I have evolved in my thinking about the CRA approach.
So what is the CRA approach?
CRA stands for Concrete-Representational-Abstract and is an instructional approach to teaching math concepts. Concrete is the stage where students hold and use manipulatives to solve math problems or work to understand the math concept. <Click here for a printable math manipulative kit> During the representational stage, the students represent their thinking through drawings or number lines. The abstract stage is where the students are able to think and solve the math problems through numbers and symbols, such as equations, formulas and algorithms.
I learned about Concrete-Representational-Abstract (or CRA) 12 years ago, when I was an instructional coach in an elementary school. Here is the picture I was shown when I first heard about CRA:
I immediately connected with this idea, and I was excited to have a solid pathway to follow when planning my math lessons. I reflected on my own math education as a child, and I felt like I was only given abstract opportunities to work with math concepts. I could have made so many mathematical connections earlier in my life if my teachers had used this model! But the arrows in the above image confused me. Why were they all pointing in different directions? How can you start at ‘abstract’? Isn’t that going backwards? Don’t we want to start at concrete and move in a linear direction to get to our final destination of abstract?
I wasn’t ready to process this image of the CRA approach. So I created a different picture in my mind based on how I processed it.
As I used the CRA approach with students throughout the next few years, I learned that my most effective math lessons were the ones where my students were using concrete, representational and abstract all in one lesson. I began to have an ‘Aha’ about that initial CRA image that I was shown. I had wanted to make it “step 1, step 2, step 3,” but I was finding that if my students used manipulatives, drew out the strategy or created a number line, and wrote an equation all in one task, they were making better mathematical connections. So I began to open my mind more about it not being a linear model. But I still felt strongly about students needing concrete manipulatives when they were first introduced to a math concept.
But in the last few years, I found that my math lessons were even better when I first hooked my students with a story or a number string. I found that it gave them a context and helped them make a connection from what they already knew to the new math concept I was teaching them. <Click here for a CRA problem solving lesson>
What does this mean to a 2nd or 3rd grader?
In this image, I see concrete manipulatives being used to help a student create arrays and understand the commutative property of multiplication. I see 4 rows of 3 or 4 x 3 and 3 rows of 4 or 3 x 4. But a student may just see a bunch of tiles… What does this mean to them and why do they need to learn this? This may be a concrete example, but without a connection to a context, I don’t know that they are making the connections to deep learning. When this is attached to a context of how many ways can a teacher arrange 12 desks so that all of her students are in rows and columns, it connects with something the students already know about and makes sense to them.
So maybe we don’t have to always start with the concrete. We could start at the abstract stage of a math concept. I worked with a group of 3rd graders on multiplication strategies, and I gave them the following number string.
I wanted them to discover the strategy of decomposing one number into smaller parts to make it easier to find the product. They had a deep conversation about the relationships and patterns they observed. They were amazed to discover that 6 x 5 + 6 x 2 = 6 x 7 because 6 x 7 was a difficult fact for them, but they knew their 2’s and 5’s facts. They wondered if this strategy worked for other facts. I told them we could test it with other numbers, and they were excited. I had hooked them with abstract thinking. Then they used concrete counters and drew rectangles on grid paper to test other facts and model why this strategy worked.
<Click here for or a FREE Number String Resource>
<Click here for more Number String Resources>
Or maybe students could start in the representational stage of a math concept. The research on fractions says we shouldn’t start with fractions tiles. We should start with the students representing fractions first. What do the students know about fractions? The students may start with drawing rectangles that are partitioned and shaded in or creating a number line with fractions.
I started questioning the CRA image in my mind.
I realized how vital it is to plan with the CRA approach because if we don’t find ways to allow students to connect the concrete to the representational to the abstract, we lose the opportunities for students to make deep and lasting connections with the math concept. But there are times when it’s best to start with the concrete; there are times when it’s best to start with the abstract; and there are times when it’s best to start with the representational. Aha! I finally understand the original CRA image I was shown many years ago. My focus is now on helping my students make connections between Concrete-Representational-Abstract for each math concept and not so much needing to start with the concrete all of the time. I realized the importance of the arrows in these models. It is not only about opportunities to explore the concrete, representational and abstract but also the connections between them.
