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__How we learn math: Using the concrete-representational-abstract instructional approach__

This podcast is going to explore how we learn math, by looking at 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, you absolutely have to visit their website, which I’ll give you at the end of the podcast.

So what is the Concrete-Representational-Abstract (CRA) Instructional Approach? It’s how we teach all students to learn math concepts like place value, multiplication, and factoring. Concrete-Representational-Abstract (it’s a series) is an intervention for mathematics instruction that research suggests can enhance the math performance of students with learning disabilities. It is a three-part instructional strategy (which you probably already guessed), 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.

· Concrete. In the concrete stage, the teacher begins instruction by modeling each mathematical concept with concrete materials (e.g., red and yellow chips, cubes, base-ten blocks, pattern blocks, algebra tiles, fraction bars, and geometric figures). Something you can put your hands on; something you would see in real life.

· Representational. In this stage, the teacher transforms the concrete model into a representational (or semiconcrete) level, which may involve drawing pictures; using circles, dots, and tallies; or using stamps to imprint pictures for counting. It’s taking that 3-d figure and puttingit on paper, usually.

· Abstract. At this stage, the teacher models the mathematics concept at a symbolic level, using only numbers, notation, and mathematical symbols to represent the number of circles or groups of circles or whatever else they are looking at. The teacher uses operation symbols (+, –, , /) to indicate addition, multiplication, or division.

OK, so why do we care? Well, first, we know for a fact that this can enhance the math performance of students with learning disabilities and we have students with learning disabilities. But, second is that it is important to understand mathematics at a conceptual level, because otherwise you don’t really understand math at all! If you weren’t around for our Equity podcast, go ahead and listen to that and you’ll hear all about why we care about all students learning math.

Concrete-Representational-Abstract supports understanding underlying mathematical concepts before learning “rules,” that is, moving from a concrete model of chips or blocks for multiplication to an abstract representation such as 4 x 3 = 12. Otherwise, how would you know what 4 x 3 = 12 really means? Let me talk you through that for just a minute.

Let’s say I’m a student who doesn’t know what 4 x 3 = 12 really means. If you tell me, “Hey! 4x3! What’s that equal to?” I might be able to say, “twelve” but I couldn’t really tell you why. So…let’s backtrack to what should have happened in my education and then we’ll see how I do. Let’s say I’m a little kid and I have piles of red chips on my desk and my teacher says that we’re going to play with chips today. So, I play with those chips and I group them into piles and I see that 4 piles of 3 counted up to 12 and 3 piles of 4 counted up to 12 and I’m learning a lot of things, including some of the important properties that I will use in Algebra. But mostly I’m learning the concept of *grouping*. Then later on, when my teacher says, “Hey! Let’s draw some of those groups on paper!” I can make little dots – three groups of four dots and four groups of three dots and I *know* that that is always going to equal twelve. So later on when I’m ready to leave this stage, I can just pull this out at a symbolic level, because I know that four groups of three is twelve and three groups of four is twelve and I’m using *times* to say I have a group of four three *times* or a group of three four *times* and that is, or *equals*, twelve.

Have I convinced you that this is important? No? Well, hang on. I’m going to give you some research, and you know how excited we all get about research, so here goes.

Research-based studies (from 20 and 30 years ago-this isn’t new to us!) show that “students who use concrete materials develop more precise and more comprehensive mental representations, often show more motivation and on-task behavior, understand mathematical ideas, and better apply these ideas to life situations” (www.k8accesscenter.org).

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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