How can we help cement students' learning without memorizing steps? In this episode Pam and Kim share the power of anchor charts and how you can make them with your students.
Pam Harris 00:00
Hey fellow mathematicians. Welcome to the podcast where Math is Figure-Out-Able. I'm Pam.
Kim Montague 00:07
And I'm Kim.
Pam Harris 00:08
And you found a place where math is not about memorizing or mimicking, waiting to be told or shown what to do. But it's about making sense of problems, noticing patterns and reasoning using mathematical relationships. We can mentor our students to be mathematicians as we co-create meaning together. Not only are algorithms not particularly helpful in teaching mathematics, but y'all rotely repeating steps actually keep students from being the mathematicians they can be.
Kim Montague 00:38
So in this week's episode, we are going to tackle the topic of anchor charts.
Pam Harris 00:43
Kim Montague 00:44
Yeah, we've talked about Problem Strings a lot, what they are and are not. How to facilitate them. We've actually done some together, but we haven't really talked about anchoring the learning.
Pam Harris 00:54
Yeah, let's be clear, as a high school teacher, this was not a thing. This could be a thing. This is a wonderful thing that we can do. kind of all over the place. Yeah.
Kim Montague 01:06
So today, we're going to talk about what they are. How do you create them? When maybe you create them? What goes on them? When would you refer to an anchor chart? And how do you get kids to use them?
Pam Harris 01:19
Absolutely. So anchor charts. So like we're saying the word anchor, like you throw over an anchor over the boat that anchors the boat to the, what does it mean to anchor learning and a way to anchor learning is to create an anchor chart. So one of the reasons we thought we would do this podcast is we have done, so we have online workshops, asynchronous online workshops. If you can check those out yet check them out. They're wonderful. One of them is free, Mathisfigureoutable.com/freeworkshop, the Developing Mathematical Reasoning workshop. In those workshops and live workshops that we do when we travel around, we will often make anchor charts. We will anchor the learning. And as we've talked to some of the participants in workshops, we realize there might be a slight misunderstanding that we thought we'd clear up today in the podcast. Also, often when we will create an anchor chart in a workshop, we will have teachers say, "Oh, hey, like I was just writing that down in my notes, but it'd be super cool. If you guys actually made cute ones, like nice looking ones, like sharp kinda, you know, and sold them. I would buy those." Totally people have like, "Put them on your store would totally buy them. "So like Kim, it's a great idea. Like we should make anchor charts and or, yeah, or not. Yeah, so Alright, let's dive into what are anchor charts? So Kim,
Kim Montague 02:45
Pam Harris 02:45
What are anchor charts? Go.
Kim Montague 02:47
So an anchor chart is an opportunity to create an anchor of the learning. Right? So in a Problem String, you're doing the Problem String, you get towards the end, and you want to in some way, kind of cinch the learning, cement it a little bit more.
Pam Harris 03:05
Bring it together.
Kim Montague 03:06
Maybe not for every kid at the exact same moment, probably not, likely not. But you want to have a poster or a chart or a paper of some sort. Something that is an example of the type of strategy that just happened. So for me...go ahead.
Pam Harris 03:25
Could be electronic, use of paper. Could be electronic. Sorry, to interrupt.
Kim Montague 03:28
Yeah, so for me in an elementary classroom, it's going to look like a piece of chart paper that has some sort of cementing the learning that just happened. And I don't know if you want to tell me, like me, to tell you what would actually go on mine, but I can.
Pam Harris 03:43
I do, but I have an order to my madness today. So talk to me about how you create anchor charts.
Kim Montague 03:51
Pam Harris 03:52
Yeah, cuz you could just order them off of your store.
Kim Montague 03:55
Pam Harris 03:56
Kim Montague 03:57
Yeah. Or you could be like some of our participants who said, "Can I just take your anchor chart?" Right? And we would never advocate that because we think that it's super important that what goes on that chart comes from the students with some facilitation, some guidance, some kind of little bit of a sneaky from the teacher. Because while you have something in mind, that's going to go on the anchor chart, some key pieces, what those look like exactly will come from your students so that they can relate to it as they use it throughout the year.
Pam Harris 04:34
And in a huge way in the intro to the podcast we say we can mentor mathematicians as we co-create meaning together.
Kim Montague 04:42
Pam Harris 04:42
And that's what I hear you saying is that you are co-creating, you know the content, but you also know your kids. So you know the content means you've got that landscape. You have a sense and a feel for how this whatever you're anchoring that day, how it fits in, what it's all about, what are the major important parts of it are. But you want to co-create that anchor chart with students. So that... Well, I'll let you, like why? Why co-create? What's the use of having students do it with you?
