Concrete Representational Abstract (CRA) Professional Development

And my thought with this presentation is I want to give everybody some common language, some common expectations. And then, you know, this is not a subject that like in 45 minutes we're going to be experts in, but we've got the common starting point. And then we will be able to dig in in PLCs with our specific grade levels so that you can see how it plays out in your particular grade level. And I am, as people join, I'm loving the outfits and the book characters. Awesome. All right. So our goals today are that you'll be able to describe the CRA model, sort of explain in your own words what that means, and then be able to give an example of the three representations, concrete, representation, and abstract for your particular math unit and grade level, something that you would use. And then I want you to be able to create and implement classroom routines that facilitate effective use of manipulative. So basically like one of the, you know, barriers sometimes using manipulative is that it's hard, right? And you kind of lose control a little bit in some cases. And so we want to talk about what are some things that are going to make that easier for you and more effective. And then facilitating the process of drawing connections between those different representations, which I'll go ahead and admit is the most difficult part, but the most important part, I think. So those are our goals. And here are our norms that you've probably seen me use before. Especially when we get into groups, I want to make sure that we use equity of voice and draw other people in into the conversation. We want to make sure that we're using growth mindset language about our own math ability and our own math teaching ability, because that's what we want our students to use too. And then just advocate for your own learning. If you can't hear me, if you think that I'm supposed to be showing something and I'm not, speak up. If you don't have the link, whatever it might be, please advocate in the chat or just unmute and let me know what you need. I'm happy to repeat or anything. Okay, so before we start, does anybody need anything or have any questions? Okay, fantastic. All right, so to just jump right in, what is the CRA model? I also want you to be aware that if you're searching for resources or see it in other places, you might be, you might see it called the CPA or the CSA, that middle letter. Different researchers call it different things. The R could be pictorial or the P could be for pictorial or S for semi-concrete. It's all basically the same thing. Some people might get all, you know, up in their feelings about how they're different. To me, it's all, you know, describing the same idea. So this particular definition comes from Dr. Berry at UVA and Kateri Thunder, who is a teacher in Virginia, and they wrote an article in a Virginia math magazine or journal article that said it's, I like this definition, it's a, the CRA is a three-phase instructional approach with each phase building on and explicitly connecting to the previous instruction. So there are these three parts and I think those, those key words about building on, you know, one, we need that foundation of the concrete and then building on top of that will be the representational and building on top of that will be the abstract. But it's not just three separate layers. We definitely have to have connection between those three layers. So those two actions seemed really important to me. So when we're talking about the concrete representation, it's really anything that's three-dimensional, that's hands-on, that you hold, you know, base 10 blocks and squares and cubes and links and pattern blocks, rec and rex, all those kinds of things. It can also be real life objects that we, that we use to model a particular idea. So today I'm going to show you an example with gummies or an example with socks. So anything that you move around that's concrete. Representational would be a two-dimensional picture or an image. It could be something that the student is drawing on their paper to represent the manipulative or to represent the object, or it could be something that you're showing on the board, on the screen. That would be an example of a representational. Also virtual manipulatives fall into this category because they are not concrete. They're not something that you're touching. They're still an important representation. And then abstract is when we take that, those mathematical ideas and move them into numbers, symbols, and words. So what we, you know, typically think of as, you know, solving math problems, plus, minus, all that kind of stuff, that's when we talk about abstract. But it's way more than that too, not just the computation. And these, these like descriptions came from a particular document from the VDOE that is linked at the very end of the presentation if you want to take a look at that. So I have asked Harmony if she would share an example of a unit where she used concrete representational abstract to kind of get us started thinking what that might look like. Yeah, so in third grade, we started with base 10 blocks, and we started at the beginning of the year with place value. And Tracy taught us this really cool thing. It was like, I see 10, group them again. And so anytime a student saw 10 or more of a certain base 10 blocks in the place value, they would regroup. And then that translated really well to additional subtraction, because