ChangED
ChangED is an educator based podcast for Pennsylvania teachers to learn more about the PA STEELS Standards and science in general. It is hosted by Andrew Kuhn and Patrice Semicek.
ChangED
Inspiring Educators to Break Educational Barriers
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Unlock the secrets to effective math instruction with our special guest, Graham Fletcher. Together, we explore the evolution of learning from concrete to abstract concepts in K-5 education, highlighting the importance of commitment over mere compliance in teaching standards. Graham also inspires us to embrace vulnerability in professional growth, encouraging educators to support each other through challenges and to cultivate an inclusive learning environment. Don't miss this opportunity to learn, grow, and be inspired by Graham's invaluable insights.
About our guest:
Graham Fletcher has served in education as a classroom teacher, math instructional lead, and currently as a math specialist. He continually seeks new and innovative ways to support students and teachers in developing a conceptual understanding of elementary mathematics. He is a coauthor of Building Fact Fluency and openly shares many of his resources at gfletchy.com.
Want to send us a show idea or just say hi? Email us at: thechangedpodcast@gmail.com!
welcome back to change ed changed change it. The number one educational podcast in all of the planet. I am your host, andrew coon, education consultant for montgomery county intermediate unit.
Speaker 2Here with me is patrice simicek, also of the Montgomery County Intermediate Unit and an EC.
Speaker 1EC.
Speaker 3And Tony Marabito out of Carbon Lehigh Intermediate Unit, sdf. I heard Patrice say EC, now I want to say SDC. It doesn't matter what I am.
Speaker 2The most important person here is our guest. Anyway, we have 12 names for the same thing. Yeah, it sounds very fluid. We work with teachers. Yeah, that's right.
Speaker 1We have an awesome job. We get to work with educators and learn from educators and talk about education. It's a really great, awesome job. So we're very fortunate to do what we do. We have a friend of the podcast on the show today, Graham.
Speaker 2Fletcher, did you know that, graham, you're a friend of the podcast? Yeah, he knows he's my friend, we're friends.
Speaker 4We emailed once yeah, we're all buddies in here. This is good. This is just excited to be playing in this sandbox today.
Speaker 2Oh wow, Thank you for that.
Speaker 1Well, welcome Graham. We're glad to have you. You mind just telling the ChangeEd Nation a little bit about who you are and your favorite football team and what you care about.
Speaker 4Ah well, I'm just kind of living in this education world. I kind of really focus in K-5 and basically like every other teacher out there, just building the plane while flying it right, like we're always changing, modifying, and I guess my job and what I thoroughly enjoy doing, is working alongside teachers and empowering them, building their conceptual understanding so that they can then, in turn, turn around and support the students that they're working with. Originally from Canada, live in Atlanta now, so you won't hear me say fix and reckon yonder y'all. You might hear throughout our little chat here. You might hear a couple A's and abouts. Don't let that throw you off. It is a Canadian accent. I love that. Yeah, that's and abouts, don't let that throw you off.
Speaker 2It is a Canadian accent. I love that.
Speaker 1Yeah, that's quite the difference. You really got pretty far away from Canada then Really don't like winter, huh.
Speaker 4Yeah well, farmersonlycom, that's what got us. No, I ended up down in Georgia on a soccer scholarship and that's where I met Mrs Fletcher. Oh good for you. So she said, you can go home and marry the cold or you can stay down here and marry me. And I chose her over the cold in Canada. And 22 years later, this is where I'm living.
Speaker 2Smart man. Smart man, very cool.
Speaker 1And I like that. You didn't hesitate, right, you're like okay, it's you, it's not like you.
Speaker 2We weren't there, he might have hesitated.
Speaker 1The way he told the story made me feel like he was all in right away.
Speaker 4Totally. All in. All the chips, all the chips. Push them all right in. You are spoken like a married man for a long time sir.
Speaker 1This man's got experience he has trained me well.
Speaker 2Well.
