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Maybe you’ve recently come across the idea of a Dan Meyer-style 3 Act Math Task and you’re intrigued to give one a shot with hopes to engage your students and help better understand the mathematics you’re teaching. Or, maybe you’ve used 3 Act Math Tasks and use them as a regular routine in your mathematics program. 

While there are many mathematics teachers out there who love a great 3 Act Math Task, the reality is that there are many more educators out there who have given one a shot and decided to move on and not look back. What gives?

In this episode, Kyle and Jon discuss why many math teachers leveraging 3 Act Math Tasks in their mathematics programs do not observe an actual increase in student achievement when compared to their pre-3 Act Math Task days.

Stick around and you’ll learn how to strengthen how you utilize your classroom resources so that students will flourish. 

In today’s episode, we’ll be digging into the “leaves” which are one (1) of the six (6) areas of every strong mathematics program.

Resources

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Learn how to run a 3 Act Math Task Effectively

R2D2 Sticky Notes 3 Act Math Task

Math Is Visual Number Talk Prompts

Take the Make Math Moments Math Program Assessment Tool [Classroom Teacher & Leader Versions]

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FULL TRANSCRIPT

Kyle Pearce:

Maybe you’ve recently come across the idea of a Dan Meyer three-act math task, and you’re intrigued to give one a shot with hopes to engage your students and help them better understand the mathematics you’re teaching. Or maybe you’ve used a three-act math task in the past as a regular routine in your mathematics program.

Jon Orr:

Well, there are many mathematics teachers out there who love a great three-act math task. The reality is that there are many more educators out there who have given one a shot, but then, decided to move on. So, what gives?

Kyle Pearce:

In this episode, we will be discussing why many math teachers that have leveraged three-act math tasks in their math programs don’t actually observe any significant increases in student achievement when compared to their pre-three-act-math-task days. Stick around, and you’ll learn how to strengthen how you utilize your classroom resources so that the students you teach will flourish.

Jon Orr:

In today’s episode, we’ll be digging into the leaves of your tree, which are one of the six areas of every strong mathematics program.

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.

Kyle Pearce:

This is the only podcast that coaches you through a six-step plan to grow your mathematics program, whether you’re at the classroom level or at the district level.

Jon Orr:

And we do that by helping you cultivate and foster your mathematics program like a strong, healthy, and balanced tree.

Kyle Pearce:

The trunk represents leadership in your organization or the classroom pillars in your math class. The roots of your tree represent mathematics content knowledge and what it means to be mathematically proficient.

Jon Orr:

Like a tree requires soil, water, and sunlight, your mathematics program requires a productive mindset and a belief system so that all students can achieve at high levels. Your professional learning structure or your professional learning plan represents the limbs of the tree.

Kyle Pearce:

The branches of your tree represent the development of educator pedagogical content knowledge, including effective teaching and equity-based teaching practices. And finally, the last section of the tree are the leaves, which represent the resources, tools, and the classroom environment.

Jon Orr:

If you master the six parts of an effective mathematics program, the impact of your math program will grow and reach far and wide.

Kyle Pearce:

Every week, you’ll get the insight you need to stop feeling overwhelmed, gain back your confidence, and get back to enjoying the planning and facilitating of your mathematics program for the students or the educators you serve.

Jon Orr:

All right, everybody. So, we want to welcome you to our newest episode of Making Math Moments That Matter Podcast, and we’re going to be diving into a style of lesson that actually got Kyle and I so geeked and excited about teaching mathematics. If you know our story, then you know that both of us taught very traditionally, very procedurally for a number of years, and I think both of us have been on record of saying, “You know what? We were close to tossing in our hat and saying, ‘Hey, I don’t know if I want to do this forever,'” until we started to experiment and dabble into some lesson structure that has given us some insights and some excitement to teach our students and engage our students. And that’s how we got started, right, Kyle, is seeing Dan Meyer’s Three Acts Of A Mathematical Story, his original blog post, and exploring how to utilize that in our classrooms. And, even though we had some engagement, we also had some failures along the way.

Kyle Pearce:

Yeah, absolutely. You’ve heard us talk about it before. We’ve talked about how we started shifting our practice over time, and, like you had mentioned there, Jon, that was probably the biggest shift that we started making, where we at least got students leaning in. So at the beginning of this episode, we talk about maybe some of the reasons why we’re not seeing much change in terms of student performance, student understanding in math class, and we’re going to dive into a little bit of that here today. But one thing that you may see, if you actually implement them well and you leverage some of the tools that we highlight in the Curiosity Path, what you’re going to see is that students actually are going to be at least engaged in your lessons.

