Representation in Mathematics

When your math students get stuck on a problem, how do they react? Are they able to apply their understanding from other topics as they try to puzzle through the sticking points? Do they guess and then ask you if they are right? Or do they just give up and ask for help? 

People are often unable to apply their reasoning skills to math concepts unless they have a conceptual foundation rooted in concrete models of the abstract ideas. However, adult learners (and sometimes educators) are often more focused on symbolic procedures than on concrete models. One of the most important ways we can support our math students' self-efficacy is to develop each concept with multiple representations. These representations are the tools needed to connect math concepts to students' innate reasoning and life experiences, which in turn supports their problem solving, conceptual understanding, and mathematical thinking. 

All five types of representations are important, and ultimately our goal is for students to be able to use them all and translate between them when appropriate. But when we first introduce an idea, we have to start with representations that connect to the students' prior experience so they can apply their reasoning skills to the new concept. That usually means starting with concrete, visual, and contextual representations.

There is a common approach to teaching math called the CRA Model (sometimes called the CVA or CPA model), which suggests supporting students' growth through Concrete models, then Representational (or visual or pictoral) models, and finally Abstract representations (symbolic and verbal). Teachers using this approach help students build connections between the different models as their understanding deepens. Learn more about this approach and the models typically used for specific math topics below.

"Research tells us that the most common cause for math struggles is students applying procedures without understanding the underlying concept. This happens when students don’t have developed conceptual models, when they have been rushed too quickly from concrete experiences to abstract symbols without building the critical bridges between different representations." ~All Learners Network Blog

Why do multiple representations matter? Because:

1. Physical, visual, and contextual representations give meaning to the more abstract verbal and symbolic aspects of mathematics. Giving students these tools to make sense of mathematical ideas enables them to check their own intuition and find connections between topics, helping them to learn not just what to do to solve a problem, but also why.  

2. Providing students with multiple representations also makes it accessible to more students by giving them multiple entry points. For example, some students are able to apply the mathematic concepts to concrete models, but are overwhelmed by the abstract representations. Other students are fluent with abstract procedures but struggle to see how they apply to various situations. And many students understand more than they can say (verbal representation is not automatic), and so being able to refer to pictures or real-world items allows them to share their thoughts, ask questions, and "hear" other students' ideas. This is especially important for English language learners, who might otherwise struggle to communicate their mathematical thinking.

3. Physical, visual, and contextual representations help us understand students' thinking, which in turn helps us be stronger teachers and tutors. The representations provide students with a natural way to communicate their reasoning, enabling the teacher to build on that understanding for students to make new connections and continue to grow. They also allow for more touchpoints to our students' understanding and help us better understand any misconceptions.

Explore the links below to learn more about these benefits and to practice using multiple representations.

Looking for a roadmap for your math classroom? Check out NCTM's Principles to Actions with eight effective practices for building a strong math community. It is worth the purchase!

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