Speaker 1Graham, we had the privilege of being in on one of your sessions a few years ago and I really appreciated your perspective and the way that you thought about it yourself and then also the way that you communicate those thoughts. And something that we're really focusing on all the time but this podcast is is how we learn, how our brains develop and how that changes. So I'm really interested in your perspective. As you said, you kind of focus on K-5. I'm sure even within that time span because I have kids within that range how different they are developmentally, but even within the way they think and how they process information. So I'm curious on your perspective of how let's just start there how our elementary students learn. I mean, what seems to shift for them, what is important for and that you know for a kindergartner, and then how it transitions for a fifth grade. I'll also add to that because I feel like that was an unclear question.
Speaker 2Plus, he just likes to keep talking.
Speaker 1I currently have a fifth grader and it is still an elementary school in our school district for fifth grade. But it got real like it really transitioned for her in fifth grade, where the concepts are just different. It seemed like it shifted from more of an abs, more concrete, and then became a little bit more abstract and just the thinking and the thought process.
Speaker 3So what andrew means is that he can no longer help with homework yeah, is what is's happening?
Speaker 1She stopped bringing home coloring books and I'm like I'm out Faithful subscriber to Khan Academy.
Speaker 4Yeah, I think that's an important thing to kind of talk about, because a lot of times we think about the content that we teach so often and I think where so often what we do is we break down our learning standards that we're teaching into such a granular, bite-sized piece, and so it becomes almost like a checklist for students and teachers as well.
Speaker 4So as they're running through that checklist of understanding, it's more of we're teaching the standards out of compliance than more so out of commitment, and so one of the things that I've really been trying to be intentional about over the last few years is finding this idea of a through line. I'll unpack that a little bit more Obviously, the content is different whether we're speaking with five-year-old in kindergarten or we're speaking with an 11-year-old or 12-year-old in fifth or sixth grade. But what are those through lines that we're building in kindergarten that actually extend? So when I start looking at that, nowhere in your standards up there in Montgomery County does it say that kids need to own 20 different strategies. It doesn't say that anywhere in the standards.
Speaker 2Thank you for saying that I need to like. Thank you. Yeah, as someone who's done this for a while, like that's what they focus on, making sure they know all of the different methods. That's right Not expose.
Speaker 4Okay, I'm done, that's right. It becomes a checklist, right? Yeah, and so? And if you think about how our resources do this is like Monday we talk about unicorns and say we're making a 10. On Tuesday, we do doubles On unicorns and say we're making a 10. On Tuesday, we do doubles. On Wednesday we do near doubles and we talk about toy cars. On Thursday, we talk about doubles plus and minus two and we talk about basketballs. And every single day we're changing the content and we're changing the context, and so kids grapple with context and they never really get to the content that we're talking about.
Speaker 4And so what if we don't say that kids need to own 20 different strategies? They need to have strategic thinking, but the idea needs to be more around numerical reasoning. And when we look at that, I think about where, in kindergarten and first grade, second grade, we have a choice. Right, if you have to add nine plus seven, you could either memorize nine plus seven is 16, or you could use like a make a 10 strategy Take one from the seven, give it to the nine to make a 10.
Speaker 4Well, what happens with that understanding when you get into first and second grade and you now need to start adding multi-digit numbers, so say 59 plus seven. Well, you can't memorize that one, so now you might have to go back to single digit, but you can still use that make a 10, that associative property. And then what would happen when you get into fifth grade and you're adding five and nine tenths plus one and three tenths, you can take one tenth away from the three tenths to make a whole. So it's the associative property. That's that through line from kindergarten all the way up. It's not that students need to know 700 different strategies, because at the end of the day every single strategy falls under a property.
Speaker 4And so they don't need all of these different names. So that becomes an equity issue for students, because one teacher might call it one strategy, but in a different grade level that same strategy has a different name and it's all the same reasoning. So I think those are some of the things that I'm looking for. How can we find those connections through the grade levels where we just don't kind of sit in our third grade silo seeing where the math is coming from and where it's going?
Speaker 2Because, let's be honest, Go ahead. Yeah, I was just going to say what I find interesting is the connection I'm making to what you're saying in the cross-cutting concepts in science. When you look at the next-gen science standards or the PA steel standards that we're moving with here, they have these cross-cutting concepts. And one thing that I, when we switched to the Common Core, Pennsylvania PA Core when we did that, they have essentially content and then they have student practices. In these new science standards, they have content, student practices and then these cross-cutting concepts. That are exactly what you're saying. There are things that we're learning in kindergarten that we can build upon all the way through 12th grade. So what I find interesting about what you're saying is why don't we have those in math? We do have those in math, they're just not called out.