So there was always this benefit that we saw, at least right away, where we saw students that were actually sort of willing to engage, some of those students that maybe were turned off from math class or came right into the math class and sort of said, “I’m not going to participate.” Some students even openly have said this. And I know they’ve said this to you too, Jon, where they’d come in and say like, “I’m not a math person. Don’t ask me to answer anything.” Basically, what they’re saying is, “I’m just going to go over there and sit down, and I’m just going to try to do the bare minimum to make sure I get through here.” And obviously, we didn’t want that for our students. We didn’t want that for our math program.

And when we saw Dan Meyer, and I know for you and I, it was around the same time, but I don’t think it was at the exact same conference. For me, it was at a conference, and I saw him speak, and I was just blown away by how we could potentially reveal some interesting things about math and actually get students excited to answer math problems.

So first and foremost, I think that’s a massive benefit to leveraging this approach to teaching, the three-act math task. Whether you do it exactly the way Dan Myers mapped it out or whether you use some of the elements, I think the elements are the most important pieces, you will definitely start seeing students looking in your direction, staying engaged longer. But Jon, as we sort of got over the hump of how do we actually deliver one of these tasks, we realized after a while that we actually didn’t see anything really happening to student achievement.

Jon Orr:

Yeah. And I think I would even go on record here, Kyle, of saying almost a drop of achievement happened initially. That’s where I think many teachers who try one of these tasks or are shown one at, say, a professional development meeting in their district, or they’ve been working with a math coach, and the math coach has recommended to use a task like one of Dan’s or one of another teacher’s, or even one of ours that we’ve created, and they might see that engagement, but then it’s like, “Well, I still feel like it’s not helping my students understand math in a deeper way.” And I think we also experienced that along that way, is that…

And I think in reflection, Kyle, one of the ways that I think we saw that was that we were so concerned about engagement, and we’ve talked about that here on the podcast before too, that we were so hung up on engaging our students, we didn’t think about how to actually deepen their sense-making or make connections with the mathematics itself. And I think that’s where we fell by the wayside on that, and that’s where we saw a drop, is that we were spending… And this is a criticism of this style of task, is that it takes so long in your class. You’re taking up your valuable time to build, say, a different culture of math class, but you’re leaving off some of the, let’s say, math that you thought you were doing in the past. And that might be true if you’re not utilizing these tasks as you should be to bring about and emerge different math strategies.

And I think that’s what we were doing. We were trying to go all-in on engagement, and while we had some engagement, we probably spent little time actually on the mathematics and understanding the mathematics and making connections to the math that our students needed. And so, come test time, they weren’t improving. Come standardized test time, we didn’t make this clear connection to what they needed to do, and all of a sudden, it’s like, “Oh, we ran out of time.” And I think that can be why some of these teachers that we said in the introduction are kind of moving on, is that they’re not feeling like it’s a great use of their time.

And you know what, Kyle, I think part of this fault that happens, no fault of the teacher, is actually thinking about the resources that we’re getting from these conferences or online places. It’s kind of like, I see this thing, or I see this lesson, and it’s presented in a way that’s like, I’m going to play this video, and then I play the next stage, and then I throw the answer up, and then it’s like, that’s supposed to teach my lesson. And I think that’s partly where we had to learn after… Kyle, it was almost 10 years ago that we started this journey. So I think it’s taken us a good chunk of time to fill in like, “Oh, we needed to fill in gaps here, here, and here to make these actual connections.” And it was a lot of pieces that had to come together. After we saw Dan’s presentations and even taught a lot of his lessons, there was a lot of gaps that we had to fill in there to make the actual connections in our classrooms with our students.

So partly as I want to talk in this episode about resources and getting the resources you need and evaluating resources you find online or conferences, or you’re sitting at your lunch table, and someone says, “You should try this task in your class,” or a coach is like, “You should try this task,” we need to think about how that task is presented to us, but also, what we need to get from that task, because too much, I think, online, and especially at conferences, you see the what of the task, Kyle. You see like, “This is what I can use in my room,” but you’re missing some big elements like the why, “Why should I be using this task in this way at this time? When should I use this task at this time. How…?” That’s an important part, “How do I use this task in an appropriate way so that I’m not only engaging my students, but I am deepening their sense-making?” All those pieces are usually missing when you are seeing a resource online or at a conference or someone just passes you one at the lunch table.

Kyle Pearce:

So I think you did a good job here, Jon, of giving us a high-level view of the challenge and, really, the issue that arises when educators start down this journey. And I want to rewind right to the beginning, because you kind of leapfrogged over this-

Jon Orr:

It sounds like a long time ago.

