Today, we mushed together four of my favorite things from #mtbos:
- Notice and Wonder (from Annie and Max)
- Krispy Kreme Me (3-Act lesson from Graham)
- The problem-solving framework (remixed from Robert)
- The Estimation barbell (from Andrew)
Since there are loads of adorable student talk in this post, I’ll take just two lines for background here: Liz and Monica (third-grade teachers and #bonitians) approached me and said, “We want to do math teaching better, is there some kind of database or model for ways to teach outside of the book?”
Here’s how it went down in Liz’s class. In the dialogue, she’s the T, as in Teacher.
T: Boys and girls what do you notice and what do you wonder?
Immediately, the murmur of 8-year-old chatter bubbles throughout the room.
I wonder how many donuts that is.
It’s a lot of donuts.
It doesn’t stop at the edge, like it keeps going underneath.
It’s like some cereal because the donuts are round like this in a box like cereal.
We already know how to solve the problem!
T: Boys and girls, I hear a lot of interesting noticings and wonderings. Let’s put some together. (moves to poster paper)
Jen: I wonder if it’s perimeter or area.
Elise: I wonder how many donuts are in the box?
T: Who else was wondering that? Thumbs up.
[Lots of thumbs]
Marie: I notice that there are more donuts inside than we can see.
Jane: There are two pairs of right angles.
T: We have a lot of notices and wonders today! Keep ‘em coming!
Tim: If you see the whole box, you can just count the donuts how high and across.
Bella: I notice that it takes more people to hold it than usual, because like on a normal box it only takes one person to carry it.
Diaz: I wonder if they’re actually fresh.
Jolene: I notice that the box is square.
T: So you’re noticing the shape of it.
Student: It’s a quadrilateral!
Jeremy: There are lots of people and not much donuts.
Christine: I wonder if there’s more than 100 donuts in there.
Isabella: Why is the box so big?
T: Thank you everyone adding, I know you have more to say than I can record—
Student: Is the box real?
T: I notice your groups had one main question: How many donuts [dramatic pause] are in the box? Did you all have that question?
T: Now I could have had you write these down on your paper, but we did it all together. Go ahead and write down at least one Notice and at least one Wonder on your papers. They’re in the middle of the table.
[PE Teacher bursts in to announce the results of the Track Meet]
Teacher: Boys and girls, if you wrote several Noticings and Wonderings, please circle this one.
Teacher: Boys and girls, look down to the barbell. Let’s try to answer the question, “How many donuts?”
What is a number that is too low? Put that number — without saying it — into this box right here on the left.
Now that you have that number, think of a number that would be too high, and write it in here on the right. This is just your thinking.
T: Somewhere on the line between them, put a hashtag and write in your guess.
Student: Hashtag school
Student: Hashtag math
T: Is there any information that you need?
Tim: We need to open the box.
T: Why is that?
Tim: We need to see if there are more on the edges.
Marco: I agree with Tim because there might be some hiding below.
Tyler: There might be some hiding on top—
T: Can you go show us on the board?
Tyler: There might be some hiding on top, like out of the green line.
T: I hear some of you saying, “Ohhh,” like you noticed what Tyler noticed. Well, other people were curious about those things, too. So they emailed Krispy Kreme and were sent this email.
T: (pauses for 20-30 seconds as students read) Oh, some of this information is missing. In this part, they’re showing blank times blank for each layer, so they’re multiplying. Is there anything here that is important that you took away? Jen?
Jen: The times signal
Tyler: 3000 millimeters
Dan: 2300 millimeters
Jolene: Three times layers
After writing their information on the board, Liz says, “Let me show you some more info.”
Tyler: That cannot be true!
T: I see you’re thinking. Let’s attack this problem; flip over your paper and let me see your genius.
I’ve been sitting in the back, pecking away at the keyboard this whole time. Monica joined me shortly after she dismissed her class to P.E.
Monica leans over and whispers to me, “That boy there is just staring. I would nudge him and ask what he’s thinking, or maybe say, ‘Talk to your partner.’ to keep him moving.”
“Maybe,” I respond. “Or maybe he needs more thinking time and is formulating his idea in his head.”
Monica giggles, “But I want talk-time right away, so I assume that all my students want that.”
“Me, too!” I whisper.
T: Tyler, why are you counting?
Tyler: … because these two sides are blank (points to the arrow going down and right)
Dave: I wanted to see if there was 25 donuts there or if only at the top there was 25 donuts.1
T: Ah, okay! Carry on.
T: Boys and girls, let’s look at some of what our classmates are doing. Michael, what did you do here?
Michael: On the box, I saw 25, so I added 25 for the top and the bottom. Then I added 50 + 32 and got 82
Christine: The 25 I counted on the other side and I added them up and it says 3 layers so I added 3 to it.
Doug: Why did you write …um… whatever that is by 114?
Christine: I wrote “not answer” to remind myself that I’m not done yet.
T: Look at Dave’s work.
T: Dave, I’d love to know what you’re thinking, Dave.
[“Please excuse the interruption. Teachers: we are on a Rainy Day Schedule for lunch.”]]
T: Now, look at Dave’s math. Dave what are you up to here?
Dave: That may look like more than ten 3s, but it’s ten 3s. …Actually, it’s more than ten 3s. I’m writing out twenty-five 32s, then I’m gonna add them all up.
Charlie: Why are you adding them all up?