Speaker 4That's right. Yeah, they're there, and I think that's where we can be more explicit in our conversations, in our planning, in the professional development that we're receiving and also that we're presenting is where are these ideas of through lines? Because, let's be honest, if you teach third grade, you don't teach third grade. What you teach are nine-year-olds who function at a kindergarten through seventh grade level. Like, wouldn't it be nice if you only had third grade thinking in your classroom? That doesn't happen, and what I find is you'll get fifth grade teachers who might say well, how are my kids supposed to add decimals if they can't add single digit facts? And so what I might ask myself is what is the understanding of single digit number that we can now apply and extend into decimals? And when we can see those through lines, we become a little bit more efficient in supporting and meeting kids where they are on that progressional learning.
Speaker 3And you are speaking to the choir right. I was a fourth and fifth grade teacher forever, and then I was a principal of a K-3 building, so I've kind of seen the whole gamut there. So if you had to, if you were talking to kindergarten teachers and you had to give them, these kindergartners need one thing to move to first grade, they need to have one piece of understanding. What would that one piece be? What would that advice be?
Effective Math Instruction Strategies
Speaker 4for them Unitizing. So, and what unitizing is is it's understanding, and so we'll. Let's just take this idea of unitizing where we can scale this up. So in kindergarten, we want kids to know that a group of 10 is also a group of 10 ones, and then a group of 10 ones is also a group of one singular group of 10. And so what happens is a lot of the times what we do is we build that understanding. Kindergarten teachers do a great job of that.
Speaker 4But then when we get into first grade, what we end up doing is we or second grade, we put base 10 blocks into kids' hands too quickly, and what kids can't do is they can't decompose that 10 rod. There's an actual abstractness of a fair trade that needs to take place there. So, and thinking about using, like linking cubes and so, where kids can take the 10 apart and put them back together. But if we take this idea of unitizing and we now jump into third grade, like when kids see a fraction, three-fourths, they only really see a three over four. We want kids to understand that three-fourths is really three, one-fourths, and so there's an idea of unitizing that's happening there, and then that's really scalable and part of the reason why so many of our students aren't flexible with fractions is because they don't see it as that idea of unitizing that they can decompose them, they can think flexibly, manipulate them and move them around like we would do with single digit operation.
Speaker 1Wow, what's fascinating about this is I know I share that I have children here in this stage. So I have four kids. They're 13 to eight and so they're, you know, and all at different grades. But I can see the impacts. As you're saying this, I'm thinking through it, I'm like, wow, the impacts that this can have. That are substantial impacts for them. Understanding kind of getting that solid base of understanding so they can build upon it. And if it's not very solid, right, like scaffolding, you want to have a good scaffold to be able to build on, versus, you know, something that is teetering or put together with tape. And I could. I appreciate your perspective because I could see how that would be building it more solid. Sometimes you have to go slower to be able to go faster. You've actually got to slow it down and build it correctly so then you can later on in their learning we can go faster because they have that good core conceptual understanding.
Speaker 4In line with that. I also think about, like the resources that we use, where resources, like good friend Brian Bushart had talked about, curriculum bloat. And so when you look at many of the resources that we use, there's 180 days worth of lessons in pretty much every core resource that we use. In my whole entire life I have never met a teacher who has taught 180 days of math. And so right from the get-go teachers are trying to play catch-up. And so sometimes I think about third grade. We'll say in that we do like three weeks of rounding, and that's a difficult concept for third graders to wrap their head around this idea of rounding. But what would happen if we kind of stretch that out a little bit more to where it doesn't become blocked or masked practice? And what that means is so often it's we teach something so quickly that it only ever stays in short-term working memory. It never gets moved to long-term memory. And so what happens is if we might do three weeks of rounding, and if you get it, you get it, if you don't, you don't. Or if we go back to that idea of strategies where we're going to do a week of make a 10, and then that's it, we're done, I'll check the box, say that you have it. But when kids come back on Monday after that week of make a 10, they now have to clean out their short-term working memory to put in doubles.
