Season 4 | Episode 2 - Dr. Sue Looney - Same but Different: Encouraging Students to Think Flexibly

Sue Looney, Same but Different: Encouraging Students to Think Flexibly

ROUNDING UP: SEASON 4 | EPISODE 2

Sometimes students struggle in math because they fail to make connections. For too many students, every concept feels like its own entity without any connection to the larger network of mathematical ideas. 

On the podcast today, we’re talking with Dr. Sue Looney about the powerful same and different routine. We explore the ways that teachers can use this routine to help students identify connections and foster flexible reasoning.

BIOGRAPHY

Sue Looney holds a doctorate in curriculum and instruction with a specialty in mathematics from Boston University. Sue is particularly interested in our most vulnerable and underrepresented populations and supporting the teachers that, day in and day out, serve these students with compassion, enthusiasm, and kindness.

RESOURCES

Same but Different Math

Looney Math

TRANSCRIPT

Mike Wallus: Students sometimes struggle in math because they fail to make connections. For too many students, every concept feels like its own entity without any connection to the larger network of mathematical ideas. 

Today we're talking with Sue Looney about a powerful routine called same but different and the ways teachers can use it to help students identify connections and foster flexible reasoning. 

Well, hi, Sue. Welcome to the podcast. I'm so excited to be talking with you today.

Sue Looney: Hi Mike. Thank you so much. I am thrilled too. I've been really looking forward to this.

Mike: Well, for listeners who don't have prior knowledge, I'm wondering if we could start by having you offer a description of the same but different routine.

Sue: Absolutely. So the same but different routine is a classroom routine that takes two images or numbers or words and puts them next to each other and asks students to describe how they are the same but different. It's based in a language learning routine but applied to the math classroom.

Mike: I think that's a great segue because what I wanted to ask is: At the broadest level—regardless of the numbers or the content or the image or images that educators select—how would you explain what [the] same but different [routine] is good for? Maybe put another way: How should a teacher think about its purpose or its value?

Sue: Great question. I think a good analogy is to imagine you're in your ELA— your English language arts—classroom and you were asked to compare and contrast two characters in a novel. So the foundations of the routine really sit there. And what it's good for is to help our brains think categorically and relationally. So, in mathematics in particular, there's a lot of overlap between concepts and we're trying to develop this relational understanding of concepts so that they sort of build and grow on one another. And when we ask ourselves that question—“How are these two things the same but different?”—it helps us put things into categories and understand that sometimes there's overlap, so there's gray space. So it helps us move from black and white thinking into this understanding of grayscale thinking. 

And if I just zoom out a little bit, if I could, Mike—when we zoom out into that grayscale area, we're developing flexibility of thought, which is so important in all aspects of our lives. We need to be nimble on our feet, we need to be ready for what's coming. And it might not be black or white, it might actually be a little bit of both. 

So that's the power of the routine and its roots come in exploring executive functioning and language acquisition. And so we just layer that on top of mathematics and it's pure gold.

Mike: When we were preparing for this podcast, you shared several really lovely examples of how an educator might use same but different to draw out ideas that involve things like place value, geometry, equivalent fractions, and that's just a few. So I'm wondering if you might share a few examples from different grade levels with our listeners, perhaps at some different grade levels.

Sue: Sure. So starting out, we can start with place value. It really sort of pops when we look in that topic area. So when we think about place value, we have a base ten number system, and our numbers are based on this idea that 10 of one makes one group of the next. And so, using same but different, we can help young learners make sense of that system. 

So, for example, we could look at an image that shows a 10-stick. So maybe that's made out of Unifix cubes. There's one 10-stick a—stick of 10—with three extras next to it and next to that are 13 separate cubes. And then we ask, “How are they the same but different?” And so helping children develop that idea that while I have 1 ten in that collection, I also have 10 ones.

Mike: That is so amazing because I will say as a former kindergarten and first grade teacher, that notion of something being a unit of 1 composed of smaller units is such a big deal. And we can talk about that so much, but the way that I can visualize this in my mind with the stick of 10 and the 3, and then the 13 individuals—what jumps out is that it invites the students to notice that as opposed to me as the teacher feeling like I need to offer some kind of perfect description that suddenly the light bulb goes off for kids. Does that make sense?

