(upbeat music) - Well, welcome everybody to my session, "Helping Students Build Relational Thinking, Not Just Computational Thinking". Now, I know this is, I set it for PreK - 2, mainly because I'm showing kind of examples from the early grades, but this idea is important throughout all of the grades. Okay, so first and foremost, if you're not out there following me out on Instagram, Twitter, come find me. I would love to hear what you are learning about from this session, and this session in sp
ecific, but the whole virtual summit in general as well. All right, so let's start off. Think about this. If you'd like to, you can throw this into the chat. How would you figure out what number goes in that empty space? We have 23 + 16 = 27 + __. How would you figure that out? I'll give you a few more seconds to think through that. Here's the two differences in these two different ways about thinking through mathematical problems. Computational thinking is probably what most of us grew up doing
. If you were like me, you were taught just the steps and procedures. So to solve something like this, I would have to do 23 + 16, I would see that it's 39, and then I would say, okay, now I need to subtract 27 from that to figure out what goes in that empty space. That is a perfectly fine way to solve the problem. So from the rest of the time going forward, we're really going to talk about how not to do this, but I don't want you to get the wrong idea that that's not okay. That is totally an ok
ay fine way to solve the problem, but what I want you to take away from this is it's not enough. This type of thinking, where it's only computational and procedural type thinking starts to fall apart as our students get into 5th grade, and especially 8th grade when they start having to think about algebraic reasoning, and algebra is so much easier if you have built relational thinking so that when you look at things, you can see relationships and not just always have to compute. So it's not that
this is bad, it's just we want our students to be able to do more than just this, okay? And the difference is relational thinking is when a student looks at a problem like this, and instead, they are looking at how the numbers relate. They are noticing what's the same, what's different, what has changed, what's happening in this problem? So you might hear a kid say something like, "Well, the 27 on the one side is just 4 more than the 23," and they can see this more as a balance, and so they hav
e to equal it out, so that missing piece has to be 4 less than the 16 to make it be equal on both sides. That is relational thinking. Now, in this article, and I've linked to this, so when you have the slides that are linked up below, the slides, anything that's underlined like that will be hyperlinked, so it will link you to the article that I got this from. But they say relational thinking is the recognition and use of relations between elements in arithmetic or algebraic expression and the pr
operties of number and operations. That seems a whole lot like mathematical speak, so my version of that is that relational thinking is using the structure of math and connections between numbers to solve problems. They aren't just reliant upon procedures. Also in that article, they go on to talk about how when we regularly and intentionally look for opportunities to surface and discuss relational thinking we provide learning experiences that move beyond calculating to produce an answer. And aga
in, calculation is not bad. I want to remind you of that. But we're moving beyond it to look at relationships and connections and help kids see how numbers relate and use those relationships to make mathematics just a little bit easier on them, okay? So think about this problem right here. And the students you work with, what do you think they would do with a problem like this? 6 + 7 = __ + 8. One of the biggest eye-opening things to me that I think I ever read, and I have never forgot it. I don
't know the whole gist of the article, I mean, I can't tell you word for word, but I distinctly remember this part of the article. It was by Thomas Carpenter et al., and that's the whole group that is part of cognitively-guided instruction. They were doing research about students' understanding of equality, and one of the things that they found was they gave a problem or problems like this to kids in 1st grade all the way up to 6th grade. And you would expect that kids in 6th grade would do bett
er than kids in 1st grade. Unfortunately, that was not the case. Kids in 1st grade performed better than 6th graders on problems like this, because they haven't been taught what this means yet. They're trying to make sense of it. What they found with those 6th graders is that overwhelmingly the number one answer that kids put in that empty space was 13. They believe the equals sign means the answer comes next. They've been trained that way throughout school, that when you see the equals sign, th