Speaker 4And so, tracy Zager, we had been working on a project the Building Fact Fluency Toolkits, and she did some wonderful research about what moves things from short-term working memory to long-term memory. And it's this idea of mixed, varied or interleaved practice to where you have something and then you leave it, but then you come back to it, and then you leave it, but then you come back to it. I think about all the years where, when April comes around that's when we would do our state assessment I'm freaking out because I got to go back and reteach everything that we had taught in August and September, because it's quickly learned and then quickly forgotten. So going a little bit slower allows us that opportunity to play with ideas, but then, if you think about it, we never really come back and revisit those ideas, unless we're cramming for that state assessment. But this idea of mixed or leaved practice is super powerful.
Speaker 2So Paul Riccomini does a lot of research around that up here. I mean he's out Penn State so he talks a lot about interleaved work strategy and how he comes at it from the angle of helping kids who have an IEP retain more information. So he talks about this interleave work strategy a lot. What are your thoughts on? Like spiral curriculum then? Do you feel like spiral curriculum helps or hinders, or is a little bit of both or a little bit of? I know we're getting off topic here, but I think the idea of like constantly coming back to it or the thought that, for example, everyday math had with their math boxes where we're always coming back to it, helped my kids when I was teaching in the classroom. I could have been wrong. I could have been all in my head.
Speaker 4Well, what I find in that spiraled learning, we can keep spiraling with just naked numbers and what happens is kids can own 20 different strategies, but if they don't understand where those strategies fit in context. So a lot of the times when we think about spiraling, we kind of spiral back with just naked numbers, like when kids come to us and like hey, mr Fletcher, is this a multiplication or division problem? If you tell me, I can figure it out. No, figuring out if it's multiplication or division is the figuring out part of the figuring out part of it, the rest of it is just the computation piece of it.
Speaker 4So a lot of the times we teach strategy and context void of each other. So spiraling is super powerful, but we just don't want to be spiraling with naked numbers. So spiraling in context through that rich problem solving that we're developing when we begin to launch ideas.
Speaker 1The way we do things matter and that, and a lot of times we end up just doing something the way we've seen it done or that you know, and the intentional thought process, which can be so hard in a classroom when you're just inundated with everything. You're like I just need toss this, you know, toss this up here. So it was interesting what you're saying about. I believe earlier even talked about, like, the resources and how we can already feel behind when we're just getting started because there's so much information. So I appreciate this thought process and even thinking, well, how does this, how should this work? Not how is it working now, but how well, how does this, how should this work? Not how is it working now, but how, how should this work. And it even got me to like the words we use.
Speaker 1They matter, right? We intentionally pick the words we do to communicate something or in a certain tone or a certain way. Right, that we can even lose that. We lose that aspect often in text messages, you know. Or it's like well, you know, then they end up having a conversation and we, what were we trying to say there? Like, that really made me angry? Or, you know, I was so confused. Or yeah, I bought this, this and this Cause I was on the list. It's just like, no, I said don't buy that, you know. So, whatever it might be, but the way we communicate with each other matters, and I guess where I'm getting at is, even with visuals, even the way, what are we communicating and what are we not communicating with what we're using? And I think it's something that we always have to consider, especially in education, when messaging is out there everywhere, not just in education, but everywhere we look. There's messaging all over. What are they saying and what are they not saying?
Speaker 4Yeah, I think being intentional and purposeful is there are two words that kind of live rent free in my head and when we're intentional and purposeful it actually allows us to become a little bit more efficient in the work. But I have a lot of empathy for many K-5 teachers because they're asked to teach 55 subjects 44 days a week and to own the content. Like to own the math content the way I do can be really difficult when you have to do math, reading, science, social studies and so how do we empower teachers to become more intentional and purposeful along that? I think pacing guides kind of hamstring teachers as well. But what I've realized is like a silly analogy.
