Episode 157: How To Flip Your Math Class The Right Way!
In this episode, we’ll unpack the real flipped classroom, how to transform a typical lesson into an engaging problem based math lesson. We’ll share how to slowly emerge shortcuts instead of giving them upfront and finally we’ll chat about what to do if students aren’t engaging with the problem based lesson we’ve just transformed!
- How to structure your lessons around the”real” flipped classroom;
- How to transform a typical lesson into an engaging, problem based math lesson;
- How to slowly emerge the shortcut instead of giving it upfront;
- What we can do if students aren’t engaging with our problem based lesson; and,
- What to avoid when planning your lessons.
Kyle Pearce: Hey, hey, Math Moment Makers. As you probably know, for those who have been listening to the podcast, I've been in a role where I get the awesome opportunity to jump into all kinds of different classrooms, across from K through 12. I've got to say though, the one thing that I still see a whole lot of is our approach to teaching math classes still looking pretty common to how I remember and how I taught for quite a few years.
Jon Orr: Yeah. I'm really excited to chat in this particular episode because I think what we also see is a ton of people asking, it might be on social media, it might be in our private Facebook group, it might be in the academy, but people are saying, they'll ask, "How do I take lesson X or how do I grab topic Y and teach it so that my students are engaged or it's a better lesson than what I've done?" It's always like, "How do I take that and change it?" And then that's what we're going to talk about here in this episode. We're going to unpack the real flipped classroom and how to transform a typical lesson into an engaging problem-based math lesson. We're going to share how to slowly emerge shortcuts instead of giving them upfront like I did for so long. And finally, we're going to chat about what to do if students aren't engaging with those problem-based lessons we've just transformed.
Kyle Pearce: All right, Math Moment Makers. I'm excited to dive into this one. I hope you are just as excited as we are. So let's dive into this jampacked episode where we're going to unpack how we can problem-based math lesson-ize our typical routine math lesson. Here we go. Welcome to the Making Math Moments That Matter podcast. I'm Kyle Pearce.
Jon Orr: And I'm Jon Orr and we are from makemathmoments.com. And we are two math teachers who together-
Kyle Pearce: With you, the community of Math Moment Makers worldwide, who want to build and deliver math lessons that spark curiosity-
Jon Orr: Fuel sense-making-
Kyle Pearce: And ignite those teacher moves. Friends, as we mentioned, one of the most challenging pieces that I find is when I'm working with educators who are super eager, like super, super keen on trying to make their lessons more engaging, make them more problem-based so that it's not them doing all the talking. One of the hardest pieces is where do we start? Where do we begin? And that is exactly what we're going to be doing here today, right Jon?
Jon Orr: Yeah, we definitely are. And we're going to unpack some important aspects of this idea of this real flipped classroom and we're not talking now, just let's just get this out of the way, right Kyle? When some people here flipped classroom, they're immediately thinking to what was it like six years ago. It was a very popular, it was like, "Let's change our lesson format so that we're teaching through video and then kids can learn the lesson at home, come in, and then we can do the engaging problem or the application problem. I can be there to help them on the tough homework." There was in math class, a number of years ago, if I'm ringing some bells for you folks, is that that was called the flipped classroom. We're not talking about that. We're going to unpack what we call the real flipped classroom, but we're going to kind of talk about you want to be in a place where you're like, "Tomorrow, I'm going to teach this topic."
And for me, for a long time, it was like, "Let me pull that one out of the binder I did from a year ago." And usually for me, it was, I'm going to show a couple of examples and then I'm going to have them do a couple of examples on their own. And then it's like, "Now it's practice time." And I walk around and help that whole time. But we're going to change that here today. And we want to show you how to do that, but we want to do that so that it's definitely changeable for tomorrow's lesson. That's an important aspect. Oh, maybe you've listened to lots of episodes before, or the first four episodes, we talk about problem-based lessons and how to modify things. And a lot of those things we talk about there, take time to develop. But in this episode, we want to give you some action items that if you're starting to start change your math lessons, what can you do tomorrow so that, that has a profound impact on your pathway forward?
Kyle Pearce: Yeah. And kind of going back to that whole flipped classroom thing, the idea was basically take your regular lesson and we're still going to do it in the same order, it's just that we're going to bump it a little bit it earlier, and we're going to get students to do the lesson at home from this video. And we don't want to do that. We actually want to take the elements of these lessons, and let's just be clear, when Jon and I talk about what we consider to be our more traditional or what we remember early in our careers teaching, or the way we were taught, is that, as Jon mentioned, we would pre-teach everything students need to know upfront. So we would introduce the concept.
