Episode 142: How To Teach Through Problem Based Lessons
In this episode Kyle and Jon dive deep into a problem based task and unpack it so you can make math moments every day in your classroom.
- The teacher moves we need to make before starting a task with our students;
- The elements that are necessary to build a great problem based lesson;
- What a problem based unit looks like over the course of 5 days; and,
- Where purposeful practice fits in to your lesson.
Kyle Pearce: Hey, hey there Math Moment Makers, it's Kyle here and Jon's hanging out with me, and actually, we are also on YouTube. So if you're watching, or listening I should say on the podcast, you can also head over to YouTube and check out. We're going to do some screen shares as we go. We're still going to be articulating everything as we go. So if you are listening while you're driving or running, don't you worry, we're going to make sure that you paint a picture in your mind. But today's episode, we're really excited about because we've got a lot of feedback after episode 139 where we talked about making Math Moments from day one through day 180. And the big idea was not only just getting that culture built in your classroom through some of the activities we've shared in the past, but also diving deeper into what is our math program look like?
So when we do this, we certainly don't want students to feel like okay, I got sucked into another day one, day two, or even day three gimmick where we do games and activities and we have some fun. And then all of a sudden math class turns into this dry, boring subject that many of us may remember from our own experiences. So that's what we were talking about in that episode. And something people were asking us about Jon was, "Can you take us through a little deeper around what does it look like when I lead my math lessons through a problem-based approach?"
Jon Orr: Yeah. So it's like, what does the problem itself look like? What do the days look like? Because we have units and we actually teach through these progressions with units. So we kick off on day one, what's day one look like? What's day two look like? Are we doing those problem-based lessons every day? Are we doing a notice and wonder and a predictions type thing every day? Or when do we build in practice? And when do we build in some of the ideas of collecting some assessment? We've got a lot of questions that people ask about, like what do our problem-based routines look like? That's what this episode is all about. So let's hit it.
Kyle Pearce: Welcome to the Making Math Moments That Matter podcast, I'm Kyle Pearce.
Jon Orr: And I'm Jon Orr. We are from makemathmoments.com and together ...
Kyle Pearce: With you, the community of Math Moment Makers worldwide who want to build and deliver problem-based math lessons that spark curiosity.
Jon Orr: And fuels sense-making.
Kyle Pearce: And ignite your teacher moves. Welcome everybody. We are super geeked to continue diving into some of really how we structure a math block. And in particular today, as we mentioned, we're going to explore what does problem-based learning look like day-to-day. It's one thing to lead maybe a media rich task, like a three-act math task, or maybe some of the problem-based or contextual tasks that we've shared on some of our websites from years ago, back on Jon's site mrorr-isageek.com, or my site tapintoteenminds.com. But for those who have sort of migrated over to the Make Math Moment's problem-based approach, you'll probably know that there are much more thorough and actually much more structured to help you do this from day-to-day. So it's not sort of a one-off thing. Jon and I used to begin Friday problem solving and we realized we weren't really getting much leverage on that. So that's what today's all about, Jon, what are you thinking?
Jon Orr: Well, what am I thinking? So let's just kick off before we dive into what we have evolved our problem-based approach to teaching, because we would say we started our lessons ... And if this is the first time you're listening to this podcast, well welcome and make sure you're subscribed. But if you have been here before, you've probably heard our stories of where we started and how we've progressed into teaching this particular way that we're sharing here in this episode. But I was a very traditional math teacher and I taught very traditionally. Imagine a high school math teacher, that was me for a long time. I taught, "Here, do this, then this. Now practice it a whole bunch of times and we'll take up the homework the next day." That was me for a long time. And when we morphed into realizing that we needed to change our practice to see engagement results, but also deeper thinking. And that's the biggest change I think we've made is getting thinking out.
