Math Teaching Tip: Build a Bridge from Concrete to Abstract with the "Concrete–Representational

When working on math skills, it can be helpful to take more abstract concepts and demonstrate them with concrete objects and pictures. This allows students to obtain an understanding of the core concepts behind the math problems they're learning (Witzel & Little, 2016) and can help close gaps in mathematics knowledge (Allsopp et al., 2008).

One way to achieve this is to teach using an instructional method called the "Concrete-Representational-Abstract" (CRA) approach.

Three Stages in the CRA Approach

With this approach, we teach each math topic with three stages:

Concrete

This is where the teacher demonstrates the math concept with manipulatives (Flores, 2010), such as adding and subtracting using cubes or small toys. The point with this stage is to use real objects to provide a hands-on demonstration of the math concept. You will also show the written numerical problem along with the objects (Miller et al., 2011).

Once the student masters this level, you can move on to the next level. It's okay to have multiple lessons at this level, as you want the student to have a thorough understanding before moving on.

Representational

This is where the objects are replaced by drawings and/or pictures (Flores, 2010), or 2-D representations of the objects such as tally marks (Miller et al., 2011), lines, or dots (Bouck, Park, & Nickell, 2017).

In this stage, you draw out the math problems on paper with pictures, dots, tally marks, lines, etc. You will also show the written numerical problem along with the drawings/representations (Miller et al., 2011).

The representational stage acts as a mental "bridge" between the physical objects and the abstract math symbols (such as written numbers, plus/minus signs, etc.) (Witzel & Little, 2016).

Once the student masters this level, you can move on to the next.

Abstract

This is where students solve problems using written numbers only (Miller et al., 2011). This is the level that many worksheets and tests are written at.

Teaching Procedures at Each Stage

Each stage can follow the same instructional sequence. You will do all of these steps at each stage, using either manipulatives, drawings, or written problems with each step.

(1) The teacher demonstrates the math concepts and models solving problems while the student watches.

(2) The student works on problems with prompts, cues, and guidance from the teacher.

(3) The student works on problems independently.

As a note, when teaching at the abstract level, you may also add a mnemonic strategy to help students memorize how to solve the problem (Flores, 2010), such as remembering the order of operations with "Please Excuse My Dear Aunt Sally."

I'll talk a little about the research support for this method, and then provide some links with some great photo examples. When I'm learning a new teaching strategy, I love to see some pictures or video to help me learn how it works, so please check out the photos if you're a visual learner like me.

Research Support

The CRA approach can be used to teach math at various levels from basic math (such as adding and subtracting) to more advanced concepts such as algebra.

It includes various instructional components that are considered effective, such as:

• Visual supports,

• Many opportunities for teacher feedback,

• Guided and independent practice,

• Systematic and direct instruction (Witzel & Little, 2016),

• Explicit instruction (teaching that is clear/unambiguous) (Miller et al., 2011).

Some research articles on this topic are listed at the end of the post under the heading, "Additional Research Articles."

Examples

The following links have some great picture examples of this approach. Don't forget to come back to this page to look at the tips and resources below!

Examples for Addition and Division: http://ebi.missouri.edu/wp-content/uploads/2013/08/EBI-Brief-Template-Concrete-Representational-Abstract.pdf

Example of Fractions: https://www.pinterest.com/pin/552324341799374151/

Place Value Example: https://www.pinterest.com/pin/81205599515985469/

Addition Example: https://www.pinterest.com/pin/559501953682141682/

Tips

Make sure that you are using the same vocabulary and procedures consistently across each stage (Witzel & Little, 2016).

To learn more about teaching with this approach, check out the articles and books in the reference list at the end of this post.

References

Allsopp, D. H., Kyger, M. M., Lovin, L., Gerretson, H., Carson, K. L., Ray, S. (2008). Mathematics dynamic assessment: Informal assessment that responds to the needs of struggling learners in mathematics. Teaching Exceptional Children, 40(3), 6-16. Bouck, E., Park, J., & Nickell, B. (2017). Using the concrete-representational-abstract approach to support students with intellectual disability to solve change-making problems. Research in Developmental Disabilities, 60, 24-36. Flores, M. M. (2010). Using the concrete-representational-abstract sequence to teach subtraction with regrouping to students at risk for failure. Remedial and Special Education, 31(3), 195-207. Miller, S. P., Stringfellow, J. L., Kaffar, B. J., Ferreria, D., & Mancl, D. B. (2011). Developing Computation Competence Among Students Who Struggle with Mathematics. Teaching Exceptional Children, 44(2), 38-46. Witzel, B. S., & Little, M. E. (2016). Teaching elementary mathematics to struggling learners. New York, NY: The Guilford Press

Additional Research Articles

Flores, M. M., Hinton, V. M., Strozier, S. D., & Terry, S. L. (2014). Using the concrete-representational-abstract sequence and the strategic instruction model to teach computation to students with autism spectrum disorders and developmental disabilities. Education and Training in Autism and Developmental Disabilities, 49(4), 547-554.

Flores, M. M., & Hinton, V. M. (2019). Improvement in elementary students' multiplication skills and understanding after learning through the combination of the concrete-representational-abstract sequence and strategic instruction. Education and Treatment of Children, 42(1), 73-100.

Yakubova, G., Hughes, E. M., & Shinaberry, M. (2016). Learning with technology: Video modeling with concrete-representational-abstract sequencing for students with autism spectrum disorder. Journal of Autism and Developmental Disorders, 46, 2349-2362.