Sue: It does. And I love that description of it. So what we do is we invite the students to add their own understanding and their own language around a pretty complex idea. And they're invited in because it seems so simple: “How are these the same but different?” “What do you notice?” And so it's a pretty complex idea, and we gloss over it. Sometimes we think our students understand that and they really don't.

Mike: Is there another example that you want to share?

Sue: Yeah, I love the fraction example. So equivalence—when I learned about this routine, the first thing that came to mind for me when I layered it from thinking about language into mathematics was, “Oh my gosh, it's equivalent fractions.” 

So if I were to ask listeners to think about—put a picture in your head of one-half, and imagine in your mind's eye what that looks like. And then if I said to you, “OK, well now I want you to imagine two-fourths. What does that look like?” And chances are those pictures are not the same. 

Mike, when you imagine, did you picture the same thing or did you picture different things?

Mike: They were actually fairly different.

Sue: Yeah. So when we think about one-half as two fourths, and we tell kids those are the same—yes and no, right? They have the same value that, if we were looking at a collection of M&M’S or Skittles or something, maybe half of them are green, and if we make four groups, [then] two-fourths are green.

But contextually it could really vary. And so helping children make sense of equivalence is a perfect example of how we can ask the question, same but different. So we just show two pictures. One picture is one-half and one picture is two-fourths, and we use the same colors, the same shapes, sort of the same topic, but we group them a little differently and we have that conversation with kids to help make sense of equivalence.

Mike: So I want to shift because we've spent a fair amount of time right now describing two instances where you could take a concept like equivalent fractions or place value and you could design a set of images within the same but different routine and do some work around that. 

But you also talked with me, as we were preparing, about different scenarios where same but different could be a helpful tool. So what I remember is you mentioned three discrete instances: this notion of concepts that connect; things learned in pairs; and common misconceptions—or, as I've heard you describe them, naive conceptions. Can you talk about each of those briefly?

Sue: Sure. As I talk about this routine to people, I really want educators to be able to find the opportunities—on their own, authentically—as opportunities arise. So we should think about each of these as an opportunity. 

So I'll start with concepts that connect. When you're teaching something new, it's good practice to connect it to, “What do I already know?” So maybe I'm in a third grade classroom, and I want to start thinking about multiplication. And so I might want to connect repeated addition to multiplication. So we could look at 2 plus 2 plus 2 next to 2 times 3. And it can be an expression, these don't always have to be images. And a fun thing to look at might be to find out, “Where do I see 3 and 2 plus 2 plus 2?” So what's happening here with factors? What is happening with the operations? And then of course they both yield the same answer of 6. So concepts that connect are particularly powerful for helping children build from where they know, which is the most powerful place for us to be.

Mike: Love that.

Sue: Great. The next one is things that are learned in pairs. So there's all sorts of things that come in pairs and can be confusing. And we teach kids all sorts of weird tricks and poems to tell themselves and whatever to keep stuff straight. And another approach could be to—let's get right in there, to where it's confusing. 

So for example, if we think about area and perimeter, those are two ideas that are frequently confusing for children. And we often focus on, “Well, this is how they're different.” But what if we put up an image, let's say it's a rectangle, but [it] wouldn't have to be. And we've got some dimensions on there. We're going to think about the area of one and then the perimeter on the other. What is the same though, right? Because where the confusion is happening. So just telling students, “Well, perimeter’s around the outside, so think of ‘P’ for ‘pen’ or something like that, and area’s on the inside.” What if we looked at, “Well, what's the same about these two things?” We're using those same dimensions, we've got the same shape, we're measuring in both of those. And let students tell you what is the same and then focus on that critical thing that's different, which ultimately leads to understanding formula for finding both of those things. But we've got to start at that concept level and link it to scenarios that make sense for kids.

