- Episode Highlights
- Transcript
How do students build true understanding in math — the kind that lets them connect ideas, solve unfamiliar problems, and explain their thinking with clarity? In this episode, we explore a powerful framework that helps make that possible: Lesh’s Translation Model.
You’ll hear how this model invites students to work across multiple representations of math ideas, and why those connections matter so much for deep, durable learning. Whether you’re new to this idea or already thinking about how students represent their math thinking, this episode offers a fresh lens to help them make meaning and not just get answers.
Exploring Lesh’s Translation Model in Math Instruction
Hello, Meaning Makers! Today we will be tackling a topic that several of you have said you want to learn more about, especially after our deep dive into the CRA model.
This week we’re going to dive into Lesh’s Translation Model. We’ll talk about the 5 different representations, what they look/sound like, how to make connections between the five representations, and investigate why those connections are so important.
This is actually going to be a two part episode, so know that in part 2 we’ll examine a second level of connection-making, what all of these representations and connections look like in the classroom, and how to plan with these representations in mind (without the overwhelm)… because the goal is always to have your new learning directly impact the work you’re doing with your students. Sound good to you? Okay! Let’s dive in!
Quick Recap: What Is the CRA Model?
In case you our episode about the CRA Model, I want to quickly recap our discussion with the CRA model, because it’s going to be important in how we approach Lesh’s translation model.
CRA stands for concrete, representational, abstract. Let’s start by talking about the concrete. 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. Simultaneously, we want students to be able to represent the concrete ideas and experiences through drawings.
For example, students might use double-sided counters to concretely model combining parts. If they have 5 red counters and 4 yellow counters in front of them, students can count each part (or color) individually, and then count all or count on to determine the total or sum. A student can then represent that concrete idea with a visual.
There are many ways that a student could represent those counters, the most basic being a sketch showing 5 red circles and 4 yellow circles. The work of transferring their experience to a drawing helps solidify the meaning of parts and wholes, and possibly even the combination of 5 and 4 totaling 9.
We also want students to connect the concrete and representational to the abstract numerals and symbols. This would include writing the numbers 5, 4 and 9 to represent different amounts of counters. This could also include operational symbols like signs for addition, subtraction, multiplication, and division, the equal sign, a fraction bar, parentheses, etc.
In this case, those symbols could be used to create math sentences like 5 + 4 = 9 or 9 – 4 = 5. These numerals and symbols have an assigned meaning in our language. But In order to truly understand what they mean, students must experience them concretely and make sense of them for themselves.
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 to yield high engagement and deep thinking and sense-making, which translates to meaningful learning.
Okay, now that we have a foundation of the CRA model, we can further elevate our work through the lens of Lesh’s Translation Model. Are you ready for it?
What Is Lesh’s Translation Model and How Does It Relate to CRA?
Lesh’s Translation Model relates to the CRA model in that it focuses on five different representations and the connections or translations students make between them. Three of the five representations you are now already familiar with, but they may go by different names.
Concrete Representation in Lesh’s Translation Model
We have the concrete representation, like working with the five red counters and four yellow counters to look at parts and wholes.
Visual Representation in Lesh’s Translation Model
We have the visual, similar to the representational of the CRA model. Earlier I shared a basic version of this looks like a child drawing five red circles and four yellow circles to represent their counters.
Symbolic Representation in Lesh’s Translation Model
We also have the symbolic, which is synonymous with the abstract from CRA, where students represent their understanding symbolically. I mentioned this could look like labeling the counter amounts with numerals like 5 and 4, or it could also incorporate symbols like addition and the equal sign to create number sentences.
Verbal Representation in Lesh’s Translation Model
Now, Lesh’s Translation Model also includes two other representations that are incredibly important to the work we do with our students. One of these is the verbal representation, or the language we use to communicate our thoughts and ideas. This can be oral, written, signed, or any way that a student would look to communicate language. James Heddens writes that students “need to be given opportunities to verbalize their thought processes: verbal interaction with peers will help learners clarify their own thinking.”
If we go back to our previous examples from concrete, visual, and symbolic thinking, we have a student with five red counters and four yellow counters. The student then sketched their counters and wrote the number sentence 5+4=9 on their paper.
So how does verbal representation come into play? Perhaps after the activity, a student shows you their sketch of the counters. When you ask them about their drawing, they may share “I had nine counters. Five of them were red and four of them were yellow, and that makes nine.” That statement is a verbal representation of the concept. They have also just translated their visual representation to a verbal representation.
Contextual Representation in Lesh’s Translation Model
Now, our fifth representation is contextual. Contextual representation is much like it sounds. We are giving context to an idea. It is how math can be applied in the real world. Given that we learn math to help us solve real-world problems, it’s a pretty important representation.
Let’s look back at the red and yellow counters from our previous discussions. The student had five yellow counters and four red counters. They sketched their concrete representation to create a visual representation. Then they used numerals and symbols, showing 4+5=9 to form a symbolic representation.
Now, our goal is to give those counters context. What could they be representing in real life? One example is, “D’Andre wrote 4 thank you cards on Monday, and then wrote 5 more on Tuesday. He has written 9 thank you cards.” This shows a translation between the concrete counters and a contextual representation.
Why Making Connections Between Representations Matters
Now, defining the five representations and understanding what those representations can look like is a brilliant start. However, the magic of Lesh’s translation model doesn’t come from knowing these representations in isolation. It comes when students are able to make connections between the different representations. This is how students deepen their understanding!
In our next episode, we’re going to take a deeper look at how to foster connections between these different representations in your lesson planning (without the overwhelm). AND we’re going to dig even deeper to show how to connect one representation to itself like visual to visual, or contextual to contextual. I know that sounds a little strange, but I promise it’s fascinating to consider how easily we can elevate student learning with one simple move.
In the meantime, We’d love to hear from you. What were your takeaways from CRA and Lesh’s Translation Model? What questions do you still have? How has this shifted your thinking about math instruction!? Check back in next time for Part 2, and until then, Meaning-Makers, have a great one!
