So sometimes when you have been doing something for so long, you might feel like everyone knows about it. Like it’s old news. But I have spoken before about how dangerous assumptions like these can be. I was talking to a friend of mine who has been teaching pre-k for quite a while and she said…hey, what is CRA? I told her, you are probably doing it without knowing what it’s called. And after I explained to her what it was, she said “Oh yeah! All day every day!” But it got me thinking I should put it out there since I talk about it so much…just for those who may not know yet!
There are a ridiculous number of acronyms in education! But CRA stands for Concrete, Representational, Abstract. It may also be called CPA, which stands for Concrete, Pictorial, Abstract. Both of them mean the same thing.
So the CRA process is a learning process that allows students to be able to visualize mathematics. Students would start by exploring and working with Concrete manipulatives. These are math tools you can touch and manipulate. Concrete manipulatives are things like beaded number lines, fraction bars, two sided counters, pattern blocks, and rekenreks. There are many other examples as well. Concrete manipulatives help bring mathematical ideas to life, allowing students to construct deep understandings of mathematical concepts, IF they are used correctly.
After students have worked with concrete manipulatives, we can transition them to drawings that represent what they have built with those concrete manipulatives! These are the Representations, and this is the second stage of the CRA process. Obviously, students may not always have concrete materials available, and once a concept is learned, using concrete manipulatives may become tedious to students, so as educators we can teach them to draw models or representations that match what they did with concrete manipulatives. The same thinking, just shown in a different way. Sometimes visual representations don’t get the credit they deserve. I have even seen students feel like drawing models is “babyish” and they won’t do it. We have to normalize making math visual because modeling with mathematics is a powerful way to communicate our mathematical thinking!
The final stage of the CRA process is the Abstract stage. When students are working in the abstract stage, they are able to show their thinking with numbers alone. This is what we may think of when we think of how math looked for us when we were in school. Students are much more successful working in the abstract stage after they have had the experience working with more visual models in the concrete and representational stage.
It is important to remember that connecting these models is super important! In fact, even when I have students working in the concrete stage, I may be drawing representations or writing the abstract numbers next to the model so they can see how those are two different ways of showing the same thing, helping them connect the two.
Finally, some may ask, does every kid need to go through this? Like what if they can already do the abstract stage and they are successful with that? How I usually respond to that is by saying that learning how to represent your thinking visually is a huge part of being a mathematician. So even if I have a student who appears to understand with abstract numbers, I may follow up by asking if they can show me a model of the math they just did. You may be surprised at the number of kids who have memorized what to do with an algorithm that actually have no concept of number.
Students will move through the CRA process at different speeds, and a student who moves through it more quickly is not necessarily smarter than a student who spends more time working with visual models. Every brain is different.
Also, it does not always work like: first you do concrete, then draw a representation, then write numbers. Then you’re done. Students may move in and out of each stage as they are working on a concept. And that is totally ok. It’s literally what mathematicians do. It is also important that as educators, we continue to work to move our kids to the abstract stage by mentoring them and helping them make those connections between the stages so that they do not always have to rely on building a model.
Finally, it is also very possible to teach students how to “answer get” by using manipulatives procedurally. Doing this almost guarantees that students will not be able to move to the abstract stage. So it is not the manipulatives themselves that are magic. The manipulatives and visuals allow teachers and students to work together to develop understandings through modeling, questioning, and discourse.
If you are curious to see what this process looks like in action, feel free to checkout our YouTube channel, which shows students at work learning in various stages of the CRA process!