If the concept is slope of a line, we'd have a title on the board. We might even define it, like the first thing we do is define what is the slope. Well, I can't teach you a lesson about slope, unless I tell you exactly what slope is. We get that all down on the board and then students are going like, "Okay, I'm not really sure what you mean." Don't worry about it. I'm going to give you some examples so that you can see what slope of the line is all about. So we're not saying not to use that lesson. What we're saying is we're actually going to try to just rearrange it a little bit and we're going to encourage you to instead of sort of pre-teaching these ideas, we're going to think about what part of our lesson, and I would argue, it's probably somewhere in those examples, or maybe even some of those textbook problems that you might assign later, there's a gem in there somewhere that you might be able to grab and you can use that as your opener for your lesson.
And it's sort of this opportunity for you to come to this lesson and almost create a reason why we're going to talk about this concept instead of me sort of saying, "Just hang on." You're telling the kids, "Just stick with me, stick with me. I promise it's all going to make sense in the end." Here, we can take something, we can almost give students a bit of this productive struggle upfront in order to, I mean, Dan Meyer would call it almost create a headache in the math classroom a little bit, create the need for more efficiency for this new concept that we're hoping to bring to the forefront. So that's really what we want to do today. And again, the one piece we really want to drive home is, we don't want this to be super strenuous on you, a huge amount of planning. We want you to literally take that math lesson that you had and kind of look and think about it and go, "What part of this can I bring to the forefront first?" Right, Jon?
Jon Orr: Yeah. And I think that's a really good point because if we can bring that to the front, it's not something that you're going to have to overhaul that whole lesson. That's an important aspect because we've talked about that here on this podcast before, in the sense that you might want to try a problem-based lesson on one day, but it's almost impossible if you're starting out and trying to do that every single day. So it might be like, "I've got this plan to do this great lesson that's engaging. It's going to use these aspects that I heard on a particular episode of the podcast or I saw at a conference, but it takes a lot of planning to get there. In the meantime, how do I change these lessons so that it's not going to take me all night to plan?"
And I think that's what we want to do here. We're going to take the lesson you already had from the previous year or it can be something that you planned this week, but it doesn't take too long. But like what Kyle said, we're going to take some elements and move them to the front. And then usually what we would do at the front, we're going to move to the back. And that's what we call the flipped classroom. We're going to bring in a problem or two or a series of problems and have our kids actually think their way through that problem. And then through what they do, you might also bring up some examples later, but also create the note later in the lesson, not say right at the beginning.
Kyle Pearce: I love it. I love it. So let's dive in here and let's talk about this and really try to help people sort of envision what this might look like. So we've already talked a little bit about the order, what we're hoping that will happen from this, by doing this little by little, each day, it's almost like setting yourself a bit of a goal. In our online workshop, there's one of our lessons where we talk about not breaking the chain. And I think this is a great place to set yourself a goal. It's like a small enough goal where, if each and every day, I don't want to throw out my binder or throw out all the resources I've used in the past, I just want to think about tomorrow's lesson. What can I do in order to bring the idea out at the beginning and push some of the teaching, which Jon and I would like to call the consolidation process, right?
The consolidation process from the three-part math lessons some of you might be familiar with, the John A. Van de Wallle three-part math lesson. We want to do the thinking upfront. And then we want to do the teaching, tying of loose ends a little later. By simply doing this thinking each and every day before you plan that next lesson, you can get yourself into this habit where things will start to come to you. And those lessons will, over time, they're going to get better and better. They might not be the best lesson the first time you try this. That's not really what our goal is here today. Our goal is to get into the habit of trying to bring this to the forefront so that we can get students thinking.
And then also, once they've done that thinking, think of how much easier it is to try to teach a concept like the slope of a line when students have been with a problem and they've all come to answers. And some students have used different elements of the slope, or some students have used a table and noticed the pattern. To be able to take this student thinking and then be able to notice and name, "That's slope right there." We're going to call it slope. Instead out of us doing all of that pre-teaching and talking about slope and here's how you calculate it, and here's where you might find it, all of these things. And then now it's like now let's go actually find where slope is in all these problems.
We're going to bring it together and we're going to be able to put a nice bow on things and go, "Did you see how Jon did that over there in the table? Wow. That's interesting. Did you see how someone over here used a graph? Wow. Look at that. Did you see how this student just used number sense? They were just using multiplication, addition, division, all of this stuff over here. I'm wondering, what do you see common about these ideas here? What do you see common here?" And they'll notice this thing, this rate, and we can name it the rate of change or the slope of the line and start to kind of build this, we'll call it, this big foundation around this idea instead of, like I say, doing all this pre-teaching upfront.