Dave: Because I heard that there was 3 layers, there could be a lot lot lot more donuts around here and I figured there could be a lot more 32s around here. Then I’m gonna add them all up and see what my answer is and see if they put more donuts here.
T: So you’re thinking twenty-five 32s.
Doug: Why did you do twenty-five 32s instead of thirty-two 25s?
T: What do we know about math? Will that work?
Doug: It’ll be done faster and you can figure out the other sides and that will bring you to the answer.
Dave: At the end, I’m gonna put the answer times three.
T: Why’s that?
Dave: Because there’s three layers.
T: What are they telling us with those numbers here?
Dave: There are 32 like this and 25 like this. (points to rows and columns) Opposite sides are equal and then times it by three.
T: Let’s take another 5 minutes of quiet thinking
Vaudrey: I notice that you were counting the dots, what made you choose to count the dots?
Doug: I thought that each dot could be a donut and if they go in this way, they could make an array.
Vaudrey: [Stunned that the third-grader knows the word “array”] Did you find that the dots were 25 just like the column of donuts here?
At this point, Liz had been deep in Act Two for about 20 minutes. Maybe a third of the class has a method that will lead them to a helpful answer. Liz crosses to me in the back of the room.
T: This might be a little high for them. We have a few more minutes, I think we’ll get there.
Vaudrey: This is the wild-and-wooly part, where they’re working in different directions. You gave them 5 more minutes, that should be enough for them to get deep into their method or to find some limitations.
Lee: Okay, good. It’s so hard to not steer them toward my method.
Vaudrey: Totally, you’re doing an excellent job of affirming all their responses and not giving any indication which one you want to see.
Lee: It’s so hard!
Vaudrey: (laughs) I know! And you even put some information on the board that’s not important and some that is. That was a good call.
T: Tim, Leadora, Jen, hand me your papers, I’m gonna show your work up here. Everyone turn-and-learn.
(Students all put down their pencils and face the board.)
T: So, walking around, everyone’s doing great math work. Here’s one, by your dear friend Kaylon. 32×25 and all this math. Do you mind telling us about your thinking?
Kaylon: I was adding up 32.
T: How many times?
T: Any questions for Kaylon?
Michael: Where did you get the numbers from?
Kaylon: I was doing 32+32 and got 64, then… um…
T: …added another 32 and got his answer, and another 32… slow and steady wins the race.
Jen: I…um… I had all that then three times.
T: Why three times?
Jen: Because there’s three layers.
Tim: I added 32 + 32 and got 64, then I added 64+25 and got 89 and 12 and got 101.
T: What questions do you have for Tim?
Vaudrey: (Waits five-Mississippi) Why 12?
Time: Um… I don’t know. I just added a 12.
T: I’m gonna give these back to you to put under your doggies. We’ll have some more think time on this later this afternoon or tomorrow.
Class: That’s dangerous; doggies love donuts.
Teacher: What questions do you still have?
Bella: How many donuts are in it?
Doug: The truck was quite wide and the box filled the whole bottom. The box was this wide and the truck was this wide and it fit perfectly.
Michael: There was three times layers, from above it kinda looks like more than one layer.
Ethan: In the beginning, I wondered if they were delivering the donuts. I wanna order that.
T: It does look like that! I have a little tidbit for the end I can’t wait to show you.
Tim: I’m wondering how much the box costs.
T: I bet we could figure it out once we know how many donuts are in there.
Tim: If we know how much one donut is worth, we could figure that out.
Monica: Those students that were sharing, are those the students who score the highest usually?
Liz: No, they’re just out-of-the-box thinkers.
Vaudrey: Liz, did you feel like the task gave all your students equal access to the material?
Liz: Oh, definitely! Like Ethan hasn’t done very well on his fluency checks this month, but he dove right in. I think he enjoyed talking about donuts!
Vaudrey: Totally. And that’s the point of class; the entree was donuts, and the math helped them answer a question they had about donuts.
Liz: So now what? We didn’t quite finish; should we come back this afternoon or tomorrow?
Vaudrey: Either one, but some kind of resolve should happen They’re gonna want to answer the question “How many?” but they might also want to get answers to some of their other “wonderings.”
Liz: (deep breath) This collaboration is so powerful. I feel like I have so many more things I wanna try already.
Monica: I wonder how a 3-act would look with both our classes together. Like a huge, co-teaching lesson.
Vaudrey: I’m down. How’s next week?
Liz: We could to it on Friday? Pull both classes into one room and do a big lesson?
Also in that conversation, Liz asked me to make a graphic so kids could describe the various types of arrays they used and . I made this.
Liz is planning to project it onto the board and let students draw their rectangular groupings onto the board around the donuts. If needed, a different student could decompose the array into
Graham’s full 3-Act lesson is here: Krispy Kreme Me. I highly recommend his database of Elementary 3-Act lessons here; Graham is a thoughtful and intentional teacher who cares deeply about kids and learning math.
Finally, if you’ve never seen this lesson, here’s the grand reveal.
~Matt “Put this under your doggies” Vaudrey
1. This was my favorite piece of student talk. I assumed that students would read the Act Two image with the red arrows as “25 rows of 32 columns.” It never occurred to me that students would seek more specificity, that they might assume it’s only 25 from the middle up or 32 from the middle to the right. As always, students find new and exciting ways to interact with content. Teaching is just the funnest job.↩