Latest / Rounding Up / Same but Different: Encouraging Students to Think Flexibly with Dr. Sue Looney

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Same but Different: Encouraging Students to Think Flexibly with Dr. Sue Looney · Rounding Up

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Students sometimes struggle in math because they fail to make connections for
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too many students.
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Every concept feels like its own entity without any connection to the larger
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network of mathematical ideas.
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Today we're talking with Sue Looney about a powerful routine called Same but
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Different and the ways teachers can use it to help students identify
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connections and foster flexible reasoning.
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Well, hi Sue. Welcome to the podcast.
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I'm so excited to be talking with you today.
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Hi Mike. Thank you so much. I am thrilled too.
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I've been really looking forward to this.
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Well, for listeners who don't have prior knowledge,
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I'm wondering if we could start by having you offer a description of the
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same but different routine.
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Absolutely.
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So the same but different routine is a classroom routine
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that takes two images or
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numbers or words and puts them next to each other and
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asks students to describe how they are the same but different.
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It's based in a language learning routine but applied to the math
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classroom.
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I think that's a great segue because what I wanted to ask is at the broadest
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level,
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regardless of the numbers or the content or the image or
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images that educators select,
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how would you explain what same but different is good for?
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Maybe put another way,
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how should a teacher think about its purpose or its value?
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Great question.
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I think a good analogy is to imagine you're in your ELA,
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your English language arts classroom and you were asked to compare and contrast
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two characters in a novel.
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So the foundations of the routine really sit there and what it's good for
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is to help our brains think categorically and relationally.
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So in mathematics in particular,
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there's a lot of overlap between concepts and we're trying to develop this
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relational understanding of concepts so that they sort of build and grow on one
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another. And when we ask ourselves that question,
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how are these two things the same but different?
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It helps us put things into categories and understand that
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sometimes there's overlap, so there's gray space.
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So it helps us move from black and white thinking into this
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understanding of gray scale thinking. And if I just zoom out a little bit,
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if I could, Mike, when we zoom out into that gray scale area,
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we're developing flexibility of thought,
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which is so important in all aspects of our lives.
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We need to be nimble on our feet, we need to be ready for what's coming.
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And it might not be black or white, it might actually be a little bit of both.
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So that's the power of the routine and its roots come
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in exploring executive functioning and language acquisition.
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And so we just layer that on top of mathematics and it's pure gold.
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When we were preparing for this podcast,
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you shared several really lovely examples of how an
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educator might use same but different to draw out ideas that
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involve things like place value, geometry,
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equivalent fractions, and that's just a few.
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So I'm wondering if you might share a few examples from different grade levels
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with our listeners perhaps at some different grade levels.
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Sure. So starting out, we can start with place value.
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It really sort of pops when we look in that topic area.
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So when we think about place value,
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we have a base 10 number system and our numbers are based on this idea that 10
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of one makes one group of the next.
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And so using same but different,
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we can help young learners make sense of that system. So for example,
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we could look at an image that shows a 10 stick.
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So maybe that's made out of Unix cubes.
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There's one 10 stick a stick of 10 with three extras next to it
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and next to that are 13 separate cubes.
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And then we ask how are they the same but different?
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And so helping children develop that idea that while I have one
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10 in that collection, I also have 10 ones.
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That is so amazing because I will say as a former kindergarten and first grade
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teacher,
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that notion of something being a unit of one composed
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of smaller units is such a big deal and we can talk about
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that so much.
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But the way that I can visualize this in my mind with the
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stick of 10 and the three and then the 13 individuals,
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what jumps out is that it invites the students to notice that as
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opposed to me as the teacher feeling like I need to offer some kind of perfect
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description that suddenly the light bulb goes off for kids.
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Does that make sense?
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It does. And I love that description of it.
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So what we do is we invite the students to add their own
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understanding and their own language around a pretty complex idea.
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And they're invited in because it seems so simple.
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How are these the same but different? What do you notice?
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And so it's a pretty complex idea and we gloss over it.
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Sometimes we think our students understand that and they really don't.
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Is there another example that you want to share?