Mike: Before we move on to talking about misconceptions, or naive conceptions, I want to mark that point: this idea that confusion for children might actually arise from the fact that there are some things that are the same as opposed to a misunderstanding of what's different. 

I really think that's an important question that an educator could consider when they're thinking about making this bridging step between one concept or another or the fact that kids have learned how whole numbers behave and also how fractions might behave. That there actually might be some things that are similar about that that caused the confusion, particularly on the front end of exploration, as opposed to, “They just don't understand the difference.”

Sue: And what happens there is then we aid in memory because we've developed these aha moments and painted a more detailed picture of our understanding in our mind's eye. And so it's going to really help children to remember those things as opposed to these mnemonic tricks that we give kids that may work, but it doesn't mean they understand it.

Mike: Absolutely. Well, let's talk about naive conceptions and the ways that same and [different] can work with those.

Sue: So, I want to kick it up to maybe middle school, and I was thinking about what example might be good here, and I want to talk about exponents. So if we have 2 raised to the third power, the most common naive conception would be, like, “Oh, I just multiply that. It's just 2 times 3.” 

So let's talk about that. So if I am working on exponents, I notice a lot of my students are doing that, let's put it right up on the board: “Two rays to the third power [and] 2 times 3. How are these the same but different?” And the conversation’s a bit like that last example, “Well, let's pay attention to what's the same here.” But noticing something that a lot of children have not quite developed clearly and then putting it up there against where we want them to go and then helping them—I like that you use the word “bridge”—helping them bridge their way over there through this conversation is especially powerful.

Mike: I think the other thing that jumps out for me as you were describing that example with exponents is that, in some ways, what's happening there when you have an example like “2 times 3” next to “2 to the third power” is you're actually inviting kids to tell you, “This is what I know about multiplication.” So you're not just disregarding it or saying, “We're through with that.” It's in the exploration that those ideas come out, and you can say to kids, “You are right. That is how multiplication functions. And I can see why that would lead you to think this way.” And it's a flow that's different. It doesn't disregard kids' thinking. It actually acknowledges it. And that feels subtle, but really important.

Sue: I really love shining a light on that. So it allows us to operate from a strength perspective. So here's what I know, and let's build from there. So it absolutely draws out in the discussion what it is that children know about the concepts that we put in front of them.

Mike: So I want to shift now and talk about enacting same but different. I know that you've developed a protocol for facilitating the same but different routine, and I'm wondering if you could talk us through the protocol, Sue. How should a teacher think about their role during same but different? And are there particular teacher moves that you think are particularly important?

Sue: Sure. So the protocol I've worked out goes through five steps, and it's really nice to just kind of think about them succinctly. And all of them have embedded within them particular teacher moves. They are all based on research of how children learn mathematics and engage in meaningful conversation with one another. 

So step 1 is to look. So if I'm using this routine with 3- and 4-year-olds, and I'm putting a picture in front of them, learning that to be a good observer, we've got to have eyes on what it is we're looking at. So I have examples of counting, asking a 4-year-old, “How many things do I have in front of me?” And they're counting away without even looking at the stuff. So teaching the skill of observation. Step 1 is look.

Step 2 is silent think time. And this is so critically important. So giving everybody the time to get their thoughts together. If we allow hands to go in the air right away, it makes others that haven't had that processing time to figure it out shut down quite often. And we all think at different speeds with different tasks all the time, all day long. So, we just honor that everyone's going to have generally about 60 seconds in which to silently think, and we give students a sentence frame at that time to help them. Because, again, this is a language-based learning routine. So we would maybe put on the board or practice saying out loud, “I'd like you to think about: ‘They are the same because blank; they are different because blank.’” And that silent think time is just so important for allowing access and equitable opportunities in the classrooms.

Mike: The way that you described the importance of giving kids that space, it seems like it's a little bit of a two-for-one because we're also kind of pushing back on this notion that to be good at math, you have to have your hand in the air first, and if you don't have your hand in the air first or close to first, your idea may be less valuable. So I just wanted to shine a light on the different ways that that seems important for children, both in the task that they're engaging with and also in the culture that you're trying to build around mathematics.