Jon Orr: Right. And I think it is important to note here about the way that Kyle has just outlined this example about using slope in one sense, that we didn't teach the students upfront about what was going to happen. And we gave them a problem to solve knowing that we have not yet introduced the idea of slope. So Kyle, I think people out there are kind of wondering, "Okay, well I normally would start my lesson and teach them about slope, but now I have to put a problem here to bring out slope." And that's an important aspect because that's tough to do sometimes. And that's what we want to kind of focus on here for the next minute. And the second thing is that when you, Kyle, when you were referencing that you're observing kids do different things, pulling from this, pulling from that, I think I didn't do that initially, Kyle, when I first started changing my lessons.
I was like what we're talking about here, I've changed when we did the thinking. We did the thinking upfront, but I wasn't at a place, when we first started doing that to be like, "I'm going to reference that person's thinking here," or "I'm going to call upon them to share." My first instance was they all did the work around the room or maybe they did it on their desks. And I watched, and I could see, but I mentally referenced what different strategies students were doing on those problems. But I wasn't at a place in my comfort level of teaching to call on that or point that out to the kids or have them explain. What I did, instead, was when I wanted to do the note, right?
Like they did this problem and that problem may have emerged a shortcut that we were aiming for, right? Maybe the topic was like terms, or we got to a point where we needed some instruction to move forward, right? We've created a need for the next step and you get to step in and go, "Okay, this is a good moment for us to talk about concept X." And so what I would've done back then, would've said, "Okay, I step in," or, "I'm ready to talk about the shortcut." And then I would've gone through an example.
And then I could have said, you know what? So and so did use this over here on their solution and I kind of just walked through it as a full example. I used some of their numbers in my examples. And that's the nice thing about having your kids do that problem first. And if you take it up and walk through it and bring out the elements of like terms or in your example, slope, you get to reference the numbers and they get to see the numbers they used in your solution as well. I'm in a different place now, I'm more like, what you said, is we'll reference different strategies and we'll pull it all together in a consolidation. But before I was ready for that, I took up the solution and then made sure that we pointed out different values that they saw.
Kyle Pearce: Love it. Love it. And what I'm hearing you say is you're giving people permission to kind of start where they are and not sort of look at this as overwhelming and go, "Wait a second. You said we were just going to take one part of my lesson and put it over here." You're saying that, "You know what? Maybe that's exactly what it will look like. And maybe there's this goal of moving towards, using student-generated solutions, but maybe that's just not where you're at, yet, right? And that's okay too." So it's like taking that step. I'm really happy you kind of highlighted that because again, we want this to be that first step for some people or that next step. So for example, if you're already doing that and maybe it's time for you to take the next step to say, "Hey, I do want to consolidate utilizing student-generated solutions." But maybe if you're just starting this process, maybe it's one step at a time.
I love that idea. So I'm thinking about this and I'm going, "Okay. I could begin by simply grabbing one of these example problems that I might use, or maybe it's a problem out of the textbook that I might have normally assigned," and maybe waited to assign, because we always think word problems, students can't do those. They have to be able to do all the naked problems first, right? All the ones that are just calculating and following the procedure. But really what we can do is actually grab one of those, bring them to the front. And then we have to sort of ask ourself a question. And the question is, "What are we hoping students will walk away with?" And that's kind of a scary question, I'll be honest and say. Some people worry that, "Well, what if I picked the wrong thing?"
And the reality is that that's going to shape over time but you have to at least start by thinking about what is the intentionality of why we're doing today's lesson? Why is this mathematics being taught in the curriculum? How does it help us? Something Jon's been using a lot is this idea of a cheat sheet. What is it that cheat sheet for, right? If we're in math class, you're basically helping students to essentially stumble upon something that's going to be beneficial for something that they might encounter. So the question then becomes, "Okay, if I can identify what that thing is, that can help me select that problem that I might use at the beginning of the lesson."
And that to me is one of those big key pieces here. And we want to think about what sort of understanding or skill are we hoping that they're going to learn throughout today's lesson so that if they were to see a problem like the one they'll solve at the beginning, let's be honest, if they solve that first problem, they're most likely not going to be using the new concept that you're introducing that day, right? Otherwise, we wouldn't be teaching it. So you're going to see them using inefficient strategies.
And this is where you have this freedom to just let students bring what they have. And we've talked about this a billion times before, in this mode, in this phase is where you get to learn a lot about what students are bringing to the table, right? Are they thinking additively? So they're skip counting, they're not multiplying. They're just adding, adding, adding, adding, adding. Okay, that's really useful information for me to know. And if the thing that we're going to be exploring today involves multiplicative thinking, I might need to take that into account when we actually do that portion of the lesson, like how are we going to bridge that gap?
