The Concrete Representational Abstract approach to teaching mathematics has a very long history. However, there are a lot of people that don’t know about it. And, I believe the people who do know about it, are probably doing it incorrectly.

Watch the video or read the transcript below:

What Is CRA And Why Is It Important

Let’s start off with what is concrete, representational and abstract? I’m going to shorten this up for us. I always call it C-R-A, it’s just easier to say. Other people might say it as Concrete Pictorial to Abstract, so you might also hear it as CPA. The general idea is the same.

The first stage is concrete. People say all the time that we need to make it concrete, get it hands on for our kids. They need to physically be doing the mathematics. This is often the case where we bring in manipulatives. We want kids to have physical objects to use. The representation stage is when we start doing away with the manipulatives and we just have kids draw, just drawing a representation. And then the final stage is this abstract stage, where we’re just writing digits. This is also when we’re doing the algorithms, it’s seen as very abstract for kids.

Now, why is this so important?

Well, number one, we have kids who are just various learners. We know kids who can jump straight to the abstract, but we also have kids who just need the visuals. They need hands-on learning. We have very different learning styles and so varying the way that we approach the mathematics with the students is helpful in that respect.

The other thing to think about is that in so many other areas, we don’t expect kids to just jump straight to the abstract. We attach a lot of visuals to it. We have kids that have experiences that help them be able to make connections. I once heard Marcy Cook talk at a conference and she gave this example. If you close your eyes and think of cat, what conjures up in your mind? It is not C-A-T, it’s an actual image of a cat, and yet, with mathematics, when we think of 37, what conjures up instantly is a 3 and a 7. It’s the abstract form of it and we need to allow kids to have the different visuals, different connections of these abstract concepts in mathematics.

How CRA is Typically Done in the Classroom

This is typically the way that this approach is done, is that we start with concrete. Then, we move kids to representational phase, and then we move them to the abstract phase. We see it all the time in the things that show up in our textbooks.

Here’s an example from a particular textbook, where it laid out what you’re doing lesson by lesson, and you can see they’re using manipulatives. Then in the next lesson, they’re using those manipulatives and relating it to a written method, which, that’s a really big jump, let’s just say. But at least they’re doing some relationships. And then another lesson they’re doing the math drawings, which is the representation and they’re attaching that to a written method.

So they have all these different lessons where kids are doing each different phase and for our kids where it’s difficult to make connections, by the time you’ve done all of those different phases, they see them as three different ways to approach the problem, instead of seeing them as connected.

To me, seeing this as a linear path, where we must start kids in a concrete phase, then move to representation, then move to abstract is the wrong way to view it.

How CRA Should Be Done in the Classroom

Instead, I want to encourage you to view it more as this Venn diagram. These overlapping circles that if we can hit an activity in what I call the Sweet Spot of CRA. If you get an activity where you can help kids see that what they’re doing concretely, with manipulatives, connects to this drawing that we’re doing right over here, and that connects to these written symbols.

When you help kids make all three of those connections within the same lesson, it’s those lessons where at the end of it we’re like “that was so awesome!” But we don’t really know why it was so awesome sometimes. Was it because kids could see all of these connections, things were flowing, or they started having those light bulb moments? When you’re in that sweet spot, it just makes a world of difference for kids. And the cool part is that you allow kids to work wherever they are at in any activity.

So if I go back here to this visual of this lesson plan, basically, you have a whole class of 30 kids and you’re doing a lesson using manipulatives to represent addition, what about those kids who are past it, who don’t need the manipulatives? And then maybe in the next couple lessons you’re jumping into the written method, but you still have half your kids who need the manipulatives.

Being able to do an activity where you allow all three of these phases to happen at the same time allows your students to be able to work where they are at within these phases.

Allow kids to be doing stuff with objects. Have them always out, it doesn’t matter if your lesson plan calls for it or not. Letting kids always have objects available so that they could work the problem out concretely. Then have visuals, help kids be able to do drawings and models.

Now, here’s one thing, change your wording from drawing to modeling because when a kid wants to draw a representation, especially in the young grades, they want to draw a very detailed drawing of it. If it’s about butterflies they want to draw the butterflies and color it, so in the representational phase is a time where you can teach kids the difference between drawing and modeling the mathematics. There’s a time and place for detailed drawings, but there’s a time and place where we’re just modeling what’s happening in the problem. And then we can move them into attaching those symbols.