Sue: I think it's really important. And if we even zoom out further just in life, we should think before we speak. We should take a moment. We should get our thoughts together. We should formulate what it is that we want to say. And learning how to be thoughtful and giving the luxury of what we're just going to all think for 60 seconds. And guess what? If you had an idea quickly, maybe you have another one. How else are they the same but different? So we just keep that culture that we're fostering, like you mentioned, we just sort of grow that within this routine.

Mike: I think it's very safe to say that the world might be a better place if we all took 60 seconds to think about [laughs] what we wanted to say sometimes.

Sue: Yes, yes. So as teachers, we can start teaching that and we can teach kids to advocate for that. “I just need a moment to get my thoughts together.” 

All right, so the third step is the turn and talk. And it's so important and it's such an easy move. It might be my favorite part. So during that time, we get to have both an experience with expressive language and receptive language—every single person. So as opposed to hands in the air and I'm playing ball with you, Mike, and you raise your hand and you get to speak and we're having a good time. When I do a turn and talk, everybody has an opportunity to speak. And so taking the thoughts that are in their head and expressing them is a big deal. And if we think about our multilingual learners, our young learners, even our older learners, and it's just a brand new concept that I've never talked about before.

And then on the other side, the receptive learning. So you are hearing from someone else and you're getting that opportunity of perspective taking. Maybe they notice something you hadn't noticed, which is likely to happen to somebody within that discussion. “Wow, I never thought about it that way.” So the turn and talk is really critical. And the teacher's role during this is so much fun because we are walking around and we're listening. And I started walking around with a notebook. So I tell students, “While you are talking, I'm going to collect your thinking.” And so I'm already imagining where this is going next. And so I'm on the ground if we're sitting on the rug, I'm leaning over, I'm collecting thoughts, I'm noticing patterns, I'm noticing where I want to go next as the facilitator of the conversation that's going to happen whole group. So that's the third component, turn and talk.

The fourth component is the share. So if I've walked around and gathered student thinking, I could say, “Who would like to share their thinking?” and just throw it out there. But I could instead say—let's say we're doing the same but different with squares and rectangles. And I could say, “Hmm, I noticed a lot of you talking about the length of the sides. Is there anyone that was talking about the lengths of the sides that would like to share what either you or your partner said?” So I know that I want to steer it in that direction. I know a lot of people talked about that, so let's get that in the air. But the share is really important because these little conversations have been happening. Now we want to make it public for everybody, and we're calling on maybe three or four students. We're not trying to get around to everybody. We're probably hopefully not going to [be] drawing Popsicle sticks and going random. At this point, students have had the opportunity to talk, to listen, to prepare. They've had a sentence stem. So let's see who would like to share and get those important ideas out.

Mike: I think what strikes me is there's some subtlety to what's happening there because you are naming some themes that you heard. And as you do that, and you name that, kids can say, “That's me,” or, “I thought about that,” or, “My partner thought about that. You're also clearly acting with intention. As an educator, there are probably some ideas that you either heard that you want to amplify or that you want kids to attend to, and yet you're not doing it in a way that takes away from the conversations that they had. You're still connecting to what they said along the way. And you're not suddenly saying, “Great, you had your turn and talk, but now let's listen to David over here because we want to hear what he has to share.”

Sue: Yes. And I don't have to be afraid of calling out a naive conception. Maybe a lot of people were saying, “Well, I think the rectangles have two long [sides and] two short.” And they're not seeing that the square is also a rectangle. And so maybe I'm going to use that language in the conversation too, so that yeah, the intentionality is exactly it. Building off of that turn and talk to the share.

The last step is the summary. So after we've shared, we have to put a bow on that, right? So we've had this experience. They generally are under 15 minutes, could be 5 minutes, could be 10 minutes. But we've done something important all together. And so the teacher's role here is to summarize, to bring that all together and to sort of say, “OK, so we looked at this picture here, and we noticed”—I'll stick with the square/rectangle example—“that both shapes have four sides and four square corners. They're both rectangles, but this one over here is a special one. It's a square and all four sides are equal and that's what makes it special.” Or something like that. But we want to succinctly nail that down in a summary. 