So there's so much that we can learn and glean when we offer students this opportunity. But then also, I think it really helps students kind of get their heads in the right space in order to go down that path to this new learning, right? It's like we've done it this way. Now let's look at how we can maybe do it a little bit, we'll call it better. We don't want to say it's better than the way they did it, but it's like a little more efficiently, a little bit faster, a little bit more cleverly, as Cathy Fosnot would say, right?
Jon Orr: Yeah. Yeah. And that's a great point, Kyle. I'll bring up a kind of a pushback. Many teachers might be saying right now, "Well, how do I do that with the time I have? I can teach them upfront and then we're over, we've given up. And then by the end of class, they're pretty much fluent with the skill we're trying to bring about." Except that, you gave kids the shortcut, right? You started with the shortcut to that lesson. That's what most of the math we teach in high school and in middle school is really. The math itself is a shortcut to solving problems. It's all a series of different shortcuts. We have so many shortcuts in our daily life. When you drive your car to work, that's a shortcut that someone created the automobile so that you could save time.
That's a shortcut. So many things that we have are really shortcuts to doing other things. This podcast, Kyle, is a shortcut to teaching math better, right? "Well, I listen to the podcast because I want to get ideas," and that's a shortcut to help speed that up for you. Math class is different shortcuts. And I think we rob kids from attaching value to learning by just handing them shortcuts without having them experience why the shortcut is needed. And I think you talked about that in the sense of we've got to put kids in these situations so that maybe we want to bring this efficient strategy to the forefront and we got to feel that, right? Kids got to feel why that's needed. And if you just hand them a shortcut, they don't feel that. And if they don't feel that, they don't attach any value to any of the learning they're doing. That's why kids say like, "Why do I got to learn this?"
It's because there's no value there. You've given the shortcut that they didn't really care about anyway. So we got to put them in a place to feel that. And I think that part is worth the time for me. I brought up this idea that we don't have time to put them in these situations where they are going to struggle through these problems. And then you're still going to help them with a shortcut later on but that struggle and that value attaching to their learning is worth that time, in my end. I think they will remember the content, they retain the ideas and the shortcuts or the mathematics that you're going to discuss in that lesson longer and stronger because they experienced that struggle, that need to have something better than where we are right now.
Kyle Pearce: I love it. I love it. So let's dive in because you're talking about want something that resonates with them. You want to give them this opportunity to problem solve and to essentially use their brain, right? And actually think in math class and who would've thought? I taught for so many years where I don't know how much they were actually thinking during the class. Maybe when they were doing the practice problems, they were thinking. But it was like I did all of that heavy lifting. So let's talk about a couple examples. I'll talk about one and this is one I used recently, so it's kind of fresh in my mind, but when exploring, "Here's this concept," sort of kind of take you down the path of what we've been suggesting here is like, "Okay, we need to understand what's the intentionality of the lesson."
Well, for this lesson we wanted students to be able to come up with an equation of a line when they're given the slope of the line, so the rate of change, and a point on the line, okay? And the way I used to teach this concept was a naked problem. And here's the thing, I used to select the values to be very, very, very easy numbers to work with. Because if the numbers are easy, this will be so obvious and everything will be fine. The calculating is not going to be distracting, nothing's going to get in the way. But the problem was there was really kind of no context and the students were like, "Well, I don't know what's happening here." So I modeled that for them. I'm like, "Here's the slope." They already know what the slope is because we've done lessons on slope where I told you what slope was, right?
This is before my time of problem-based lessons. And now I start to realize, and I go, "Okay, well what could I do?" And you flip to that textbook and you kind of look at some of those word problems and you think are there any scenarios here that are realistic or maybe you can come up with one yourself. And for me, I just thought about packs of paper sitting on a desk. And the relationship was how many packs of paper and what's the height, the total height, of the packs of paper while they're on this table, or on this desk. And for those who may not be kind of in those middle grades where you're playing in this land, just to kind of give you a heads-up, basically what we're looking at is, the equation would, be whatever the height of the table is, that's what we call the initial value, plus the height of the stack of papers. And basically that would give you your relationship.
