Episode 182 – How To Reach More Students After Two Tough Years
As we continue another school year during this pandemic, educators everywhere are asking themselves:
What should I focus on most?
How can I help my students get ready for the next grade level?
Do I have time to teach through problem solving or should I just teach with direct instruction?
What resources can I use so that I cover the curriculum and still teach deep learning?
Let’s take a deep dive into what we can do to reach more students and reduce our own teacher stress and anxiety at the same time.
- What is worth focusing on and what isn’t;
- What you can do to best help your students get ready for the next grade level;
- How you might go about structuring your lessons to reach more learners regardless of readiness; and,
- Where you can go to access resources to help you uncover your curriculum without leaving students behind.
Squares To Triangles [Problem-Based Unit]
Make Math Moments Framework [Blog Article]
How To Transform Your Textbook Into A Curiosity Machine [Free Video Course]
Make Math Moments Problem-Based Lessons & Units
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Jon Orr: As we continue another school year during this pandemic, or maybe Kyle, we are coming to the, dare I say it, close to ending the pandemic. I don't know. That still has yet to be seen. But educators, I think everywhere ask themselves, "What do we focus on most next?" And it could be next of this year, the remaining of this year, but it also could be like, "What do we focus on next year, knowing what we've gone through in the last two years? How can we help our students get ready for that next grade level?" We're going to answer that question here in this episode.
Do I have enough time to teach through problem solving or should I teach through direct instruction? That's a question we're going to tackle here. What resources can I use, so that I can cover the curriculum and still access deep learning? That's what we're going through. Kyle.
Kyle Pearce: Jon, we are going to be diving in here into actually a recorded webinar that we did recently and we are releasing it for you, friends, to dive into. For friends who want to actually see the visuals, dive in with us and pretend like you were there, make sure you hop over to YouTube. This is episode 182. If you just look for 182 on our YouTube channel at Make Math Moments, you could dive in because we're going to be looking at how we can reach more students. And at the same time, reduce some of that stress, that anxiety, that weight that you might be feeling on your shoulders in today's webinar, which I hope even if you're listening, you're going to get some big value from.
So, Jon, what do you say? Let's do this?
Jon Orr: Let's do it.
Kyle Pearce: Welcome to the Making Math Moments that Matter podcast. I'm Kyle Pierce.
Jon Orr: And I'm Jon Orr. We are from makemathmoments.com and we are two math teachers, who together-
Kyle Pearce: With you, the community of Math Moment Makers worldwide, we want to build and deliver math lessons that spark curiosity-
Jon Orr: Fuel sense making-
Kyle Pearce: And ignite your teacher moves. That is right, my friends. Right now, we're almost going to be starting from this igniting your next moves idea. That part of the framework, where we're going to be talking about some of the things that we want to do and continue doing as we try to reach more students after two really, really tough years in education.
Now, in particular, Jon, we're going to make sure that we don't fall into maybe some of our old habits, because that's what we tend to do when things get stressful, when things get hard, when that pressure is on, we tend to resort to the things that we know best. And I don't know about you, Jon, what I knew best is how I learned math, and we know that that isn't necessarily going to help reach more students.
So, Jon, in today's episode, we're going to be diving into some of the strategies, some of the tools that educators can rely on. And I hope a big message, we'll talk about a big message at the end that people will take away from this webinar, that's being released today on the podcast.
Jon Orr: Yeah. I'm excited to dive into this with you, Kyle, and share this with the Math Moment Maker community because I think after two years of teaching in such a unique situation, we are left with this, where should we go next? And I think this is going to be very helpful for everybody. So, let's not waste anymore time. Let's dive in.
Kyle Pearce: Here we go. All great.
Jon Orr: Awesome stuff there, Rose. Thanks so much for the introduction and thanks to all of you for joining us here this evening on this Monday, 8:00 PM time slot. We are excited to be here. We're going to talk a lot about how to reach more students after these two tough years that we've been going through. We all know that they've been tough. We're going to put that aside and go, "Hey, what should we be doing next?" Kyle, let's get in here. Let's do a couple normal things that we do when we do our presentations here online.
Kyle Pearce: All right. Awesome stuff, friends. Awesome. Great to see who is with us here. The room is filling up, which is great to see. At the end, we've got, if you stick around and you think of your biggest takeaway, we've got something to give away at the end for all you awesome, awesome Math Moment Makers out there. For those who are new to Jon and I and the things that we try to do and share in the math community, we have a podcast that we've been doing for quite some time. It would be something that actually today being a Monday, every Monday, we release an episode.
Jon Orr: Any listeners out there?
Kyle Pearce: And we just... yeah, I'm wondering in the chat, if there's any listeners out there, let us know. If not definitely head on over, we have a blast. I got to say we probably learned the most from doing the podcast because we're right in there. Get to reflect on things. We do some mentoring moment episodes, all kinds of great stuff. And we bring amazing, amazing friends onto the podcast to chat all things math.
Now, today, we're going to be doing a bunch of different things, but in general, something that we focus on every single day is our thinking about how we can make more math moments in our classroom. That is something that Jon and I have been on this quest for quite some time. I would argue, every math teacher is on this quest where you're trying to find ways to engage your students in order for them to learn the content that we are passionate about sharing with them. And over time we realize that it actually, it's not luck. That we can actually create these moments.
There are ways. There's elements that we can leverage each and every day in order to do that. And today, we're going to be trying our best to help you with some strategies that might be helpful given the circumstances that we've been in over the past couple of years. Right, Jon?
Jon Orr: Yeah, totally, totally. So, in the session that we're going to go through here tonight, we're hoping to pull some of these ideas that help to create these math moments in our class. Like Kyle said, it's not luck. There are things that we can do in our classroom. We're going to subtly mention those things and we're going to actually not subtly mention some other things and talk about what we can do in our math class to make these math moments that matter.