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Yeah, I'd love the fraction example. So equivalence,
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when I learned about this routine,
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the first thing that came to mind for me when I layered it from thinking about
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language into mathematics was, oh my gosh, it's equivalent fractions.
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So if I were to ask listeners to think about,
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put a picture in your head of one half and imagine in your mind's eye what that
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looks like. And then if I said to you, okay,
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well now I want you to imagine two fourths, what does that look like?
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And chances are those pictures are not the same. Mike, when you imagine,
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did you picture the same thing or did you picture different things?
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They were actually fairly different.
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Yeah. So when we think about one half is two fourth,
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and we tell kids those are the same, yes and no, right?
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They have the same value that if we were looking at a collection of m and ms or
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Skittles or something, maybe half of them are green, and if we make four groups,
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two fourths are green. But contextually it could really vary.
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And so helping children make sense of equivalence is a perfect
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example of how we can ask the question. Same but different.
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So we just show two pictures.
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One picture is one half and one picture is two fourths,
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and we use the same colors, the same shapes, sort of the same topic,
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but we group them a little differently and we have that conversation with kids
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to help make sense of equivalence.
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So I want to shift because we've spent a fair amount of time right now
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describing two instances where you could take a concept like equivalent
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fractions or place value and you could design a set of
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images within the same but different routine and do some work around that.
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But you also talked with me as we were preparing about different
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scenarios where same but different could be a helpful tool.
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So what I remember is you mentioned three discreet instances,
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this notion of concepts that connect,
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things learned in pairs and common misconceptions or as I've heard you describe
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them, naive conceptions. Can you talk about each of those briefly?
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Sure. As I talked about this routine to people,
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I really want educators to be able to find the opportunities on their
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own authentically as opportunities arise.
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So we should think about each of these as an opportunity.
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So I'll start with concepts that connect when you're teaching something new,
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it's good practice to connect it to what do I already know?
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So maybe I'm in a third grade classroom and I want to start thinking about
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multiplication.
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And so I might want to connect repeated addition to
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multiplication.
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So we could look at two plus two plus two next to two times
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three. And it can be an expression, these don't always have to be images.
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And a fun thing to look at might be to find out where do I see
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three and two plus two plus two?
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So what's happening here with factors, what is happening with the operations?
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And then of course they both yield the same answer of six.
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So concepts that connect are particularly powerful for helping children build
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from where they know which is the most powerful place for us to be.
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Love that.
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Great. The next one is things that are learned in pairs.
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So there's all sorts of things that come in pairs and can be confusing.
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And we teach kids all sorts of weird tricks and poems to tell
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themselves and whatever to keep stuff straight.
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And another approach could be to let's get right in there to where it's
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confusing. So for example,
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if we think about area and perimeter,
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those are two ideas that are frequently confusing for
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children. And we often focus on, well, this is how they're different,
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but what if we put up an image, let's say it's a rectangle,
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but wouldn't have to be. And we've got some dimensions on there.
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We're going to think about the area of one and then the perimeter on the other.
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What is the same though, right? Because where the confusion is happening.
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So just telling students, well perimeters around the outside,
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so think of P for pen or something like that and areas on the
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inside. What if we looked at, well,
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what's the same about these two things? We're using those same dimensions,
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we've got the same shape,
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we're measuring in both of those and let students tell you what is the same
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and then focus on that critical thing that's different,
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which ultimately leads to understanding formula for finding both of those
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things.
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But we've got to start at that concept level and link it to scenarios that make
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sense for kids.
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Before we move on to talking about misconceptions or naive conceptions,
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I want to mark that point,
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this idea that confusion for children might actually arise from the fact
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that there are some things that are the same as opposed to a misunderstanding of
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what's different.
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I really think that's an important question that an educator could consider when
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they're thinking about making this bridging step between one concept or another
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or the fact that kids have learned how whole numbers behave and also
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how fractions might behave,
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that there actually might be some things that are similar about that that caused
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the confusion,
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particularly on the front end of exploration as opposed to they just don't
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understand the difference.
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And what happens there is then we aid in memory because we've developed
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these aha moments and painted a more detailed picture of our
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understanding in our mind's eye.
