The Connection Between Students Understanding and Doing Math

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If you are a modern-day math teacher, then you know how important the conceptual understanding of new topics is. Teaching conceptually means you are giving students the chance to explore math physically and with manipulatives before teaching the algorithm. One model that I like to use when I am teaching math is using the CRA model. The CRA model allows for students to explore math concepts conceptually before introducing them to algorithms. 

Infographic describing the CRA model.

What does CRA stand for?

CRA stands for concrete, representational, and abstract. The CRA model gives students the chance to explore math with manipulatives, which leads them to representational and abstract strategies.

Concrete models include manipulatives and other math tools to help students feel the math they are learning. Tools that help students to physically do the math are base ten blocks, fraction strips, play money, counting cubes, counters, place value disks. Basically, anything the students can feel in their hands to help them do the math. 

The second part of the CRA model is representational. This is where students take the physical objects they are using to solve the problem and draw pictures and visuals. This might look like students using base ten blocks to multiply with the area model, and then transferring that skill to drawing an area model sketch on a piece of paper. 

Finally, the “A” of the CRA model is abstract. Here, students take their knowledge of the math concept and solve problems using the appropriate math structure and algorithms. This looks like students taking the area model and relating it to the standard algorithm and regrouping. 

Click here to grab a free CRA Model Cheatsheet! This includes 2 planning guides and a multiplication CRA example!

Why do we need to use the CRA model to teach math?

Using the CRA model to teach math gives students a chance to really understand what they are doing. Students will be able to bridge the conceptual understanding of the math skills they are working on and relate them to the abstract algorithms. This will, in turn, help them to understand and remember the algorithm more efficiently. 

Teaching with the CRA model in the early years will also lead to a better understanding of the higher-level math concepts when students are to learn Algebra, Geometry, and Trigonometry. 

How can we teach using the CRA model remotely?

Teaching with concrete models is definitely a challenge when teaching remotely. Something that you can do with your students is to have them create their own manipulatives or use everyday objects around the house as manipulatives. For example, if they have toothpicks, cotton balls, and paper they can use them as makeshift base ten blocks, students can even use toothpicks to make bundles of 10s. If your school has the funding for it, is to give each student some manipulatives to take home. 

Last, but not least, my favorite way to get students exploring manipulatives without physically having the manipulatives is to use virtual manipulative sites like and Those are both free and have awesome manipulatives that give students the chance to explore with models without having the concrete models in front of them. Click here to read posts about how I use the websites.

Overall, teaching math while keeping the CRA model in mind gives students the opportunity to conceptually understand the math. It gives students a chance to make connections from physical math (concrete) to algorithms (abstract) and structure to solve problems algebraically. When planning a lesson, think about how you can teach the concept physically, pictorially, and algebraically to help students better understand what exactly it is that they are doing. Grab my free planning guide to help plan using the CRA model here!



P.S – If you are looking for resources to help students at home make the connection from concrete to abstract concepts, look in my TpT store! There I have study guides that walk students through the conceptual understanding of various topics.

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Hi, I'm Alexandra!

I am a fourth-grade math teacher turned elementary tech teacher. I help upper elementary math teachers like YOU get organized digitally and engage students with digital tools. When I’m not teaching, you can find me taking long walks with my dog, Frannie, or travelling (especially to Disney World)!