And Kyle's got a link on the screen. We'll throw that in the chat as well about some of the tasks and ideas and the lessons that we've created over the years. We're going to share some of that here in this session as well. But we like to point out some of the things that if you stick with us in this session, what you will learn, what you will get by staying with us. And one thing in particular, since we've been talking about, and the session title is How To Reach More Kids After A Two Toughest Years So Far, we're going to talk about specifically what is worth focusing on moving forward and what is it?
I think we all know that there are some things that we're like, "Oh, I notice a difference in what's happening in my classroom now versus what was happening before all of this." And we want to talk about what should we be focusing on now and what we shouldn't be, right, Kyle?
Kyle Pearce: Mm-hmm. Absolutely. And this is something I hear a lot from educators. We hear it on Twitter. We've heard it on the show, on the podcast quite a bit. People are saying there's gaps. Students are coming to us with unfinished learning. They maybe didn't have or maybe didn't adapt well to the online structure or maybe they just got lost in the shuffle between online, back to face-to-face. We've been on and offline a few different times over the past couple of years. And how do we actually help our students get ready for the next grade level? That seems to be on our minds.
We take this responsibility that if we're going to send students off, whether I'm teaching Grade 1, whether I'm teaching Grade 7, whether I'm teaching Grade 9, whatever the grade level is, how do I help prepare them, so that they're successful next year. We're going to talk about some of those things. So, not only what do we need to focus on, but then how do we help them so that they're still progressing. Right, Jon?
Jon Orr: Right. And then even deeper than that, Kyle is we're going to think about how do we help them there, but also, how do we go about structuring our lessons to help these students and reach more learners regardless of where they are in their readiness level. We've got students all over the map. We all can agree on that, depending what grade level we teach, but we can structure our lessons in a certain way to help all of our students and we want to talk about how to do that here this evening.
Kyle Pearce: So, then we're going to give you some access to some resources that might help you in that journey to get you started on that journey. Not to fix all the problems, but to hopefully help you with thinking through what this might look like and sound like in your classroom. But Jon, before we go any further, we often reference all kinds of different reads and all kinds of different learning that we, our own learning journeys. And actually recently we read this book by Adam Grant called Think Again. And there's something that I think is worth us highlighting here before we dig into this two tough year session that we're going to be doing tonight.
Jon Orr: Yeah. We read this book, I think last year and I recommended it to Kyle or Kyle recommended it to me. This is not a math teaching book. This is not a teaching book at all. This book is called Think Again, which is about the power of knowing what you don't know. And in a sentence, this book is a mass market book. It's for anybody of any topic or anything. It's a self-help type book in the realm of social science.
But the book is primarily about thinking about our understanding of what we know and what we don't know and how to rethink our beliefs, so that we are on a better trajectory of learning. Because the book is about understanding that learning is not about confirming our existing beliefs about evolving new beliefs. And we can only do that by constantly reevaluating the beliefs we have by thinking about them again. And that's will be the premise of the book, but the book actually highlights as a framework for all the stories it shares about how to rethink.
There's a story about why Blackberry came to the rise, but then fell and why it didn't rethink some of its thinking structure when the iPhone came up in the markets. But every story in this book is framed with four mindsets. And we want to outline these four mindsets to help frame our thinking moving forward in this session. And hopefully, you can use these mindsets to when you go back to your classroom and think, "Am I thinking like this or am I thinking like this?"
So, there's four mindsets. And the first one is when we think about any interaction we have with people, when we're talking about things we believe or a belief system or something, we hold true, sometimes we act as a preacher. We act as a preacher when we're sharing ideas, saying, "You know what? I really think Ozark is the best show on TV. And I'm going to tell you exactly why I think that." And that's being a preacher.
When you preach to someone else or a group of people or wherever you are in a social setting, you're telling them all the things why you think this is true. So, you're preaching to the people to convince them to get your argument out there. This is one of the roles we take on when we think about or interact with others when sharing or discussing beliefs. Kyle, what is another mindset we take on when we're interacting with people?
Kyle Pearce: Yeah, so the first one is more or less you're out there and you're spreading the word. Right?
Jon Orr: Right.
Kyle Pearce: You're spreading your belief. You're trying to share that what you believe and oftentimes you wear that like a bit of a badge of honor, oftentimes. And then there's this other one where when let's say there's something out there that you disagree with. So, that thing that maybe I might be preaching over here, you might hear somebody say the opposite.
And you might come at them with this mindset, the prosecutor mindset, where you're actually looking for flaws or those flaws actually jump out at you because you're like, "Wait a second. That doesn't work. That doesn't match my beliefs. And therefore I'm going to prove why you are wrong. And we're going to go for you to better understand what I believe." And that is that prosecutor mindset.
And when we think about this, this can actually hurt us in the classroom a little bit where we come at it and we look at things as if we're using data such as how students perform on tests in order to prove why something doesn't work in the classroom. Instead of maybe realizing that, "Hey, maybe data's actually coming from some other aspect of the teaching practice." So, that prosecutor mindset can be looking for those flaws and that's something that we're actually going to try to move away from in this session here. Jon, how about our third mindset, what's the third one.
Jon Orr: The third one is a politician. And so, a lot of times when we are interacting with people at about our beliefs, we'll try to get them on our side and this is the like, "I'm going to persuade you to be on our side. I'm going to campaign for you to be with me." And sometimes when you're acting as a politician, you might say something to gain some popularity with a group, but you might not actually change your belief in that conversation. So, that's what the politician is.