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And so it's going to really help children to remember those things as opposed to
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these mnemonic tricks that we give kids that may work,
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but it doesn't mean they understand it.
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Absolutely. Well,
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let's talk about naive conceptions and the ways that same and difference can
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work with those.
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So I want to kick it up to maybe middle school and I was thinking about what
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example might be good here and I want to talk about exponents.
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So if we have two raised to the third,
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the most common naive conception would be like, oh, I just multiply that.
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It's just two times three. So let's talk about that.
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So if I am working on exponents,
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I notice a lot of my students are doing that,
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let's put it right up upon the board, two rays to the third power,
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two times three, how are these the same but different?
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And the conversations a bit like that last example,
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well let's pay attention to what's the same here,
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but noticing something that a lot of children have not quite
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developed clearly and then putting it up there against where we want them to
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go and then helping them. I like that you use the word bridge.
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Helping them bridge their way over there through this conversation is especially
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powerful.
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I think the other thing that jumps out for me as you were describing that
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example with exponents is that in some ways what's happening there when you have
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an example,
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two times three next to two to the third power is you're
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actually inviting kids to tell you, this is what I know about multiplication.
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So you're not just disregarding it or saying we're through with that.
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It's in the exploration that those ideas come out and you can say to
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kids, you are right, that is how multiplication functions,
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and I can see why that would lead you to think this way.
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And it's a flow that's different.
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It doesn't disregard kids' thinking it actually acknowledges it.
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And that feels subtle, but really important.
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I really love shining a light on that.
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So it allows us to operate from a strength perspective.
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So here's what I know and let's build from there.
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So it absolutely draws out in the discussion what it is that
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children know about the concepts that we put in front of them.
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So I want to shift now and talk about enacting same but different.
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I know that you've developed a protocol for facilitating the same but different
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routine, and I'm wondering if you could talk us through the protocol, Sue,
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how should a teacher think about their role during same but different and are
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there particular teacher moves that you think are particular important?
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Sure.
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So the protocol I've worked out goes through five steps
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and it's really nice to just kind of think about them succinctly.
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And all of them have embedded within them particular teacher moves.
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They are all based on research of how children learn
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mathematics and engage in meaningful conversation with one another.
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So step one is to look.
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So if I'm using this routine with three and four year olds and I'm putting a
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picture in front of them, learning that to be a good observer,
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we've got to have eyes on what it is we're looking at.
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So I have examples of counting,
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asking a 4-year-old how many things do I have in front of me?
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And they're counting away without even looking at the stuff.
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So teaching the skill of observation, step one is look,
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step two is silent think time.
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And this is so critically important. So giving everybody the
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time to get their thoughts together.
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If we allow hands to go in the air right away,
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it makes others that haven't had that processing time to figure it
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out shut down quite often.
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And we all think at different speeds with different tasks all the time,
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all day long.
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So we just honor that everyone's going to have generally about 60 seconds
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in which to silently think,
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and we give students a sentence frame at that time to help them. Again,
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this is a language-based learning routine.
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So we would maybe put on the board or practice saying out loud,
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I'd like you to think about they are the same because blank,
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they are different because blank.
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And that silent think time is just so important for allowing access
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and equitable opportunities in the classrooms.
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The way that you described the importance of giving kids that space,
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it seems like it's a little bit of a two for one,
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because we're also kind of pushing back on this notion that to be good at math,
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you have to have your hand in the air first,
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and if you don't have your hand in the air first or close to first,
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your idea may be less valuable.
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So I just wanted to shine a light on the different ways that seems important for
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children,
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both in the task that they're engaging with and also in the culture that you're
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trying to build around mathematics.
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I think it's really important. And if we even zoom out further just in life,
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we should think before we speak, we should take a moment.
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We should get our thoughts together.
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We should formulate what it is that we want to say and learning how to be
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thoughtful and giving the luxury of what we're just going to
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all think for 60 seconds. And guess what? If you had an idea quickly,
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maybe you have another one. How else are they the same but different?
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So we just keep that culture that we're fostering, like you mentioned,
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we just sort of grow that within this routine.
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I think it's very safe to say that the world might be a better place if we all
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took 60 seconds to think about what we wanted to say. Sometimes.
