__How we use the CRA Approach, Part I__

This podcast is going to explore how students can learn math through the concrete-representational-abstract instructional approach as it is presented by the U.S. Department of Education’s Access Center. If you don’t know about the Access Center, then you must not have listened to the last podcast! That’s okay, because I’ll give you the web address at the end of the podcast, and you can have a good time looking around it then.

So, let’s have a brief overview of the Concrete-Representational-Abstract Instructional Approach. In a nutshell, it’s a three-part instructional strategy, with each part building on the previous instruction to promote student learning and retention and to address conceptual knowledge. It’s got three stages: the first is concrete, the second is representational, and the third is abstract. Let me tell you a little bit about each one (I know some of you already know this-hum to yourself for a minute while I review it).

· The concrete stage involves the teacher beginning instruction by modeling each mathematical concept with concrete materials like counting bears, or algebra tiles. Kids put your hands on the materials and can actually see the concept modeled.

· In the representational stage, the teacher transforms the concrete model into a representational (or semiconcrete) level. A lot of the time, this is done by drawing a picture of what the concrete modeling stage looked like. Representational stage--get it? The picture *represents* the model. It’s taking that 3-D figure and making it 2-D.

· By the time we get to the abstract stage, students have been playing with the concept for a while. At this stage, the teacher models the same mathematics concept at a symbolic level, using only numbers, notation, and mathematical symbols to represent the representation of the concept. We use operation symbols (+, –, , /) to indicate addition, multiplication, or division. We also rely on numbers and variables to show what we mean. It’s the *abstract* level because these symbols have been selected to describe more than just a specific instance and instead indicate a general principle.

Whew! If I lost you there, let me give it again in a plain English example: I went to Costco and bought 3 cases of 25 boxes of breath mints (Don’t ask.) So, how many boxes do I have? In the concrete stage, I just stack them in front of me and count the actual boxes. Point my finger at each one and count, “1, 2, 3,…75”. I might also group them and count by 25’s, if I’m good at that: “25, 50, 75.”

In the representational stage, I don’t bother to go out to the garage and count the boxes at all. I’m able to be a little lazy, and I just sketch out what I bought on paper. Lots of little boxes, each inside a bigger box. They don’t even have to look like the breath mint boxes, just maybe little tic marks (or tic-tac marks, maybe? Ah, I slay me!). Anyhow, I can use the same counting strategies (one-to-one or by groups) as I did in the concrete stage.

At the abstract stage, I’m a little more distant from my subject. I don’t really care what’s in the box, I only want to know how many case there are and how many boxes are in each. As long as the items are the same, they could be boxes of dead snakes or crystallized jellyfish-I don’t care. All I need to know is how many groups of how many items in each group: because I’m using these two numbers and the multiplication symbol to answer this. It’s a formula that I know will describe more than just this specific instance…it will work for any instance of *x* number of groups of *y* objects. It’s a general method in mathematics. That’s some heady stuff, huh?

Now I’m going to slip in a reminder from the last podcast. We know from research conducted as far back as the 1970s and 80s that when kids use concrete materials to explore mathematical concepts, they develop more accurate mental representations of what that concept really means which means they building new numeracy skills and strengthening their existing ones. These kids often are more motivated and stay on-task, which is not surprising because they have the skills to participate. You can’t talk about what a concept is if you don’t have the words and pictures to do it! Put it all together and you get students who can apply these ideas to real-life situations.

If we go back to the first podcast, about the Equity principle of mathematics, being able to jump into the conversation and really be a part of the discussion at some level—that’s equality. And it’s going to lead to improves outcomes for our marginalized populations. But it’s going to take a lot more understanding and hard work from a lot of people; building numeracy is not going to be any easier than building literacy—and look how many folks are involved in that!

OK, that’s enough for right now-in the next podcast we’re going to start to discuss how to choose the right manipulatives for the concrete level with an example about fractions.

In the meantime…for more information about the research-based use of concrete, representational, and abstract representations in math instruction, visit the U.S. Department of Education’s Access Center at www.k8accesscenter.org.

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