And basically, the book says that at any time we're interacting with people about our beliefs or our values or just things in general, like sitting at the lunchroom table and having a conversation, we act like these three. Because sometimes we can act to all three at the same time. We can act at one or the other. But you can imagine that you probably have found yourself in one of these roles at one time when you're talking about something. But when we think about the classroom, Kyle started to think about the classroom itself. And this is where we want to frame what we're talking about here this evening is when I think about my old teaching self and I found myself primarily in these three categories quite a lot.
So, when I think about the preacher, I taught from this front. We call that like the sage on the stage, right, Kyle? It's like I was the bearer of information. I would tell, tell, tell the students how to do things. I was a rush to an algorithm type of teacher that would say, "You know what? Here is our lesson. Here's let me show you how to do it." That was me preaching the math content at my students and getting it out there. You can imagine that maybe you have found yourself in a preacher mode as well in the classroom.
Kyle, I know you talked a little bit about the prosecutor. But the prosecutor, also, is I think thinking about when you're in the classroom with students, I found myself as a prosecutor on thinking like judging the students, thinking as either right or wrong. It's like you had talked about in the prosecutor, it was, they were shooting down ideas. But when I think about a prosecutor in the classroom, acting like a prosecutor is like trying to make sure that the kids do it your way. I used to find myself saying like, "Well, okay, you did it at that solution, but why don't you try this one over here?" Or, "Let's make sure inaudible-"
Kyle Pearce: Yeah, this is going to be so much faster and more efficient. So, I'm going to prove to you why you want to do it this way because your way is going to take too long or what if the numbers get too complex?" And when you're a student and a teacher is coming at you in this way, you can imagine that for some students that could actually shut them down. We want them to think. We want them to share. But if we're then going and saying, "No, no, that isn't the correct way," then we're not in a great spot.
And with the politician piece that is like trying to convince our students of things. We can try until you're blue in the face to say, "Students, here's why you should do this or do that," but ultimately at the end of the day, being the politician isn't necessarily going to do anything. It's almost like we have to change how we approach this in the classroom in order for students to see the need to complete their work or see the need to put that extra effort in. Correct?
Jon Orr: Yeah, yeah. And when we act in these mindsets, whether at the lunchroom table with your peers or in the classroom like we've described here, what happens is we can't move forward. We can't grow like the book was suggesting. That we're thinking about how we can use learning to not confirm our understanding or confirm our understanding of what students know. We want to use these ideas to evolve what we know. We want to evolve what we know about our students.
Because if you stay in these roles, what you're doing is you're really just living in this echoing chamber. And you get this confirmation bias that happens when you are at the lunchroom table and you're trying to preach these ideas and prosecute other ideas and politician and the people around you. In the classroom, we're doing that as well. We're reconfirming things we already know and we're not learning anything new.
So, that's how the book gets set up. But the author, Adam Grant talks more about a fourth role that we should take on more consistently, which he calls the scientist. If you think of the scientist, the scientist will make a conjecture and then they will go and test that conjecture. That's what a scientist would do. And so, he says in the book that about our beliefs, about our values, when we're interacting with people, we should have an open mind. We should think like a scientist. We could think, "Hey, I don't know what I don't know and it's possible, I'm wrong. I should gather evidence and let the evidence tell me what's the truth."
And that's true, I think, in my life that I want to do that on a regular basis when I interact with my friends and my family, but I also want to be a scientist in the classroom. I want to learn about what my students know or don't know. And I want the evidence to tell me that. And I don't want to think about being a preacher anymore. I want to be a scientist. I want to see what my students know and gather evidence to help them on their learning journey.
Kyle Pearce: Awesome stuff. So, to set this up for today, we really want to start diving in and we want you to start thinking like a scientist with us. Okay? As much as maybe some of those other mindsets might be there. We're human. Some of those things you're going to have beliefs in your mind. You're going to have some of those things and maybe some of those beliefs are helpful. We don't want to ignore everything that we bring to the table, but we want to keep an open mind. So, what we want to do right now is give you an opportunity to do some sharing in the chat. And we want to ask you, because this is a question that keeps coming up.
Teachers are saying, "We don't have enough time to get the curriculum done. Students have gaps." There's all kinds of reasons why. Whether it's because of a new curriculum, because of a two-year COVID pandemic, because of any other number of things. Socioeconomic status, anything could be affecting here. What we want to know from you is what do you think really matters, especially given your circumstance?
Now, here's the thing though, as a scientist, I have to read what you're saying in the chat and I have to respect the fact that your context may be very different than my context. And somebody else's context may be very different from your context, so we have to keep our minds open here. But I want to get you guys going in the chat right now, share, what are you thinking really matters for this school year? And I know that this school year, there's not a ton of time left, but there really still is at least a month and a half where we can still do some pretty important work. And let's be honest, whatever issues we're dealing with now, aren't going to vanish over the summer.
One thing and one message that we're hoping that people will take away from our chat here today, and we will dive into some math. I think if we don't do the math, then it's really difficult to take this information and really put it to use to implement it. But when we look at this and I see all of these things. A lot of times we get teachers, and we've been there, we know how this feels, where you're like, "Ah, I just don't know if I'm going to have enough time to do it all. Look at all of that stuff."
And really the messaging that we're feeling like what really matters here, what really matters to us are really two strands that all the other strands are going to feed into. Right, Jon? We're going to take this and we're going to look at it and say, "If we look at those social-emotional learning skills and mathematical processes and number sense and operations, those to us are what really matter." Now, we're not saying algebra doesn't matter at all, but what we're saying is we want to use those to feed into these ideas.
Jon Orr: Exactly, exactly. So, it's not saying one comes before the other, but what we're saying is, and what we've found in the last year or so of working with these strands is that we can use these first two strands to fuel the discussion. Fuel the learning of the other ones and we can go through those to get to the other ones. So, right, the question is how do we focus on that, Kyle?