Speaker 4I remember when I first got married we had a cookbook and I would hold on to that cookbook and if it said 21 grains of salt, I would put 21 grains of salt in the meal. But over the years, as I learned to cook more and more and more, I don't need the cookbook anymore, I can step away from it. I might look at the cookbook to say, oh, what am I cooking this at? Oh, 350. Okay, perfect. And so what happens is because we're asking teachers to be cookers, bakers, all of these different cooks in the kitchen. They have to continually go back to that resource, and so being wise consumers is a challenge as well. But I think, at the end of the day, teachers want to be the chefs in their classroom, just the resources don't allow them to do the cooking they want.
Speaker 1I love that cooking analogy.
Speaker 3I like that analogy.
Speaker 1It's very applicable to a lot of people.
Speaker 3Graham, can you talk to me about fact fluency? Your take on it? Is it about speed? Is it about reciting your times tables? Can I use my fingers to count what's allowed? What's your take on it?
Speaker 2Automaticity versus fluency.
Teaching Math Strategies and Context
Speaker 4Yeah, and it's always kind of a hot topic and I think every teacher has either said or heard themselves like my kids don't know their facts Right. It's one of those things. And what tends to happen when we talk about fact fluency then becomes a gatekeeper to rich, meaningful problems. So that also creates an equity issue, because what we're saying is our kids, who tend to struggle a little bit more in school, shouldn't ever really have access to the highest quality tasks Like I can't give this to this kid because this student doesn't know their facts. But what the research says quoting here the National Research Council and the National Council of Teachers of Mathematics and lots of other research is I think a lot of times we focus on the accuracy, the efficiency and the flexibleness, but when you think about those three pieces, those are all taught through naked numbers. But what the research says is that fact fluency should be an outcome of meaningful problem solving, not a gatekeeper to meaningful problem solving. And so what happens if we say let's jump in, tackle a rich problem and then from there students might? Here's a perfect example.
Speaker 4I was in a first grade class this was probably about maybe eight, seven, eight years ago and we were doing a number talk. Any teacher who's been around in a K-5 space for over five years you know a number talk where you're doing like a series of equations or expressions that are purposefully connected. So we were doing make a 10. And during that 15 minute, 10 minute warmup kids could make a 10 beautifully during number talk, adding a nine plus seven. But when we get out of the warmup part of it and we go into the core part of our lesson the meat and potatoes of the lesson and I drop a nine and seven into a story problem or a word problem, kids would go right back to drawing all, counting all, and I'd be like what are you doing? Why aren't you using your number talk strategy? And it's in that moment where I begin to realize we're teaching strategy void of context. So when we think about problem we think about fact fluency. That is often like a standalone conversation that never really gets associated to problem solving. But the two should be so interconnected that kids know when to use what strategies at an appropriate time. It's just not naked number computation.
Speaker 4And when we get into that idea of multiplication one of the biggest jumps that kids need to make in all of K-12 education I really honestly believe this is moving from additive thinking to multiplicative thinking, which is like where, where, where kids are seeing groups of groups and we have so many students that are struggling in high school, not because of an algebraic reasoning or a functioning issue, it's because they're still stuck in additive thinking.
Speaker 4So, like where kids are, like they just where students might just skip count or use repeated addition or they just know their facts and there's nothing in between there and I. It is is when you actually have multiplicative reasoning. If you're skip counting like, say, for instance, if you have four peaches in eight baskets and you go four, eight, 12, 16, 20, you're only keeping track of the number of peaches. But if you start thinking multiplicatively, what you're keeping track of is the number of peaches. But if you start thinking multiplicatively, what you're keeping track of is the number of peaches and the number of baskets. And that is a really, really difficult concept for students to grasp and it's a really difficult concept for teachers to support, and I think that's where the context comes into a lot of that.
Speaker 3Well, and that leads actually right into my last question. A lot of that, well, and that leads actually right into my last question. In your in your typical, you know, pearson Envision whatever textbook no free ads. Sorry, no free ads. Come on, tony, no free ads, there are so many. I was literally just in an elementary networking group yesterday and the frustration that was being brought up was we're teaching our kids five different ways to multiply, and so as soon as they grasp one concept, I'm teaching them yet another one because the book says so, and then I'm confusing them. Right, and I was trying to stress the point that I personally think it's valuable to show multiple ways, but their frustration was they're getting confused then. So I kind of wanted your take on that. Should we be showing multiple ways? Should we be allowing students to choose the one that works best for them? Are we doing too much when? What should we be doing? Showing just the algorithm, you know?
