If your class is anything like mine you know that kids come in at all ability levels with gaps in a hundred different places within math domains. You might have a kid rocking the number and operations domain but working below grade level in algebraic thinking sitting right next to a kid who is working above grade level everywhere except geometry. That’s why flexibility is so important when we’re creating small math groups.

In this post, we’ll show you how to simplify planning for small math groups by using the CPA phases of math mastery:

• Concrete

• Pictorial

• Abstract

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Step 1: Which Math Domain are You Teaching?

First and second grade math is broken into four big domains. If you’re not familiar with them here’s a quick rundown of the skills generally included in each grade.

• Operations & Algebraic Thinking
  • 1st Grade: understanding equations, addition & subtraction word problems, fluently adding & subtracting within 20
  • 2nd Grade: addition & subtraction word problems, fluently adding & subtracting within 20, equal groups
• Number & Operations
  • 1st Grade: read & write numerals to 120, understanding place value,
  • 2nd Grade: read & write numberals to 1000, properties of addition & subtraction, mental math, +/- 10 & 100
• Measurement & Data
  • 1st Grade: expressing length, ordering objects, time to hour & 1/2 hour, introducing graphs
  • 2nd Grade: measurement & estimation with standard units, representing & interpreting data (graphs), time, money
• Geometry
  • 1st Grade: 2D & 3D shapes and their attributes, partitioning shapes into equal parts
  • 2nd Grade: 2D & 3D shapes and their attributes, partitioning using rows and columns & into equal parts

Once you know which domain you’re teaching, you can narrow your focus even more. Instead of trying to reteach everything, you’re zeroing in on just the essential skills within that domain. That’s where a good screener becomes your best planning partner. It helps you figure out exactly which parts of that domain your students have already mastered—and where they need support.

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Step 2: Use a Screener to Identify Skill Gaps Within the Domain

This screener has 30+ subtests to help you pinpoint learning gaps. Once you identify gaps in your students’ understanding, you’ve got learning targets for your groups, which means half the battle is won!

Now you know which domain you’re teaching and what specific gaps your students have. So, it’s time to decide how you’ll teach those skills in a way that sticks. That’s where the phases of math mastery come in. Whether a student needs help visualizing place value or solving two-step problems, using the CPA approach helps you meet them exactly where they are—then move them forward with confidence.

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Step 3: Use the CPA Approach to Differentiate Instruction Within Small Groups

Second grade really begins that big shift from concrete to abstract math concepts so differentiating instruction from group to group is essential! If we can help them move through the phases of mastery we can set them up for success in upper elementary and beyond.

The phases of mastery are referred to by a few different names:

• CPA (concrete/pictorial/abstract)
• CSA (concrete/semi-concrete/abstract)
• CRA (concrete/representational/abstract)

For the sake of simplicity, I’m going to focus on CPA. Using the CPA approach has been shown to improve mathematical reasoning skills. 

The CPA approach was developed in the 1960s by American psychologist Jerome Bruner. If you’re curious about its history this paper includes a short survey of its origins. It’s pretty interesting! 

1. Concrete Phase of Math Mastery

When you’re working with students who are just beginning to understand a concept or showing confusion in your screener results, start with the concrete phase in small group.

The concrete phase of mastery is when students are using physical objects to represent mathematical concepts. This is when students are using rekenreks, unifix cubes, clocks, shape blocks, beads, base ten blocks, ten frames, or double-sided counters. Math that can be held in their hands and physically manipulated.

If you find you have students who are struggling with a math standard like regrouping – backing up to concrete might be the first step. Having them physically compose and decompose numbers using base ten blocks is going to be really valuable. If you don’t have actual plastic base ten blocks or other manipulatives check out the printable manipulatives in the Small Group & Intervention Math Kit.

Once students have mastered using manipulatives in any given skill then it’s time to move on to the next phase – pictorial. 

2. Pictorial Phase of Math Mastery

Once your students are confident using manipulatives, your small group time can shift toward the pictorial phase. This builds the bridge to mental math and visualization.

In this phase of mastery, students will be representing their math problems using drawings. This includes things like tally marks, number bonds, part-part-whole drawing, or just drawing a picture to represent their thinking. This is a great time to pull out the instructional mats in the Small Group & Intervention Math Kit. Some teachers skip this step but it’s an important bridge between concrete and abstract. Your students are practicing the skill of visualizing a math problem.

In our regrouping example, students may be drawing base ten blocks on their paper to help them solve problems. Once students are confident and accurate, it’s time to move on to the abstract phase! 

3. Abstract Phase of Math Mastery

For students who show readiness on your screener or who’ve shown success with visual models, your small group can focus on abstract thinking and efficient strategies.

Your kids have made it! The abstract phase of mastery is when students use abstract symbols to represent math problems- aka numbers and addition/subtraction/equal/inequality signs. During this phase your students may still need to revisit the concrete and pictorial phases, and that’s fine! The abstract phase is about putting it all together and touching base with the previous two stages to reinforce learning.

Going back to our regrouping example this phase would look like using the standard algorithm that we all know and love. This would be a great time to introduce your students to the Build a Robot math center. It’s a fun way for repeated practice that can be use independently or with a small group. 

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Step 4: Deliver Differentiated Instruction at Your Teacher Table

Now that you know how to support students at different phases of mastery, let’s talk about what this actually looks like during small group time.

Check out this training on math interventions for more info about how to run small groups by targeted skills.

If you find you need additional resources, our Small Group & Intervention Math Kit can fill in the gaps!

Can you Teach Phases of Math Mastery Virtually?

Teaching in a classroom makes using the CPA method feel natural—but what if you’re teaching online or in a blended setting? Can students still move through these phases without all the physical tools? The Lucky Little Toolkit has a collection of digital math manipulatives so kids can practice math skills on devices. Take a look at the video below for a closer look.

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Small group instruction doesn’t have to be overwhelming. With flexible groups, clear learning targets, and the CPA approach, you can meet your students exactly where they are—and help them build the confidence and skills they’ll carry with them into upper grades.