Kyle Pearce: Yeah, absolutely. And I think it comes down to, so for us, we're thinking, "Okay, I want to take every opportunity to help students build those math process expectations to build those SEL skills." I want them to have opportunities to do that in my math classroom each and every day. And these other strands are a means, they're almost like an opportunity for us to engage in them. So, when I think about it and I go, "Okay, what is the intent? What is the intent of my learning goal for tomorrow or my learning for tomorrow?
I think about this. And I go, "Okay. Let's say that I'm working in spatial sense." I'm thinking about my spatial sense expectations. And I want to figure out how do I take those expectations and use them as almost like an excuse to bring out these ideas. Not the other way around. We don't want to be like, "Okay, we did number in operations or we did SEL or we did the math process expectations." Those should be happening all the time. And of course, spatial sense, it would be great if that's happening all the time as well.
But when I look at those specific expectations, I want to try to find ways that I can bring out these pieces. And before we tell you specifically what we're going to get after, I think this is a great time for us to actually dig in and try to show how we can focus on these big ideas.
Jon Orr: Yeah. We're going to outline a task that we've used. Kyle actually used it today in one of the classes here.
Kyle Pearce: By chance, yeah.
Jon Orr: I used it a couple weeks ago in my class. We want to walk you through what this could look, but with the intent, what Kyle is saying is that we're going to use this as a vehicle to talk about some of these ideas that we should be doing on a regular basis about number in social-emotional learning with skills in mathematical processes, which is what we focus on as much as we can and as a priority. And then the specific learning goal pops out as almost like a byproduct. Right, Kyle?
Kyle Pearce: I love it. Yeah, exactly. And that is so key. It's like my end goal here isn't like that students know that specific expectation. It's that we use that expectation as a means to build their fluency and flexibility with number operations, right?
Jon Orr: Right. So, it's like, my old self would've said, "Today, I'm going to make sure my students can do this skill, which was 2.1 from the textbook in this unit. And the skill I'm thinking of is one of those very specific expectations and then going, "Oh, yeah. I probably will make sure that I've got these other things in here like my problem solving processes." They all come along for the ride.
But no, what we like to do is, "How do I develop and start with the problem solving processes? How do I develop my fluency around number sense first? And can I incorporate this specific learning goal in here as well?" So, we're going to pop that in here and we'll revisit afterwards and reiterate what we're talking about now. See if you can pick it up.
Kyle Pearce: Awesome. So, what we'd you to do is we actually would love you to wear two caps throughout this. But you're going to participate as a student, which means we don't want you robbing the thinking of others. And I actually will say this to the class as well. I'll say this and I'll say, listen, if you think you know, if you know what's going on here, I want you to try to hold yourself back from robbing the opportunity for someone else to come to this same conclusion, have the same epiphany.
All right. As educators, it's highly likely that some may catch on to where we're going here. So I want you to be the student, but I also want you to continue being the scientist and thinking about this and thinking about the teacher moves that we're using and as we go here. So, we're going to try to run this as we would with students, likely going to be faster than the process that we might use with students, so do keep that in mind. But we believe that our learners, our group, that we're working with, are likely going to be at a different place than some of our students who may have been in the classroom when we're doing this.
Jon Orr: Right. So, we're going to show you a very short, short clip. And at the end of the clip, it's actually going to ask you to notice and wonder a few things. So, get your typing fingers read. Kyle is ready at saying ready, write in the chat. Make sure you can see the chat.
Kyle Pearce: Yeah. I'll definitely play it a couple of times. Anything and everything goes here.
Jon Orr: Yeah. Notice.
Kyle Pearce: No judgment. Nothing is too obvious. Nothing is bad. Nothing is not worth sharing here.
Jon Orr: Exactly.
Kyle Pearce: All right? Our math community is very accepting and very, very open. Here, we go.
Jon Orr: inaudible ready.
Kyle Pearce: Whew.
Jon Orr: Well-
Kyle Pearce: All right. Some friends are saying like, "I wonder what the area or the perimeter might be of each? How is the area divided, the angles? And how do they relate to one another?" Awesome, awesome noticing. Awesome wonders. Some of the students today were saying things like, "I think I see a square in a square." And I loved that the student articulated they think.
And I asked them like, "Why are you not certain?" And they said, "Well, I don't know for certain. There's nothing to tell me for certain that it is squares." And I was like, "That's actually, that's a really great point." Whereas a lot of people would make the assumption and that's okay too, if they did. But I really want to dig into that with those students, lots of stuff. We have people talking about spatial sense. All kinds of awesome stuff. Getting kids to just notice.
And something I told the kids today, and I do this with any new classes that I do lessons with, is I say, I intentionally tell them why we do this, and it's to turn your brain on and prime it for learning. Because your brain is primed. It's trained to try to ignore things that don't matter. There are so many things your brain has to ignore in a day to save energy. And it will only remember things that it feels is important. So, what we're doing is we're priming the brain for learning. And a lot of students tend to really like that idea. That they're like, "Okay, I get it now. It's not like a waste of my time." We're just noticing them more.
Jon Orr: Right. And I was just going to jump in here, Kyle. If I have been a preacher in my class up to this point, and I'm doing this, then this is also priming your brain to get ready. Because at note, in my preaching days, I would just say, "Today, we're going to learn about this." And then I would show everybody how to go about that particular learning goal or this particular learning goal. And no time was I asking my students to voice what they thought about any of the things that were going on in that particular lesson. So we're priming our brain to get ready, to use it, to do a little bit in depth here.
Kyle Pearce: I love it. So, we've got all kinds of noticing and wondering going on here. I have a challenge here for you, friends. I love these wonders here, and these are all great things that we can get to with students as well. If you want to explore them with students as well, amazing, awesome. But I do have a challenge for you here. It's a very specific challenge. And I'm wondering just based on what you see up on the screen, I'm wondering about how many of these, right here, this little orange square, I wonder how many of those will it take to fit in this square right here?
