After Mia Buljan’s third-grade students work individually on a multiplication/division equal-grouping-scenario problem, Buljan brings them together on the carpet. She invites them to choose among several cards to find the one that describes in words “what’s happening in your problem.” Each student matches his or her math story/contextual word problem to an explanation in words.
Lesson Part 2b
MIA BULJAN: How many people think they have an answer to their problem? Okay, I see lots of pictures. Go ahead and sit flat. Okay, boys and girls. Some of you noticed when you were doing your problem, that other people have your same problem. So, in the case of Enmy and Esbin, they had the same problem but got different answers because they did it differently. So they just had a little talk and figured out how they wanted to do it. Some of you are telling me I found a friend who has my problem, and they have the same answer as me. And we know that that feels good. But we also know that that answer isn't right until we can prove that it is right. So what I want you to do right now is, um, there are... What I want you to do is leave everything at your desk and come down to the carpet. Quick, quick, quick.
MIA BULJAN: And I saw pictures. I saw words. I even saw number sentences. How many people used all those? Both of, at least two of them? Celine, can you just sit anywhere sweetheart? [laugh] Thank you. Okay. So. I was going to wait for you to sit flat. Sit flat. Thank you. Okay, so one of the things that we did this morning, um, was we were going to write down these, these, this five times ten. And some of you on your white board wrote it in words. Amari? So I saw that Adil had written like five groups of ten. Instead of using the number sentence, he wrote it in words. Five groups of ten. And so each one of these orange papers, one of them is describing your problem. One of them describes what's happening in your problem. So I want you to think about how you did your blocks. What you started with, how you built it. What you ended with. Can everybody picture that in their heads right now?
MIA BULJAN: And then you're going to come up and you're going to look at these. Some of these are the same because you know more than one of you can pick it. Some of them, you'll notice, have nothing written on them. So we didn't get all the cards. So if, if you're missing a card, I found some blank ones and you can write what you did on here. If you don't find what you did, you can take a blank one and write it yourself. Does everybody understand that?
STUDENT: The one that we have, we could just write the sentence?
MIA BULJAN: So the first job is to check these cards and see if one of them is using words to describe what you did in your problem. Daniel? So first, check all the words. But it's possible that your words aren't here. That your problem hasn't been described. And you can grab a blank one and write your words. I’m not telling you which one it is. I'm telling you, you have a choice. First thing you do is what? Who can tell us what the first thing we do is?
STUDENT: Look for your card.
MIA BULJAN: Aliza, what's the first thing you do?
STUDENT: Look for your card.
MIA BULJAN: Look for a card that describes what you did. What happens, Daniel, if I can't find a card that describes my problem?
STUDENT: You get the, one of the blank ones and you write it down.
MIA BULJAN: And you can write your own. So go with it. Does everybody understand that? This part, I know that some of you, I know that some of you worked with...see that you have the same problem as your friend. But this part you have to do on your own. I don't want you to work with your friend on this part. I want you to pick the card that makes sense to you. Does everybody understand? Do you understand? Should I be working with a friend right now?
MIA BULJAN: Should I be thinking quietly?
MIA BULJAN: Should I see a bunch of yelling and running around?
MIA BULJAN: Is it quiet think time?
MIA BULJAN: Okay. So I'm going to move these apart so we have a little bit of space. And um, you can come up and just look at them. Do not get too close because everybody has to be able to see. So these people over here, go ahead and stand up and you can move over here and start looking at these problems. And see if you can find the one that describes. See if you can see your one. And then I want you to sit down or back up so people can see. Okay, I want you to either sit down or back up so people can see. You think you found yours? If you think you found yours, sit down. So that other people can see. If you think you found yours, sit down. Good job, Ramon. Do you see yours, Chase? Okay.
STUDENT: Found mine.
MIA BULJAN: Do you need a blank one? They're all the same.
STUDENT: I'll get that.
STUDENT: There’s three at once.
MIA BULJAN: It happens. Okay, you want a blank one to write yours down? Okay, who's next? Come, do you see one that describes what you did?
STUDENT: I found mine.
MIA BULJAN: Okay, have a seat. Huh?
STUDENT: ...wants to talk to you.
MIA BULJAN: They have to wait, honey.
STUDENT: The person from room seventeen needs scissors. A box of them.
MIA BULJAN: Okay, can you get the box of scissors for Jonathan? Okay. Do you see one?