Kyle Pearce:

… period in… You leapfrogged over this period in time where you looked at it and you talked about the intentionality of the lesson, the making the connections, like the doing the math. And I think that is a major, major challenge, and I think we need to dig deeper into that here. But before we do, one thing, what I know we did, and I know a lot of educators tell us they do, is they tend to start… Like anything, they start slowly. So this three-act math task becomes kind of like a field trip. What I mean, by that is, they treat it as a special event, as if that’s not really what math class is all about. It’s like, we’re going to do Monday, Tuesday, Wednesday, Thursday is going to be me using gradual release or direct instruction or whatever lecture style that maybe you were using previously or very teacher-centered approach, and then on Friday, we’re going to whip out the three-act math task.

And then what ends up happening is, students sort of go like, “Oh, it’s a break day.” So it’s almost like the mindset, the approach, the head space that students are in, is that, “This isn’t real math class, and we’re actually not going to necessarily learn anything new.” And oftentimes, it gets delivered in that way. So, there might be an interesting context. Students are engaged in the lesson. Maybe there’s a lot of great math discourse happening, so students are talking about it, making estimates, doing all these great things, but then, all of a sudden, we get to the end of the quote, unquote, math task, where we’re going to now reveal the answer. Students use their calculator and figure it out using the formula the teacher already pretaught. And then boom, here’s the answer, and then we watch the result, and everyone’s like, “Yay, we got the right answer.”

But the problem there is, all they’ve done is they’ve spent almost an entire class solving one problem, and they’re using a procedure, which I’m not going to say it was necessarily developed through procedural fluency, meaning students actually worked towards it. They just likely memorized an algorithm or a formula or the steps in order to get the answer, and out pops the number, and then, weeks later, we get our assessment back, our standardized test, or whatever you’re using to gauge student understanding or student achievement levels, and you’re going, “Huh, I’m not seeing a change from what I was doing before.” Or, like you said, Jon, maybe I’ve even seen it decrease.

Well, if I think about that for a second, running it in that way, and we did it, we’re calling ourselves out, we did this, we’d say, “Near the end of the unit, we’re going to do a three-act math task, because now the kids are ready for it.” So, kids are ready to solve this problem, this long, drawn-out lesson where students already know the formula for volume, or they already know the formula for solving a proportion or setting up a proportion and solving with some sort of trick, whatever that was that we were using, and then we dedicated an entire period, an entire class to our students solving one problem. They were curious. They may have talked about it. They might have been excited. But think about how much math they actually did if there wasn’t any intentional use of conceptual understanding or models or strategies or any of those things. They were all absent.

So what did you and I do there, Jon? Well, what did we do? We were like, “Well, we’re going to do this more often. Instead of just Fridays, we’re going to try to do this as often as possible.” It wasn’t every day, but we were still pre-teaching steps, formulas, algorithms. We were rushing to algorithms and then basically committing a big chunk of time for students to simply plug and play into a formula that we had already pre-taught them. And then that is when I specifically remember in my groups where I saw, actually, a decline, like you were saying, Jon.

I was trying to teach the same way. I was trying to teach through memorization, which I had been doing… Actually, I’ll say, I had it down to a pretty good science. We used to do pretty good on standardized tests. I knew. I’m like, “If we just bang out all these problems, everything’s going to be okay.” And it worked great. All of a sudden, now, I’m like, “We’re going to do less problems, but I’m still expecting you to memorize these steps and procedures without understanding what’s going to happen.” Well, they were more engaged, but they did less math, or, I don’t even want to call it math, they did less regurgitating of the steps that I wanted them to memorize, and therefore, the test results actually decreased. Holy smokes. We had a problem on our hands. And I think, Jon, that’s when we started to make the connection between this, “Are we actually engaging and making connections in math class, or are we just memorizing, regurgitating, and we’re just wrapping it in an engaging lesson structure?”

Jon Orr:

Yeah, and I was just trying to think about when was the moment that you realized that we need to actually flip the model around and teach through a task first to develop the sense-making and the connections and get our students actually thinking. It was a natural progression for us to go through that. And I think we needed to go through that. Today, and this is partly why we record the podcast, is so that teachers don’t have to go through what we went through to fast track math education. And why people write books about teaching math education is so that you can communicate things you’ve learned along the way so that we don’t repeat these learnings or this teaching style that we know there’s better ways to do that. But Kyle, can you remember a moment or a time where you’re like, “You know what? We need to pivot and teach through these”? Was there a lesson you did where you’re like, “I did it backwards, maybe by accident, and it worked out better”?

