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Unlocking the Power of the CPA Approach in Mathematics: A Simple Guide to Math Mastery

What is the CPA approach in mathematics? If this is your first time hearing this term, you’re probably thinking it has something to do with accounting or taxes. Rest assured, you are not lost and this is an educational blog for elementary school teachers! The CPA method moves students through 3 learning stages as they work towards math mastery: Concrete, Pictorial, and Abstract.

This math method is a powerful learning strategy to help your students master math concepts easily and efficiently, while grasping the WHY and the HOW. In this blog post, we’ll take a deep dive into all the details about the CPA approach in mathematics.

CPA approach in mathematics

If you’re more of a visual (AKA, Pictorial) learner, you can check out this deep dive in my YouTube video below!

What is the CPA Math Method?

I recently put up a poll in my Instagram stories and asked teachers if they knew what the CPA approach in mathematics was, and I was surprised to find out that a majority of the 300 teachers that responded did not know about this math method! Now, maybe that makes me a math nerd, but that’s okay! After seeing the results come in, I just knew I had to share this powerful approach to learning with you!

The CPA approach in mathematics was developed by American Psychologist, Jerome Bruner in 1966 and stands for Concrete, Pictorial, Abstract. It is a method for learning that emphasizes the use of hands-on and visual learning to help students understand mathematical concepts. This approach is also known as scaffolding, which is what you may know it by. 

The CPA approach in mathematics builds on a student’s prior knowledge by introducing mathematical concepts in a concrete and relatable way. The CPA method is a dynamic approach that begins with the use of hands-on materials and manipulatives and gradually moves to pictorial representations and diagrams. Ultimately, it leads to abstract symbols and equations. 

CPA approach in mathematics diagram

3 Stages of the CPA Method for Math Mastery

The CPA approach in mathematics is divided into three stages: Concrete, Pictorial, and Abstract. Let’s break down these three stages and look at some examples of what this might look like when teaching basic addition and subtraction strategies

The Concrete Stage

The concrete learning stage is the first stage in the CPA method. In this stage, students use physical objects or manipulatives, such as math cubes or base ten blocks, to represent mathematical concepts and problem-solve. This allows students to see and physically touch the concepts they are learning, making them more concrete and tangible.

As teachers, we know the value of using a hands-on approach to help meet different learning styles, especially for our struggling students. This is why concrete learning is so crucial for understanding mathematical concepts, especially more challenging ones like addition with regrouping

To solve basic equations using concrete objects, students might use: 

  • Math cubes
  • Counting bears 
  • Counters and a ten frame
  • Beans
  • Mini erasers
  • A physical number line
concrete math method

The Pictorial Stage

The next stage of the CPA method is the pictorial stage. This is where students use diagrams, pictures, and other visual aides to represent mathematical concepts and help solve problems. Some researchers define this as the “seeing” stage of learning, as it allows students to see the concepts in a different way and make connections between the concrete manipulatives and the abstract mathematical symbols.

Pictorial learning is important because the ability to draw pictures to represent problems with pencil and paper is a tool that students will always have access to, whereas they won’t always have access to manipulatives and concrete objects. 

To solve basic equations using a pictorial representation, students might draw: 

  • A number bond
  • Part-part-whole 
  • A number line 
  • Dots or x’s to represent a problem
pictorial math method

The Abstract Stage

Finally, the last stage of the CPA approach is the abstract stage. This is where students work with mathematical symbols and equations, with no concrete object or picture. Abstract learning is more mental math.

By this point, students have a strong foundation in the mathematical concepts and are able to understand and apply the abstract symbols and equations effectively. As teachers, it is our goal to get students to the abstract learning stage so they can be successful without the use of manipulatives or drawings. 

To solve basic equations using an abstract process, students will: 

  • Solve using symbols 
  • Use mental math
  • See information presented in a bar model, number bond, on a number line, etc., and understand what to do without the use of any other tools 
abstract math method

Why You Should Use the CPA Math Method

Let’s be honest, math can be hard, especially for our struggling math learners! I was one of those students who found math extremely challenging growing up. I never would have thought I’d be where I am today, a total math nerd with a huge passion for helping teachers fall in love with teaching math and making it fun and impactful for their students… but here I am. 

One of the key benefits of the CPA approach in mathematics is that it helps students understand math concepts at a deeper level. By starting with concrete objects and physically interacting with them, to visualizing the concepts with pictorial representations, students are able to work with concepts in a real-world way. This helps them progress to the abstract learning stage where they are able to build a strong understanding of the concept, rather than just memorizing math facts or procedures. Instead, they are taught the why before the how, which allows students to reach math mastery that much quicker!

Another benefit is that the CPA math method is highly adaptable and can be used to teach any math concept at any grade level. Therefore, it’s a great choice for teachers who are looking for a flexible and effective way to teach math and differentiate instruction. I recently wrote a blog post all about how you can plan and differentiate your math stations with the CPA method, which you can read right here

The CPA approach in mathematics is valuable for all types of learners, including those with math learning disabilities. No matter what a student’s learning style is, this will help them feel confident in their skills and achieve math mastery.

Overall, the CPA approach in mathematics is an effective and engaging way for students to learn mathematical concepts. By using hands-on tools and manipulatives, followed by visual learning representations, students are able to build a deeper understanding of the concepts they are being taught, which in turn, can lead to greater success in math.

If you are looking for more resources that follow the CPA method to support your math instruction and help your students achieve math mastery, check out my guided math units and math stations for 1st, 2nd, and 3rd grade.

I’d love to hear your takeaways from this post in the comments below!

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