Kim Montague 05:12
Well, I think that it just makes it really nice for them to go back to the experience that they had. If they recognize making those pieces with you, right? I could create a chart. And I could say, "Hey, we just did this thing. And so this is the chart that I made for you to remember it." But if they have the experience, and have to kind of verbalize what they would put on their charts, what they want to represent the experience you just had, that it is so much more meaningful. They are so much more connected to what is on that piece of paper or that digital piece.
Pam Harris 05:45
Well, let me just repeat, like often we will have, we have this kind of traditional setup in a class, especially the older the student gets, where the teacher writes everything down, maybe they have a PowerPoint slide. But in some way, the teacher hands the students something, and the student puts it in their notes. So the teacher writes something, gives it, it's on a poster, it's in their notes, it's on a PowerPoint, and the student puts that then in their notes.
Kim Montague 06:10
Pam Harris 06:11
What we're suggesting is something completely different.
Kim Montague 06:14
Pam Harris 06:14
Because we want to anchor the learning through this co-created chart. Think about where we talked about do, say, represent. So if you haven't listened to that podcast, we'll put that in the show notes where we talked a lot about that.
Kim Montague 06:27
Pam, I don't think we've done that yet. We should do that.
Pam Harris 06:29
Are you serious?
Kim Montague 06:30
Yep. Okay, well, let me just briefly talk about the fact that we can do more in our head than we can say. Llike, we own stuff up in our brains, and it's hard to sometimes articulate that stuff. And we can say, we can articulate things more than we can represent on paper. It's hard to sometimes put our ideas on paper, even when we can talk about them. So if that hierarchy exists, one way to help get stuff on paper, is for the teacher to say, "Tell me what you're thinking about." Yeah.
Pam Harris 06:57
If we skip from the students not thinking about it at all, not talking about it at all, and the teacher just handing them a representation. That's not learning. What happens better is when we have the experience. The students are creating the relationships, and now they have, if you could see me, my fingers, like, what, how do I describe this? My fingers are like, circling around my brain, like, my brain is like, go and go and go. And so the relationships are being created. Then the teacher draws out the words. So do, say, represent. The 'say part', the teacher draws out the words from students. Let's anchor this learning. Let's put this up on this anchor chart. What would, when you walk in the door tomorrow, what would ping for you? What would you, "Oh, yeah, that's a thing we did yesterday that I remember those, that it worked like this, like we could remember? Oh, yeah, we were." And as you pull that out, as you say to students, "What could we put up here that would help with that? What kind of words have you been using to describe, say, this strategy that we're working on?" As students grapple with verbalizing that, learning occurs.
Kim Montague 08:00
Pam Harris 08:01
The very grappling, the very act of trying to decide how to anchor the learning, how to generalize what was happening, how to put words to the relationships that were going on, how to put a model on that anchor chart. Where like, what should the number line look like? Or what would an array, what should we use an array or ratio table? What would help us? Should we draw the equation of the line, and also should we have the graph of the line? Like what would go on the anchor chart students? What is meaningful to you about what we just did? As students grapple with that and decide how that looks, learning occurs. They actually sharpen their ideas in the process of co-creating that anchor term with you.
Kim Montague 08:44
Pam Harris 08:44
And therefore means so much more to them, like you said, Yeah.
Kim Montague 08:47
Pam Harris 08:47
Yeah. So what kinds of things go on an anchor chart? I just kind of mentioned a couple things. But what?
Kim Montague 08:54
So I think there's three big pieces that I have in mind that I want to put on an anchor chart with kids.
Pam Harris 09:01
Oh, I'm super curious. I've never heard you say that three, okay.
Kim Montague 09:05
Three big things. And so I think it's going to look a little bit different from class to class. And it's going to look a little bit different from year to year and from poster to poster. I'm a poster writer, because some students might want to add a little bit more. But I'm aiming for three things. And one of them is I'm going to name whatever that strategy, that big idea is. We're going to give it some sort of name. And in early years, right, we probably did a little bit more of a loosey goosey naming kind of thing. Maybe I might have honored a kid's thinking, called it 'Mike's Strategy', whatever pick a name.
Pam Harris 09:40
Mark, Mike, ahh.
Kim Montague 09:42
I have a brother named Mike. And so I might have called it 'Mike's Strategy" and then kind of like maybe put a little mathiness to it later. Now I know that it's really more important for me to give it more of a mathy kind of feel. So that's one thing is we're gonna name it so that later we can refer to it.