we just continued that thing. So when we were doing additional regrouping, anytime we were doing base 10 blocks, as soon as they saw 10, they're like, oops, I gotta regroup. And then with subtraction, we did the same thing. But it was, I need to trade them in. And so we would, they needed 10 ones, they would make a trade with a rod for 10 ones. And then that just like gave them, I felt like a great visual understanding of the math behind it. Like we're not borrowing and trading just random numbers like this. We're actually making a trade, you know. And so I was so grateful Tracy gave us that language and we were able to use it. Thanks, Harmony. That was perfect. And I like, I could hear the connection part in what Harmony was describing, because I feel like once they were working within the abstract, they could connect it back. Hey, remember, we're not just, it's not just random crossing off numbers. Think back to when you were actually trading in or regrouping or grouping together. That's what we're talking about when we represent it abstractly. It was awesome. I read recently about another good example, and this one had to do with the graphing, which I don't think is something I usually think about in the concrete representational abstract, but I'll try to describe how this teacher did it. So they wanted, I think this was probably his second or third grade example, and they wanted to investigate how students were getting home from school. So they first just had the kids like group yourselves together in a mob and talk about it. And then they realized, okay, if we're going to talk about how we get home from school, like car, bus, that kind of thing, we have, we can make groups. So they, then the next thing that they did was separate their actual selves into these groups around the room. So the walkers stood here, the car riders did here, they had some bike riders went here. And then this idea of, well, it's kind of hard to compare. So they lined up in row, like one in front of the other. I'm almost picturing like on the tiles on my floor, like everybody's standing in a tile. So they lined up. So they kind of had these bars sort of being created with their bodies. And then you could compare, right? Well, then they sort of went back and they had everybody put their name on a sticky note and how they got home. And then, so they're all a mess, right? Together on the board. Then they sorted again. So you're still, you know, the, it was really concrete when it was yourself. I think it's, to me, I might call it semi-concrete when it's a, you know, a sticky note. It feels like it is still a 3D, but it's also a representation of you, you know? And so then they sorted them into groups. And then they, again, how could we know which one is more? And so they, they thought of this idea that, you know, the kids kind of coming up with it, that they need to line them up one on top of the other. You can't overlap them because you can't compare them as well if they're overlapped. So we're all going to start our bar, you know, our, our line of sticky notes in one place and, and line them up on the board like this. So then they sort of created these bars and you could compare them. But then like, this sort of seems mind-blowing to me. They like literally drew the bar around the sticky notes, right? So it went from like, hey, each card represents a piece of data to, look, it's a bar. That's where that bar idea comes from. And, and to me that, that, then that's in the representational phase, right? And that seems like such a seamless transition or connection from concrete, your body, kind of concrete with the sticky note, to, and then representational and abstract once we made that bar. I just loved that transition. And I had never thought of it before until I read that. So it's important to keep in mind though, that it doesn't always have to be linear. And what I mean by that is it doesn't need to be like, Monday, we're going to do concrete. Tuesday, we're going to do representational. Wednesday, oh, you're ready for abstract. You know, there's lots of going back and forth, moving forward, moving up, going back. I think the big idea though, is if you are teaching a new skill, certainly don't start at the abstract. Certainly start back at the concrete, you know, a couple of steps forward, maybe a step back, a couple more steps forward, oh, go all the way back to remember, a couple of steps forward. You know, this, this idea of like, you've got to start with the concrete before they ever make it into the abstract is kind of the big idea there. Any, let me pause for questions. Anybody thinking anything or have any questions? I was just going to say that even though you said that more of a second or third grade example, I feel like that's something that we could translate to first grade when we teach crafting, because we do lots of like survey type questions. And so we could do all of those steps even in first grade. Totally. Even, even kindergarten, they do the same thing. And I saw like an example outside of Maddie's room that, you know, I was thinking about too. Yep. Perfect. So Tracy, I do have a question. It's something that Harmony said about regrouping with addition. So can you clarify that a little bit? I didn't know you regrouped with addition. Is that just meaning like going from 10 to 20, 20 to 30? What, what, what do you mean by that? Regrouping in addition? Harmony just took a big old bite. Harmony, do you want to answer? Do you want