And that's really what we want to talk about here with problem-based lessons because we should start off talking about what that is and what that isn't. And then we can show you what we've been doing to help what goes into a problem-based lessons. We want to share in this episode what it look like, what are the things we should think about when designing a problem-based lesson or getting ready to teach a problem-based lesson. But let's first talk about what like what it is and what it isn't. But basically, we feel like a problem-based lesson is any situation where you're asking students to engage in thinking and strategies to solve a problem that maybe they didn't know how to solve in the beginning. That's a pretty general definition.
And you can do that in lots of ways. Some people are like, "Well, you got to show a video for that kind of thing. That's a three-act math task." Well, no, for a problem-based lesson, you don't have to show a video. It could be a picture. It could be just a context where you set the stage, tell a story, or you're doing a demonstration in front of the class. It involves a piece of curiosity that gets our students curious about what an answer might be. We want to do some predictions. We want to do some anticipation. We want to build that in. We call that the curiosity path, but really there's lots of different ways to do that. But really what we're getting students in the door to help solve a problem, then we ask them to solve the problem and we try to elicit what information they need to know. There's a lot of thinking that goes on in this, and you can learn more about that, we call that Make Math Moments Framework, a three-part framework, and you can learn more about the actual framework themselves at makemathmoments.com/framework. We talk about all the pieces there.
But I'm kind of thinking about this general idea of problem solving and what tasks look like. But the biggest idea for us is we get thinking from our students to start, and then we're going to do some consolidation later. But this purpose of this episode is what's that look like from start to end? And what's that look like the next couple days? So what we're going to do here, and if you're watching both of us over on YouTube, you're going to see what goes into the problem-based lessons that we teach with? What we think about when we teach them? And if you're going to make your own, or if you're going to design your own, because you definitely should and can, then you're going to want to write down some ideas of what you should be thinking about.
So we're going to dive into that and to help do that, we're going to showcase one of the lessons that we've created over on our tasks page, which actually it's a full unit. So we do want to talk about what it looks like on day one through day, I think what five here, Kyle?
Kyle Pearce: I am on this one. Yeah.
Jon Orr: Yeah. So we're excited to dive in to all of these elements with you so that you can make Math Moments That Matter everyday.
Kyle Pearce: Yeah. So like Jon's saying, you definitely want to check out the framework. I think that gives you sort of like the pedagogical sort of underpinning to get a sense of what it is that we're trying to achieve on each day. And like Jon said, you could do this in many different ways. We've got a downloadable guidebook that you can take with you, print, bring on vacation, whatever. Because it is quite lengthy, but the whole idea here is that what we don't want to create is we don't want to create a classroom in the longterm where students see it as sort of like flip-flopping between approaches. And what we mean by that is often how you start shifting your practice is by doing just that. Like I had mentioned earlier in the episode, we used to do a problem solving Friday or the review day of a unit, we would take a contextual task and we would try to have an engaging lesson, but it was almost like students were already pinned down by all these algorithms that they weren't really willing to take a chance.
And what we realized we had to flip that around and we had to incorporate this into our main teaching practice and make any sort of direct instruction happen after students have been given some time to explore, inquire and share their thinking. And at that point, that's where we kind of take what they're up to and we try to bring it together and we try to help, nudge them all forward by offering some of our perspectives based on their thinking. So from a nutshell, that is sort of the big idea. And the three-part framework can really help us with that. So, if in your mind you're thinking, "Okay, I'm going to pre-teach division." Maybe long division or maybe some division strategies. The idea is trying to get division to happen and that we help to consolidate. When we say teaching, when we're going to teach students something, we try to do it after an experience has happened. It's like a big takeaway. It's like a reflection, an opportunity for them to go like, "What did we learn here?" Think about your favorite stories. There's always a message in the story, and they don't tell you the message at the beginning of the story, it's at the end. You get the moral of the story once it has happened, once that experience has happened. And we often reference the hero's journey as sort of a nice, clear connection to that.
So today what we're going to do is we're actually going to flip over to our problem-based math units section of the website. If you head over to makemathmoments.com/tasks, that'll bring you right over here where we have 30 plus units that are created to really provide you with an opportunity to, I guess, maybe fast forward through some of the struggles that Jon and I both had, where we were like ... I was one day doing I do, we do, you do, and then I would do that again the next day. And then maybe I would throw in a three-act math task. And I would wonder why sometimes students didn't fully engage, they were like, "Well, what are we doing here? I thought you were going to tell me everything, and now you're asking me to think on my own."
