- Episode Highlights
- Transcript
It’s one thing for students to memorize a math fact, but it’s another thing entirely for them to understand it. When students rely on abstract symbols alone, their learning can feel disconnected, temporary, and fragile.
That’s where the CRA model (Concrete, Representational, Abstract) comes in. In this episode, we explore how this powerful approach helps students build deep, lasting understanding by connecting hands-on experiences, visual thinking, and symbolic reasoning.
You’ll hear how CRA can shift the way students think about math, and why this isn’t just a “primary grades” strategy. We’ll talk about what it looks like in practice, how to get started without feeling overwhelmed, and how this one framework can transform the way your students make sense of math.
Links mentioned in this episode:
Understanding the CRA Model: A Closer Look
Hello, Meaning Makers! Today, we will be tackling a topic that several of you have said you want to learn more about, and that topic is the CRA model, which stands for Concrete, Representational, Abstract.
You might remember that I touched on CRA back when we discussed Math Practice 2, which is where students “reason abstractly and quantitatively”. CRA plays an important role in Math Practice 2, but there’s a lot more for us to talk about with it.
I’m excited to take a closer look at CRA with you. Let’s get started!
Visualizing Meaning
Close your eyes for a second (unless you’re driving, then of course please don’t close your eyes). I’m going to read two sentences to you. After each sentence, I’m going to pause and give you time to picture this sentence in your mind. Here’s the first sentence:
- The black and white dog runs through the yard. (pause)
What came to your mind when you heard that sentence? What did you picture? Now, set that picture aside for a second.
Here is the next sentence:
- Four plus eight equals twelve. (pause)
What did you see in your mind when you heard this sentence? What did you picture?
Visual Thinking in Literacy and Math Instruction
From Words to Mental Images
Let’s talk about the first sentence. I bet you pictured a black and white dog running through a yard, right? Maybe the dog looked like one that you own or know or have seen before. I bet the dog had four legs, two ears, and a tail.
Maybe you saw a Dalmatian, or maybe you saw a dog with bigger patches of black and white. And the dog was running, so you probably envisioned her legs moving.
What did the yard look like? Was there grass? A sun? Trees? All of these details will vary from person to person based on our own interpretations, but at the core, we all pictured a black and white dog running through a yard.
When I read this sentence, I’m guessing you didn’t picture the actual written words that I said: the black and white dog runs through the yard. No, you pictured the dog, not the word dog.
Why is that? Well, let’s think about literacy for a second. The purpose of reading is to comprehend text. Yes, we work on phonics, phonemic awareness, and oral reading fluency, but at the end of the day, we do those things so that we can comprehend and make meaning from what we read. And that is exactly what you did when I read you that sentence – you made meaning from it and pictured that meaning, not the written words.
From Words to Abstract Symbols
Let’s think about four plus eight equals twelve.
Whenever I talk to teachers about this sentence, most of them say that they pictured the numeral 4, a plus sign, the numeral 8, an equal sign, and the numeral 12. Is that what came to mind for you?
Occasionally teachers will tell me they saw two ten frames, one with 4 dots and one with 8 dots. Or they saw 4 base ten cubes and 8 base ten cubes. But for the most part, the majority of teachers picture the number sentence. Why is that?
Well, traditional math instruction has always focused on those abstract numerals and symbols and not the meaning behind them. Most of us grew up doing computations with numerals and symbols with little to no concrete work to help us understand them.
We memorized math facts and procedures for solving more complex problems with little to no emphasis on the meaning behind them. Picturing, for example, the ten frames or base ten cubes, or even something else like 4 pennies and 8 pennies or even 4 red counters and 8 yellow counters, would be something concrete that shows the meaning of those numerals and symbols.
The Purpose of CRA: Making Math Meaningful
The difference between what you probably pictured when you heard those two sentences earlier is exactly why CRA is a best practice for math instruction.
CRA gives meaning to the math students are doing, just like comprehension strategies give meaning to text. What is the point of reading words if you don’t understand what they mean? By that same token, what’s the point of doing computations if you don’t understand what they mean?