And I'm going to confirm to you right now that this is a square inside of a square. And there is no tricks here. There is no lie. I do not lie to students. So, my wonder for you, I'd love to get some friends sharing their estimates. Many of you have been with us before and know that we like to frame it just a Dan Meyer at Andrew Stadel might with this low or high, like what's too low, what's too high. Today, because of time, I would love it if we can actually just zoom in on your best estimate, based on just what you see here, which is your spatial reasoning and a bit of your intuition. So. we're going to give you a little bit of time to see what we can come up with.
Jon Orr: Yeah. Feel free to throw a sentence as well on the end of your estimate, just to hint at where your thinking might come from on your estimate.
Kyle Pearce: I saw 11-teen, so when I see 11-teen. what that makes me think of right away is a little bit of James Tanton. But it also makes me realize, "Well, wait a second, 11-teen that would be like 110, I guess? 11-teen, so I'm going to do that, 110 is one. What else do you seen in the chat, Jon? Let's get a few of these estimates down here.
Jon Orr: I see a few 400s, a 625, a 400, a 1500.
Kyle Pearce: Whoa, 400 and then 1500, like 1500 is way over here. That makes me, like I'm always wondering how this gut shot and this gut shot are so different. It's interesting. This is four times bigger-ish, but this is basically three times bigger-ish. So, we've got a big range so far. Would you agree?
Jon Orr: Yeah. We got 80 as the smallest so far. We've got a 225, a 100-ish from Heather. It is to be an X square, where X could be between 10 and 12, so a square with 10 and 12...
Kyle Pearce: So, between 100-
Jon Orr: ... and 144.
Kyle Pearce: And 144.
Jon Orr: Do some math there.
Kyle Pearce: Okay. Let me do that. That's an interesting approach. I like that. So, we can see we've got quite the range here today. And of course, something that we to announce to students is when we estimate, that range might be really wide, but I'm wondering if we can actually get a little bit closer. Now, Jon-
Jon Orr: Yeah. Let's pivot a little bit. Right, Kyle?
Kyle Pearce: Yeah.
Jon Orr: If you were on your email this evening, we were like, "You know what? Let's give them a little template that we use with our students." And you used it with your students today. We'll email it out. See if anybody prints it out. If not, no worries. Kyle is going to put it up on a little bit bigger on the screen so you can see it. But if you have printed this out, now, let's give them a little bit of insight, Kyle. This diagram is not exactly the same as the diagram we have up on the screen, but it's pretty much the same, same layout.
Kyle Pearce: And I'm going to confirm that, we still have a square and we still have another square inside. So, we have the same situation. But something that I will mention is that actually here, our unit of measure is not the same. So, this square may be a little more tilted this way or that. Those pieces I cannot confirm or deny for you right at this point.
Jon Orr: Right. So, it's almost like we're going to do two rounds, so we're going to give you this. This is what we did with our students. We're give them this printout at this point and say, "Okay, you know what? We're going to give you this. And we are going to ask you to represent your thinking. You can take this page. You can draw on it. You can use the filled in square.
Kyle Pearce: Like cut it out.
Jon Orr: You can cut it out. You can use the one that's blank. You can cut pieces out. You could cut the along the dotted lines. You could do whatever you want to do to help demonstrate your thinking on helping decide how many little squares, which are now the grid lines on this grid. How many squares would fit in the middle square? And then, what we'll do is we'll pivot back to the one that we showed you already on the screens. We're doing a two rounds here that look similar.
Kyle Pearce: So, there's some people in the chat sharing that they clearly have some prior knowledge that maybe students may not have. Because when I did this with students today, they did not have the prior knowledge. They had basically what they brought with them, which was not necessarily a very full plate of their past couple years, which was evident. So, I want you thinking about what might students in your classroom do and maybe think about the different students in your class. What might you see? Because we're not pre-teaching them here. We're not saying, "Hey, if you do this, this and this, we're not preaching to them."
Jon Orr: No. We're going to be scientists.
Kyle Pearce: We're not going to prosecute them either. We're not going to be the prosecutor and say that the way you did that is not good enough. We want to see where students are. And as a scientist, that will help us take information and then say, "Okay, where do we need to go next?" So, I'm wondering, what would students actually do? We'll give you about, I don't know, another minute, and then we will start sharing some of what we saw.
Jon Orr: Yeah. And also, what you could do is turn your mic on and talk and verbalize.
Kyle Pearce: Yeah. And I'll try to facilitate what you're saying and what you're doing. So, in the class, just to help for those who are here and start thinking about this, in the classroom, students are working together. Students are raising their hands saying, "Ooh, ooh, ooh, I want to tell you." And I'm saying, "Oh, don't share with me, share with a neighbor." And I'm walking around listening to their discussion, but I don't want to be the person in the discussion. I want to be listening to their discussion.
So, I say, "Oh, try to convince a neighbor. Oh, you think you've got an idea? Share it." Do they understand your idea? If they don't, then maybe you need to work on how you're going to make that more clear for them to understand your thinking. And I'm trying to just listen and observe and think about which solutions should I be bringing to the front as strategies to share with the room as we go.
We're going to start with the most accessible. The most accessible method. I saw this one today and I actually had a student. In this, it was a grade nine classroom. Took some of the extra grid paper after cutting out their template. And they actually cut out one square and they started iterating it over top.
Jon Orr: Just one and then they just kept moving it?
Kyle Pearce: Just one. And that told me a lot about where that student was because that student was iterating one square and they were using more of accounting approach in order to find the area, which at that point, I don't think the student made the connection that that's what we were actually finding was the area here, because we did not make that explicit yet.