Kyle Pearce:

Yeah, actually, Jon, this is such a great question, because I think when I really dig into it and I start thinking about myself, when I started to realize that there was something missing from what I was doing in my class, I was, again, teaching. We had a note that we’d want students and… For many years. I actually started… The students I needed to help the most were the ones who were not going to actually copy my notes, so I was like, “Hmm, this is problematic.” The students who are already doing well will copy anything I put on the board, because that’s who they are. They’re going to get that work done. The students that I was concerned about, and the ones who typically already had labeled themselves or maybe were explicitly labeled as not being math people, those are the students who were like, “Well, I’m not going to copy this down. I don’t understand it anyway, so what’s the point?” or whatever their mindset was.

So then I started doing handouts. So I was like, “All right, here’s all the stuff I want you to know,” which now, thinking back to it, I wrote it for them. I basically went, and it’s like, “Well, wait a second, you already have it in that textbook right there, so it’s already written there,” but then I was like, “I’m going to take it, and I’m going to summarize it even more, and I’m going to make it more compact. And now, you have this boiled-down version of what was in this textbook, which was already written down, and now, I want you to fill in blanks and stuff.” So again, that was the model I was using. And, again, if I’m only teaching for regurgitation, memorization, memorizing steps, not necessarily understanding why things work, that actually is more helpful in terms of getting those marks up.

Now, I’m not advocating we do that, because what I realized over time is, a lot of that information was very temporary in their minds. It was like short-term memory stuff. They didn’t actually take that information with them. The very next year, they’d come into my class, and they wouldn’t know it. So now, I’m starting to realize, I’m going, “Wait a second, when did I actually make the connection that there was something more to it?” And it’s when I started playing with animations. And I know you were animating too at the time. And we started playing with making animations in order to create interesting three-act math tasks. But then I started going further in actually playing with like, “Well, I wonder how I could make this more visual,” hence the website mathisvisual.com. “How can I make this more visual so the student could see why it makes sense?”

And I know a lot of people are listening, going like, “Well, duh, of course you want them to make it make sense.” But this was like a revelation for me. Imagine if I could help my students understand why something works. And I think it was through tinkering through these animations, and as I would play and I would try these animations, I was like, well, that doesn’t really make any sense. I spent hours and hours making these animations. I started to realize that there’s more to this and that visuals are actually really helpful to make understanding the math easier.

And from there, getting to go to more conferences and then bumping into learning with… Cathy Fosnot’s probably my number one, where I started to go, “Oh my gosh, there’s stuff happening here, and there’s a reason why it’s happening in this way.” As I started to engage in more mathematics understanding for myself, I started to come to realize that, first of all, I didn’t have this understanding. I wasn’t given this opportunity as a student. And I realized, I’m like, “Wow, if I could understand this better, maybe I would then be able to use this as a tool to help more students in my classroom.”

So, it’s interesting, because we didn’t prepare to talk about that portion, but your question totally triggered a bit of an epiphany for me of when the math connections piece started making sense, because we often talk about the beginning of our journey when we started and were inspired by Dan’s work, but we often, all of a sudden, talk about sense-making, and we talk about math and conceptual understanding, but we’ve never really talked about when did we come to realize that conceptual understanding’s important and that, you know what, models and strategies are a huge, huge reason or a huge, huge tool to help us get there.

Jon Orr:

Yeah. And I’m hearing you say that when you tinkered with animation and helping kids make sense of the mathematics led you down that pathway. And I think when I asked you the question about when you decided to flip things around on this, you started to focus on sense-making first. And my flipping things around is a little bit of a different pathway to get to the same spot. But I remember starting the school year off, I taught with these tasks, like the way we described them, let’s save them for later, save them for the middle, save them for the end. But I decided one year to teach… On day one, I’m going to do a three-act math task. But I wasn’t thinking about flipping it around to do at the beginning of a unit. I was thinking, “I’m going to engage my students right from day one.” I want to show them what our math class is going to look like on the first day.

And when I did that, I had no opportunity to pre-teach, right? There’s no pre-teaching of anything on day one, because you’re getting your students. But I was thinking I can do a task with my students that they should already know about, because it’s really from the grades prior. So I did the R2-D2 task, Kyle, which is about area of rectangles and space on the inside and how many of these fit in here.

And what happened, and this is what I think convinced me to teach first through task and then work on strategies after, is that I was blown away by what happened next, is that I always had taught, and we’ve talked about this in the podcast too, is that I’d always said like, “This is the way you do a math problem, and I’ll show you how to do it.” Right? This is the way we taught for a long time. But when I taught them the first day, we predicted how many sticky notes would cover this space. And then, we showed them a little bit of how many were covering at that point. And then we showed them the dimensions of a sticky note, the dimensions of the board. And what happened was, some students… This is what I always did. I always assumed that students would calculate how many sticky notes fit in a space by finding the area of the big space, find the area of the sticky note, and then divide by saying, “How many of these fit in here?”

Kyle Pearce:

Yeah, doesn’t everybody know that?