me to go? You got it, Tracy. Okay. I think she is meaning if you are adding two, two digit numbers, let's say 15 and 16. When you put all those ones together, the five ones and the six ones, it becomes 11 ones. And we don't use the word carry anymore, because that doesn't actually mean anything mathematical. We regroup that 11 ones into a 10, one 10 and a one, and we regroup it over into the tens place. Okay. Harmony, did I represent that correctly? Yeah, exactly. Yep. Okay. Thank you. Yeah, good question. All right. So let's take some time to talk about what, or to think through specifically, what does this CRA look like in your, in your particular setting? Where does it come in? What are the concrete things you would use? What are the ways you represent them in pictures? And then what are the ways that we represent a mathematical idea abstractly? So what we're going to do is break up into groups in a minute. And each grade level is going to group, you're going to choose, like I'm going to open up small breakout rooms, and you're going to choose which group you are going to go into. If you are a special ed teacher or somebody else who doesn't have a grade level, choose what makes sense for you, who you work with the most. And you all are going to navigate to your particular slide that has your grade level at the top. And then I want you to choose a skill that's coming up. It could be what you're working on right now, but it would probably work best if like the next thing that you're getting ready to start, think through what that is, and what are some ways that you can represent, represent it concretely, representationally, and abstractly. And when I was, when Elizabeth and I were talking about this, kind of planning it out, she was like, you know, maybe clarify that we're not, we're not trying to be like super creative here. She gave an example of a teacher who was like doing place value, and they, they kind of got off, a little off base by like the ones are marshmallows, and the tens are pretzel sticks, and the hundreds are graham crackers. We don't need to be creative. We need to think through, like, this is kind of 101 level, how to, what, what materials match it concretely, concretely, representationally, and abstract. You know, think basic, and you could, you can think of actual real life objects. Like, I'm gonna, you know, share an example of where gummies really isn't appropriate. My son just looked up like, oh, gummies. Gummies is an appropriate thing, but yeah, you don't have to be like, oh, let's think of the most creative thing I can ever think of. So, for example, let me do a quick example, because that might help. I know that many, many grade levels started out with place value, but let's think for second grade, because I have it up here. So, just counting, like, I'm going to think about what could we do to help them with place value? Well, counting collections, you know, is a fantastic place to start, because they are not, they're not pre-grouped materials. They're groups, materials that you have to actually group into tens. You know, like, sticks and rubber bands, because you can, like, take 10 popsicle sticks and actually group them together, and then we work into pre-grouped models, like base 10 blocks. Okay, and there might be more, but we'll stop there. And then ways to represent it. So, literally, just like drawings, right? I don't know why I didn't like that spelling. You know, drawings at base 10 blocks, where the rods are just a line and the units are just a dot. I think that, like, base 10 mats, where you have a space for hundreds and a space for tens and a space for ones, help with that. And then the abstract is, could be numbers, could be, like, writing expanded form, I'm thinking of. So, there's way more. You, you, I want you to brainstorm how you would represent it and talk about it and show it. And then if you finish, if your team finishes before the time is up, probably going to give you, like, three, four, five minutes, somewhere in there. Pick another one and, and fill that out here with a separate skill. Okay? Before I open the groups, what questions do you have? Oh, I have, yeah, let me pause for questions. And then one more thing. Maybe you're going to be like, uh, Tracy, I don't actually know. So, on the slide right after that second grade example one are some ideas. If you, if you're not sure what tools are available or what representations are available, if you open up the, the log and click on the curriculum framework document, the one that's linked, many, many, many times, probably almost every single time, it gives you, it says concrete materials, it gives you ideas, and it oftentimes can also give you representation ideas, like what to draw, what to show, what kind of models to use. So, if you're not sure, look back at that page to see what examples are there. You probably, I don't know if you'll be able to pull it out right now, but every classroom should have a hands-on standard book. That's a good place to look. And then this particular, this links to a particular document that just might jog your memory because there's pictures of manipulatives on it. So, that is a tool that is available. All right, I'm going to put the directions, the directions are in the chat right now, as well as the link to the presentation. And then here come the breakout rooms. Heard some great discussions happening in those rooms. Thanks for thinking through those. And this can be, certainly be, what I like