So we really want to help you get to this place where you can feel confident that you could give students the opportunity to think, and then we can help to consolidate that. And that's what we do here through these problem-based units. And actually, we continue to release new units pretty much every week, every other week. And we are just so, so dedicated and committed to making this happen for the community so that we can take a lot of that planning off of your hands. So as of this recording, one of our most recent ones is planting flowers, right Jon?
Jon Orr: Yeah. And I was just going to say, let's dive into one and thinking about when I say one, we're talking about one unit. So this also addresses this very common question is like, what are we doing day-to-day? Day one versus day five? Are we always doing that notice and wonder on each of these days, teachers will say, "Well, how do I have time to get practice?" And we're going to bring that in here. So we're going to open this up and we're going to jump, I think, right over to the teacher resource guide. We have the teacher resource guide there purposely because it teaches you how to move through a task. And that's what we want to do here in this episode. How do we move through the task? What are the things we should think about? And then what should we be thinking about after a task is done? Because we are going to move to the next day where practice might come in.
So Kyle, let's talk about what are the first things we should think about when we decide to run a problem-based tasks? Because we're talking about problem-based tasks, many teachers think, "Well, I'm just going to toss this curiosity prompt up there." But I think there's a lot that goes into before we even attempt that, right Kyle? So let's talk about what needs to happen on the teacher move side before we even present a problem to the students?
Kyle Pearce: Yeah. You really nailed it. I mean, you'll notice that when you go to, let's say, a typical three-act math resource site, and I'm going to use my site as an example, many people who are listening to the podcasts have maybe been there. And you'll notice there's a task, there's a question, obviously there's intentionality possible there, but it wasn't super crafted. So there was the risk of just grabbing the task because it was cool and doing it with my class, but who knows whether the intentionality would actually come out through that task. So what we started developing, as Jon said, was multi-day units with teacher guides that are really clear and intentional. So you'll notice, up at the top, that you'll get in this task a general sense of what's going to happen.
So in this particular one called Planting Flowers, we talk about exploring the ratio of flowers to flower boxes, and they'll scale the composed unit in tandem and reveal a rate. Now, right there Jon, I don't know about you, but about 18 months ago, before I went down the rabbit hole of proportional relationships and how complex and how muddy it really is, I didn't know what a composed unit was. I didn't even really ... If you said to scale in tandem, that wasn't really common language for me. And when it says reveal a rate, you'll notice that throughout, we use this idea of revealing a rate oftentimes because again, a lot of people just say, "Well, isn't this a rate and isn't that a ratio." So the whole intent here is not only to provide you with a resource to use with your students, but also to give you some built-in, professional learning. And you'll notice in the intentionality section that we actually dive in very particularly into what it is that we're hoping to achieve in this particular unit?
Jon Orr: And I just want to touch on this, Kyle, because the intentionality is so important and that's the first thing we need to think about when we decide which task we're going to use. Because I think we used to get caught up in this FOMO, right Kyle? It's like, "I wanted to teach with this cool resource because someone else said it was great or I saw it and I thought it was going to be super awesome for my students and fun, engaging." And then we're like, "I'm just going to toss that in there without actually thinking about what the learning goals should be." Or the learning goals you want to bring into your classroom for that lesson. We have our standards, so we got to make sure that we are bringing those standards out with our students so that they can learn them deeply. We got to make sure that it matches what the task is going to be.
So if you're going to make your own task or you're going to get a task, definitely you want to think about that intentionality piece because that's the whole thing. At the end of the lesson, we want to make sure that we hit the nail on the head. We want to make sure that we brought the learning goal out so that we can talk about it. There's nothing worse than going through a lesson and going, "I thought this was going to be super engaging." But at the end you're like, "Well, did my students get it?"
Kyle Pearce: Or what were they supposed to get in the first place?