Jon Orr:

Right. And so, when I taught on the very first day, in this particular task, I had some students do that, but I actually had more students figure out what the dimensions of the board were in terms of sticky notes first. So they took, say, the width of the board, which I think was 42 inches long, and then they took the width of a sticky note, and it was two inches wide, and then said, “Oh, 21 are going to fit in here,” because they did 42 divided by two and said, “Oh, 21 sticky notes are going to go across.” And then they did the same for the height. And then they said, “Okay, well, that’s length times width,” and they get so many sticky notes.

Kyle Pearce:

Now, can I pause you and just say, how blown away were you when they did that and you were like, “That seems easier to do than the way I did it in my calculator”?

Jon Orr:

And it also seems way more intuitive-

Kyle Pearce:

It’s like genius.

Jon Orr:

… to do it that, right? It’s so much more abstract to do it the way that I was doing it, versus this way, where it says, “Let me just figure out how many go up, how many go across, and I can use my length times width formula to figure out how many are in the space.” And so, when I saw these two come out, I was, in the moment, forced to go, “That solution is right,” and we all got the same answer, but we now have two battling different strategies to solve this problem.

Kyle Pearce:

How uncomfortable was that for you the first time that happened? Maybe you’re like, “Wait a second”-

Jon Orr:

I think the first time I… Because I was like, “Wait a minute, does that work?”

Kyle Pearce:

… “That’s not what math class’s supposed to do.”

Jon Orr:

And I’m like, “Does that work? It works. Yeah. That’s smart. That’s very smart.” And we celebrated this moment, and I’m sure the kid who presented or shared that the first time felt like a million bucks, because I was clearly making it look like, “I’ve never seen this solution before.” And I actually had never thought about doing it that way. And I think, Kyle, what convinced me was that we were preloading our kids with what we thought they needed. They come with a lot of stuff they already know, and we were looking at different strategies in that lesson and going, “Look, how does that strategy relate to this strategy?” We’re using the same ideas, but we used them in different ways.

And the power of connecting those two strategies together in the class in the moment is what sold me on, we got to teach this way first, and once we make the connections of the strategies on how it relates to our intentional learning goal for the day, then, hey, we can start to make some serious mathematical understanding here. And that’s what led me down that path to go, “All right, now, how do I make that a little bit deeper? How do I use models and strategies now to help do that?” And that’s the path that I took to get to this way. And I think that was my mistake, but also so important for us.

And part of that journey, and I remember this happening later, is that we were doing this before things were written about how to structure these things and what order do you do these in at the time, and I think we were paving a way for ourselves that made sense, and then when we discover the 5 Practices for Orchestrating Productive Mathematical Discussions and we’re reading that, we’re like, “Hey, this is what we’re doing in class. It just is a nice way to back us up on what we’ve already figured out works here in the classroom.”

Kyle Pearce:

No, totally. I love how you mention it’s almost like it happened by accident, much like my experience, although different. Mine sort of happened by accident as well. But yours was… You had the opportunity to go, “Holy smokes.” And I think the part I want to be very explicit about, you basically learned in that moment, and I’m guessing in the follow-up opportunities every time you used a task to introduce an idea, and I know for a while too, I should also mention, that for a while, you and I tended to do a task… Even when we started using tasks to start the introduction of a concept, we would go to a point, and then we would be like, “Okay, now solve it this way.” We did that a lot.

But this triggered you to go like, “Wait a second.” What I’m hearing you say is like, “I can give students a task without pre-teaching them concepts, and they will have some intuition.” You said intuition. I love intuition, because it’s like basically a combination of your prior knowledge, your experiences, just like you’re using your gut, right, like what makes sense. So when we talk about sense-making, it’s like, when you can get kids using their intuition, it’s magical. It’s amazing.

But you basically are proving that if we select tasks that have a low enough floor where students can engage with the task, there’s something else I don’t think we realized at the time, but the concept being introduced contextually and visually, whether it’s concrete in the real world or whether it’s visual through images and video, are great opportunities for students to be able to grapple with a concept and, ultimately, at the end of the day, come to a solution through maybe an inconvenient strategy, where that tends to happen more often than not, where they use… Let’s say you’re trying to introduce multiplication, and students are just repeatedly adding over and over and over again. It’s like, the goal here is we’re going to try to get them to see that that’s not as helpful or efficient as potentially trying to understand multiplication and leveraging, maybe, partial products or some other strategy. So it’s really cool that your experience happened in a different way.

