Graham Fletcher, Why Progressions Matter
ROUNDING UP: SEASON 1 | EPISODE 18
Many educators were first introduced to the content they teach as a series of items on a checklist.
What impact might that way of thinking have on a teacher’s approach to instruction? What if there were another way to understand the mathematics our students are learning?
In this episode, we talk with Graham Fletcher about seeing mathematics as a progression and how this shift can have a profound impact on teaching and learning.
Mike Wallus: Many educators were first introduced to the content that they teach as a series of items on a checklist. What impact might that way of thinking have on a teacher's approach to instruction? And what if there were another way to understand the mathematics that our students are learning? In this podcast, we talk with Graham Fletcher about seeing mathematics as a progression and how this shift could have a profound impact on teaching and learning.
Mike: Welcome to the podcast, Graham. We're glad to have you with us.
Graham Fletcher: Yeah, really excited to just kind of play around in this space with you here talking about math and supporting teachers so that they can, in turn, support kids.
Mike: You bet. So, just as a starting point, we're talking about progressions, and we're talking about some of the work that you've done, building progression videos. I have, maybe, what is kind of a weird opening question: How would you define the term “progression” so that we're all starting with the same understanding?
Graham: So, when I think about progression, I think a lot of the time, as teachers, we can become, like, hyper focused on one grade level. And within that one grade level, there can be a progression of where things are learned in a sequential order. It's probably not as linear as we'd like it to be, but I think that little micro progression, or sequence, of learning that we see in one grade level, we start thinking about what that might look like over a grade band, over, like, K–2 or even K–5. So, there’s things that happen within certain grade levels, and that's kind of where progressions happen. How do we move kids through this understanding of learning? And it's that progression of understanding that we tend to want to move kids through, where everything's kind of connected. And that's really where I see progressions.
Mike: So, I think you're kind of leading into my second question, which is—I love the work that you've put together on your website. I'm unabashedly going to say that this is a great place for teachers to go. But part of what strikes me is that there are a lot of things that you could have done to support elementary math educators, and yet you chose to invest time to build this series of videos that unpack the ideas that underlie processes, like counting or addition and subtraction or fractions. Like, why that? Why was that a thing where you're like, “I should invest some time in putting this together.”
Graham: So, I guess we're all teachers at heart, and so I start thinking about how I'm in a place of privilege where I've had an opportunity to work with some really amazing educators—that I've stood on their shoulders over the years. And I think about all the times that I've been able to huddle up in a classroom at the end of the day and just listen to those people who are brilliant and really understand those progressions and the smaller nuances of what it is to just understand student thinking and how to keep moving it forward. So, I started thinking about, “Well, what does this look like in one grade level?” But then, when I was starting to think about that whole idea, the big piece for me is: not every teacher has a person that they can sit next to. And so, if I've had the opportunity to sit down and make sense of these things where, like, on a Friday night (laughs), maybe I'm sitting down with some math books, which most people don't choose to do, I enjoy doing that.
Graham: And so, if I've had the opportunity to do that, and I'm able to make these connections, I start thinking about those other teachers who, teachers that teach 75 subjects, 54 days a week, right? And we want them to focus solely on math. So, maybe just sharing some of that knowledge to kind of lessen the burden of understanding that content. So, giving them, like, a 60,000-foot view of what those progressions could look like. And then them saying, “OK, well, wait a minute. Maybe I can do a deeper dive,” where we're giving them those [aha moments] that they might want or need to kind of do that deeper dive. And the big piece for it was, there's always talk about progressions. There's always talk about, “This is the content that you need to know”—content after content after content. But very seldom is it ever in a coherent, consumable manner. So, when I start thinking about teachers, we don't have that time to sit down and give hours and hours and hours to the work. So really, just what is a consumable amount of time to where teachers won't be overwhelmed? And I think that's why I tried to keep them at about 5 to 6 minutes, to where you can go kind of light that fire to go and continue building your own capacity. So, that's kind of where it was. My North Star: just building capacity and supporting teachers in their own growth. For sure.
Mike: You know, it's interesting, ’cause when I was a classroom teacher, the lion’s share of my time was kindergarten and first grade, with a little bit of time in second grade. So, I was thinking about that when I was watching these because I watched some of the ones for younger kids, and I was like, “This makes a ton of sense to me.” But I really kind of perked up when I started watching the ones for kids in the intermediate grades. And I think for me it was kind of like, “Ah, these ideas that I was working on in K and 1, so often, I wasn't quite sure what seeds I was planting or how would those seeds grow in the long term—not just next year, but in the long term.” I wonder if that's part of what you think comes out of a teacher's experience with these.