Jon Orr: Yeah. So the intentionality, you definitely have to start there. So if you're making your own task or you're picking a task, what is that intentionality? Write it down. And that's what we've done in our tasks. We start with the intentionality.
Kyle Pearce: Absolutely. And as you'll look in this particular unit, some of the big ideas that we share is that there's two types of ratios, a compose unit, and a multiplicative comparison. I want to be honest here and say that I never used that language in my own classroom until I was out and I was doing my consulting role that I'm in right now because I didn't understand it. I didn't know that they existed. So what we're trying to do here is we're not trying to complicate things, right? It is complex, but we're not intentionally complicating things. We want you as the educator to kind of understand what's going on under the surface. Because if I go through and I teach ratio and rates at a very, we'll say, high level, and we sort of just solve problems involving ratios and rates and just kind of hope students can cross multiply, we're going to run into some struggles because there's so much going on under the surface.
For example, this idea of a composed unit is often, but not always a ratio with two distinct units. You might be going like, "What does that mean?" As we go through the unit, you start to make your own connections and go, "Holy smokes, I see what you mean here." We can scale them in tandem. Equivalent ratios are derived from the same rate. If you think about that and you go, "What does that mean?" You definitely want to do this unit yourself because in the consolidation steps, which we're going to get to here, you'll start to go like, "Oh, I didn't even realize that was going on there." And then finally, when you divide a composed unit ratio, you reveal a rate and we are using partitive division every time when that happens. And again, I never knew that. And we're not here to say like, "You don't know that."
So there's so much work you have to do. If you're wondering why students might be struggling in this multiplicative land, where students are spinning their wheels, you're spinning your wheels, this will help you kind of get to the nitty gritty as to what's actually going on in the background? And of course, we provide a contextual and often very visual, maybe even animated way for you to bring this to your students so that the floor is low. Of course, I think it goes without saying that we're not going to put on the board the intentionality of all these things. This is for you as the educator at this point. We want to emerge these ideas throughout this unit. We want to come back to them as well because let's be honest, you are dividing all the time in math class. Once you hit division in grade three and four, division never goes away. It's happening and this is something that we need to continually kind of go deeper with. So Jon, now that we have the intentionality, where do teachers go and how does the guide help them get there?
Jon Orr: Exactly. So if you're thinking about running a problem-based task is what we would encourage you after you kind of nailed down what you want to accomplish is thinking about how you can use the curiosity path to spark curiosity, and spark engagement in your students. And then that will help set the stage for the lesson. And we do that in all of our tasks on our tasks page and we've talked about that here often in the podcast. If you head back to episodes one through four, we definitely outline what that looks like, what we're talking about about the curiosity path. But we're doing a sparking curiosity with withholding some information. If you're here with us on YouTube, we're showing you what this Planting Flowers lesson looks like to help spark curiosity. We got a little animation here. You're going to show your students. You're going to ask them to notice and wonder, and what we're doing here is we're holding information so that it drives some anticipation in your students, and then we're going to work towards having them make estimates and predictions here to kind of hook them into wanting to learn more about this problem and then solving this problem.
So that's the part that's next. If I'm going to teach a problem-based lesson, I'm not just going to present information and then show them how to do this. This is we have to open up the floor so that they can step into this task. And we use the curiosity path by asking the notice and wonder so that they can get in easy, and then we'll amp it up a little bit by introducing a problem.
Kyle Pearce: Awesome. Yeah. So as you just saw those on YouTube, you saw we sort of had this context sort of mapped out here, there's some flowers, there's some planter, pots. And then we sort of just very quickly gave you a flash of a bunch of planters and a bunch of flowers. And the intent there was to really allow students, again, low floor, high ceiling, to notice and wonder. We haven't asked them a question yet. We were asking them to kind of just share what they saw. They'd probably say things like planter, pots, flowers, these are all listed here in the teacher guide. Some of these ideas students might've said. And then we're going to eventually land on a prompt and we were going to have them actually estimate how many flowers do they think they saw?