Now I’m wondering, though, Jon, what sort of tips do we have for people? Where can they turn to if they’re maybe relating to this? Maybe they are like me, and they were like, “Well, I just used three-act math tasks as sort of like a break. We use it as a fun thing.” And students almost turn off their mathemagical brains for that, because they know it’s not real math class. Or, maybe it’s the other way around, where we just saved it, and we only use them once in a while or once the kids knew how to do everything and had the steps and procedures memorized. How do they, as educators, start shifting towards this idea of leveraging a task to introduce a concept and then actually ensuring that we take the math as far as we can so students actually develop conceptual understanding so we can work towards procedural fluency?

Jon Orr:

Yeah. And I think an important point about what we experienced and what we continually do with our students is we have a shared experience, and I think we have this experience where students are all working through something together at different points. They’re trying different things. And it’s the fact that we can come together and go, “We all had this, and now we’re going to tie it all up. We’re going to see where we’re in alignment, where we’re different, and then we’re going to try to build those together.” And I think that part is where the deep sense-making happens. This is where we’ve seen the gains in our own classrooms with students and their strengths, is because we had something that we shared that we can now tie together. Think about our old classrooms that we didn’t have that. It was like, we came in, we said we’re going to solve this problem. Nobody has any shared experience at all. Again, “I’m solving this problem I don’t really have any interest in solving. I’m solving a problem you asked me to solve, not something I wanted to solve or was intrigued to solve.”

So think about that, Kyle, if I’m going to teach regularly so that I start with a problem and ask students to work through a strategy that they may have at their disposal, we have to do a few things. This is where the drop-off happens at three-act math tasks, is because… Or, in general, teaching through problem-based lessons like on our website, is that we have to have a few things in place. One is, teachers have to have a comfort level with the content knowledge itself in a sense of where the progression of ideas are coming from and where they’re going to go to. So we have to know our curriculum. We have to know our standards from the before, after, where our students progression’s going to lie, and that takes, say, maybe experience teaching at a particular grade level. It’s also experience with understanding the mathematics itself at a deeper level. So, we have to know that, so that when we ask students to get into a problem, we are flexible enough to know where they are when we see work and then where they need to go. So we have to know that progression.

And not everybody has that, right? So drop-off might happen on a problem, our teaching through problem-based lessons, because we don’t have that knowledge ourself. So we have to gain that knowledge ourself. The other thing, Kyle, is, that if we’re going to make real dent with our students and effectively use that so we save time, is that we have to make sure that we’re essentially teaching… And this is where the connections get made, is we want to bring out strategies utilizing models in our math class so that students have the tools they need so that we can build towards procedural fluency, right?

Procedural fluency isn’t just memorization. There’s a few things. We talked with Jennifer Bay-Williams all about fluency on the podcast prior, about being flexible, about interchanging different ideas, about efficiency. There are lots of elements to being fluent in something, and so, we have to be knowledgeable, again, as a teacher, in those same things. We have to be fluent ourself with different strategies for different topics, and that is not something we are equipped with coming out of school, because our schooling system, by default, didn’t focus on these things because of the history of schooling up to that point. We’re hopefully creating a system of teachers eventually where the next generation has better tools at their disposal to teach the next generation. We’re working towards that, but we didn’t have them, so we have to develop them. And that’s not something we have, usually, coming in.

So that’s like, “What kind of professional development can I figure out? How do I do this on my own? Where do the resources…?” This is something I wanted to talk about here as well, is that, when you find resources online, when I was talking about someone passes you a three-act math tasks and say, “Try this,” is there any support there for you to do these things and gain the progression knowledge, gain the knowledge and understanding yourself about the conceptual understanding and the models and the strategies? We need to get that so that we could be successful and flexible and confident in the classroom. It can’t happen in the classroom unless we are confident in ourself.

So, that’s a really important next step, is, how do we get that? And when you’re looking at resources, you got to ask yourself, “Is that resource I see on some website…? Does it give me that stuff?” Because if it doesn’t, my three-act math task’s probably going to fall flat. It might be engaging, like we said at the beginning, but it’s not going to help me become a better teacher in the sense that I’m going to be building strategies, models, and fluency with my students.

Kyle Pearce:

You bring up something that I think is worth mentioning, is that there are so many amazing educators out there who have developed really engaging three-act math tasks. So, of course, Dan being one of them, we’ll call it the godfather of three-act math tasks, came up with the idea based on how three-acts of a story work, and it really is genius. Just, the whole idea is amazing. And then you have all kinds of people sharing amazing tasks out there. And I’m not going to mention names, simply because I don’t want people to take this the wrong way, because they have all kinds of great stuff out there, including your old website, mrorr-isageek.com, and my old website, tapintoteenminds.com. Those two sites are still up there with three-act math tasks up there in a very similar fashion to what I’m about to describe.