Graham: Yeah, I definitely think so. I think finding that scalability in reasoning and relationships is key for students, and it's key for teachers as well. So, for instance, when we start thinking about, in kindergarten, where kids are sitting and they're practicing counting and they're counting by singular units, singular units of 1—where it's 1, 2, 3. Well, then when we start making that connection into third grade, where kids are counting by fractions instead of going ahead and saying, like, “One-fourth, two-fourths, three-fourths,” really focusing on that iteration of the unit, that rote counting where it's 1 one-fourth, 2 one-fourths, 3 one-fourths. And then, even that singular unit that we're talking about in kindergarten, which now is in fractions in third grade, well that begins to connect in sixth grade when we start talking about unit rate, when we start getting into ratios and proportions. So, that scalability of counting is massive. So, that's just one little example of taking something and seeing how it progresses throughout the grade level. And making those connections explicit becomes really powerful because I know, just in my own experiences, in talking with teachers as well, is when they start making those connections. Bingo, right? So, now when you're looking at students, it's like, “OK, they're able to count by unit fractions. Well, what now happens if we start grouping fractions together and units, and we start counting by two-thirds?” So, now you start moving from counting strategies to additive strategies and then additive strategies to multiplicative, and seeing how it all kind of grows together. That scalability is what I'm really after a lot of the time, which falls in line with that idea of teaching through progressions.
Mike: Yeah, I think one of the things that's really hitting me about this, too, is that understanding of children's mathematical thinking as a progression is really a different experience than thinking about math as a set of procedures or skills that kids need to leave second grade with. It feels really different. I wonder if you could talk about that.
Graham: Yeah, absolutely. So, working with Tracy Zager—good friend of mine—we've done a lot of work around fact fluency here over the last three, four years, per se. And one of the biggest things that we have spent a lot of time just grappling and chewing on, is when we have students in second grade, and they move to third grade, how do we move students from additive thinking, which is adding of singular units, to multiplicative thinking? So, seeing groups of groups of groups. And so, I think when we start thinking about third grade teachers, I'll go ahead and throw myself under the bus here. Like, as a third-grade teacher, when we start thinking about that idea of multiplication, it becomes skip counting and repeated addition. But then no kids ever really move from skip counting and repeated addition to knowing their multiplication facts. Like, I could sit there and do jumping jacks in class, but kids aren't going to know their facts.
Graham: So, then what I would do is, I would jump to having kids try to memorize their facts. And just because kids can memorize their facts doesn't mean that they can reason multiplicatively and are seeing those groups of groups. So, I think, thinking of that, what [are] those big jumps in the progression from grade level to grade level? That's probably one of the ones for me that really stands out that I know I struggled for. And we always look back and say, “What are the things I wish I knew back then that I know now?” And I think that jump from additive thinking to multiplicative thinking is a really big jump that is often overlooked, which is now why we have kids struggling in fourth and fifth grade and middle school. ’Cause they're still stuck in additive, but we want them to think multiplicatively and proportionally. But yeah, that's one of those big jumps in terms of a progression that we want kids to make.
Mike: Yeah, this is a great transition because I think, like, what we've been exploring is, how if I understand what I'm helping kids think about in the context of a larger story rather than a set of discrete things that I need to check a box on, that has impact on my practice. But I almost wanted to ask you, just on a day-to-day basis, what's your sense of, if I'm a teacher who's absorbed this sense of progression either across my grade level or across a larger band of time, how do you think that changes the way someone approaches teaching? Or maybe the way that they set up tasks with students?
Graham: Well, I start thinking about learning objectives as they're handed down, and standards. And a lot of the time standards can become, or learning objectives can become, more of a checklist. And so not necessarily looking at these ideas of learning as a checklist, but how do they connect between the grade levels? And so, I think it's important as much as on the day-to-day practice that we're really down in the trenches and we're doing the work and we're making sure that we're meeting those learning objectives, I think it becomes really important that we provide ourselves that space and grace to zoom back out to that 60,000-foot view and say, “Wait a minute, how are all of these connected?” And I think that's a really big piece that maybe we don't always do when we start thinking, even planning, on a day-to-day or a week or a unit. “Where am I going to be able to zoom out and maybe connect some big ideas around an understanding or around a piece of learning?” And I think it can become cumbersome when we start looking at those learning objectives and they're so granular. But I think when we can zoom out and make connections between them, it lessens a little bit of the burden from having to go ahead. “Well, there's just so much to teach, trying to make those connections.” There is a lot to teach, don't get me wrong here. But I think going ahead and making those connections just lessens that burden for us a little bit.
Mike: It's interesting, because I think part of what is coming to mind for me is this ability to zoom out and zoom back in and be able to say, “In what way is this relatively granular learning objective or learning goal serving to advance this larger set of ideas that I want kids to understand about, say, additive thinking as they're making a shift to multiplicative thinking?” And the other connection I'm making is, in what way can I ask a question in this moment that's going to actually advance that larger goal rather than—again, guilty as charged—rather than what I've done often in the past, which is how can I help them just complete the task or get this particular thing right? And if by them getting it right in the moment, I failed to advance their thinking, that's a place where I'd want to take it back. Does that make sense to you?