Pam Harris 10:00
Okay, let's pause on that for just a second. Because the mathy name, like, I remember one year we called something a "lasso strategy."
Kim Montague 10:08
Pam Harris 10:09
Cuz you throw it out and bring it back. The problem with that is, it doesn't allow you to, well, first of all, it's not very mathy. But it doesn't allow you to, what's the word I want? Travel, it doesn't allow you to travel with that. So when you go talk to a teacher next door, or next year, and you say, "Oh, yeah, I'm using the lasso strategy."
Kim Montague 10:29
Pam Harris 10:30
"The what? Mike? Who's Mike?" So if you can say, "Oh, yeah, I'm adding a friendly number, or I'm using equivalent ratios, or I'm looking at this as a proportional relation that's been shifted." If you can talk about the, so it's really less about a name, and it's more about a short, descriptive, sort of description, name, title kind of thing. That again, helps sort of sharpen generalize what's going on. Okay, so one thing is we're going to have to describe in a short, brief name like way, okay.
Kim Montague 11:03
We're gonna name it. And then I'm always gonna have some sort of example. And so when we're talking about it, we're wrapping up a string, I'm having students discuss. And in that discussion, I'm going to say, "Hey, what, which problem from our Problems String, which problem feels to you like a really nice example?" It might be more than one example. If I'm doing a string that has whole numbers and decimals, I might choose one or the other. But I'm asking them, what feels like when you look back at this again, it'll be a really nice example for you to remember the strategy that we're using here?
Pam Harris 11:36
Kim Montague 11:36
Pam Harris 11:36
Yeah, it'll ping for you. You'll look at that example. And you're like, "Oh, yeah, that's that one. Yeah."
Kim Montague 11:42
So the third thing is, we're going to put a little bit of words to it at the bottom. So in my head, my posters being created, as we're talking. I'm gonna have the name at the top. I've got some nice, really large examples. And then at the bottom, we're going to have a little bit of language to it, a little bit of words about what is happening with that strategy. So if it's Add a Friendly Number Over strategy, I said that at the top, we've got some examples. And then at the bottom, I'm having the students come up with some words that really describe, you know, because for some of the students, it's going to be a little bit more tenuous. They're not, they're not perfectly cemented. But as they stare at that poster over time, I want them to have some words that they can read and go, "Oh, yeah. Oh, yeah. Like that's starting to make a little bit more sense." And as we do more Problems Strings. And as we have more experiences with them, they read that and go, "Oh, down at the bottom, it says, you can add a little bit too much. And then backup some." Whatever language, whatever words the kids come up with, and again, we're co-creating. And so I'm have a little bit of a hand in summarizing.
Pam Harris 12:47
Yeah. So it's not like really whatever they come up with, because you're kinda nudging. You're nudging kinda hard. You know that, you know, the content. So,
Kim Montague 12:54
Pam Harris 12:55
You're nudging that well. And again, in that creating, as students are putting those words, and you're kind of helping that make sense, those words at the bottom, that's helping sharpen their understanding. Because in order to create those words, they're kind of generalizing. They've now use that relationship over a few problems, or they're beginning to. That generalizing is a form of Algebraic Reasoning, which is also desirable. If you've looked at the Developing Mathematical Reasoning, graphic. Counting strategies is inside of Additive. We build on that to get Additive Reasoning. We build on that to get Multiplicative Reasoning. We build on that to get Proportional Reasoning. We build on that to get Functional Reasoning. But more longitudinal is Algebraic Reasoning. That sort of skirts all the way through there. This is an example of building Algebraic Reasoning, as you have students generalized relationships that are being used, generalize a strategy, generalize a way of using patterns to solve problems. That's an example of Algebraic Reasoning. And later, we just put variables to it. We're getting lots out of anchor charts that we don't get if we just hand kids, "Here's the thing put in your notes."
Kim Montague 14:09
Pam Harris 14:09
It's coming out of experience, where they're actually doing the relationships. And then in the creative process, the learning is occurring. Yeah, nice. How do you refer? When do you refer? No, maybe? Sorry, I'm gonna ask you this one first. When are you creating anchor charts?