And oftentimes, what people are doing is they’re giving you the tools to engage your students and get them into that moment, and then it’s up to you, though, to make the math connection. So, what I mean is, and again, this isn’t a negative towards any of the three-act math task creators out there, it’s just, what they’ve done is they provided you with the tools to get students to lean in on your tasks, or on the math, but then, the baton is clearly in your hand to figure out, “What do I do next?”

So, I want to give an example, where I can see, some educators sometimes say like, “I understand. You said things. You were like, ‘Do you understand the mathematics? Do you understand conceptually how the mathematics works? Do you understand different models and strategies? Do you understand a lot of these things?'” And a lot of times, we tend to go, “Well, yeah, of course I do,” especially high school teachers, right? High school teachers oftentimes have math degrees, and they’re qualified, quote, unquote, to teach math. So we’re like, “Yeah, of course we know how it works.”

Well, I just want to give an example to give you almost like where the flag might go up that maybe there’s more work to be done. So I’m going to give you the example of one fourth is equivalent to x over 12, or x twelfths, and basically, it’s a proportion. And a lot of people would say, “Well, you could,” and the way I did it for many years was, I gave them a trick, whether it was cross-multiplying y thingy thingy, which was a big thing at one of the first schools I ever taught at, and magic circle was another one. There’s all these tricks.

Okay, so you’ve got this trick, great. Now that you’ve got this trick, the question is, why does it work? And here’s what I want to say. I used to do things like, I would say, “Well, because one fourth is equal to some number of twelfths, I might say, ‘Well, 12 is three times bigger than four, so therefore, x must be three, because three is three times bigger than one.'” Awesome. That’s great. That is one explanation. Do you have a deep conceptual understanding of…? That is the only way that you can help someone to understand that concept, because the next part is, okay, so you explain that to a student, and then a student goes, “I don’t get it.” What’s my next move? And the only way that we can help more students is by having such a deep understanding of the content that if that doesn’t work, then it’s like, “Oh, okay, maybe this, or maybe that.”

And the reality is that, actually, developmentally, there’s kind of an appropriate way. I say kind of. There’s actually a more appropriate way that would be developmentally helpful for students to work from more concrete towards visual, towards abstract. When we’re talking about fractions and proportions, using symbols, like numbers with a fraction bar and the equal sign and all of those things, that’s very abstract for a lot of students. So what we’re going to have to do is we’re going to have to find more ways to help these students. And the only way we can do that is by us doing the work. So I would ask yourself, before tomorrow’s lesson, “Okay, how are you going to explain this idea when a student bumps into it and they go, ‘I’m not really sure how it works.’ How many different ways do you have? How many tools in your tool belt do you have?” And if it’s only one, maybe it’s zero, for some people who taught cross-multiplying, if you didn’t have a way to help students understand conceptually how it works, then I would argue that we’re not making those mathematics connections.

So, my thought is, Jon, for people who are wondering, “Where can I go?” to start upping their game, because I’m going to guess there’s at least one concept, and I’m being very kind, because there’s lots of concepts that I don’t have a really deep understanding in. I’m still working on them. So, where can people go so that they can actually start working towards building more conceptual understanding so that they can help more students develop the procedural fluency that we’re after?

Jon Orr:

Yeah, there’s lots of places you can go. Obviously, you can read books, get yourself hands on some books that develop deep understanding. I think was Teaching Developmentally, Van de Walle’s book, Jennifer Bay-Williams’ book. There’s a few books like that that we’ve learned from. You mentioned Cathy Fosnot’s has helped you make some understanding.

But you’ve taken it in a step further. When we create lessons, so we talked about three-act math style lessons or problem-based lessons. When we create lessons, we actually create the teacher guide that goes with those lessons. And sometimes, Kyle, we’ve had people say like, “I haven’t read the teacher guide, because it was lengthy. There’s a lot there.” Well, there’s a reason that we created our teacher guides to be somewhat lengthy. It’s because we can’t be beside you while you look at that online for us to explain how the math works. But that’s exactly what we try to do in the teacher guide.

So, Kyle, you’re asking like, “Where do I go to actually learn about how to understand different representations, different strategies of mathematics?” when we make our tasks, we outline what that looks like, how to actually teach that task, but also, so that’s kind of like the how part of a math lesson, we actually tell you the why, why are you using this lesson at this time, not just the what, even though we give you the what, what are you going to use in your classroom. But the how not only shows how to teach that lesson, but we try our hardest to explain the mathematics and give you insight on developing your own mathematical sense-making and give you tools and representations in the teacher guide itself.

Sometimes there’s videos in there for you to watch. Sometimes there are student solutions that are using the models and the strategies, so you can outline… We’ve built them in a way to educate the reader on the math ideas, which is one way that we’ve taken to help teachers understand their mathematics a little bit better, and that’s not to mention the courses that we have inside the Make Math Moments Academy.