So you could say it or you could just have a visual. You could have a visual of five reams of paper on this desk and you tell them the total height and you tell them how many packs of paper, but you don't tell them how tall the desk is. And right there, I don't have to say anything. I don't put a title on the board. I don't do anything. I've just asked them, "I wonder how tall this table is." And before I even give them measurements, we could just have them estimate, like just you're looking at it, you've seen a ream of paper before. I don't know exactly how thick they are, but I can make a guess. Students can make a guess. I don't know, 30 centimeters, 50 centimeters, whatever their guess is. And then they could use their spatial reasoning to kind of scale that and go, "Ah, I don't know, it's maybe 150 centimeters."
Get them all estimating. And then after I could reveal the total height and I can reveal the height of one package of paper and then I ask them to update their estimate. And for those who are sitting there, I know many people are listening, some people are watching on YouTube, but if you're listening, if I know the height of each pack of paper, I could very quickly figure out the height of the five packs of paper. And if I know the height of the entire packs of paper and the table, all I need to do is subtract those two values and I've found the height of the table, which is the initial value for this relationship. And what we've just done is we've used, without kids realizing it and for many years I didn't realize it, that basically that process is how you would intuitively use your reasoning skills in order to battle your way through that problem.
And then once they share that out, it's now my turn to kind of do the beginning of the lesson, my old beginning of the lesson where I go, "All right, here's this thing. Remember slope from the other day? Bam. There it is. Here's this, bam. There it is." And then I can serve up that next example and I could give that to them and instead of, again, me doing the example, I could say, "I want you to try this one. I think you can do this one." Now they're amped and ready to go because they just saw that they solved a problem without me telling them anything.
And now I've just made some sense. We just named some things like, "What you did here," like, "What Jon just did is this and what this student over here used a really cool strategy and that what you did is called that." And then all of a sudden, now we go, "Here's this other problem. I am wondering, is it going to work the same way or is it going to be different? How do you know? Are you going to be able to convince someone? Get to it and then share with a neighbor." And next thing you know, it's less of me and more of them. And that's exactly what we want when we see students thinking in a math classroom.
Jon Orr: Yeah. And I think what's really important to point out here, Kyle, is that your kids are doing the thinking and then you are doing the summarizing at the end, in naming the things that you want to name and then making that consolidation or the note at the end of the lesson. This is where this flipped idea comes from, is that you did problem eight from the textbook first and then you unpacked it and then you've done examples one, two, and three afterwards. And again, at that point, if kids are already showing that they can do that, they can then do examples one, two, and three on their own. And then you really probably didn't have to do any examples in front of them. And I think that's what happened to me as we did that process.
It's like well, we did a hard problem. We put them in a position that they could solve using inefficient strategies. We tied it together so that we talked about efficient strategy. Now it's like, "Okay, well now do this one. You did this one." But we've talked about efficient strategy. It's almost like I used to do like four examples before setting them to do the homework or the practice questions. And then you're trying to cover your bases to get all the pieces in there. But then if you pick your problems right at the beginning of the lesson and have them do it, you don't have to do anything. And they do all the thinking and you're doing the guiding after that. So another example I'll toss in here quick that I referenced earlier about like terms and I think really a big idea in order to pick the problem, right Kyle?
Picking the problem is important because it has to be open enough for them to try different strategies without knowing the goal or already understanding and knowing the goal that you're going towards, because maybe they can't solve it without that. You've got to pick that problem that allows them to bring in those inefficient strategies so that you can talk about it. For example, if I'm talking about like terms, understanding the big idea behind these topics is so important because it will help you pick that problem that allows it, right? If you're like, "Well, like terms are about just adding Xs together and collecting those like terms and adding these." If that's my understanding of why like terms are important in mathematics or the big idea of what we're really doing there, then it's going to be tough to pick that problem because I only see like terms that way. And when you see like terms as creating equivalent expressions or grouping items together to shorten up the calculations, right? That's an important aspect. It's like I could put these comp... You can hear my cat. Right here. crosstalk
Kyle Pearce: I'm just smirking because-
Jon Orr: Well keep it in, keep it in.
Kyle Pearce: Yeah.
Jon Orr: It's Taboo. He's going crazy. This is the one you took a picture of on the weekend.
Kyle Pearce: For the Virtual Summit I-
Jon Orr: Yeah.
Kyle Pearce: Just passed it on the couch?
Jon Orr: Yeah.
Kyle Pearce: Totally Math Maker.
Jon Orr: So back to the idea of the big idea is that you have to know what that is. And so in like terms here, if I can group things together, I can shorten up calculations and create equivalent expressions that are nicer to work with. Those are big, important ideas. Equivalence, that's really what's the huge idea there. One thing that's easy to do is, is instead of just saying, "Let's add these like terms together," I've seen it done lots of different ways, but you could throw up this shirt cost this and this shirt cost that or these pants cost this and then just throw them all up on the board and then say, "Total it all up."