Graham: Yeah, absolutely. I think about tasks and really about when I first would start to use problem-based lessons or three-act tasks and start thinking about those lessons. Normally it would be, like, “OK, I just taught the task for no rhyme or reason just to see if kids could get the right answer.” And so, for me, the big piece with that is a shift in my own craft, is looking at that task placement. And so, thinking of, “Are you a teacher who learns math to solve problems or are you a teacher who solves problems to learn math?” A little play on words there. And I think by default, many of us were taught to learn math to go ahead and solve the problems. But when I start thinking about this idea of using tasks and why we use tasks, it's to use—well, to quote Dan Meyer, talking about this headache and aspirin analogy where you have a problem that's your headache, and then from that problem, the math serves the headache—that's the aspirin that you need.
Graham: So, when we talk about zooming back out, instead of saving the really good tasks for the end of the unit, what would it look like if we put it on day one of a unit? Knowing that the goal on day one isn't for kids to get the right answer, but it's for us to just pull the veil back and see, “Hey, where are my students thinking?” And what I've realized is that when we don't front-end load or pre-teach things, students will usually fall back to the strategy that they feel safe enough. And if you have a student who, say we're in fourth grade and we're playing with two- by two-digit multiplication. If you have a student on day one of a unit who's doing draw all, count all, great, right? That's what they're doing on day one? But if they're still using that same strategy at the end of the unit, that falls back on me.
Graham: Like, what have I done to be intentional enough about moving that student's thinking forward? So, even in the moment when students might not be getting the right answer, it might be the wrong answer, but it might be the right thinking. And I think at that moment I need to zoom back out and say, “They don't have the answer yet, but I've still got three or four weeks to get there.” So, now that I know what students are thinking, how can I be intentional? How can I be purposeful about asking the right questions, presenting the right activities and tasks to continue to move that student's thinking forward to the end goal? The end goal isn't on day one of a unit. So yeah, I think that's such a great question because I think a lot of the time we feel as if we fall short or we failed as a teacher if kids aren't getting the right answer. But so often there's beautiful thinking that's happening—it just might not have the right answer. So yeah, big, big change in my practice.
Mike: We've been talking about the use of the progression videos that you've built, and I think in my mind I've imagined myself as a classroom teacher, as the consumer. And I think that's a really powerful way to use those. My wondering is, if you have any thoughts about how someone who might be an instructional coach or an instructional leader in a building or a district, if you could wave a magic wand, how do you wish folks who have that type of role might take and use the things that you've built?
Graham: I can share how I've used them in the past. I don't know—I'm sure there's coaches out there that are probably using the progression videos way better than I'm using them. But many times, I've found that when we start looking at individual standards, it's standards out of context. And granted, the progression videos, if I could go back and redo them, I would love to embed much more context into those progression videos. It would definitely lengthen them, which kind of defeats the original purpose of keeping them short and compact. So, now when we show those videos, what's nice is it's not really a coach in that moment talking with the teachers. The coach can now, after the video, say, “Hey, what was new to you? What was something that, that maybe you didn't recognize?” And also, like, “What are you doing well?” There's so much goodness that's already happening.
Graham: I think as coaches, we have to be really mindful, like, there's great things that [are] happening with teachers. Let's support and lift up those great things that are already happening with our teachers that we're supporting, just like teachers do with students as well. So, I think showing the videos and asking, “Hey, what's the same, what are you comfortable with? What doesn't sit well with you?” Thinking about kindergarten teachers when they see five frames—it's like, “Whoa, wait a minute. I've never really thought about using five frames.” So, just different ways of thinking it to kind of be a catalyst for the conversation, just a launch point.
Mike: Totally makes sense. So, I suspect there are some folks who are going to be listening to this who are, like, “Oh my goodness, I want to go check these things out right now. Or I want to think about sharing them with my teammates that I'm working with on a daily basis.” Walk me through how to find these and any kind of advice that you might have for people as they start to initially poke around and look at what's there.
Graham: Well, you can jump on my website, gfletchy.com, with my full name, Graham Fletcher. Just one of those things that we kind of went with growing up. I was called “Fletchy” as a kid. So yeah, at gfletchy.com you can look on progression videos, and then right there you'll see five of them. But as you start poking around, I'm going to harness my inner Brené Brown here and just say, “Vulnerability is the birthplace of professional growth.” And so, no one is ever going to get a new idea and go ahead and try it and then it be successful right on that get-go. So, when you poke around there, give things a try. I love reaching out on Twitter, sharing on Twitter, and just kind of growing in that space. Find a colleague. Or if you are a coach—one of the things I love is when coaches ask for ideas, go muck about, find a good task, and then muck about in a third-grade classroom with that task and make yourself vulnerable around the teachers you're supporting.
Graham: And that really helps build and solidify that relationship where, “Hey, we're in this together, and I'm trying to fumble through this just like you. Let's kind of work here together. Give me feedback and, and in the end, I think kids win.” I'm a firm believer that all of us are smarter than one of us. And so, I love finding new things, testing new things with a friend, and trying not to lock myself in a silo. So, that would kind of be it in terms of poking around there. Yeah, find an idea and go share it with a friend and see how it works and keep on tweaking and revising.
Mike: I love that. Graham, thank you so much for joining us. It's really been a pleasure.
Graham: Yeah, it's been great. I appreciate it. And thanks for the opportunity.
Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling individuals to discover and develop their mathematical confidence and ability.