Concrete, Pictorial, Abstract: Using the CPA Method in Upper Elementary Math

Four years ago, I was given a fantastic opportunity to be a math coach for a primary years program. But when I started, there was no framework to work within, no script to follow. I was starting from scratch. I had to decide what my focus would be.

I researched best practices, and time and time again my research brought me to the value of the instruction style known as CPA (concrete, pictorial, abstract). In CPA, student understanding is gradually built up in stages, from physical (concrete) to visual (pictorial) to the traditional, symbolic (abstract) stage that most people think of when they think of math—numbers on a page. I became convinced that CPA should be a pillar of my math lessons. 

Based on the work of educational psychologist Jerome Bruner, CPA was adopted in the Singapore math curriculum in the 1980s and was introduced to the United States in the 2000s with the help of math reformers like Jo Boaler. Over that time, studies have shown multiple benefits of CPA, including not only better test scores but also deeper conceptual understanding and boosts in mathematical confidence—an important predictor of persistence when problems get tough.

Much of these benefits are attributed to how CPA creates built-in scaffolding—giving students cognitive time to better comprehend and work with the math being presented. For your race car drivers, this could mean slowing down to make more sense of the concepts that they are quickly executing but maybe not absorbing more deeply. For your hikers, the CPA method is a chance to process the math and learn it slowly. In the end, both styles of learners benefit.

CPA in Action

Let’s look at some concrete examples of CPA in action, based on how I teach decimals. The first lessons focus on a concrete understanding of what decimals represent: a partition of one whole. In order to grasp that, we look at the breakdown of one whole into thousandths.

For the concrete stage, we use base ten blocks—physical, tangible blocks. Typically, when working with integers, a small, single cube represents one, a column represents 10, a 10x10 “sheet” represents 100, and a large block represents 1,000. But I teach the students to look at the base ten blocks almost backward from how they have used them before. In fifth grade, we do it like this: The big block that has typically represented 1,000 now represents one whole, the sheets are now tenths, the columns are now hundredths, and the small, single cubes are now thousandths. This helps the students see that one whole can be created with 1,000 thousandths or, likewise, 10 tenths. Using base ten blocks, we practice creating decimal numbers for a lesson or two, depending on the class.

Next, we practice pictorially. We do this by using a hundred chart and a thousand chart. These charts each represent our one whole. Students use colored pencils to color in the charts to represent different values. This can be one to two lessons as well, depending on the class.

After the concrete and pictorial lessons, students are ready to learn how to add or subtract decimals. First, we return to the concrete stage, and I ask the students to concretely add decimals by combining two decimal values, each represented by a combination of blocks. I like to use a place value chart and have the students work out the equation with the blocks on the flat surface of the chart. Then we do something similar with the hundred chart. When we add pictorially, I like to have the students use two different colors of colored pencils to visually see the addition happen.

Once the students have learned how to work the equations concretely and pictorially, I teach them the steps of the traditional, abstract method of adding decimals, with careful attention to aligning decimal points.

Integrating CPA

Using CPA is a practical way to support your students to become more comfortable and capable with math. But you don’t need to overhaul your curriculum to use the CPA method—you can start by integrating it where it fits. It does take a bit of extra time and planning, but that time gives your students the chance to explore and understand math more fully. Over time, you’ll likely see their confidence and competence grow.

As for some concrete proof? Here are some real student reflections about CPA:

“Because many students learn abstractly, people don’t know what division or multiplication actually does. And only knowing how to solve it won’t help when you get older to solve harder problems,” one student said. “So I like the CPA method because it helped me to know what the question actually means.”

Another student put it this way: “[CPA] taught me a new way by… going step by step and making sure that I really understand and remember what I’m learning. I think that this method will stick with me forever.”