at means compute, compute, compute. So the number one answer by 6th graders was 13. The next common answer, 21. They just added up all the numbers because they weren't quite sure what to do with it. So I want you to start right there. That study alone really showed me how much emphasis we put on computation and not them looking at relationships, and really what the equals sign means as well is part of it. I want to show you a little bit of what it looks like when kids have this understanding. We
just talked about what it looks like when they don't, because they'll just compute. But this problem right here, could a five year old solve this? And I'll give you a hint. He does. So instead I want you to think about how would a five year old solve this? Now, five year olds, this is not in their standards at all. It is not an expectation that a five year old should be solving this, and I didn't even ask the five year old to solve this. In fact, this comes from my seven year old. They're not t
his age anymore, but when he was seven, one of my sons brought home this worksheet, okay? And so this is a bigger picture of what that worksheet looked like. The first problem was 49 + 20. Then it was, and let me move around, 21 + 49, I had something blocking it, 21 + 49. And then he was working on 49 + 19. And his five year old brother was, you know, hopping around, looking at what he was doing, kind of paying attention, kind of not paying attention. And now I'm gonna play a video for you so yo
u can hear what my five year old said. - What do you think this last one is? (pencil tapping) - 68. - Why do you think it's 68? - Because this one has 20 and this one has 19. - He's correct. - So how does? (child speaking indistinctly) - Because this one only has 49 and this one has 49, but this one is one less. - That one has one less so the answer's gonna be one less, you think? - Yes. - No way. How old are you? And how old is your brother? What grade is he in? - 2nd and. - 2nd, and he's seven
years old. And you're doing his homework for him? - No, I already knew the answer. - No, he wasn't doing my homework. I already knew the answer. He has to say that. So a five year old doesn't know how to add 49 + 19. But he sees relationships. If you didn't notice, he said they both have 49, I'm summarizing it here, both the problem, the original problem, 49 + 20, has a 49. They both have 49s. But the first one had 20, this one has 19. He understood the relationship, that it's one less, so then
he knows the answer's gonna be one less, and he knows what's one less than 69. He knows it's 68. That is relational thinking. We need to help kids see how numbers relate to each other and look for those relationships when they're trying to solve problems, okay? One of the key ideas that is inherent in all of this relational thinking is helping your students understand that the equals sign is a balance. It is not a sign to calculate. Ingrain that, take a screenshot, whatever you need to do right
now, write it down, highlight it. Go for it, whatever you need. The equals sign is a balance. It is not a sign to calculate. Now, we may say that. You may know it in your heart of hearts that that is true, but I want to highlight a couple of things that we do as teachers that help students believe that the equals sign is a sign to calculate, okay? And I'm calling these out because I was guilty of them. I did them all the stinking time, okay? So first off, how do you say this to your students? W
hen you have an equation or expression, whatever you call it, I know some people will be very particular. It's an equation when it's this, it's an expression when it's this. I'm not talking about vocabulary in this one, so forgive me if I say the wrong thing, okay? But how would you say this when you're talking about it with your students? If you'd like to, you can put it in the chat. But if you don't want to because you're afraid that maybe I'm gonna call you out, don't worry, I won't call you
out, but you know, people do see the chat and I know what it's like. You're like, oh man, I'm guilty of that. So I will raise my hand and say this is the way I used to say it. Three plus six equals. That's it, right? How else are you supposed to say that? That's the way I was taught. That's the way I said it to my students. However, it is a common, common mistake that I really want to encourage you to avoid. Not just avoid, I shouldn't say avoid, because you could say that, but don't just say th
at. Like say that when you're also saying it this way, three plus six is the same as, and then whatever they are, once they get to that answer. Three plus six is the same as, and then you pose it to your students. Three plus six equals. And then when they say it, you can say, yes, 3 + 6 =9 because three plus six is the same as nine. Use those terms interchangeably, okay? All right, now let's go to another thing that I was guilty of. When a student is solving a problem like 36 + 58, how would you
