# How to Teach Math Effectively Using the Concrete Representational Abstract Model

### WHAT IS THE CONCRETE REPRESENTATIONAL ABSTRACT MODEL?

The CRA Model is an instructional approach for teaching math. It consists of three phases:

1. Concrete
2. Representational
3. Abstract

In the concrete phase, we focus on using hands-on manipulatives. Students should be able to move and manipulate 3D objects to represent their thinking. An example of this might be base ten blocks to represent an addition expression.

In the representational phase, we draw representations. For example, we could represent the base tens from the previous picture with a drawing of base ten blocks.

In the abstract phase, we represent our thinking with digits and symbols. For example, the base ten blocks could now be represented as an equation.

### THE “GAP”

It’s no secret that there is a huge gap in a lot of our students’ mathematical understanding and fluency. Why do some students just seem to “get” math and some never do?

One reason for this gap is a lack of focus on concrete learning.

I know that I have been guilty of rushing through concrete activities to get to abstract activities faster. Have you? It is easy to see the abstract phase as that end goal that we rush to get to – but is it really our end goal? Or is the goal to help our students construct their understanding and become flexible thinkers?

If you have students who are struggling with math, consider that the reason for their weakness could simply be that they don’t “see” the math in their heads. Instead of seeing 25 as two tens and five ones, they see it as literally a “2” and a “5.” This makes it very hard for them to make connections and see relationships.

We can help bridge this gap by giving our students opportunities with concrete materials so they construct the understanding that is essential to their future success in math.

### DESIGNING LESSONS WITH THE CRA MODEL IN MIND

When you teach a math lesson, make it your goal to incorporate concrete, representational, and abstract into the same lesson. This way you can be certain that you are differentiating for all your students, regardless of where they are in their understanding.

It helps to think about the CRA model as a Venn diagram rather than a sequential series of steps.

Here are some tips to seamlessly incorporate the CRA model into your lessons:

• Don’t store your manipulatives in a drawer and only bring them out for special occasions! These should be a regular part of your math instruction.
• Make manipulatives available in students’ table groups so that they are easily accessible for those who need them.
• During a whole class math talk, represent thinking in a variety of ways. Remember that not every student thinks the same.
• Change your thinking – if the goal is flexible thinking, then the bulk of time should be spent with manipulatives. Once students can “see” the math in their heads, the abstract phase will be a natural, simple progression. It will also mean that less intervention and re-teaching is needed.

### TOOLS FOR TEACHING WITH THE CRA MODEL

Here are some of my favorite ways to incorporate the CRA model into a variety of math tasks.

How do you use the CRA model in your classroom? Let me know in the comments below!

## FREE MULTIPLICATION ACTIVITIES

• I noticed that number talks seem to utilize only representational and abstract thinking and ignore the concrete learning that some learnings still need to access the number talk. Any thoughts on how to make number talks more concrete?

• Hi there! Yes, many number talks focus particularly on abstract thinking. You can make this into more of a math discovery time by giving students kits with manipulatives. Have them each show their response in their own way and then share. This will help other students see different ways of visualizing the problem as well.

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