And again, this is very low floor because it's very visual. Some students may have looked at all the flowers, other students may have really quickly supetized using conceptual supetizing, they were like, "Every flower pot had six flowers in it." I believe. I'd have to double check that now, but I think that's what I saw. And that's what students are going to be thinking too when they make the assessment, "I think I saw six flowers, maybe it was eight flowers." And how many flower pots did they see? And this is really going to this idea of using multiplication and start thinking multiplicatively, however, some students might use counting or additive thinking in order to access.
So this is how we start off with a really low floor, get everybody talking, get everybody as a part of the lesson. And this is going to use some time, but I'm going to argue if all students are benefiting, here we're giving students an opportunity to leverage mental math skills and really emerge an opportunity for them to leverage their mental math skills, building models and strategies. We're going to ask them to actually share their thinking and they're going to have to convince us. In everything we do in our problem-based units, you'll notice we have messages all over that are saying, "Promote students using a mathematical model, not a calculator." And the intent there is we want all students to be able to create an argument for why they came up with what they came up with. And then we're going to work towards more precise calculations by revealing more information. And really these are all set-up leveraging the curiosity path, which we try to share, obviously through many of our webinars and through this podcast.
But as we map out these units, I think something big for educators to see as well is that currently, the units don't cover every possible concept out there and every possible grade level. But how might an educator use these units as a means to help them transform their teaching, even if they were to just use their own resources that they've been using, whether it's a textbook or maybe it's the binder that they've had, how they might be able to apply this idea to some of the stuff that they've already been doing versus throwing out the baby with the bath water.
Jon Orr: Sure. Sure. We're going through some elements here about ... Kyle talked about the curiosity path, we strongly encourage that you can do that with any resource about how do I strip information away? How do I withhold some information? We can do that with lots of different resources. How do I slowly bring that information back so that we can build some sense-making? We can do that with not just our tasks, you can do that with your textbook, you can do that with resources you find on the internet. It's thinking about how do I craft a productive struggle with my students? If I can get them into a productive struggle, then they're going to be doing some thinking and that's what we want, right? We want our students to do that thinking. And on our tasks and in the tasks that we talk about, we call that the sense-making, how can we get them into them solving a problem?
And then the moves that we're making while they're solving a problem, we usually typically rely on the work from Peg Smith, Mary Kay Stein, and the 5 Practices for Orchestrating Productive Mathematics Discussions is we're doing some selecting, we're doing some sequencing or monitoring our students work while they're in this productive struggle. Because remember, we have this intentionality in mind and the tough part is how we take that intentionality and see what our students are coming up with in this productive struggle part, in the sense-making part, and we have to kind of connect that together. That's talked about in the 5 Practices. And if you are a fan of the 5 Practices and you're trying to use that in your classroom, our tasks are built to mimic some of the work that Peg Smith, Mary Kay Stein have developed there in a sense of like we definitely provide you the connect stage of the task.
And that's so important I think. And it's the part that when you're making your own task or deciding to do a problem-based lesson, that consolidation part, that connection part of like, "What my students have done? How do I tie what they've done to the learning goal that we want to make sure it comes out?" Maybe you've seen the strategy that you want to make present from your students. You can highlight them. You can ask them to explain. You can ask different strategies to explain so that you can connect all the strategies together. And if you don't see them, then you can feel free to interject at the end and say like, "Look, we've got these strategies that were all great. Let me show you this other strategy." And then you definitely want to connect those strategies to the strategy that you're looking for, or that learning goal that was the intention of this problem. You're going to do that with any of the problems you teach through problem-based lessons. On our problems in the task area, we've outlined how to do that and giving you some resources to do that.
Kyle Pearce: Yeah, absolutely. So with this particular problem, so we did an estimate, those watching on YouTube saw the reveal of our estimate, there were 72 flowers. Students will have different strategies. You'll let them share. Some students are skip counting, other students have it as a known fact. Other students might be using other type of scaling scenarios like doubling and that sort of thing. As we go into this particular task, the actual struggle prompt is about taking those 72 flowers and planting them in four boxes in the front yard. This is all very visual, those who are watching right now can see this. We're asking how many flowers will be in each box? And then if we wanted a consistency in the backyard and we had six boxes to fill in the backyard, how many flowers do they need to buy for the backyard?