notate what they say of how they solved it without using visuals if this is how they solved it? I took 4, added it to the 36 because that would make 40, and I had 54 left to add, so that makes 94. If that's how they solved it in their brain and you were doing a number talk or a number string and you're trying to show it up on the board, but you've moved into this spot where you're wanting kids to see it symbolically, not just visuals, and you're trying to write it out using equations, right, ho
w would you write that? And I'm gonna show you the wrong way to write it that I was very, very, very guilty of all the time until I read these documents by Carpenter et al. Common mistake number two is writing mathematical run-on sentences. So when they tell you this is their strategy, this is the way I used to write that. I would put 36 + 4 = 40 + 54 = 94. That is not equal. When we write these mathematical run-on sentences, it is inherently telling the students to just compute and put the answ
er after the equals sign and then just keep computing and move on. It is negating what the equals sign actually means, which is the same as. So please don't write it that way. So when you get to that symbolic representation, period, when you're working with your students, and you want to write it how mathematicians would write it, which is, you know, sometimes mathematicians use visuals. Sometimes mathematicians use symbols. So when mathematicians want to use symbols, this is how they could writ
e that. There are lots of activities that help kids build this relational thinking, and that's where we're gonna spend the rest of our time in our little mini session here, okay? So some of my favorite experiences are not mathematical at all, but it is focusing on helping kids see things that are the same and are different, because that's what you're wanting them to do to look relationally, when they're looking at what's on one side of the equation and the other side of the equation. You want th
em to pay attention to what's the same and what's different. So doing lots of same but different routines is really, really helpful, and kids don't have to "know the the math" because there's not math involved. There's pictures like this, where you put those up there, and all you're talking about is what's the same and what's different. You're gonna get kids who will say things like they both have two in each picture. They're both dogs. In one of them, they're wearing little handkerchiefs, in th
e other, they're not. They're all these things that have nothing to do with mathematics, but we're helping them to start paying attention to things that are the same and different. If you're not familiar with Same but Different, Same but Different website is linked up. My friend, Sue Looney, runs that site, and there are tons and tons of things that you can use on that website for free. She also has a book, "Same but Different". She has also done past virtual math summits about Same but Differen
t, so those of you who are members of the Build Math Minds site, you can just search inside the site for Sue Looney's name, or Same but Different, and things will pop up. So Same but Different, really low floor type of activity to get all of your students started on relational thinking. The other thing I want to encourage you to do is to vary the way that you write the equations. So we get so ingrained in just writing it this kind of typical way. 3 + 6 is the same as 9. Writing it always like th
at, with the answer coming after the equals sign. So even if I say it as 3 + 6 is the same as 9, we tend to always have the answer on that right side of the equation, so it comes after the equation. Now, some textbook companies and worksheets have gotten a little better about making sure that sometimes the total is on the left side of the equation sign to vary that up a bit. But if yours does not have that, then you need to be intentional about it. The other thing that kids get thrown for a loop
with is when you show them things like this, because they're so used to it having some kind of an operation involved with it, that when there's an equals sign, there needs to be an operation. But no, we can just have things like that. The other thing I want to encourage you to do is when you are giving kids practice problems or homework or whatever you want to call it, and if it's a homework sheet or just practice in your classroom, have them be related in some way. It doesn't always have to be
this way, but when you're intentional about picking your problem sets, kids will start to see relationships. If the numbers are all just random numbers, they don't get to see that, so their only option is to use computation to solve those problems. This is a screenshot from that video of my son's worksheet when he was in second grade, and if you notice, the entire worksheet is that way. So if you look at problem set B, we have 23 + 7, or 23 + 70, sorry, then 23 + 71, and then 69 + 23. They are
meant to be looked at in relation to each other. Now, will every kid see that? No, but at least we have the opportunity for kids to see that. Another simple thing that you're probably already doing, but I want to help you remember that this is really important and remind you that it's beyond just decomposing numbers, right? When you do all these part, part, whole activities where you take seven, and you're talking about all the different ways that you can make seven. It's not just about understa