STUDENT: Yeah, I got mines.
MIA BULJAN: Do you see yours? Do you guys see yours? How about Marlene, do you see one for you? You do? You got pretty excited there. Aliza, do you see one for you?
STUDENT: Miss B, uh mine, mine is...
MIA BULJAN: No? Okay, so you might want a blank one.
STUDENT: It doesn't have mine, does it have to be like this one? Does it have to be like here?
MIA BULJAN: So, go ahead and grab a card if that doesn't describe it for you, okay. Oh, oh, oh, oh! I'm dying to hear this, but what I want you to do right now is to quietly, quietly, when I call your name come and take your card. Either the card you think matches. Get your hands off of him please. Or a blank card like Alex just did. Okay? So Kristof, go get yours. Andrea, go get yours. Hmm? You already have yours? Okay. Navreet, go get yours. You can go. Just grab it and sit down. Don't worry about the tape. Get yours, go get yours Ramon, I know you found it. Go, go. Did you see yours? Go get it, yep.
STUDENT: Can I go get my paper?
MIA BULJAN: Mm-hmm. You can't find yours? So you got to write your own. Go do that now. Ah, excellent. I'm not sure, I just think it's great that you found something you can use. Go back to your desks. Come over here and look. Why am I listening to that?
STUDENT: It's not me.
MIA BULJAN: You don't see yours?
STUDENT: No, mines is, um.
MIA BULJAN: Shh. Come over here, look over here. Don't talk to her.
STUDENT: Do these match?
MIA BULJAN: You have to tell me, I can't tell you that. Go sit at your desk.
STUDENT: What do we do again?
MIA BULJAN: The second thing I notice is how I use this “match making” idea during math class. As predicted, we encountered some controversy on what is happening in the math stories. Esbin and Enmy both have the following problem: “Sam’s dad buys 24 hot dogs for Sam and his 3 friends. How many hot dogs can they each have?”
Again, in terms of reading comprehension, that tiny word “and” can throw a wrench into the works. Is it 24 divided among 4, or is it 24 divided among 3? Sure enough, Esbin is one of the first students we see, explaining his way of dividing 24 among 4. Enmy explains to me that she did it as 24 among 3. I am not surprised by her interpretation, but I know that at some point she is going to be talking to other kids who solved the same problem, and she would have to confront the issue of including Sam or not. Sure enough, Esbin comes by and shows me his work, and I immediately ask them to go to the carpet and talk to each other about what is happening.
I cannot overemphasize the power this teaching choice has had in my practice. Rather than group kids at the start of a session, when their ideas are still forming, I wait until they are deeper into their thinking, and then I curate groups that will have interesting conversations. Did these two have the same idea, but different strategies? They should talk. Did those two have different ideas and come up with totally different answers? They should talk. It takes some work, creating a class culture where students learn to listen to each other, where they focus on understanding each other versus “being right,” and where we have fostered the idea that it is a totally acceptable (and coveted) mathematical behavior to let yourself be convinced by mathematically persuasive arguments. Having said that, the work it takes to create that environment is one of the greatest investments I have made, as a teacher, in my classroom.
As I watch their conversation unfold here in the video, I’m reminded of how aware I felt that Enmy may be changing her mind because it was expected, rather than because she was legitimately changing her mind about whether the problem needs her to include Sam. She originally built the problem by laying down 3 “hundreds” flat place value blocks, as 3 children, and giving them 8 unit cubes (“hot dogs”) each. When Esbin explains that he included Sam, Enmy counters with it wouldn’t work to put down another flat. She knows that the problem works for three kids, so it doesn’t occur to her to dismantle her work and restart distributing the unit cubes among four. Since it works among three, she assumes that adding another flat means that the fourth flat (“child”) will have zero cubes. I encourage her to test it out, show us what would happen. When she sees that it does work, she immediately switches to saying that’s the right way.
But, again, I was asking myself, does she really think that’s the right way to do it now? Or does she just think, “oh it works, that must be what they want me to do”? I question her about the wording in the problem, and right when they both seem satisfied with 24 among 4, I can’t help myself, I ask one more question: How would the problem change if it were supposed to be 24 among 3? What words would be used? Enmy nails it with “Sam’s dad buys 24 hot dogs for Sam’s three friends” and, funny enough, when I ask Esbin the same question, he also gets there but sounds less confident, to my ear, than Enmy does.