Have you ever been in the middle of teaching a math lesson and noticed your students look like this?

A man looks confused as he counts on his fingers and math equations float through the air.

Math can be stressful for students and teachers...but it doesn't have to be!

Two girls are sitting at a table in the library. They look up from their books and high five.

Use the Concrete, Pictorial, Abstract (CPA) approach to help students build a deep understanding of math concepts and develop problem-solving skills.

Concrete Stage

What is it?

  • the "doing" stage

  • students model math problems using concrete objects

Example:

  • laying out two mittens and adding three more mittens to find the sum

  • cutting a watermelon into twelve pieces to model twelfths

Concrete representations are bestfor:

  • when students are learning a new concept

A man slices a watermellon into ten parts.

Pictorial Stage

What is it?

  • the "seeing" stage

  • students model math problems by drawing

Example:

  • drawing two mittens and drawing out three more mittens to find the sum

  • drawing a picture of a watermelon and dividing it into twelve pieces to model twelfths

Pictorial representations are bestfor:

  • concepts that don't have a concrete representation (like negative numbers)

  • when students show mastery of concrete representations

Abstract Stage

What is it?

  • the "symbolic" stage

  • students model math problems using symbols like +, -, x, ÷, =

Examples:

  • writing the equation:2 + 3 = 5

  • using the equation:1 ÷ 2 = 1/2

Abstract models are bestfor:

  • concepts that don't have a concrete representation (like negative numbers)

  • when students show mastery of concrete and pictorial representations

A Simpson holds a note pad that reads 3-2=1. The caption reads,

Why use the CPA Approach?

Ben Stiller points with his finger as the caption reads,

The CPA approach is beneficial because:

  • students achieve a deeper understanding of math concepts instead of using memorization

  • teachers can scaffold abstract math concepts to make them more accessible

  • students and teachers can go between stages as needed

  • it can be used with students in both primary and secondary school

  • you can use any of the models alongside each other to provide multiple representations of class material or to double-check student knowledge

Quiz

Mrs. Ramirez just began a multiplication unit with her class. Her students have minimal experience with multiplication. She wants to show her students how to solve 3 x 10. Which of the following methods would be most useful in introducing this concept?

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