nding seven. It's about helping them build how 7 relates to other numbers, so that when 7 or 37 or 397 is inside an equation, they know a lot about 7 and the different ways to make 7 that they can use their understanding of seven to think relationally and not just have to compute with those numbers, okay? So I love doing lots of decomposing with just the symbols, but it needs to be visual as well, and those are some of my favorite visuals there. Doing true, false number sentences with your stude
nts. This is a great thing that you could do like a number talk, is put up an equation like this, and just ask is this true or false? And they shouldn't always be true. This example does happen to be true, but you want to throw some in there that are false. Don't always use true ones. Now, just because we're doing a true/false number sentence here doesn't mean kids are gonna think relationally. You will have kids who look at it like this. 10 + 6 is 16, because they're adding those together, 4 +
12 is also 16, so yes, those are equal. That's another reason why, back on a previous slide, I said you want to show it sometimes as 9 = 9. So to finish this, if I was doing it with the students and they were talking this through with me, I would then also write 16 = 16, and we would say 16 is the same as 16, so this number sentence is true. But you might have some kids who think relationally, and this is super cool. When the kids are looking at how the numbers relate across the equals sign, the
y are understanding how numbers relate to each other, and they're actually understanding what the equals sign means. Another one is like this in the box, like the beginning slide where I was talking about the cognitively-guided instruction study where you have that missing piece. So throw those up there and then have a talk about what goes in the box. Now, you can use just one empty spot, or you might have two. The nice part about the second one is that there are infinite answers to that problem
. What goes in that box is dependent upon how your students think about numbers. They could put any number they want in there, but the key is to talk about the relationship between this. If you decide to put 15 in that first empty spot, how do you know what that other number's gonna be? Do you have to compute to figure that out, or can you use some relationships that are happening across the equals sign to help you figure that out? And then also just a reminder, don't always put the empty spot o
r the box in the same location. Vary it. It can be on the front side of the equals sign. But I would encourage you to put some right after the equals sign, because this is where kids will kind of give you their true colors. This is kind of one of those that's like a way to kind of really check to see if kids are confusing what the equals sign means. So if a kid puts 9 in there, you know that they are just computing and they're not thinking about that equals sign as being is the same as, and it a
ll needs to balance out, that they need to pay attention to that seven as well, okay? All right, one of my favorite visuals ways to build relational thinking is these solve me mobiles. They're just so fun. I like doing them even as an adult. But they're for young kids as well. And you can even create them yourself. I love this site. It is linked up. Even though it's not underlined, it will be linked up. I'll make sure of it. So that screenshot that you see right there is just the first page when
you go into the solve me mobiles. And then I brought up a few examples. So here's an example of just puzzle number three. It shows that that symbol, and they don't even have to know the names of them. You can just say the yellow shape. The yellow shape is worth two. You don't have to use the vocabulary, but if you are at that stage and you want to use the vocabulary of what those shapes are, then go right ahead, but you can just call them the yellow shape or orange shape, whichever color you th
ink that one is. It's kind of orangey-yellow, so it could go either way. So that shape is worth two, and then you have this discussion about how they're figuring out, and basically it's an equals sign, right? The mobile has, and the cool part is if you put in, let's say you put in 10, I'm just gonna throw that out there, right? You put in 10 for the blue moon shape, right, the kids are gonna say moon. It's a moon. You put in a 10, guess what happens when you hit enter? The mobile unbalances. It'
ll show you which side is heavier, and so you have to really think through that. Just the visual also helps them see oh, it's too much on that side. And then they have to think about well, what should we put in there? If they don't know exactly, they can use that relationship of what they're seeing the mobile do to help them out. I also like it because they move into ones like this where there's no number up there. There's nothing that it equals. It does equal something, but it's not equals a nu
