One of my favorite topics to teach is ten more and ten less with 1st graders. We start out by building a 2-digit number with base ten blocks and then add or subtract a ten stick to determine the number that is ten more or ten less. I mix in adding or subtracting one to ensure that my kids don’t overgeneralize the skill along the way.
As my students gain confidence in the skill, I start asking them to build the 2-digit number but to predict what the number will be when they add another ten. Over the course of our first lesson on the topic, many students (students in an intervention setting!) start to add or subtract ten, with confidence, mentally!
This is the power of the CRA Math Model.
What is the CRA Math Model?
The CRA math model refers to the three levels of support or modes of communicating math ideas to students. You begin with concrete (hands-on & tangible materials), move to representational (drawings & visual models) and finish with the abstract (numbers & equations).
When you introduce a new idea to your students, starting with the concrete allows your students to understand the idea more easily and completely than they would if you began with numbers and equations straight away.
Starting with the concrete also ensures that more of your students will be able to access and explore the concepts you are teaching.
How Do I Use the CRA Math Model?
Keep these two thoughts in mind as you implement this strategy. First, it is imperative that you link the three levels together. Second, you need to understand that the three “levels” aren’t necessarily linear (more on that below)!
To use the method to teach, for example, basic addition:
- Start with a hands-on tool such as linking cubes. Use these blocks to show how you could add two and three together (concrete).
- Link to a representational model. Could you draw (representational) a picture with circles that matches your blocks?”.
- Ask linking questions to draw an explicit connection between concrete and representational “Show me the part in the picture that matches your 3 blocks” “Can you tell me where the total is in your blocks? And where is that total in your picture?” You are quite literally asking questions that require your students to link their concrete model to their representational drawing.
- Link to the abstract. In this case, an addition equation. Write an equation that shows how you put 2 and 3 together to create a total of 5 blocks.
- Ask linking questions to draw explicit connections between the abstract equation as well as representational and concrete models.
The CRA Method is NOT Necessarily Linear
A powerful math activity will link the concrete, representational and abstract together. Moving back to the basic addition example, this would look like an activity where students choose two numeral cards and add them together by build linking cubes to match the numbers on their cards, drawing circles on a whiteboard to match their cubes and write a matc, finally, writing a matching equation.
If your students aren’t ready for an activity with that many steps just yet there is no reason why you couldn’t have your students build with blocks and write a matching equation (concrete & abstract) one day, have them build with blocks and draw a related picture (concrete and representational) another day! On a third occasion, ask students to add numbers using pictures and an equation (representational and abstract).
While there is an order in terms of concrete being most supportive and abstract being least supportive, you don’t (and shouldn’t!) have to move strictly in a “one-at-a-time” linear fashion through the three levels. Your goal is to build connections between concrete, representational and abstract so that you can build a web of understanding for your students.
Anything Else I Need To Know?
Yes! As you are asking your students to use a variety of hands-on materials and representations, remember that they aren’t all created equally!
Some concrete materials are more supportive than others and it is not at all wrong to use two or three different concrete materials to illuminate a concept. The more tools and representations you use, the richer your students’ web of understanding will become.
You can read more about how different hands-on materials lend different levels of support when teaching place value here.
What Topics Can I Teach Using the CRA Math Method?
All of them!
Are you looking for math intervention units that follow a simple and systematic pathway from concrete to abstract? Each and every one of my math intervention units have been written following this method of beginning in the concrete and then linking to representational and ultimately abstract models as well!
Want to read more?
Each of these blog posts details how the CRA method can be used to teach specific topics:
- Missing Numbers in an Addition or Subtraction Equation
- Understanding Teen Numbers
- Addition & Subtraction Fact Families
- Comparison Word Problems (Addition & Subtraction)
- Subtraction with Regrouping
- Rounding Whole Numbers OR Decimals
- Fractions on a Number Line
- Teaching Line Plots
- Multiplying Fractions