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Yes. So as teachers,
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we can start teaching that and we can teach kids to advocate for that.
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I just need a moment to get my thoughts together. All right,
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so the third step is the turn and talk.
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And it's so important and it's such an easy move.
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It might be my favorite part. So during that time,
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we get to have both an experience with expressive language and receptive
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language, every single person.
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So as opposed to hands in the air and I'm playing ball with you, Mike,
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and you raise your hand and you get to speak and we're having a good time.
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When I do a turn and talk, everybody has an opportunity to speak.
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And so taking the thoughts that are in their head and expressing them is a big
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deal. And if we think about our multilingual learners,
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our young learners, even our older learners,
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and it's just a brand new concept that I've never talked about before. And then
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on the other side, the receptive learning.
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So you are hearing from someone else and you're getting that opportunity of
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perspective taking. Maybe they notice something you hadn't noticed,
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which is likely to happen to somebody within that discussion. Wow,
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I never thought about it that way. So the turn and talk is really critical.
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And the teacher's role during this is so much fun because we are
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walking around and we're listening and I started walking around with a notebook.
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So I tell students while you are talking, I'm going to collect your thinking.
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And so I'm already imagining where this is going next.
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And so I'm on the ground if we're sitting on the rug, I'm leaning over,
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I'm collecting thoughts, I'm noticing patterns,
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I'm noticing where I want to go next as the facilitator of the
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conversation that's going to happen, whole group. So that's the third component,
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turn and talk. The fourth component is the share.
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So if I've walked around and gathered student thinking,
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I could say who would like to share their thinking and just throw it out there.
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But I could instead say,
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let's say we're doing the same but different with squares and rectangles.
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And I could say,
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I noticed a lot of you talking about the length of the sides.
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Is there anyone that was talking about the lengths of the sides that would like
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to share what either you or your partner said?
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So I know that I want to steer it in that direction.
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I know a lot of people talked about that, so let's get that in the air.
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But the share is really important because these little conversations have been
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happening now. We want to make it public for everybody,
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and we're calling on maybe three or four students.
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We're not trying to get around to everybody.
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We're probably hopefully not going to drawing Popsicle sticks and going random
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at this point. Students have had the opportunity to talk, to listen, to prepare.
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They've had a sentence stem.
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So let's see who would like to share and get those important ideas out.
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I think what strikes me is there's some subtlety to what's happening there
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because you are naming some themes that you heard.
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And as you do that, and you name that, kids can say, that's me,
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or I thought about that or my partner thought about that.
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You're also clearly acting with intention as an educator.
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There are probably some ideas that you either heard that you want to
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amplify or that you want kids to attend to,
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and yet you're not doing it in a way that takes away from the conversations that
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they had.
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You're still connecting to what they said along the way,
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and you're not suddenly saying, great, you had your turn and talk,
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but now let's listen to David over here because we want to hear what he has to
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share.
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Yes. And I don't have to be afraid of calling out a naive conception.
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Maybe a lot of people were saying, well, I think the rectangles have too long,
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too short, and they're not seeing that the square is also a rectangle.
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And so maybe I'm going to use that language in the conversation too,
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so that yeah, the intentionality is exactly it.
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Building off of that turn and talk to the share,
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the last step is the summary. So after we've shared,
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we have to have put a bow on that, right? So we've had this experience.
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They generally are under 15 minutes, could be five minutes,
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could be 10 minutes, but we've done something important altogether.
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And so the teacher's role here is to summarize,
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to bring that all together and to sort of say, okay,
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so we looked at this picture here and we noticed I'll stick with the square
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rectangle example that both shapes have four
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sides and four square corners. They're both
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rectangles, but this one over here is a special one.
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It's a square and all four sides are equal and that's what makes it
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special or something like that.
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But we want to succinctly nail that down in a
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summary.
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If you do a same but different and nobody gets there,
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and so you chose this with intention, you said,
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this is what we need to talk about today. And all of a sudden you're like,
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oh boy. Then your summary might not sound like that.
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It might sound like some of you notice this and some of you notice that,
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and we're going to come back to this after we do an activity where we're going
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to be sorting some shapes.