Speaker 4Yeah, I think. I think that comes back to like what we do is we teach a bunch of different strategies and so a problem we don't. So what happens is if we just teach strategy, strategy, strategy, that's very dive, sorry. Convergent thinking to where kids are, we're now expecting that all kids think the same and we know that that doesn't. That doesn't really work.
Speaker 4But once students have an up and that's where a really rich problem can come into play, because that now gives us divergent thinking where you can have multiple solution paths. It has a fixed beginning and a fixed ending, but the way you get there is up to you. And you can see students who might draw all, count all, some kids might add, some kids might be able to use multiplication. But I guess that's where the idea of understanding the progressions of learning really come into play, because if I have a kid who's drawing all, counting all, then I want to move them to additive thinking. I'm not looking at a specific strategy, I'm thinking of zooming out more. How do I move you from counting to additive? And if you're additive, how do I move you from additive to multiplicative? And it's not about which strategy students are using, it's more about the reasoning that's happening, which comes back to we shouldn't really be teaching 400 different strategies within one year.
Speaker 3Yeah, I like that learning progression. That makes a lot of sense.
Speaker 1Graham, believe it or not, we're at the end of our time together for our podcast, so we wanted to give you the honor of the second to last final word, anything you'd like to say, to send off our audience from this episode.
Speaker 4I am a huge fan of Brene Brown and she talks a lot about vulnerability and for me vulnerability is the birthplace of professional growth and so many times we're okay being wrong in front of students, but we're not okay being wrong in front of our peers. So I've really challenged myself to just embrace the suck If I don't know the answer to something. I would much rather screw up publicly with a group of teachers than before I ever screw up privately in front of students. So I think, as teachers, making ourselves vulnerable around each other and knowing that we're all really kind of building the plane while flying it, is something that's helped me tremendously over the years.
Speaker 2I think Andrew already embraces the suck.
Speaker 1Yeah, publicly all the time, Every podcast On the daily Graham.
Speaker 2I do really like that.
Speaker 1The theme that I heard throughout this podcast was and you wrapped it up really well was flipping our ideas of, like, what we know the world to be versus what the world can be, and how do we flip that lens. So it even came down to when you were saying how do we organize something? A chart maybe will make it make sense with the way it actually flows, the way our mind thinks about it. But also it's a very interesting perspective that you just offered about when are we willing to be vulnerable and when aren't we willing to be vulnerable, and if we are lead learners, we should be willing to model that all the time.
Speaker 1In all, things so I also really appreciated that how you're talking about educating the whole person, not just teaching the content, and you can have different times where you're focusing on different areas, but you're still working with the whole person. So you know, challenges, issues, concerns, struggles, victories that you have in one space will carry over and could be a great integration and bridge to bring you back together. But they also the. The thing that I like circled multiple times was when you talked about fact fluency becoming a gatekeeper, and the reason I circled so much is because we there are gatekeepers in every area of education and they're not even intentional gatekeepers. So in science, the gatekeeper was vocabulary. Once you knew the vocabulary, then you can get into the science because you could have the conversation. And what's so interesting about these? The new steel standards in Pennsylvania, but the NGSS standards is we're like let's flip that script. So I feel like Graham Fletcher and NGSS are like good friends.
Breaking Down Educational Barriers
Speaker 1They're like let's flip the script, let's do it differently, let's have a different conversation to allow all to get in. Let's remove the toll gates so you get everybody's allowed on the highway and to get in and be part of it. So I think for all of us in education whether in the classroom, whether you're an administrator, whether you're on a podcast is to mind the gatekeeper. How do we be mindful of the gatekeeper and how do we eliminate those barriers?
Speaker 1for students and their education. So, graham, thank you for coming on. We would love to have you back and for all of our listeners, thank you, as always, for tuning in and don't forget to follow your favorite podcast.
Speaker 2Oh, we switched it yeah.
Speaker 1I nailed it this time.
Speaker 2I can subscribe work.
Speaker 4I appreciate it, friends. Thank you so much.
Speaker 2Thanks.