to, what I've heard called rough draft thinking, right? We can revisit these as we're planning, and then definitely, again, in PLCs as we're talking about it. But I like, I liked what I was hearing, the communication and the recording. Thank you so much. So, I put this here because I just wanted to remind you that maybe you were like thinking of some manipulative that you were like, this would be great to use for concrete, but I don't have enough. I heard first grade we were talking about something, and they had a pretty basic manipulative that they were like, but we don't have enough to give everybody 20. Okay, there's like jillions of dollars of money floating around out there to be used. If there is a math manipulative that you need, just speak up, and we can make it happen, you know? So far, I have not been told no. Kelly's not looking up, but I think, I think she's agree. Yeah. So, like, don't let that be what keeps you from, from doing something, you know? And then also, if you don't, I, I think, for me, it's kind of fairly easy to pick the right concrete manipulative, you know? Like, I feel like I have some, some decent ideas, and maybe you do too. I think the representational is maybe what's harder to think about. What, how am I actually going to draw it? How am I actually going to show it? If you're stuck, let me know. I get all geeked out and excited about that stuff, and we can, we can digest that one-on-one. I can, you know, draw it on a note card and send you a picture. Like, I can help you with the representational piece, and then the abstract, and then how to connect them. I mean, that's the kind of stuff that, that I am here for to help you with. So, if you're stuck on how do I represent it best, I feel like Tammy and I have had a really good conversation one time in that area. I don't remember. Maybe it was subtraction or something. Like, a way to, a way to show it so that, or maybe it was third grader. I don't know. But, but, you know, those conversations of, like, connecting is where I'd love to help you. Okay, you got the point. So, moving on to that idea about routines, how I talked about, you know, it can get scary to put objects in the hands of every kid in your room, because there's a little bit of, like, losing control, or it can feel like you're losing control. So, when it comes to organization, I think the, it, that's super important, right? The, kind of, the first step is making sure that you've got them set up in a way that are accessible to you and the students, a way that they can get them quickly. In my room, they were never in the closet. They were out on the shelves right where they can get them, because I want them to use them. If they're in the closet and they have to ask permission to get them, or you have to reach them because they're up high, they're not going to use them. So, putting them in a place that's easy to get to, I think that labeling them is important, of course, but then also labeling where they go, and I'm not that good at organization. You might know that from looking at my desk from time to time, but when, when I did get it right in my classroom, I had three shelves of manipulatives, and kids, kind of, roughly knew how to put things back where they go, because each bucket, they were, the shelves were colored. So, this is, like, the pink shelf, the green shelf, and the yellow shelf, and my labels, I just, like, drew on it with a highlighter of, kind of, what shelf it would go on. Just drew pink, all these labels had a pink line on them, so they go on the pink shelf, and if they roughly can get them back in that place, that would be great, and then prep ahead of time, if needed. If everybody's going to need 20, and you, they don't need to count out the 20, they just need 20 of something, then go ahead and have the 20 counted out, ready to go, so that you can pass them out quickly. That's not always appropriate. Sometimes, it's, the part of the task is the kid figuring out how many of something that they need, but do that if you can. One of the biggest things that helped in my room was once I finally figured out that if I took one of those flat bins, like this, with the green top, and put, like, a table's worth of base 10 blocks in them, then I could just put that on a table, and that was enough that, you know, two, three, or four kids could use, basically, all the time, and I wasn't having to, like, separate everything all the time. So, think through how to organize it, so that they're not, you're not losing them as you're passing things out, or they're getting them. I think it's really important to think about that math manipulatives are a tool, but they can, they can also be a toy. Kids loved playing with snap cubes in my room. Kids loved playing with Cuisiner rods in my room. They're fun to play with, right? That's kind of why they're effective, because they're interesting. So, we had, when I taught fourth grade, we had times where they were toys, and we, a lot of times during quiet time or indoor recess, they played with them, but then in math class, they are a tool, and if you're getting something out for the first time, I think it's important to let them just play. Have five minutes of play time first. Okay, everybody pause. Now, they're a tool. From here on out, we're going to use them as a tool, and if you, you know, set up what that expectation looks like, and if you don't, if you go back to it being a toy, you might lose the privilege for that particular time or for a