And naturally kids are going to do a strategy that already makes sense to do, is group all the things that speed that calculation up. The last couple of years, I've created a big page that has a ton of different shapes all over it. Rectangon is building on our work from area of rectangles and squares, in trapezoids, like disc shapes, but there's a whole bunch of the same types. So there's a whole bunch of triangles that are exactly the same size. There's a whole bunch of squares that are exactly the same size. And I just say, "Find the area of all these things." And I pretend that I've got to do something else. So like, "Hey guys, I need you guys to do this problem while I get ready for the real problem," which is a lie, right? That's the real problem.
And so I'm pretending to fiddle with stuff at my computer and students are like, "Okay, so we're going to do the area this, so then we take this up," and they've all grouped all these same shapes together and found the area of one and then said, "Well, there's seven of them, so we're going to multiply by seven." And so we then take that work and be like, "Well, let's write a formula because oh wait, I forgot, sorry guys. I got the dimensions wrong." And then I changed the slide and it's the same shape, same orientation, but all the dimensions are changed. So then they're like, "Ah," they got to do this again. And so they're like do this again. And then some kids are like, "I don't think we have to do that really again. We just have to change a couple numbers here."
So we're building an equivalent expression for the area of all these shapes and we're understanding there's power in grouping and there's power in simplifying, in creating this equivalence. So if you can understand the big idea behind the math that we're trying to teach and you got to go big, right? You can't just go granular. Think where does this fit? What's important about it? How does it fit in the grand scheme of things? That can help you immensely in choosing what is the problem to start with.
Kyle Pearce: I love it. I love it. And something that I think you touched on a little bit, so I won't go too deeply in it, is a pushback we often get or just a question. A common question you get is like, "What if I can't think of a good initial problem or the reason for this concept to be taught?" Some people are like, "I don't know, we always push for context wherever possible." But really context is only important if it helps you with the thinking. So we don't want contrived context. The one with the area that you just described, there wasn't really context. The context is the fact that there were visuals and there was area. So the context is we know what area is, they have an understanding of that, and they can build off of that.
But in reality, what you've done, yeah, at super low floor and you've created almost a bit of a puzzling prompt. Your puzzling prompt was almost a non-prompt. You basically said, "Hey, just stay busy for a couple minutes." And that was your way to kind of get them into entering into that problem. And that's using some of the elements of that curiosity path, which we don't want to mar down today's podcast on making the prompts curious, because that's something I think you can work towards over time. Withholding information is always helpful. So like Jon, you had kind of withheld information in that particular scenario. I withheld information with the stacking paper problem by not having measurements on the image at first and had them estimate. So we could do all of those things but the reality is that as long as it's puzzling enough or challenging enough, again, those are two things that help to make it curious enough for students to enter into a problem.
The example I love is, you were describing it a little bit earlier today, is this idea of finding the distance between two points. We could just start by just having two points on the screen and nothing else and ask them, "How are you going to measure how far they are?" And there's no context there. Sure, you could say, "Hey, this is this town and this is that town. You can add that if you'd like." But we're trying to give you a scenario where it's sort of because it's such an easy entry point, it's almost irresistible not to participate because you could have kids estimating, some kids think standard measure and they'll be like, "That's like a foot apart." And we're like, "Oh, okay, that's a foot apart."
Yeah, what if we didn't have a measuring tool, right? And then all of a sudden, maybe we throw a grid on there and the grid has a scale on it. "Oh, those are actually kilometers. Okay, great. Now how far do you think?" And getting closer and closer to build the need for a way, a means for us to actually get a more precise calculation or a more precise quantity that we're trying to unveil through that problem. So that's another big one.
So again, don't put too much pressure on yourself for every single initial prompt in your class to always be a grand slam because also, think about it. We're telling you to keep your lesson that you were going to use and use an element of it in a different spot. So that's what your lesson was going to be anyway, right? So don't overwhelm yourself and maybe not do this process because you're worried too much about making that intro problem perfect. Let's just get that order swapped and get comfortable making that the routine in your math class, where when we do math, we start with math and then we make sense of what we've done afterwards. And that's kind of, I think, the big sort of takeaway we're hoping people are hearing out of this episode here today.
Jon Orr: Good summary there, Kyle. I think you've really nailed one of the big ideas we want to drive home here in this episode. I think before we wrap things up here, there's a couple things we should bring back up too, about what happens if this happens. When you're trying something new, you start to think about what happens if this is going to happen? And then we've got some common things that have happened to us, like what happens if I change this but students aren't engaging in that initial problem, right? Or that initial task? And one thing we can do is, if that's true, if you've set the bar low floor and you brought that task and that built on maybe what they did the day before that, or maybe it was like my example with the area we had used area before, but we were really kind of putting it in a different context, if that's not getting kids to step into the problem and get thinking, and that's probably going to be like that the first time you try this.