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So it's an opportunity for formative assessment.
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So summary isn't say what I really wanted to say all along,
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even though I do have something I want to say,
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it's a connection to what happened in that conversation.
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And so almost always it comes around to that.
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But there are those instances where you learn that we need to
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do some more work here. Before I can just nicely put that bow on it.
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You're making me think about what one of my longtime mentors used to say,
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and the analogy he would use is you can definitely lead the horse to water,
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but it is not your job to shove the horse's face in the water.
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And I think what you're really getting at is I can have a set of mathematical
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goals that I'm thinking about as I'm going into a same and different.
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I can act with intention,
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but there is still kind of this element of I don't quite know what's going to
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emerge. And if that happens,
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don't shove the metaphorical horse's head in the water.
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Meaning don't force that there.
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If the kids haven't made the connection yet or they haven't explored the gray
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space, that's important. Acknowledge that that's still in process.
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Exactly. There is one last optional step which relates to summary.
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So if you have time and you're up for an exploration,
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you can now ask your students to make one of their own.
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And that's a whole other level of sophistication of
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thought for students to recognize, oh,
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this is how those two were same but different.
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I'm going to make another set that are the same but different in the same way.
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It's actually a very complex task.
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We could scaffold it by giving students, if this was my first image,
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what would the other one be?
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That would be like what we just did very worthwhile.
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Obviously now we're not within the 10 minute timeframe. It's a lot bigger.
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What I found myself thinking about, the more that we talk through intent,
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purpose, examples, the protocol steps is the importance of language.
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And it seemed like part of what's happening is that the descriptive language
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that's accessed over the course of the routine that comes from students,
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it really paves the way for deeper conceptual understanding.
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Is that an accurate understanding of the way that same and different can
25:00
function?
25:02
A hundred percent.
25:03
So it's really the way that we think as we're looking at something,
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we might be thinking in mental pictures of things,
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but we might also be thinking in the words and if we're going to function
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in a classroom and in society,
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we have to have the language for what it is that we're doing. And so yes,
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we're playing in that space of language acquisition, expressive language,
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receptive language,
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all of it to help us develop this map of what that is really
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deeply all about so that when I see that concept
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in another context,
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I have this rich database in my head that involves language
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that I can draw on to now do the next thing with it.
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That's really powerful. Listeners have heard me say this before,
25:52
but we've just had a really insightful conversation about the structure,
25:57
the design, the implementation, and the impact of same and different.
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And yet we're coming to the end of the podcast.
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So I want to offer an opportunity for you to share any resources,
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any websites,
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any tools that you think a listener who wanted to continue learning about same
26:14
but different, where might they go? What might you recommend Sue?
26:18
Sure. So there's two main places to find things,
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and they actually do exist in both. But the easiest way to think about this,
26:25
there is the website,
26:28
which is same but different math.com,
26:31
and it's important to get the word math in there.
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And that is full of images from early learning,
26:37
really even up through high school. So that's the first place,
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and they are there with a creative common licensing.
26:45
And then you mentioned tools.
26:46
So there are some tools and if we wanted to do deeper learning,
26:49
and I think the easiest way to access those is my other website,
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which is just loony math.com.
26:56
And if you go up at the top under books,
26:58
there's a children's book that you can have kids reading and enjoying it
27:03
with a friend.
27:04
There's a teacher book that talks about in more detail some of the things we
27:09
talked about today.
27:10
And then there are some cards where students can sit in a learning
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center and turn over a card that presents them with an opportunity to sit
27:19
shoulder to shoulder.
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And so those are all easily accessed really on either one of those websites,
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but probably easiest to find under the loony math.com.
27:29
One for listeners,
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we'll put a link to those resources in the show notes to this episode. Sue,
27:34
I think this is probably a good place to stop,
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but I just want to say thank you again.
27:38
It really has been a pleasure talking with you today.
27:40
You're welcome, Mike. It's one of my favorite things to talk about,
27:43
so I really appreciate the opportunity.
27:49
This podcast is brought to you by the Math Learning Center and the Meyer Math
27:52
Foundation dedicated to inspiring and enabling all individuals to
27:57
discover and develop their mathematical confidence and ability.