Kim Montague 14:28
Okay, so that answer is probably not as hard and fast as maybe somebody would want to hear because I think it depends on the students. I'm definitely almost never going to create it after the first Problem String that we do. So I have a series in my mind of Problem Strings that we're going to tackle a particular strategy. And I'm really like eyeballing my students, and I'm watching what's happening. And I'm saying to myself, "Oh, we lobbed out this Problem String that's about Add a Friendly Number Over and a couple of kids are hanging on to it." And then I'm going to come back, I don't know, maybe two days later, and I'm going to do another Problem String for Add a Friendly Number Over. And I've got a couple of more kids who are tinkering with that strategy. And maybe one more time I'm going to come back around. It's when I see a, I don't know, is there a good percentage? Like 75% of my students, maybe who were tinkering with this particular strategy that I go, "Okay, now I'm ready to put some words to it a little bit harder. I'm ready to exemplify it." So I don't have a firm answer. But it's not after the first one. And that's what we've seen a lot of people do, "I'm doing a Problem String, one time. We're going to anchor it. We're going to move on to a new strategy." And that's it.
Pam Harris 15:41
And they do that maybe because they've been in a workshop. So when I'm doing a workshop, say it's a one day workshop, I've got one shot to teach you a strategy. So we do a Problem String, it's usually longer than ones that you would do in class, which also gets us a little trouble. Because if people are like, "Your Problem Strings are long." I'm like, "Well, I know because I have one shot to teach you the strategy, to help you develop these relationships." So I should be more clear about that when I work with people that you know, you probably have to do a little bit of a shorter version with students and then do it more often. So I think you could do an anchor chart after one Problem String, if most of your students are, maybe not even tinkering. But I'd have to see what your definition of tinkering is because I want them actively using the strategy. They're actively making sense of it. They're actively, I'm not saying that they own it yet. But when the majority of your students are actively using the relationships that you're trying to build, then I think you can make an anchor chart, because the process of making it is going to sharpen that thinking. It's going to bring some more students even more on board, and it's going to help everybody get better at it.
Kim Montague 16:50
Oh, see, we could do a whole podcast about that, because I have more to say about that. But that's not that's not the point right here.
Pam Harris 16:59
Okay, write that down.
Kim Montague 17:00
Okay, got it.
Pam Harris 17:00
So we can talk about it. So, golly, where are we?
Kim Montague 17:07
So I think I create them when I see students actively using it. But it's not where 100% of students are using the strategy. Right, right. But I think even more important to that piece is when we refer to them, because the anchor chart, like you said, does not go in a notebook on a piece of paper, that then they just turn the page and they're taking notes that somebody's telling them to take. That poster for me is up on the wall, and then it stays up on the wall for a really long time. My classroom walls at the beginning of the year are pretty blank, because I'm leaving space to create together and it goes immediately up on the wall. And then that's something that I refer to as often as I possibly can when I see it naturally occurring in my classroom.
Pam Harris 17:57
So you see a possibility to use that strategy. And you might say, "Hey, like, I think we have something that we've done that would really work well here." Or you might see somebody using that strategy. You might go, "Oh, y'all, check out what so-and-so just did like look." And you're standing in front of that anchor chart, and you're pointing to it, you're like, "Yeah." Hey, if I may, let's say that you're a secondary teacher, or just a departmentalized teacher, someone who teaches math more than once a day. Just a little bit of a kind of trick to getting these made. Here's what we're not suggesting. So say you teach sixth class periods. Don't create an anchor chart for each class period and hang up six on the wall. "Well, Pam, shouldn't it be the ones that the kids suggested that we want to co-create it together?" Absolutely. So what you do is during first period, you go, "Hey, y'all, let's anchor this learning. What should we call it? You know, brief name, brief description? What examples should we put up here that say to your heart and your soul that will ping for you tomorrow? What are some words we could put down here? I'll put, as we're creating this, I'll put it up here on the whiteboard. I'll put it on something not permanent. Don't worry. I'll put it on chart paper later." Then second period, "Hey, let's create an anchor chart. Don't worry, I'll put it on chart paper later." Fourth, fifth, sixth, seventh... seventh period, you're like, "Hey, guys, let's create this anchor chart. I'll put it right on chart paper." So yeah, that very last class period, just make it on chart paper. By then you're pretty good about what that anchor chart should look like. You've had some examples and time to kind of like, I'm not suggesting totally winging it from the first, but you've gotten like, "Okay, like this makes sense." When the kids come in the next day for first period, and it's a different sample problem, but it's a really good one because you've nudged it all the way along, you've made sure that the anchor chart is a really good sample problem. And they're like, "Hey, that's not the one we chose." You're like, you could just say, "Yeah, I remember I had to transfer to chart paper by that point, you know, like seven periods, which kind of created this like, you know, combo anchor chart." So that can be a little bit of a trick, teacher, what's the word? Hack, a teacher hack, so that you don't like waste poster paper, and we don't need multiple of them. We just need one for each major strategy or big idea or model that we're working on.