Jon Orr: No, we didn't. We said, how many of these little squares. Kyle, did they count or did they eventually have a row and then go, "Hey, you know what, I've got another row?" Or did they just keep putting the one everywhere?
Kyle Pearce: I think once they got to the end of one row, then they did another row and then they started to count by 10s, so they had their tens there. And came to the conclusion that it was 100. Something else that I saw that I thought was actually, I did not anticipate a student had cut out this square and as they cut it out, I was watching and I was ready to be so excited. And then the student proceeded to put it here like this. And then this has tried to estimate using these measures.
Jon Orr: So, they placed it in the exact same positioning.
Kyle Pearce: Exactly.
Jon Orr: Melanie in the chat has said, "Trace the square." You said they did that, too, I think when you showed me earlier. They had actually traced the edge of that square and then Melanie is saying, "And counted the squares." So, that seems similar. Now, I'm wondering if Melanie means exactly the way Kyle's student did it today like placed it in the same position. Or do you think Melanie, that maybe you're anticipating that the student or maybe your solution is not that exact placement. She can show us a picture perfect. Melanie, feel free to toss your camera on there.
Kyle Pearce: Yeah. A student in the classroom did this and traced it and then went out and started approximating. I had another student who did something similar to this and started making rectangles.
Jon Orr: When I see Melanie's, it looks kind of like what your student had done. Started counting the squares inside that exact placement of that diagram. Interesting stuff. Interesting stuff, Kyle.
Kyle Pearce: Interesting. Absolutely. And what we did see, some students, one that I thought was really clever was a student actually determined that, well, if they took this. And let me get another one here, they took this and they took this square and they placed it here to determine that this was a 10 x 10, 100 cube or sorry, 100 square area. And then they proceeded to use their neighbors that looked this.
And they actually went and they said, well, they knew that if this is 100 and they figured out that, "Well, wait a second." This was, I believe it was 14 x 14 and students, we did this, no calculators, by the way. So, we had some students, we said, how are you going to figure out then what 14 x 14 is? And then we had an opportunity to actually look at the distributive property with partial products by decomposing our 14 into 10 and 4 and 10.
Jon Orr: Now, Kyle, you did this during the lesson, correct?
Kyle Pearce: This was during the... yeah, exactly. Students were sharing and I was trying to facilitate their thinking.
Jon Orr: Right. I love it because I know that some folks. And my old self would've said this, too, Kyle. It said like, "Look, why not allow them to use calculators." Because Kyle, aren't you going to derail the lesson because now you have to talk about how to multiply it like you're showing. Why would we not allow calculators? And we would think this is not actually derailing the lesson.
Kyle Pearce: Yeah, no, absolutely. And I would say for me, once again, going back to the reason that we're doing this learning is to give students the opportunity to get better at the things that we are constantly saying they are not fluent and flexible with. So, in this case, if you've ever said, "Students do not know their math facts, or they aren't flexible with numbers, or they don't know this, or they don't know that," this is an opportunity for us to give students the ability to build these skills. And to see that, "You know what? I can decompose these numbers in any way I choose. It just seems that 10 and 4 seem to be very helpful."
And not only that, but if I want to get them ready for the next grade, this was a Grade 9 class we were working with. If this is the Grade 9 class, and today, I'm working on a spatial reasoning expectation and I give them the opportunity to build their number and operational sense and basically set them up perfectly. Because what we just did here is we took 14 x 14 and we turned it into something that's easier for them to solve, but not easier for them to solve next year algebraically, unless they've already done this work.
So, next year when I go into Grade 10 and I'm asking them to multiply binomials and students already understand how partial products works, there is no need for foil because it looks this.
Jon Orr: Exactly.
Kyle Pearce: I'm building my area, my measurement and my number sense, which will extend to algebra because next year, instead of 10 and 4, it's going to be X+4 and X+4, or it's going to be X+42. And I'm building towards fluency for the next grade level. So, for teachers who are saying, "I need to get them ready for next year," this work is necessary. It's not just something that we want to do or we don't want to do because we don't want to derail. It's actually, to me, this is the important part of the lesson.
So, these students were able to put this together and say, "Well, okay, if this is 140, 80, 96. This is a 196." Students took this and they said, "Okay. Well, if this is 196,"" we see that right here. There's my 196. We know that this is 100 and you can see right there, there's my 100. All right? So, I have 100 right here. And some students simply subtract it. But other students, I then asked them, I said like, "What's the area of just one of these triangles?"
Just to see, what are they going to do? What's their strategy now? And one of the most remarkable things happened. I had one student who was just quietly in the back and they just folded just like this.
Jon Orr: Now, I think Ian was trying to show you that he did that, Kyle.
Kyle Pearce: Oh, oh, Ian, you're the best. I wasn't working in Ian's virtual class today, but I have worked in Ian's virtual class a few times, which is pretty awesome. But Ian, were you doing something similar to like these-
Jon Orr: There it is. He's showing you.
Kyle Pearce: Oh, I love it. I love it, Ian. That's fantastic. I love it. And how many were missing there, my friend? How many were missing, Ian?
Ian: I approximated four.
Kyle Pearce: I love it. I had a student who then did this and they said, "Yeah, it looks like four." They all eyeballed it. And then another student just took a piece of the grid paper and just put it up against. I thought I had some. Here it is. They put inaudible there.
Jon Orr: Slide it in there.
Kyle Pearce: Look at that. And they said, you know what? Yeah, that's four. And they said, "Okay." So, I have 100-4, that's going to give me 96 for each of these triangles. And then of course, they used part of the division to divide it amongst the four triangles and now, they have the area of each triangle. Other students, Jon, took these triangles, inaudible-
Jon Orr: Yeah. There were some people that cut some triangles out, right?