They're going to be like, "Wait, you usually show us what to do. If that's the case, I'm still going to wait here for you to show me what to do." That happens to us a lot. Especially when you start a new class, the new year and you ask kids to do this, they'll kind of hold back. I know something that has worked well for me, Kyle is, you referenced the curiosity path and how to adjust for that. And we're not going to go down that path, but that can work. But the other thing that has helped me so much in the last few years is getting our students standing and getting our students to discuss with each other. And me putting the whiteboards up on the walls has changed so much about how much engagement initially happens when I pose that problem.
I'll say, "Let's go to the walls," and students are right away working out the walls to solve that problem. I think just the fact that they're standing has changed that part. So that's something that folks might want to give a shot to. It's one of the easiest things you can do and not change too much about what's happening on your day to day planning by having them move to a wall space, to work in partners.
Kyle Pearce: I love it. I love it. And the other thing that came to mind, as you mentioned that is something that we used to do. And again, you might feel comfortable doing this and maybe this is your comfort zone, but I would encourage that if you do want them truly problem solving, try to give them as little as possible upfront. And that includes a handout. It was almost like I wanted to make everything so easy for everyone. I thought that that was what they needed. So I'd have a template and there'd be a graph and a table and fill this in and fill that in. But that template sort of approach, especially with this upfront problem, I'm not saying you can never give a handout during your lesson. Maybe at the end of this problem solving, you might have a summary that you want to work and build something together, just for time efficiency and things like that.
We're not saying to never do that. But what we are saying is that if you want them to truly come at this problem with their own ideas, their own strategies, their own tools, and models, you don't want to give them a paper that has a table on it and a graph on it because then in their mind, they're going like, "Mr. Pearce wants me to do a graph." And maybe that wasn't where their brain was going. And maybe the point of the lesson is to graph, but remember this intro prompt, this intro problem, isn't for them to show how awesome graphing is. We're trying to almost do the opposite, show them how helpful graphing can be afterwards.
Some students might choose to do a graph and that's super cool. And you'll get to highlight that. But when we sort of send them down this path, you're making them stop thinking a little bit because they see what's on this template and whatever they had in mind, they're like, "Oh, well, that doesn't fit on here. So it must not be good or it must not be valid." I wish that I would've known that ahead of time. I would've saved a lot of, I guess, trouble with trying to get my students to start thinking.
Jon Orr: Yeah, good point there, Kyle. And I know that I still use templates, but like you said, I introduced them after the fact, after they've tried their thinking. Two examples come to mind in the same topic. One is when we've done your task on the candle burning task... The candle burning task is you can see this candle burning. And then you're trying to figure out when, how long does it take to burn completely out. And for us, we've used that as an introduction to scatter plots and using lines of best fit to model the data. And in the past, if I gave them the template first, like you suggested Kyle, there's no thinking. We just fill in the table, we fill in the graph. Now you still have to go, "This is the line of best fit." But when I change that lesson to not give them anything and ask them to answer, most kids actually won't use a graph in the beginning.
They will take the data points and they start to average them. Usually kids will start to build a linear equation actually out of it and start to go, "Okay, well, if I average these and it's going down by this much every minute, well, how many minutes until it runs out?" Which is amazing because there's so much to unpack there. And then when you bring in, on your consolidation, if you're still going to introduce lines of best fit, what a perfect way to talk about what a line of best fit. It's an average line. What did you guys do? You made the average. What average did you get? Oh, it's this. This is our slope of our line. Oh, it's all about the same. And you can draw a line and you can say, "Look at the power of the graphical solution here is that you can visually see where this is going to end up and it's not, so you're not so bogged down in some of the algebra that you guys were in."
But if the kids were doing the algebra, hey, that's awesome too. And then I've also used that when we did Barbie Bungee activity, for the exact same skill in doing that again. So important to bring up these ideas. Let's do a quick recap here, folks. We wanted this big idea to come out, being that something you can do tomorrow is take your lesson and just reverse parts of it, change a couple elements, put one element you might have done later at the front, get our kids to think. That's an important aspect because creating that thinking will do lots of side benefits here for you with getting them to see the shortcut, which is a lot of power. And also getting them to attach value to learning.