Kim Montague 20:03
Pam Harris 20:03
Cool. Okay, Kim, what else do you want people to know about anchor charts?
Kim Montague 20:11
So I think besides me mentioning the anchor chart on a regular basis, you know, when I see it occurring, I want kids to know that they're there for them to use. It's not a thing that we created, and it sticks on the wall. And it's just cute and decorative. And it's, you know, I made it really pretty because mine never really were. It's there as a tool. And so besides me referring to them, I'm also going to celebrate when I see a student glancing at it, looking at it, wondering about it, and I'm gonna say things like, "Hey, remember, we've talked about a variety of strategies that you can use to solve multi digit addition problems. You're welcome to take a look around and notice some strategies and wonder to yourself if they're ever going to be any of these strategies that you're welcome to use." So I encourage students to use them as a tool, because that's exactly what we created them for.
Pam Harris 21:02
Kim Montague 21:04
Pam Harris 21:04
Which could add also, maybe you tell me if I'm wrong, but if you see a student continually using a less sophisticated strategy. Like for example, it's a multiplication problem, and they keep using Partial Products, you might say, "Hey, is that one of our major like, is there for these numbers, we've got these four up there. Like which, do any of these feel like they would be..?" And then walk away, right? You're like, lob it out. Like, "You are working hard. You are working so hard. Let's, let's see if we can save you some effort, man, well, way to persevere. Are any of these fit better for those?" That can be, it can be a way of kind of pointing it out.
Kim Montague 21:37
Pam Harris 21:38
Hey, I did want to mention, you mentioned something about how this is not where we just go, like I had said, where the teacher hands you something that kids put in their notes. And it wasn't co-created together based on experience. I do want to say secondary teachers often will say to me, "Well, then what goes in notes? If if I'm not having students, like write down the example problems that I'm doing on the board, because we're not doing I do, You do, We do any more then what does go in students notes?" Well, this could be an example of what meaningful notes could look like. So I'm actually quoting Peter Lilienthal. There who says we need meaningful notes. Meaningful notes should be created by the student. And so as we co-create these anchor charts, that could be the thing that I'm like, "Hey, All right, y'all." Let me backup a little bit. As you do a Problem String or Rich Task, we don't have students have their notebooks open, and they're copying all the problems down for the Problem String. We don't do that. But as we create the anchor chart, that's the kind of thing that we do want students to sort of anchor for themselves as well. So if you are having students keep a notebook where they're putting notes, quote, unquote, those meaningful notes, anchor charts would be a great example of a meaningful note that a kid could put in a secondary notebook.
Kim Montague 22:54
Yeah, I'm gonna jump in real quick, because maybe I didn't hear you correctly. But you just said, we're not going to have students have their notebooks out and record and I don't want people to get the impression that they never record. What they're not doing is mimicking, copying every single problem and every example that you write. They're going to take, as we do Problem Strings, they're going to make note of the things that they need to write down,
Pam Harris 23:17
They're going to keep track of what they need to in order to solve the problem. Yes, what I meant, thank you for helping me clarify, what I meant is that's not in their formal, this is your notebook. Yeah, we do that more on scrap paper. Or you can even have sort of like a journal where you kind of whatever. The reason I say that is we have had teachers in the past, where it's a real formal thing, "Hey, we're gonna do a problem stream today, turn to page 19. And I expect you to have every problem that we just did copied down." That's mimicking, that's put students in a mimic mode, and they're not thinking. They're just trying to get stuff down in a notebook. So that's what I'm trying to discourage. I'm not trying to discourage students from having a pencil in their hand. Ah, Kim, I just had a pencil!
Kim Montague 23:57
Yes, you did.
Pam Harris 23:59
Having a pen in their hand and recording whatever it is that they need to be able to solve the problem. So we absolutely believe in Cathy Fosnot, mental math does not mean you do it all in your head. Mental math means you do it with your head. So kids can be the sketching down their thinking, writing numbers down, keeping track, as they solve. Yeah, thanks. Thanks for helping me. All right, y'all, so anchor charts. We think they're useful, handy things to help students anchor, to solidify the learning and also to create Algebraic Reasoning. Thank you for tuning in, and teaching more and more Real Math. To find out more about the Math is Figure-Out-Able movement, visit mathisFigureOutAble.com. Let's keep spreading the word that Math is Figure-Out-Able