The Concrete – Representational – Abstract (CRA) pedagogical model for teaching mathematics is an incredibly accessible model for any student. We often think of using a concrete representation, manipulatives, with younger children. However, any student in grades K-12, along with any ability level, can better understand the why behind the mathematics with a concrete or visual model.

The ‘high ability’ student

While students can carry all types of labels, I am referencing the student who might possess the following characteristics:

• Likes the procedures in mathematics. Might say in their mind, “Just tell me how to do it and I’ll do 50 of them correctly.”
• Doesn’t necessarily like to ‘draw out’ the mathematics or use manipulatives.
• Does the math ‘in their head’.
• May not always know why a rule or procedure works.  “It is the rule” might come to their mind.

For this type of student, they don’t always know why they are doing the steps they are doing, but they sure love to do them. These students often like math because of its structure and repetition. They see math as a set of rules that you follow to get the answer.  By the way, this isn’t necessarily a bad thing. However, for students to be able to connect mathematics and see the big picture for future mathematics topics, they need to understand the why, not just the how. Most of these students don’t want to draw a picture or use manipulatives, they just want to do the math in their heads or show little work. Mental math is fantastic and we can encourage students to use mental math. Additionally, modeling with manipulatives and visual images demonstrates why the procedure works or how the concept is connected to other mathematics. For example, if a student in 7th grade can explain, using two color counters, the rules of operations with integers that demonstrates an understanding of the mathematics. If they can only ‘do’ the operations, that understanding may not be present.

The student who might sometimes struggle

For students who might struggle to remember the procedure or rule, the use of visual models such as manipulatives or drawings, can be a major shift in their understanding of the mathematics and their positive perception of themselves as mathematicians. As we think about the 7th grade example of two color counters to demonstrate integer operations, we can show students why the rules work.  We can allow students to develop the rules in their own words. Then they can move to drawing the counters, then generalize integer operations into rules. Students are much less likely to forget how to do operations with integers when they have developed an understanding of the mathematics. When teachers jump to the rule or procedure, we are missing the visual and physical connection the brain needs to make to retain the information. Then we wonder why students forget these meaningless rules after the test on Friday.

When educators make a point of finding the visual or concrete model for all of the mathematics we teach, students have more access to learn math concepts. Students are more likely to feel successful in math and then more likely to like math. A student who feels successful in math will want more of that feeling of success. That student will engage in class, share their thinking, and grow in their math ability. Let’s dust off those counters and algebra tiles to help students experience math like a mathematician.

Research Basis
Boaler, J. (2015). Mathematical mindsets: Unleashing students’ potential through creative math, inspiring messages and innovative teaching. John Wiley & Sons.
Gallo-Toong, N. (2020). The extent of use of concrete-representational-abstract (CRA) model in mathematics. International Journal for Research in Mathematics and Statistics, 6(5), 1-25.
Witzel, B. S., & Kiuhara, S. A. (2017). Overcoming Mathematics Difficulties using CRA Interventions.