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Understanding CRA

If you teach elementary math, then you have probably heard plenty about the concrete, representational, abstract sequence of instruction. It’s usually shortened to CRA. The idea is that concrete and representational (pictorial) experiences are required to make sense of abstract concepts.

CRA instruction

But I have found that CRA is not always well understood and is sometimes misunderstood. Read on for a roundup of important concepts related to CRA.

CRA is not based on age

A very common misconception is that manipulatives are for little kids, and once students get to 3rd grade or so, manipulatives are no longer needed. Nothing could be further from the truth! Concrete learning is necessary whenever students are introduced to new concepts. Third grade is when the foundation is laid for both multiplication and fractions. They dabble in both in 2nd grade, but the real learning begins in 3rd. And both are very abstract concepts. It’s a huge shift going from additive thinking to multiplicative thinking, and students need concrete experiences to make that shift. And fractions! When you have 5th graders telling you that 1/8 is greater than 1/4, it’s a sure sign that they haven’t seen fractions enough to develop a mental image of what those two fractions look like. Which means they didn’t have concrete learning in 3rd or 4th.

Demonstration lessons are not concrete for the students

For many reasons—time, classroom management, lack of materials—teachers are often the keepers of the manipulatives. That is, they demonstrate lessons using manipulatives rather than passing out manipulatives for students to use. While it certainly requires good classroom management to keep manipulatives under control, it’s important that students use the manipulatives. If lack of materials is a problem, consider having students work in pairs, so only half as many manipulatives are necessary. Or, better yet, move more of your instruction to small groups. Working with manipulatives in small groups makes management easier and you can get by with very limited materials.

CRA is part of your differentiation

As with anything else related to instruction, one size does not fit all. Some students will require concrete and pictorial support longer than others. Knowing where your students fall within the CRA stages for a given skill is part of your differentiation. Do not rush all students to the abstract phase.

Student choice is important

Of course you will have lessons where you introduce and use specific manipulatives, but at some point students should be encouraged to choose the manipulative that makes the most sense to them. Place value and fractions are great examples because both can be modeled with a variety of manipulatives. Students should be introduced to multiple representations and then be allowed to choose from the different models.

More abstract practice doesn’t build understanding

A great illustration of this point is subtraction with regrouping. If a child doesn’t know how to use the standard algorithm to subtract with regrouping, continued practice with the algorithm is likely to be fruitless. Even if you explain the place value understandings behind the algorithm. Get out the base-10 blocks and let the child model the process. Put the algorithm right along side the base-10 blocks so the child can make the connection. Check out this post for a video of using base-10 blocks for subtraction.

Let’s talk cut and paste…

I always feel a little silly talking about this, but cut and paste is not concrete learning. I’ve had teachers tell me, My students have hands-on activities all the time. We ALWAYS do cut and paste. Remember that for learning to be concrete, the manipulatives have to be used to model the math they are representing. Cutting out numbers and moving them around is not concrete learning. As you see below, a cut and paste activity is at best representational.

Hopefully, this has helped you understand CRA just a little better! You can watch a free video lesson on CRA by clicking here.

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6 Comments

  1. I have a question please. Is CRA only pertain to Tang Math or is the concept…which I totally agree with across the board teaching math no matter the curriculum. Thank you!

    1. I’m not sure about your reference to Tang Math. CRA dates back to the 1960s and the work of Jerome Bruner. It is widely considered to be best-practice for teaching math.

    1. Trust me, that’s a huge debate! For me, the answer is somewhere in the middle. Not totally concrete, but not totally pictorial. From what I’ve read, most experts agree that it should not be used exclusively to replace concrete materials.

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