Kyle Pearce:

I love it. One of the key pieces that I think is of utmost importance in the guides… And again, like anything, I get it, you’re busy. Things are hard. I’m going to admit that, every day, right, we, as educators, some days, we have less time to prepare than we might want. So, we get it. That’s going to happen sometimes. But knowing that it’s there for you when you are able to commit that time and energy in order to try to prepare yourself as best you can.

And actually, Yvette Lehman, who, again, great colleague, great friend, and also a content writer for some of our units here at Make Math Moments, something she always says is like, “You’re going to be way better off entering into a classroom if you can remove as much of the ad-libbing as you possibly can.” And what she means by that is if you have a great guide there so that you can go through and you can actually get yourself comfortable, confident, and fluent with that lesson yourself, then you’re going to be in a situation where when students do show a different strategy, like Jon experienced with the R2-D2 task there with the Post-it notes, when that does happen, because it’s going to happen, you can’t predict everything, but you will predict more things, and therefore, you will ad-lib less in your class.

If everything in your class when you’re trying to do a problem-based lesson, which, again, a three-act math task, if done well, is falling into that category of a problem-based lesson, where you’re trying to get students to engage in mathematics and bring what they have in their tool belt so that we can then take them to the next place, then you’re going to see things that you didn’t anticipate. So, if I can remove some of that out of my day, it’s going to make my life easier in the moment, and it’s going to make sure that I’m more helpful to more students.

So, the part of the guide, though, that I think you never want to skip is checking out the intentionality section of the guide. We highlight the big ideas that we’re trying to help get across to the students, but then there’s also some big ideas in there that we’re making… They’re statements that we’re making to you, the educator, to think about. So some of those don’t have to necessarily come out to the students, but we want you thinking about them, because when you think about them and then go, “Huh, that statement, when I read that statement, is it really true?” What we’re hoping to do is for you to go, “I want to test that. I want to try that. Is that always true?” And then it gives you the opportunity to emerge these ideas with your students.

So, again, do your best, pick resources that are going to set you up so that you can develop your content knowledge that’s helping you build your repertoire of strategies, models, and really, understandings of big ideas, and therefore, you’re going to, in turn, be able to provide more opportunities for students to understand those big ideas, to emerge those strategies and those models so that students can do some deep conceptual thinking, and over time, build procedural fluency.

Jon Orr:

At the end of our episodes, we like to highlight some of the big ideas we talked about here. We’ve often been saying lately, there are six areas that everyone can improve in their mathematics program. And we actually hinted and chatted about two big ones here today. So we talked about the leaves of your tree, which is the classroom resources, so this is the lessons that you’re using in your classroom. It also can be the environment and the structure of your classroom. But in this case, we talked about the lessons and choosing a three-act math task or a problem-based lesson that you could use in your classroom. Kyle just mentioned the intentionality of that lesson is so important when choosing the right lesson for the right time.

But we also talked about another area of the tree, Kyle, which is the roots of your tree, which is your content knowledge and what it means to be mathematically proficient. We have to develop our own content knowledge so that we can be flexible in the classroom, so that we can help our students develop their own understanding of mathematics and develop their content knowledge and for them to be mathematically proficient. So that’s two big areas of our tree that we often talk about here at Make Math Moments.

Kyle Pearce:

I love it. I love it. Friends, if you would love to dig into some of those problem-based units as we’ve been describing them here, we encourage you to do so. Just a reminder to those who haven’t been there in a while, over at learn.makemathmoments.com/tasks, you’ll see over 60 units of study. Most of these units are five-or-more-day units of study. Just a reminder, day one, all of day one, that whole lesson is open there for you to explore, and just any of the additional materials are for Academy members. But you can run a fantastically well-intentioned lesson by hitting up and checking out day one.

Also, keep in mind, we’ve got our math program assessment screener available for you, both the classroom version and the district leader version. You can find them both over at makemathmoments.com/report, and that will bring you to a screen where it will essentially ask you whether you’re a district leader or whether you’re a classroom educator, and that’ll send you off to the appropriate report. So once again, head over to makemathmoments.com/report and make sure you check out the math program screener, and you’ll get your customized report right upon completion.

Jon Orr:

In order to ensure you don’t miss out on our new episodes, we push them out on Monday mornings. Be sure to subscribe on your favorite podcast platform. And if you have subscribed, great job. Welcome back. Show notes and links to resources from this episode plus complete transcripts, you can find over at the show notes page, which is makemathmoments.com/episode227. That’s makemathmoments.com/episode227.

Kyle Pearce:

Thanks, my friends, again for listening to the Making Math Moments that Matter Podcast, where we help you grow your mathematics program like a tree so your impact can reach far and wide. 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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