few minutes to actually use it. Some people get all in their feelings about, no, they are not toys. They are tools, but I feel like, I think you can set the expectations, and they'll be fine, and then student choice is important. I, like I said before, I really think that if a kid needs it to solve something or to solve a task or a computation, they should be able to have them. I really don't think that there should be like, nope, we are done with manipulatives. You can't use them. Let them have them, and let kids know that if you need them, here are the ones that you always have access to, but then there are certain times where you want to go ahead and guide them in a certain direction of what materials are a best choice, and then let them choose between a few things. Fourth grade is planning on a particular task that's about, they're going to work on one that's comparing distances from like somebody's house to another friend's house, and so we're talking about fractions, like parts of a mile, and we talked about how there are tons of fraction manipulatives, but tons of them are not appropriate. It wouldn't make sense to use fraction circles because miles are linear, so they're going to set out fraction bars, fraction towers, and number lines because those are three linear models, and so you're kind of directing them in a particular direction, but they still get choice, which is really important for kids and developing their mathematical identity. Okay, doing good on time. Then the piece that I think is the hardest, which is facilitating the connections. We don't want it to be like chunk, chunk, chunk. We want to be, like I said before, moving backwards and forwards and connecting those ideas for the kids because without those connections, though it could just become two more algorithms, right? Like you could just be teaching an algorithm, a step for how to do it concretely. That's not anything like how we do it with a picture. That's not anything how we do it with picture, with numbers. You want them to see those connections to really build that conceptual understanding, so, and I'm not getting this out of thin air. This is, you know, I want you to see the connection to the bigger like mathematical research, so the National Council of Teachers of Mathematics is like the, you know, the group to listen to when it comes to teaching math, and they wrote a book called Principles to Actions, which it doesn't matter, but it's right here somewhere. Anyway, it's kind of like the guide on how we should be moving forward in math, what we should be trying to do, so like the book that we should be working toward as our goal, and a lot of our, tons of what are in our Virginia standards are based on this particular book, Principles to Actions, so one of the things that they share in there are the mathematical teaching practices. There are eight of them, and I love this graphic, how it shows that they are all connected, you know, starting with the goal, and then we have these two big parts of math, which is implementing tasks and procedural fluency, and then all of these are part of it, and then I like how these teaching practices are kind of connected. You can't really separate one from the other, so we are talking, when we're talking about the CRA model, we are specifically working on this one, using and connecting mathematical representations, which is one of the eight most important things we should be doing as math teachers. That's kind of my point there, so what I'm going to do is kind of show you a little bit of that progression with even and odd, which is something that, you know, officially gets started in second grade. I think that, you know, third and fourth certainly talk about it again because of multiplication, and then fifth grade they revisit again to particularly talk about, like, adding odd and even and multiplying odd and even, but it's an idea that really any grade level can understand. I think it's important in a lot of places, so I'm going to walk through what that could look like in using the concrete representation in abstract and try to focus on those connections as I go through. All right, so if we are talking about odd and even, I might start with socks, right, and so I might say, hey guys, we are going, there's like fuzz everywhere, we are going to see can we pair these socks up and see if we have any extra or if we have pairs, so then the task would be for the kids, not me as the teacher demonstrating it, like literally give them socks to pair up and see if they can match them, and of course they're going to be like, there's no matches here. Okay, well in our house we wear non-matches, so it doesn't matter that they, they don't have to be matching. All right, so there's a pair, there's a pair, there's a pair, there's a pair, there's a pair, and then I have this one left over. Isn't that odd? How weird, I'm missing one, right? I don't have a partner for this guy, so we would do that multiple times, and this is very concrete, and it's also very real life, like a context that they might have had experience with, okay, so this is starting with the concrete. Okay, so then you could, I, this is still concrete, but maybe a little less real life, right? These bears, oh and let me, let me comment here that I purposely had eight different socks out of our, you know, lone sock basket. We have like 10 times this many, but I purposely picked 11 different socks so that they weren't looking for the same. I don't want them to start