourse to describe the problem rather than having us (the teacher) “give” them the problem.
What did you notice?
We kept this short. A quick Pair/Share and then some whole class discussion. Once students all agreed that “somebody filled up a container with Skittles” and “the Skittles are different colors” we moved forward on. As you can guess, second graders notice some crazy other stuff too! To help move the conversation forward we focused on “useful” math noticings by paraphrasing the the noticing, asking how many other students noticed the same thing, and then scribing it on the board. We did not scribe the “un-useful” noticings.
I’m really curious. What did you wonder? What questions do you have?
Again, many students wondered about interesting, but un-useful things. For example, some students wondered if you could power a lightbulb with Skittles or if it was a glass bulb or a plastic bulb. We would nudge them back toward the noticings we had scribed. Eventually the key question comes out:
How many Skittles are there?
We scribed this key question on the board and asked students to write it on their graphic organizer. Then we watched the Act 1 video again and asked them to make a prediction (estimate) about how many Skittles there might be.
We gave them a moment to discuss in groups, make an estimate, and write it on their graphic organizer. Teacher walked around and scanned student work for estimates to get a feel for their values. Then we asked them to share their estimates and we recorded a range of them on the board (usually about 30 to 80 in most classes).
Wow. That’s a wide range of estimates. We have estimates less than 30 and all the way up to 80 or 100. If you wanted to make your estimate about the number of Skittles more accurate, what information would you like to know?
“How many of each color are there?”
We displayed the Act 2 image that answers this question. We allowed them some discussion time to make sure they all understood what the image was saying.
Through (Act 2):
Students got to work, and it looked like this (except the students had faces):
This problem sets up an open opportunity to conceptually and/or procedurally explore how to add 19 + 15 + 19 + 17 + 21 and find the sum of 91. We let students dive in for about 10-15 minutes. During this time we monitored the room and gathered as much data as we could. How are they making meaning of the problem? What approaches are they using?
We provided some more intervention for really struggling students. The teachers had posted the different strategies they had used for addition, and we often asked students to reference that for possible starting points.
Then 15 minutes turned in to 20-25 minutes. Students who arrived at 91 quickly were asked to convince themselves (and us) that they were correct by using a different addition strategy. Students who used a strategy but made a small error were asked to share their work with each other and try to resolve the conflict.
One of the interesting struggles that students faced was keeping their work organized. There are a lot of “ones” in this problem to organize and regroup for students who are using concrete manipulatives. Students who chose abstract procedural methods didn’t face this organizational issue.
Here’s a snapshot of some work we saw from students at all levels.
We saw students building the problem and finding the correct answer.
We saw students building the problem and finding the incorrect answer. In these cases, we had students with similar strategies (like these two) try to convince each other who was right.
This student drew 91 Skittles. While arriving at the right answer, we noticed that he had yet to develop strategies to make his calculations more efficient.
Students drew other models with varying degrees of success. Notice he drew 90 and said 89. We asked him to check his calculations to see which answer he thought was better. He did not end up arriving at 91.
Other students used place value disks. Check out this video below. She and her partner communicate about building numbers and adding them. She uses regrouping strategies with the manipulatives. She uses her fingers for single digit addition. She makes and remakes changes and calculations. She catches several mistakes. She struggles with some organization. And then there’s the tragic (mathematically speaking) ending. I invite you to watch. I couldn’t quite follow everything. What do you notice? What do you wonder? After watching the ending, what would you say to this student?
Did you catch her mistake at the end? She’s off by 1 somewhere along the line, but let’s let that go. What interests me is that she catches her own mistake as she counts 73. Then she builds 17 more and starts to add, but she uses the 10 of the 17 to regroup her existing 3 ones with the 7 ones from the 17 instead of grabbing another 10 from her pile. She does not catch that 73 plus 17 should not be 80.
There’s so much to this video. What did you see? What would say to her and her partner?
All the examples above involve students using concrete or visual models. In general, these students made mistakes more frequently than students who used more abstract procedures. In other words, there was a fluency gap in both classes. We discussed this fluency gap in our reflections and how to address it moving forward.
How do you handle fluency gaps in your classrooms?
As for procedures, we saw students using the arrow method:
We saw students decomposing numbers in to 1s and 10s and adding strategically from there. Notice that this student makes a number bond with 9 and 1 to make a 10.
This student wrote about his decomposition.
Sam used a formal algorithm and backed up her work with a drawing.
Some students used a variety of strategies as they wrote down their thinking.
Richard was an interesting case. After revealing the image in Act 2, he eyeballed the list for about 20 seconds and then said “91.” I asked him to explain how he got there. And he gave me a rambling explanation. Afterward, I asked him to write down the order of his calculations. Here’s what he wrote.
He strategically added 19 and 21. Then added the other 10s and then has a fuzzy moment with notation as he writes “=14” before adding 7. (Sloppy and imprecise, but not a dealbreaker.) Many students strategically made a 10 out of the 9 and 1 in 19 and 21, but Richard was the only one that leapt right to 40.
Beyond (Act 3):
When productive struggle had been maximized, we selected a few student work samples to share with the class and to connect to their learning strategies. We tried to focus more on the abstract strategies.
What strategy did this student use? Can you describe the steps this student took? Pair/Share first and then a whole class discussion.
This lesson provides so many pathways to the solution. It’s also highly unlikely you’ll have the time (and the students will have the mental fuel) to go through each solution method. Be choosy and strategic.
We played Graham’s Act 3 video. For the most part, students were leaning in and eager to know what’s going to happen next. In one class, there was an outright cheer, the type you’d expect in a sports bar as the local football team makes a big score! It was awesome, and I wish I had recorded it.
We concluded by asking students to think about the addition method that they liked most. And then had them reflect on a method that they liked but wanted to practice more so they could get better at it. These types of reflection questions are helpful for students to connect the work of others to their own thinking.
You can find our 3-Act template and our exit ticket here.
Reflections, Takeaways, Analysis:
At the end of the day, we reflected on our objectives and student learning. We noticed that this 3-Act Math lesson allowed for students to talk to one another about math because:
- The task was appropriately challenging. It was in the sweet spot of accessible and challenging for almost all students.
- Students had a choice and a voice in the whole learning process from start to finish.
- The numberless introduction (Act 1 video) was low-floor so all students were on board right from the start.
- We used Pair-Share and other conversation structures to increase student talk time. It also gave students time to think, work, and ponder.
- The task creates a motivating sense of earned accomplishment as they start adding numbers and getting closer to a solution.
Other professional learning included:
- There’s no need to tell students what they can tell us or tell each other.
- Problems like these help students abandon “learned helplessness” as a way to “survive” math class.
- Productive struggle takes time…but it’s worth it.
- How can we step away from our textbook? It is too much like a “workbook.” How can we open up other math concepts for students to explore?
What we are wondering about how to do better:
- Being more clear around expectations for notice/wonder.
- Refine sentence/question frames that further student thinking without restricting student thinking. What might those be?
- What’s the best balance between students expressing their thinking in “2nd grader language” and “academic language”?
- How can we involve more movement in the classroom? Are there ways to make the carpet a useful space? Should we do Act 1 and Act 3 on the carpet? And Act 2 at their desks?
What could we have done differently to make student discourse more successful/productive?
As always, feedback and comments are welcome. What inspired you? What opportunities did we miss?
Help us get better together.