So this is fairly complex for those who struggle with multiplicative thinking. And the intent here is that everybody has an entry point and we make it very visual to encourage. And you'll notice the boxes here are from a top view, which is really going to hopefully promote this idea of arrays. And then maybe, in the consolidation we can help them get closer to maybe a linear-type model that they can leverage with any type of multiplicative problem, any type of ratio problem. And as you'll see in our inaudible moves here, we do reference the monitoring and selecting and sequencing, and we give some guidance on how you might do that with this particular problem. What you might see, some of the tools and representations. And then we don't stop there, we actually give you some possible student approaches. Sometimes it's actual student work. Other times it's crafted by us as the educator jotting down what we might anticipate seeing. By all means, here this problem has three different approaches, these are not the only possibilities. So keep that in mind.
Jon Orr: If you are planning out this, because that's the purpose of this episode is to help you plan these yourself and see where you can make improvements or help yourself. And I think one of the most important things you can do, which is from the 5 Practices, which is anticipation. That's the first practice from the 5 Practices is that once you have the task, you want to anticipate what students are going to do. So even though they come down lower in our teacher guide, these are the approaches that you're going to want to try before you even tackle this problem with your students, right? It's in the teacher guide. You're supposed to be reading the teacher guide from us before. But if you're making your own, you definitely want to plan out what you think your students will do on this problem. Map that out so that you're ready to go when the students are solving the problem. Because if you can anticipate what strategies that they're going to tackle, then you can be ready and feel confident that when you see that strategy, you can then have this good map in your mind of how you're going to connect it to the strategy that you're intending. So it might not even be the strategy you're intending, but at least you'll know.
So we've provided you them here so that you can see them, but definitely you're going to want to create and think about what strategies your students might come, it might be some of these actually, it actually might be more.
Kyle Pearce: And something else to consider as well. If you are making these on your own, or if you're using ours, it's always a good practice for you to do them first and then check and see. Or if you're doing your own, maybe getting some colleagues involved too, and you all say like, "I want you to pretend you're that student in your class that you're concerned about, that might not have as easy of a time accessing this task. How will they enter this problem?" And if they can't, "What can I do to adjust the problem so that they can access it?" So what do I need to do in order to make sure that that accessibility is there for all students to engage without lowering the quality of the mathematical experience for students who might be at a different place.
So I'm hoping you can see this here. You'll notice we go from more of a manipulative using counters and students are making stacks of counters. And then we move on to looking at a bar model and using repeated addition on this bar model. And then finally, the last one for our YouTubers especially, you'll see here we have a double number line, which is sort of like an evolution of this bar model, which used additive thinking. But then we go to multiplicative thinking on the double number line, and we always ask educators to kind of try to envision the proportion that you see in this double number line. This problem we knew there was 72 flowers and four boxes, And we knew we had six boxes to fill and we were looking for how many flowers we needed to purchase. So that is your equivalent ratios. You're seeing it not perfectly to scale, but with some scale. And without this over-reliance on just setting up your proportion and then cross multiplying, here students are using reasoning to benefit them with procedural fluency, and fluency and flexibility with numbers and multiplication and division facts.
So this for us is a huge, huge part of this particular unit. And ultimately, you'll notice in our consolidation, what we're trying to help students see is that this ratio, this 72 flowers to four boxes is what we call a "composed unit", as we referenced in the intentionality. And when you divide those two quantities, this type of ratio will reveal a rate. It will reveal a rate. And at the end of the day, we see that rate on the double number line, 18 flowers per one box. But I want to be specific here because 18 to one is different than 18 per box, all right? One is a rate, the other one is just a ratio that has one of the quantities as one or a unit.