to develop this wrong, this misconception that it's only a pair if they are exactly the same. All right, so then these bears are going to do a water balloon toss, and they need to pair up to see if they can each have a partner. Can you make the bears have a partner so they can be a pair? They can be a pair. They can be a pair. They can be a pair. How fantastic! No one was left out. They, they were able to group into, into four groups, and then we can start to develop the, the language of this being even, and we can relate it back to the idea of the socks. Remember when the socks, we had, you know, one sock left over that we called that odd, but here, because we have all of, we're able to make pairs, this is an even number. Okay, so that's like kind of the first half of even, but even an odd is a whole, is a whole nother idea too. Not just making pairs, but making two equal groups. So, oh wait, hold on, this is first, real life example first. So, my two kids want to share, and you can actually give it to students, right? Here's a, here's a container, Tripp's shaking his head like, absolutely not, not sharing. Here's a container of gummies, and here, two kids in my class share these evenly, and so I actually have no idea how many is going to be in here. I'm curious to see. I eat my gummies in pairs of two all the time. All right, so one for you, one for me, one for you, one for me, one for you, one for me, one for you, one for me. Oh, fantastic! We were able to split them evenly into two equal groups. This is an even number. Some kids might end up with an odd number, and then you have to talk about that, right? But we're building that idea with something that they've done before, which is splitting into two equal groups. It's an even number, and then you can kind of work a little less real life. Here, Tripp, come get these gummies. You can work a little less real life and say, hey, this, you know, I'm giving you this building material today. Can you, you and your friends, see if you can split these evenly, just like we split the gummies evenly? Making that connection back, and so they could, yeah, that made two equal groups. You take these, I take these. Okay, see what cool thing you can build out of these now that you know you have an even number of pieces, and yeah, you've built that idea that even is two different things, making pairs or making two equal groups. So all of those four were concrete, and I did kind of real life, and then more mathematical tools, and if you're familiar with building fact fluency, they do the same thing, right? They use real life pictures of tennis balls or something, and then they go into tools, okay? So then you would go into the representational, and you might, you might ask them, take a number like nine. The number, let's take the number nine, and I want you to show me two ways that you know that nine is odd or even, and so sometimes kids just, you know, might do this. This is a wrong answer I've seen before. Okay, nine is odd. This picture doesn't have anything to do with odd or even. It is nine, and it is an array of nine, but you have to connect back to the two things that we did. Hey, remember when we did socks? Pretend these are socks. Well, how would you show that it was odd or even based on socks? Okay, one, two, three, four, five, six, seven, eight, nine, pair, pair, pair, pair, odd. That's a picture that shows nine is odd. The other way, so connecting back to the concrete, hmm, I remember we were doing gummies. What if these were gummies? Okay, let's see. One, two, three, four, five, six, seven, eight, ugh, nine. So group for you, group for you. Nine is an odd number. So this, this is two different ways and representational to show, and, and when they were stuck, we were making those connections back to what we did concretely. We did this one, how we know, so these are some representational. I think skip counting, um, I actually, I would call this abstract, probably a hundred highlighting on a hundreds chart. I think I would probably call representational. And then when you get to, you know, a question just like this, this would be the abstract level and not where we would start, you know, just circle all the numbers. That would be the abstract level. So each time, if a kid was getting stuck, make a connection back to what you did concretely, or, you know, make a connection back to concretely and then say, can you draw it like we did with, you know, the, the, that step. Okay. So, um, in the document, I have listed some, um, links here. This is that one from UVA. Uh, I found this one very interesting about how, like, it's not all about the manipulatives. It's about what you do with the manipulatives. Um, and then a document from the VDOE that I thought gave some good basic information. And then if you need a reminder of what tools we have here, this is back to our RSP lending library of things that you can borrow, or you can just look at it. You know, maybe you have something in your room and you forgot you have it, but if you look at the, all those manipulatives there, you might be like, oh, that's a good one to use. Let's try that one. Um, so I really appreciate you being here. Sorry for going over a minute, but, um, I just want you to know that now that we have this, um, like language to start with, and maybe we can continue this, you and your groups and your planning, but in PLC, we can continue. So thank you guys. And I'll stick around if you have any questions, but you're free to go.