If you do a same but different and nobody gets there, and so you chose this with intention, you said, “This is what we need to talk about today,” and all of a sudden you're like, “Oh, boy,” then your summary might not sound like that. It might sound like, “Some of you noticed this and some of you noticed that, and we're going to come back to this after we do an activity where we're going to be sorting some shapes.” So it's an opportunity for formative assessment. So summary isn't, “Say what I really wanted to say all along,” even though I do have something I want to say; it's a connection to what happened in that conversation. And so almost always it comes around to that. But there are those instances where you learn that we need to do some more work here before I can just nicely put that bow on it.

Mike: You're making me think about what one of my longtime mentors used to say, and the analogy he would use is, “You can definitely lead the horse to water, but it is not your job to shove the horse's face in the water.” And I think what you're really getting at is, I can have a set of mathematical goals that I'm thinking about as I'm going into a same and different. I can act with intention, but there is still kind of this element of, “I don't quite know what's going to emerge.” And if that happens, don't shove the metaphorical horse's head in the water, meaning don't force that there. If the kids haven't made the connection yet or they haven't explored the gray space that's important. Acknowledge that that's still in process.

Sue: Exactly. There is one last optional step which relates to summary. So if you have time and you're up for an exploration, you can now ask your students to make one of their own. And that's a whole other level of sophistication of thought for students to recognize, “Oh, this is how those two were same but different. I'm going to make another set that are the same but different in the same way.” It's actually a very complex task. We could scaffold it by giving students, “If this was my first image, what would the other one be?” That would be like what we just did. Very worthwhile. Obviously now we're not within the 10-minute timeframe. It's a lot bigger.

Mike: What I found myself thinking about, the more that we talk through intent, purpose, examples, the protocol steps, is the importance of language. And it seemed like part of what's happening is that the descriptive language that's accessed over the course of the routine that comes from students, it really paves the way for deeper conceptual understanding. Is that an accurate understanding of the way that same and different can function?

Sue: A hundred percent. So it's really the way that we think as we're looking at something. We might be thinking in mental pictures of things, but we might also be thinking in the words. And if we're going to function in a classroom and in society, we have to have the language for what it is that we're doing. And so yes, we're playing in that space of language acquisition, expressive language, receptive language, all of it, to help us develop this map of what that is really deeply all about so that when I see that concept in another context, I have this rich database in my head that involves language that I can draw on to now do the next thing with it.

Mike: That's really powerful. Listeners have heard me say this before, but we've just had a really insightful conversation about the structure, the design, the implementation, and the impact of same and different. And yet we're coming to the end of the podcast. So I want to offer an opportunity for you to share any resources, any websites, any tools that you think a listener who wanted to continue learning about same but different, where might they go? What might you recommend, Sue?

Sue: Sure. So there's two main places to find things, and they actually do exist in both. But the easiest way to think about this, there is the website, which is samebutdifferentmath.com, and it's important to get the word “math” in there. And that is full of images from early learning, really even up through high school. So that's the first place, and they are there with a creative common licensing. 

And then you mentioned tools. So there are some tools, and if we wanted to do deeper learning, and I think the easiest way to access those is my other website, which is just looneymath.com. And if you go up at the top under Books, there's a children's book that you can have kids reading and enjoying it with a friend. There's a teacher book that talks about in more detail some of the things we talked about today. And then there are some cards where students can sit in a learning center and turn over a card that presents them with an opportunity to sit shoulder to shoulder. And so those are all easily accessed really on either one of those websites, but probably easiest to find under the looneymath.com.

Mike: Well, for listeners, we'll put a link to those resources in the show notes to this episode. 

Sue, I think this is probably a good place to stop, but I just want to say thank you again. It really has been a pleasure talking with you today.

Sue: You're welcome, Mike. It's one of my favorite things to talk about, so I really appreciate the opportunity.

Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling all individuals to discover and develop their mathematical confidence and ability.

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