So these are things that we explore in this particular unit, and then we set students off. So we set you up with, "What are you going to consolidate here?" We're going to ask students things like, "What do all these ratios have in common?" We're going to give them all the different ratios we sort of revealed from student thinking. And then we're going to have a clear connection, we want students to walk away with this idea that sometimes ratios lend themselves to revealing a rate. And sometimes, scaling in tandem might be the easier way to go versus just dividing straight up if you don't have a calculator. We have a reveal, and then finally, a reflection where we can determine who heard what?
So we have two consolidation prompts. This gives students the opportunity to independently work through a context that's very familiar. It's still with flowers. And we ask them to essentially tell us what they heard. What do we need to continue working on? And we're going to use this information to lead us into day two. Jon, I know we don't have a ton of time on this episode-
Jon Orr: Yeah, I was just going to comment on what I really love about problem-based tasks, and especially the tasks that we've set up is that not everybody can make their own, right Kyle? So we provided you basically a building-block where you can grab resources from us, because you also get all the video files that you need and you can get the images from us and you can run these right from the website. But what I really love about the tasks here is that our goal is to not only give you the resources for your students and you to use, but if you go through the teacher guide, it will help you. Kyle has been using a lot of language here. The teacher resource guide is set up to actually help you teach the math to you as well because a lot of teachers are like, "I really love the idea of problem-based tasks, but I'm scared about this connection and consolidation stage where I have to connect their ideas to the main idea." And it's like, "But I don't know enough. I'm not confident enough on the math part to really do that on the fly." And I feel like we do a lot of that by teaching you the math here in this resource guide.
And before we jump over to day two, Kyle, I wanted to point out if that's what you're looking for, if you are also looking for ... This topic that we're showing here on YouTube and talking about, this Planting Flowers is all in the proportional reasoning realm. And if it's something that's like ... We know the proportional reasoning has a huge impact on so many different ideas in mathematics. We actually built a course called the Proportional Reasoning course, it has a fancier name than that, which is called The Concept of Holding Your Students Back. Because we know that if kids are struggling with proportional reasoning, it holds them back from measurement, it holds them back from linear relations, it holds them back from so many future and current topics that you might be teaching your students. And that's our 12-week course that we run every fall.
So if you're listening to this or watching this over on YouTube and it's summertime still, you have some time to also register upcoming for that 12 weeks course where we talk about proportional reasoning from start to end. It spans grade levels. And we not only kind of give you the resources to teach different ideas and proportional reasoning in your classes, but we show you the math behind it. So I wanted to bring that up before we move on to day two because I think that's our goal here, right? Our whole goal of this podcast and everything Kyle and I do is to not only give you resources to use in your classroom, but teach you how to use those resources and feel confident doing it.
Kyle Pearce: And two things I'll mention too, and I'm glad you reminded me of this is one thing is day-one of every unit, we have the full teacher guide open. We wanted to be able to give you an opportunity to do some learning without having to become an academy member. So that is all there and open for you. As you move through out or deeper into the unit to gain access to some of those teacher guides, not all of them, but some of them, that is reserved for academy members. So that's something for you to kind of consider is the fact that, wow, one thing we wanted to maintain was the fact that we've been sharing problem-based lessons on our personal websites for years, where you can just run them right from the web. We wanted to maintain that, we didn't want to change anything about that.
But the part about trying to really build the intentionality, provide you with principal downloads, for example, on day-one here, you can see those on YouTube, our consolidation prompts here, you can share them right from the web, just like this on the screen, or you can actually open up a Google doc and you can print it as is. You can share it on your LMS or you can even make a copy of it and modify something. Maybe you don't like the language. Maybe you want to add a question, remove a question, whatever it is, you can do that. And that's all included for you.
So that, again, we want to take a lot of that weight off of your shoulders because we know that we couldn't do things as well as we might've wanted to in our minds, in the classroom. So we've got a team of awesome educators who are helping us to build out this curriculum for you friends. But again, the main goal of why we do this is not to make necessarily the complete resource, but to give you tasks and units that will help you so that you can shift your teaching practice regardless of the tasks that you're using.
