I don’t know about you, but I really struggled with developing an understanding of fractions. Even as an adult, I did not have fraction sense, and it wasn’t until I had experiences that involved the CRA approach around fractions that truly developed my understanding. This week we’re going to be taking a look at using the CRA approach to the teaching of fractions.
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What Exactly Is The CRA Approach
Now, if you don’t know what CRA is, let me give a quick refresher. So, the C stands for Concrete. This is typically where we have kids doing hands-on stuff with manipulatives. And then the R stands for Representation. This is typically where we have kids draw a model or a representation of the concrete things we’ve been doing. And then the Abstract is when we attach symbols, so it’s the abstract world of mathematics.
Typically, these are done kind of in isolation, and I’ve done a whole other blog post about why I think we need to work in the sweet spot, where students get an opportunity to work in all three of these areas at once.
Example Of Doing All Three At The Same Time
So today, I want to show you an example using fractions to show you how it really deepens their understanding. What we’re going to take a look at is a game called Race to One. This is a pretty popular game. It’s in a lot of curriculums out there. I don’t even know who started it. But basically, you have a strip that is the whole. And you can make these using your own fraction strips with construction paper. I’m going to show you an example here in a moment. You can also do it with pattern blocks, you can do it with fraction strips that you already have.
The idea is you start with the one, whatever the whole is, and then you’re rolling dice to try to fill the whole. Now, as I said, typically this is done just with concrete materials. And so I want to show you a video. I created this video showing how to play the game with the concrete materials, but then, partway through, I talk about attaching the representation and the abstract to it.
If you’ve ever played this game, there are kids who will play it, and they can just mindlessly play it. There’s not a whole lot of mathematics that you actually need to do. The only thing you’re doing is rolling the dice and then trying to figure out which piece you’re going to use, based upon what you roll for the dice and you’re putting it onto the whole, and then you’re trying to fill it up. You’re not adding fractions together, really. There’s not a whole lot of thinking that kids are actually doing around fractions. If you aren’t sure, try playing it yourself without the extra steps here. Just play the concrete game and see what mathematical thinking you have to know about fractions. There’s really not that much.
But, when you attach the representation where you have to draw and model what’s happening with those concrete manipulatives and then you have to attach the symbols, oh my word, the level of understanding and the connections that it builds for kids just goes through the charts! Man, it is so cool to watch.
[To watch me play the game, please watch the video above starting at 3:52.]
In the game Race to One, you need five different colors of construction paper, all cut to the same length. One of them you will keep whole, one of them you cut into half, one cut into fourths, one into eights, and the last one into 16ths.
Now, of course, this is not something you need to do. This is something you can have the students do, and there’s a lot of learning that happens as you do that.
The other thing you need is a wooden cube that you will write a half, a fourth, a 16th, and an eighth, with an extra 16th and an extra eighth on one of the sides. So once all the pieces are cut, the game is you roll. Whatever fractional amount you roll is what you place onto your game board. And the next player’s turn, and then when it’s back to my turn, I get to place that amount.
The goal is to be the first one to get your game board filled or all the way to one. Now, as you’re playing this game, there’s really not much that kids are actually learning except recognizing the fraction and picking which strip is that fractional amount and placing it on here. They’re not doing any addition whatsoever. In the end, you could have them try to add this all up, but the kids who actually get to one, just already know that’s it’s one, so they don’t have to do a whole lot. So instead, one of the things I like to do with this game is basically to have the kids physically working hands-on, putting stuff on there.
I’m a big believer in trying to incorporate all three parts of Concrete, Representational, and Abstract all in one. If this is our concrete, I’m going to have a tape diagram, also known as a bar model, as my Representational. And as I’m going along, I’m going to have the kids draw a representation of what is happening on their game board right on their tape diagram.
So, in this case, I would need to put a 16th on there. Now instantly, I have to figure out approximately where a 16th is. The idea is not to have them actually cut this into 16ths, but to do a little bit of estimation and visualization to figure out about where that would be.
And here’s my abstract, where I’m attaching the symbols to the representation. I want to get a fourth. And now, here comes the part when they are going to have to figure out how to add because they have to figure out approximately where they are along their game board now. So I have a 16th, and I have a fourth, but how far am I?
Now, for most of our kiddos, they’re gonna be at a spot where they don’t actually know the algorithm yet for doing this. So the goal is to help them learn the algorithm by using their manipulatives. If I have a 16th and then a fourth, they can use their manipulatives to figure out how far along their game board they are by attaching the strips on there, and they see that 1/4th is the same as having 4/16ths, and they are now 5/16ths along the path. And then the game continues until someone reaches one.
But the real power happens when the kids have to do the concrete, the representation, and helping them figure out along the way this idea of common denominators and how that really means just having the same-sized pieces when we add and subtract.
And just a little side note: I know on here, I’m showing it with the construction paper and we made a strip to represent one. But if you want to do this with pattern blocks, you can do the same thing. It’s just you probably need to provide them with the hexagon. If you’re using the hexagon as the whole, provide them a template of that hexagon and then have them draw in the pieces and attach the symbols as they go along. The same thing if you, whatever fraction pieces you have, you can use this.
However, I do prefer using these strips and having the kids make them because there is such powerful mathematics that kids develop as they are creating these strips and helping them realize that a half and a fourth and how they are related. This happens when they get to fold and they get to see that a fourth is half of a half and that an eighth is half of the fourth, and so on. They develop such cool understandings when they get to build their own version of these strips.
So hopefully you see through that, that a lot of times, we think kids aren’t quite ready for the representations or the abstract, and so we just kind of stay in that concrete world, but when we attach the representations and the abstract to it, all three of these work together to build this cohesive understanding.
When they’re trying to draw this representation, they’re attaching the symbols, the abstract world, to it, they need to go back down to the concrete. The concrete can help them develop a better understanding of the representation and the abstract. These should not be done in isolation. They can work together in the same lesson to help kids build their deep understanding of fractions and what it means to operate with fractions. There is so much understanding that gets built in just that one game. That’s why it’s one of my favorite games for fractions because you’re talking about equivalent fractions, you’re being able to visualize fractions, and you’re doing addition.
There are also things you can do with subtraction, where you can fill your whole and then race to zero, so you can go down, and that builds the idea of subtracting. Like there’s so much you can do with just that one activity. But the understanding that gets built really happens when you are doing the concrete, representation, and abstract all together.
Now, remember, this is just one example. What I want to do is, through that example, encourage you to look at the other activities you’re doing with fractions. If it’s a concrete activity, is it okay to just be working in concrete? Because sometimes, that’s totally fine.
But what would happen if we attached the representation to it? And what if we did the symbols with it as well? What would that build for kids, right? I want to encourage you to take a look and see where the activity is at, because sometimes they’re really abstract activities, and maybe we need to add in the concrete and representation to help the kids better understand it, right?
So, this was designed to encourage you to look at the activities and see if there’s a way that you could use the CRA (Concrete, Representational, and Abstract), to build a more cohesive understanding about fractions for your students. Alright, I hope that this video helped you build your math mind so you can go build the math minds of your students.
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