And students are going to deepen that understanding. Deeper, because they've made actual personal connections too. And think about this, Kyle, we didn't talk about this in this episode specifically, but when you get your kids to think you are telling them they're worth the time you allow to have that thinking happen in class. There's value you in your thinking and I want to see it. Think of the message your students get when you dedicate time at the beginning to go, "I want to see what you're going to come up with. I want to see your thinking here." And kids will feel empowered and they'll feel worth it.
Kyle Pearce: Absolutely. And you just kind of gave me another epiphany here. Sometimes educators, we tend to tell students not to do things certain ways, because we want them to get to the thing we want them to be using, like the graph. In your lesson you could have said, "Don't average them all. Don't average all those data points. I want you to use the graph today." And there's a time and a place for that. So in some of the practice problems, you can specifically ask them to utilize a graph in this scenario. But again, in that upfront learning part, valuing their voice, valuing their thinking is so key. And the one big takeaway I hope you walk away with is, again, this idea of getting in the habit of how you start your lesson. Now, mind you, I'm talking about a problem-based lesson. Some people might do a number talk or a math talk before.
But I'm talking about when you introduce this new concept that you normally do your note on the board for, we want to just get you in the habit of getting students thinking first. It will give you so much space and it will give you so much more attention because students have already grappled with the problem. And then bringing it together is so much easier for you so it doesn't feel like you're talking and students are kind of looking at you like you have two heads, right? They were all a part of the process and now we can use that thinking in order to drive home the big idea, the big point at the end. Another key piece that I wish I knew was, when you do a problem-based lesson, we don't just do the problem and then just hope everybody got it. I did that a lot.
And that's where I think we get a bad rep. When we talk about problem-based learning or inquiry-based learning, or discovery-based learning, it's like this thought that students will just do it however they choose to do it and that's just it. And that's the way it goes. And it's like no, we have to be really clear to bring it together and bring that new learning to the forefront. So you're still getting that opportunity. You're not throwing out all of that good stuff that you would normally share with students. We're just going to do it a little bit later in that lesson and then give students other opportunities, other prompts or problems to work on, to kind of drive home that point. So, wow. This one's been a big one as they all are there.
Jon, I don't know about you, but I love talking about how we can slowly transform these lessons and I'm hoping that everyone feels like this is something that they can actually do, that you can actually attempt to do in tomorrow's class. And, I don't know about you, I want to hear from some friends, whether you're responding on the podcast episode show notes page, or whether it's under the YouTube video on YouTube, we'd love to hear how this went for you. And remember it might not be perfect the first time, but get in that habit, try not to break that streak, because if you do this once in a while, only on Fridays, you're not going to get that consistency from students knowing that this is how we learn mathematics in this classroom. So build that routine, your students will follow. I think at the end of this, you're going to feel like wow, you're seeing a ton of thinking that you never realized was there.
Jon Orr: Yeah, exactly. Thanks Kyle for that. And as a reminder, please do what Kyle said. Share your big takeaways with us. That might be on our Twitter page or Instagram or in our private Facebook group, Math Moment Makers K-12 over on Facebook. Share what you've been learning, share all of the things that you thought. Maybe you disagree with us. Hey, share that too. We want to hear from you. Don't miss our next episode. We're putting out episodes every Monday morning and to be sure you get that next episode, subscribe. Hit the Subscribe button, right now, if you have not yet done this. Maybe this is the first time you've listened to us. You can head to the back catalog and get all the episodes that we've ever had in the last three years. So, Kyle, we're coming up, I think, our anniversary. It's three years, right? Coming up, I think, yes.
Kyle Pearce: This is crazy. Holy smokes! This has been fantastic. I'm just looking at the episode number even there, Jon. I think it would've been what, last week's episode that went out, that would be like the official, even though we jammed a couple extra episodes in at the beginning. But holy smokes, three awesome years! Friends, go check out those show notes and links to resources on our wonderful, wonderful show notes pages on makemathmoments.com/episode156 for this particular episode. Or is it 157, Jon?
Jon Orr: Yes. 157, kyle. Thanks for correcting that.
Kyle Pearce: There it is.
Jon Orr: This is episode 157. crosstalk
Kyle Pearce: Yeah, there it is. Okay. Makemathmoments.com/episode157 is where you can find transcripts, you can find links to resources, all the show notes. And obviously, you've got the gateway to all kinds of other ideas like our problem-based math lessons. Once you're there, click on the Tasks and, you know what, let us do some of that hard, heavy thinking for you by using some of those problem-based units? All righty, my friends. Well, until next time, I'm Kyle Pearce
Jon Orr: And I'm Jon Orr.
Kyle Pearce: High fives for us.
Jon Orr: And a high five for you.
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