Kyle Pearce: Yeah. Absolutely. Some students cut the triangles out and they did this and they said, "You know, if I take this and I take this." They were like, "I could just use the formula for area of a triangle." Right?
Jon Orr: Mm-hmm. Mm-hmm.
Kyle Pearce: But other students said, "Well, I could find the area of a rectangle by just finding the area of that rectangle right there." Which I think was 6 x 8.
Jon Orr: Yep. I like it.
Kyle Pearce: And interestingly enough, some students were like, "Oh, 6 x 8," they didn't know what that was. I was like, "Hmm."
Jon Orr: That's a tough one.
Kyle Pearce: Six groups of eight. I was like, "I wonder, do three groups of eight?" And they were yeah, "It's 24." I'm like, "So, do six groups of eight?" And they said, "Oh, 48." I'm like "Exactly. Nice work." So, 48 here. Wait a second, 48 with these two triangles, I wonder where's my other triangles, huh? Actually, I don't know where they are. Jon, I don't know where I put them. Oh, maybe they're under-
Jon Orr: Are they underneath?
Kyle Pearce: Yeah, they've got to be. I thought I h ad some extra...
Jon Orr: I hope so.
Kyle Pearce: ... but maybe not. Ah, there's just so many around here. Oh, well, I think I've lost them. That's all right.
Jon Orr: Oh, bummer.
Kyle Pearce: But, yeah. So, students had all these different ways and we got to build and actually allow students to leverage the area model in multiple ways to build their number fluency. They also are building their spatial reasoning skills by being able to take these shapes, decompose them, and recompose them in different ways.
And then we actually asked them and we said, "Okay, well, listen, if you know the relationship between this big square," which was 196, "and this square here is 100. And you also told me, 'Wait a second, these were 96,' which holy smokes, you're telling me that these four triangles actually take up this amount of space over here? That's interesting. If I decompose these in a different way, that's really interesting." And then we went back and said, "I'm wondering, what about this one here? What if I give you the length of this side and the length of that side? And we were to ask you, can you update your estimate now?"
And this is the opportunity for students to now go ahead and actually leverage some of what they've learned to apply it to a similar situation, but with different values and really not as accessible because now they don't have the actual physical manipulatives to be able to simply cut it out and look at a grid.
Jon Orr: And I know that you had some students sliding some triangles around. And seeing Chris is saying, "Turn the triangles," because we had lost two. But if we go back to that slide, Kyle, we could also progress and push kids once they've practiced those skills and figured out how many are in that big square, knowing that each of those are 12 and 16 side lengths for those triangles. But I know that a little bit later we were sliding some triangles around and also, we saw some really cool and really interesting properties for this diagram.
Kyle Pearce: Absolutely. So, while some people may have noticed this initially other people may not have noticed it yet. There's something magical hidden here that students will after we give them the opportunity to work, that they can actually look and see, that. "Well, wait a second. There's something going on here that they can play around and they can start manipulating here." And if we actually look at this and we were to let's go ahead and recognize-
Jon Orr: But the green areas are the same, no matter where they are on that space.
Kyle Pearce: Exactly, exactly.
Jon Orr: And I had a student in my class move them to the left side and go, "You know what? When I do that, then the areas of the two empty squares on the left side of the screen 16 x 16 is 256." And they're saying, "You know what? That's part of the area if I didn't rearrange them." Because we gave them that two diagram side by side. And I said, "You know what? The other empty square now on the left side is a 12 x 12 square." And that's filling up as we speak. And then that part is going to be 144 in just a moment, but that's going to be the rest of that big square.
And then all of a sudden, there's some really interesting things here. Obviously, you're adding those two smaller squares, add to the larger square and you do get 400 little tiny squares on this particular example. And as I said, there's some really cool things that happen here if you look closely on this triangle.
Kyle Pearce: Absolutely. And if I slide one of these out and I actually were to take this and go, "Well, wait a second." If we look at this and we slide this over here and I say, "Well, hang on a second. What do I have here?" I've got this one right here and I'm dropping...
Jon Orr: Your squares.
Kyle Pearce: ... all my pieces all over the place here. This one goes right here and-
Jon Orr: Yeah. That's square.
Kyle Pearce: Okay. And then let's take my other two triangles and let's-
Jon Orr: Oh, you found them?
Kyle Pearce: I did find them. Yeah. I found them. They fell. Here they are. So, let's have a look here and I'm just going to put this over here. And if I just take one of these-
Jon Orr: Yeah. Look at one triangle.
Kyle Pearce: If I could just take one of these and I put it there and then I noticed that this, I can almost take it with it, right? I see that right there and then like I look at this one here and I go, "Well, wait a second. Those are the same side length." So, this one goes here and we knew that by rearranging that we got this 100 here. And actually, look at that, this longest side length gives us this right here. We get this really interesting relationship and we get to introduce this through the work that students did.
So, once again, as the scientist, we're now giving the students the opportunity to discover by asking them very specific questions that they're able to answer. We didn't say right from the get-go like, "Tell us the side length of the hypotenuse." We didn't say that. We asked them for different measures. And then through those different measures and those relationships, we're able to manipulate what we have to build that number sense, that fluency problem solving, modeling, strategizing. All of those pieces in order to develop this expectation, which is the Pythagorean theorem.
Now, this might not happen in one lesson, but this is where we want to go with this thinking. So, this was the expectation that we were exploring, but notice how it wasn't the light wasn't shining only on Pythagorean theorem. It was shining on these other pieces over here. These are the important pieces. And it was an excuse for us to dig deep in this thinking right there.