mber. It's you want it to balance. You've got to keep it balanced. So let's say they put the 4 into the green shape, they don't have to know the name for it, they know that that 4 is the green shape, and then they put the 3 in for the triangles, or the pink shape. You may have some kids who will add it up and say 3 + 3 is 6, and we have 4 over there, so 6 - 4 is 2. However, you may have some kids who will say, oh, well 4, the green shape is just one more than the triangle, so it's kind of like y
ou took one from the pink shape, and you gave it, and that made it four on the top, so now it's only gonna be two. That's relational thinking. So just because you're doing these types of activities does not guarantee that your students are going to think relationally. But we are giving them an opportunity to see what kinds of relationships they are noticing, and talk about those. Okay, now this stuff that we've been talking about is not just for the early grades. As I said, it's not just for ele
mentary school. The way that we get kids thinking about numbers and mathematics and the equals sign in particular is going to impact how they think about algebra. This problem right here, I only ever knew one way to solve this, one way. And it wasn't until I was an adult and I saw some videos from the cognitively-guided instruction group that had, I think it was a 4th grader, 5th grader maybe, I can't remember the exact grade, but I know exactly how that kid solved it. That has stuck with me for
ever. It was not a kid who was doing algebra, guarantee you that. He was not in high school or doing algebra, but he was solving this algebraic problem right here. I want you to think about the way you were always taught to solve this. And then how do you think a kid who has been, who can think relationally, and looks about what's the same and what's different on both sides of this equals sign, how they might think about that. And let's talk about the way we were taught, which is basically compu
tational thinking, right? When we see an equation like this, we were taught, number one, combine like terms. And then if I want to isolate that variable, now I have to do the opposite operation, right? If I want to get rid of that 5 so I can get the variables all together, I've got to do the opposite operation, which is subtract out that 5. And now I'm left with K + 22 equals 3k. Well, now I've got to get all those k's together, right, so I've got to think about what's the opposite operation for
that k, and I've got to subtract these ks. And then I'm left with 22 = 2k. And then I've got to divide by two because I've got to do the opposite operation, because 2k means 2 * k. All this stuff that we learned, all these steps, and then I finally get to understand, well maybe not understand, but I get to my answer that k equals 11. And then our teacher would make us go back to the original and input the 11 to prove that 11 is the correct answer, right? Did anybody else learn this way, right?
Okay. Versus the way I saw this 4th or 5th grader, I can't remember what grade, think about that same problem. And he has it on a piece of paper, they give it to him, and he's got paper, pencil there, and he's thinking out loud. And he said, "The answer's 11." And he like did nothing. And they're like, "Well, how do you know it's 11?" He's like, "Well because there's a k over here and a k over here, so those are the same, it doesn't matter." And then he said, "And there's a 5 on that side and th
ere's a 5 on the other side," and he goes, "In the 27 is 5, so really if I take 5 out, then I have 22, and I have a k plus a k. And a k plus a k to make 22 has to be 11." And I was like, "What did that kid just do?" It just blew my mind that you could think about a problem that way. I was only ever taught computational thinking. That was it. I did just fine in mathematics, I had As all the way through until geometry, got a B in geometry because you had to prove things and I was not a good prover
. I was a good rule follower, I could do the steps, but I was not a good prover of things, and really understanding what was happening. I followed steps to a T. We want to get kids beyond just following steps and computing. We want them to think relationally so that when they do get to algebra, they see things differently. But it's not just for algebra. It's life, guys, it is life. Life is relational. What happens in one situation, we need to relate it to another situation. Relational thinking i
s not just for mathematics, but it is a way that we can bring what matters in real life into the world of mathematics to make mathematics a whole lot easier for our students. And I just want to thank you again for joining us for the 2024 virtual math summit. I'm always honored that so many educators sign up for the entire conference, and I'm especially honored that you chose to come to this session. So thank you so much, and start building those math minds by building their relational thinking.
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