That’s one of the downsides that I see is when kids are working hands on and they’re doing manipulatives, we aren’t connecting it to a drawing and we’re not connecting the abstract symbols, and so when they get to those phases, they don’t connect it back to the hands on stuff that we did five lessons ago.

So, here’s a little example, just starting out with letting kids build a certain number. Here’s a couple examples of moving them from building it concretely to representational to abstract. We want them to be able to see those visuals, and connect it to the abstract because there’s so many more ideas around that number that they build besides just seeing it as a four and a three for 43. When they can build it hands on, they see the drawings, and they connect the representation to it, the abstract representation to it, so much more number sense gets built.

Then as they start moving into working with numbers, doing operations, we still want to do concrete, representational to abstract. We want kids to physically build and model the problem.

Here’s an image of a rekenrek. They’re building seven plus eight on a rekenrek. And then they can use the number path to represent what they saw on the rekenrek, so it’s the same visual that these kids created. They both had seven on the top and eight on the bottom, but one kid saw the fives within there and so on the number path, they’re circling the fives and then the group of two and three. While another kid saw the groups of sevens within, that seven plus eight is just like having seven plus seven. So on their number path, they modeled what they did on their rekenrek there. And then we attach the symbols to it.

That’s all it takes is when you’re doing the concrete stuff, layer in the representational, layer in the abstract piece so that we can help kids start making those connections and go quicker throughout these phases. We want them to go from concrete and representational and finally get to abstract, but it doesn’t happen in a linear path. We need to be doing it together so that when they eventually get to the point where they don’t need the concrete, they can jump straight to the drawing piece and connect that to the written piece. And then when they get comfortable with that, they will naturally not want to do the concrete, they won’t want to do the representation because they see it just takes way longer than that.

I see far too many worksheets that have kids draw out all of the 10’s and ones when a kid could do it quickly. If a kid is at that abstract stage, and they’ve built that representation, they understand the representations, they understand the connections and now they’re just able to do it abstractly, that’s okay, even if your textbook says they have to do it with a drawing, they don’t. And even if your textbook says they should do the written method and you’ve got kids who need to draw it out, let them draw it out. The concrete, representational and abstract, kids will be working in those phases in a messy way. It does not happen the way that our textbooks want it to happen.

Here’s another quick example using multiplication. We do a lot with building area model when it comes to multi-digit multiplication and we use base 10 blocks to model that. So the concrete phase we’re modeling with base 10 blocks. Then we move into the representational phase of drawing an area model and then we move kids into what’s known as a partial products or even the traditional algorithm. But we don’t let kids see the connections between those.

How often have you done a lesson where kids do all three of those within the same lesson and then, the key point here is, at the end, is you ask how these are connected?

If somebody just did the concrete, they did just the base 10 blocks and they didn’t make it all the way to the partial products algorithm, can we help them see connections between those? Or do they see these as three separate algorithms to do? We want kids to see that you’re doing the exact same things, it’s just one was physical, you’re doing the concrete. One was a drawing and one we’re doing it without any pictures at all, but we’re doing the exact same mathematics in every single one of those.

Now, take a look at your standards, because I can bet at almost every grade level, you have at least one standard in there that has it specifically called out that kids need all three of these stages.

This is one example from second grade where it’s talking about adding and subtracting within a thousand and they need to have concrete models. They need to do drawings and they need a written method. It doesn’t say these need to be separated though.

We can do all three of those at the same time. That’s my reminder to you is the concrete, representational and abstract is a very powerful thing in mathematics, but it’s not a linear progression. These things overlap and there are definitely times when you might just work in the concrete phase. Or you might just be doing drawings or you might be doing a drawing and the abstract, right?

That’s the cool part about this Venn diagram, sometimes you’re in that spot where just two of those overlap. Or sometimes you’re at the spot where you’re just doing abstract, you’re only working on the written method. That’s okay. They’re all spots that work for kids, but when you get all three of them together, it helps build some magical connections for students that are just so, so, powerful.

I hope that this helped you build your math mind, so that you can go build the math minds of your students.

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