Jon Orr: Yeah. And I think that's that when we think about what really matters in our class, we're going to go back to that. It's this idea that we want to develop fluency as much as we can. We want to develop a number sense. We want to develop this social-emotional learning with our students and we can get to the concepts from the specific expectations, like using the Pythagorean theorem or the triangle relationships through strengthening these skills that we know students need more practice with.
Kyle Pearce: Absolutely. So, really the goal here is we want to begin. How do we do this every day? We want to begin with a problem. Right here, we didn't really have a story, but there was a provocation there, right? There was definitely something there. We wanted to make sure that there was a low floor and a high ceiling everyone could access, that student who counted one by one all the way to that student who is able to apply formulas that they had brought with them. We want to make sure that we're revealing big ideas and truths about the math and we want to emerge strategies and models throughout the thinking. We don't want it to just be a light shining on one specific expectation.
So, that for us is one of the biggest pieces here. So, when we go back to think about what really matters, this is something that you as the scientists have to do. You have to reflect and think, "What can I do from now to the end of the school year? And what am I going to do next year in order to reach those goals or help my students reach those goals?"
So, Jon, let's go through a few things that we hope we gave friends with. We'll drop some links in the chat for anything that they need. And let's first of all, start with our first what you'll get? Make sure you let us know in the chat if you feel like you're getting a little bit of this from today's session.
Jon Orr: Yeah. Tell us, Kyle, did we do this? We wanted to talk specifically about what we should be focusing on and what we should focus on now, and what we shouldn't focus on. Let us know in the chat. Give us a yes, if we did that here in this session. We said we were going to. We hope we pulled that out for you. Give us a yes in the chat.
Kyle Pearce: Well, we a "Wee." I love it. I love it. Awesome. Hopefully, you feel like you saw some ways that you can help students get ready for the next grade level. Today, we gave some examples of what that might look like. Let us know. It looks pretty good in the chat. Jon, keep going.
Jon Orr: Well, we wanted to help you understand how you might go about structuring your lessons. Hopefully, you saw how we could do that with this one particular, but thinking about all your lessons. How can you reach more learners regardless of the readiness level, by structuring them this way.
Kyle Pearce: Awesome. And we're going to drop some links in the chat for how you can get that task and many other tasks and dive into them. My friends, it has been such a pleasure. We're going to hang out. We do have something to give away, but the recording is going to end in just a moment. So, thanks so much for hanging out with us, friends. Anyone who's watching the recorded session, hopefully, you'll check out some of the links that are included in the Google Doc after this session here. So, thanks everybody for hanging out.
Well, there, Math Moment Makers, I hope you enjoyed today's episode. I know that both Jon and I really enjoyed actually putting it together and thinking through some of these challenging problems, these challenging questions that might be going through your head. I know that a lot of educators I've worked with feel like, "Should we just be doing?" And then fill in the blank. Should we just go back to direct instruction, because we'll cover more, "cover more."
Jon Orr: Yeah. We got to get them caught up.
Kyle Pearce: Yeah. But at the end, I hope that what people are taking away from this is really, we want to help students become fluent and flexible. We want them to become numerate, right? We want them to be comfortable with number and operations. So, look at your curriculum expectations or your standards, depending on where you are in the world and look at them through that lens. How can you reach more students and help them build that number sense and build that fluency from... I don't know if you heard, Paisley there, but Paisley's in the background.
Jon Orr: Yeah, I heard him. I heard him. It's a her.
Kyle Pearce: It is a her. So, I hope that you take that message away and you start realizing that, "Wait a second." I know that maybe things aren't the way they were before these two years. But let's make sure that we don't start getting back into old habits, even though we're dealing with a few struggles.
Jon Orr: Yeah, yeah. And I think one of the biggest things that I think we talked about in this particular episode that I've stuck with and trying to implement as much as I can is thinking about those big ideas. What are the things we want our students can walk away with and how can we focus on those using our curriculum, our standards as a way to get at those.
Hopefully, you saw an example if you're here on YouTube with us or you heard about the example that we talked about if you're listening over on our podcast, about how do I bring in the ideas of number sense and problem solving. And reasoning and proving and posing important problems while you're using your content in your standards to get at these other big ideas. So, hopefully, we've got some messages there and the key takeaways for you. Those are our key takeaways.
And hey, in order to ensure you don't miss out on our new episodes as they come out each week, be sure to hit the Subscriber button, wherever you are listening to this right now, so that you can get next week's episode automatically delivered right to you. We put them out on every Monday morning, early and also over on the YouTube page.
Kyle Pearce: Awesome stuff. Awesome stuff. Yes. And this episode in particular is really helpful on YouTube. You get to see and experience the examples and some of the visuals that we put together to hopefully reiterate some of the messaging that you heard here today. So, definitely hop on over there.
My friends, we are looking for more math mentoring moment episodes for you. So, if you have a pebble in your teacher's shoe that is shaken around and you want to discuss it on an upcoming Math Mentoring Moment episode, make sure you head over to makemathmoments.com/mentor, because we are scheduling for the next couple of months, in particular over the summer months. We love to try to dig in with as many educators as we possibly can to get us ready for next school year. So, start thinking ahead.
I know that some people are thinking spring. They're thinking or at least here upper in the Northern hemisphere are thinking summertime is coming. And sometimes, we want to take a little bit of time off, but let's start thinking ahead. So, hop on with us over at makemathmoments.com/mentor, so that we can dive into an episode with you and try to shake out that pebble that might be in your shoe.
Jon Orr: Show notes and links to resources from this episode, and all the examples that we shared here can be found over at makemathmoments.com/episode182. Again, that's makemathmoments.com/episode182.
Kyle Pearce: Well, until next time, my Math Moment Maker friends, I'm Kyle Pierce,
Jon Orr: And I'm Jon Orr.
Kyle Pearce: High fives for us.
Jon Orr: Hey, and a big high five just for you.
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