Traditionally, it didn’t matter if we understood what those computations meant. As long as you could get the right answer, you were fine. But math today demands more from our students. Math is NOT just about finding the right answer. It’s about sense-making, critical thinking, problem-solving, and communicating ideas with others.
Understanding Each Part of CRA
CRA stands for concrete, representational, abstract. Let’s start by talking about the concrete.
Concrete
Your prior knowledge and experiences with dogs helped paint the picture in your mind of the dog, not the fact that you previously learned how to read and spell the word. You were able to picture a dog because a long time ago, you met a dog for the first time, and you have continued adding to your bank of knowledge about dogs ever since.
We want the same for our students when it comes to math. We want students to have concrete, hands-on, real-world experiences – think manipulatives, objects, and real-life situations – that they can think back to as they build their understanding of math concepts.
Representational
Simultaneously, we want students to be able to represent the concrete ideas and experiences through drawings. Just like you could draw a picture of a dog based on your concrete experiences of owning one or meeting one, students draw a picture of the concrete experience they had with a mathematical idea.
For example, they might use 3 tens rods and 8 ones cubes to build the number 38 (that would be the concrete) and then draw a picture of those blocks to represent what they built. The work of transferring their experience to a drawing helps solidify the meaning of 38.
Abstract
Finally, we want students to connect the concrete and representational to the abstract numerals and symbols. This is what comes to mind when we typically think about math – the written numerals and symbols.
Just like you could write or decode the words in the first sentence I read earlier, this would be writing out the numeral 38.
It also refers to operational symbols like the signs for addition, subtraction, multiplication, and division, the equal sign, a fraction bar, parentheses, etc. These numerals and symbols have an assigned meaning in our language. In order to truly understand what they mean, students must experience them concretely and make sense of them for themselves.
Common Misconceptions About CRA
People mistakenly think of CRA as a progression where students move through phases, when in fact, CRA is NOT a gradual process. It’s not breaking up concepts by doing a few days of concrete, a few days of representational, and a few days of abstract.
It’s also not a progression across grade levels where students have concrete experiences and use representations only in the primary grades, and only focus on the abstract in the upper grades. Manipulatives, hands-on experiences, and drawings belong in ALL classrooms from kindergarten through twelfth grade and beyond.
CRA involves all three of its components working together at the same time. This “trifecta” yields high engagement and deep thinking and sense-making, which translates to meaningful learning.
Questions to Guide Your Instruction with CRA
It’s important to remember that when doing CRA, you will cover fewer problems, but you will go deeper with them – prioritize depth over breadth!
It’s also important to note that it’s not realistic to suddenly do CRA in every single lesson every day. That’s too much pressure to put on yourself. Instead, as you plan your lessons, consider these prompts as you think about how to start incorporating CRA (these are the same prompts I gave in episode 2, but they are worth repeating):
- What manipulatives can my students use to help them understand this concept? If I don’t have access to manipulatives, how can I get creative with what I do have access to? For example, if you don’t have base ten blocks or pattern blocks, can you download, print, and cut out templates of these blocks?
- What hands-on experiences can I offer my students? In what real-world scenarios or contexts might they encounter this problem, and how can I connect them with these scenarios or contexts?
- Am I providing time and space for my students to draw so that they can represent the concrete? This act of having to translate their concrete actions into drawings deepens their understanding.
- Am I modeling what I want students’ drawings to look like and expecting them to replicate my representations? If so, how can I turn this over to my students and let THEM make the decisions about what these drawings should look like?
- Am I giving students opportunities to work with others during the CRA process so that they have opportunities to share their thinking and learn from others?
Final Thoughts and an Invitation to Reflect
Meaning Makers, I’m so happy we took more time to dive into CRA. What were your big takeaways? What are some ways you’d like to incorporate CRA into your classroom? What questions do you still have? Thank you for taking the time to listen in today. If you enjoyed this conversation, don’t forget to subscribe so you don’t miss an episode, and we’d love it if you took a small moment to leave a review. Thank you, meaning-maker, and have a great one!