So the last thing I think we should talk about here, Jon, we're not going to be able to make it through all five days here. But on day two, this is I think the big question, the burning question teachers have, which is like, "So like day two, do we just do another problem-based lesson. And then day three, another problem-based lesson." So how do we tend to structure some of the in-between days when we're not doing a problem-based lesson?
Jon Orr: Yeah. So if you're with us her on YouTube, you can see what day two looks like. But basically what we do on day two is we want to extend the same context, but now we want to bring in a math talk to kind of kick things off. It doesn't have to be another problem-based lesson where you're noticing and wondering and estimating and all that. Usually we do that on day one, but on days, 2, 3, 4, these intermediate days, we definitely want to have a math talk. We provide you the questions for the math talk. If you're making your own, you're definitely going to want to think about like, "How can I extend what we did yesterday to kind of keep that idea going. And what's my next learning goal?" So we still need that other learning goal, but how do I extend it into the next learning goal?
Oftentimes, we use a math talk to do that and then we set up some purposeful practice. Typically, in our old way, we would just teach the lesson and then say, "Here's a whole bunch of practice problems." That's what the purposeful practice is. It's the practice problems, but it's extending the context that we had. And then also introducing some new contexts in that purposeful practice. So if you're looking over here on YouTube, you can see that. But like Kyle said, we also have a downloadable resource you can use and print out and hand to your students so that they are doing some individual practice time. And you can also arrange these the way you like, you can have that your students leave them on the screen, up on the main screen in your room. And then they can be doing this on the boards around the room or in groups, but you get to decide that part yourself.
So that's kind of what the days two and three and four might look like on these intermediate days. But keep in mind, our unit day one through five is a progression of ideas. So it's not just all this one idea for five days, because you're like, "Man, how am I going to cover everything if I spend five days on one idea?" From days one to five, you progress through a number of your standards through those five days. So some days, they look like this estimation type problem, and some days they are the math talk and the purposeful practice. So I hope that kind of clears things up about like, "What do the other days look like for you guys?" Because a lot of people ask us that, like, "Where do you fit in practice?" We definitely fit in practice.
And for some of you, even though we might have days one through five, you might be doing days one and two on the same day. It depends on how much time you have in your math block. If you've got a longer math block, you can probably get through days one and two in one day versus say maybe just spending day-one on one math block.
Kyle Pearce: Yeah. And the last thing we'll say and then we're going to wrap things up here because of course, as usual, things take us longer than we anticipate them to. You'll see day three is another problem-based day. We had a purposeful practice on day four. And then day five is what we call a "show your growth day". So a combination of a math talk, but then also kind of like an opportunity for you to get a sense of where students are at. And we say "show your growth", You get to decide how you use this, whether use all the problems or some of the problems, but it gives you an opportunity to get a sense of where students are, take that temperature and then decide where do we go next?
So Jon, I think that's it for us on what does it look like for a problem-based unit? And we're going to continue. When Jon and I do our solo episodes, we've got another one coming up about how do we structure these units? If we're going to be spiraling, doesn't this seem like we're not spiraling? And we're going to say that the answer to that is no, there is spiraling going on and we'll talk more about that in just a couple of episodes, right Jon?
Jon Orr: Right. Yeah. We will for sure. It's coming up soon. So in order to ensure you don't inaudible on that episode as we release new episodes every Monday morning, be sure to subscribe on your favorite podcast platform.
Kyle Pearce: Awesome stuff. And friends, if you like what you're hearing, let us know and help your colleagues by sharing with them and do us a favor, give us a rating and review on iTunes on Apple Podcast and smash that subscribe button on YouTube so that we can build more video followers.
Jon Orr: Yeah. Show notes and links to resources from this episode, plus complete transcripts can be found at makemathmoments.com/episode142. Again, that's makemathmoments.com/episode142.
Kyle Pearce: Well until next time my Math Moment Maker friends, I'm Kyle Pearce.
Jon Orr: And I'm Jon Orr.
Kyle Pearce: High-fives for us.
Jon Orr: And high-five for you.
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