Visual Models in Math: Why They Matter in the CRA Progression

This multi-week series is all about representational models. We are talking about where representational models live in the sequence of math instruction, why they are important and useful to students, and where these models can lead our students in the future.

When we think about a CRA math progression, we are starting with hands-on tools and concrete manipulatives. Next, we move to representational models. There can be a wide range here, and this is exactly what we are examining this month. This includes drawings, diagrams, charts, and any visual model that supports our students’ math thinking. Finally, students arrive at a place where they have the ability to reason about numbers and ideas at the abstract level.

Understanding the CRA Progression

Let’s think about this progression outside of math. If I wanted to explain the concept of an apple to an alien, the most effective method would be to give them an actual concrete apple. They can touch, taste, and smell the apple. They can hear the crunch. This concrete experience gives the alien the most robust understanding of an apple.

A few days later you mention to the alien that you had an apple for lunch but that the skin was bruised on one side. They look at you with a quizzical expression. You do not have an apple nearby, but you show them a diagram of an apple. Reminding them of the core, the flesh, and the skin through the diagram, they quickly remember their experience and understand what you are saying about the bruised skin.

A few days later you mention that you had an apple cut into slices for lunch. Given the previous experience with the apple and the diagram, they ask, “Were there seeds in the slices?” They are now able to reason at an abstract level about apples because they were provided with examples at the concrete and representational levels.

Why Visual Represenational Models Matter

This is the experience we want to build in math. We want to start with concrete models where students can touch, feel, and move the math themselves. At some point, you will want to move your students away from the need for hands-on materials, and representational visual models are a way to help you do that. They are the stepping stone between concrete and abstract ideas.

Additionally, representational models allow you to explore math that might be too cumbersome to explore with hands-on tools alone. Adding two-digit numbers using base ten blocks or linking cubes is a reasonable and powerful exploration. When we move into six-figure numbers, linking cubes are simply too cumbersome to be useful. Drawing dots on a place value chart, however, still allows students to have support as they explore larger numbers. Representational models allow students to extend beyond the concrete level while still maintaining structure and support.

Types of Visual Models

Across the weeks of this series, we will look at four different types of visual models and trace how those models build and support students across grade levels from kindergarten to fifth grade.

1. Discrete models
   These countable sets show up as early as kindergarten. Students might be counting a picture of bunnies hopping in a field. The visual model is representing the numbers they are exploring for the first time. As students get older, discrete models continue to live on. This might look like an equal groups drawing in third grade or fractions of a set in upper grade levels.
2. Counting models
   As early as kindergarten, students are looking at numbers on a number path to relate the counting sequence to ideas such as one more or one less. Students in second grade might move along a number line as they develop mental strategies for adding and subtracting two-digit numbers. Students in fourth grade might look at fractions on a number line as they search for equivalent fractions. The tools are the same, but they grow with students over time.
3. Part whole models
   These models can be as simple as a number bond showing partners of ten in kindergarten or 1st grade and as complex as multiple bar models representing a system of equations in upper grade levels. These tools build and grow with students across elementary school and beyond.
4. Area and array models
   One of the first exposures students have to multiplication is looking at rows and columns in an array. This model supports fourth grade students as they multiply multi-digit numbers for the first time and appears again in fifth grade as students multiply and divide fractions.

In the final week of the series, we will look at how to choose the right model for your students and for the topic they are studying. This series includes a great deal of information, and this final week will allow you to distill it in a way that leaves you feeling confident in your decisions around visual models.