- Episode Highlights
- Transcript
Too often, we move students through manipulatives, drawings, and number sentences without ever slowing down to help them make connections between these different representations. In this episode, we go beyond identifying the five representations in Lesh’s Translation Model and talk about how to build bridges between them.
You’ll hear how translating within and across representations can deepen understanding, spark rich math talk, and support all learners.
Deepening Understanding with Lesh’s Translation Model in the Classroom
Hello, Meaning Makers! If you missed our last episode, part one of connecting the 5 representations, be sure to go back and have a listen before starting this one.
In our previous episode, we laid the foundation for today’s work. We reacquainted ourselves with the CRA model and used that knowledge to help understand the 5 different representations of Lesh’s Translation model. We looked at examples of each representation and how connecting them can deepen a student’s understanding.
In today’s episode, we’re going to dig a bit deeper. 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. Are you ready? Okay, let’s dive in!
What the Diagram of Lesh’s Translation Model Reveals
Representations Are Connected, Not Linear
If you ever look at an image of Lesh’s Translation model, you’ll see the five different representations, each in their own bubbles: Concrete, Visual, Symbolic, Verbal, and Contextual. The bubbles are organized in a circle. It is not linear.
If you look closer, you’ll notice that there are 2-way arrows connecting each representation with the other four. For example, there are paths connecting Concrete to the bubbles on either side: Visual and Contextual, but also across the circle towards Symbolic and Verbal. The paths go both ways, indicating a two-way connection, not a direction.
I hope you’re not sick of the five red counters and four yellow counters yet, but to respect your time, I want to continue using this scenario so we can build off the representations we created in our last episode.
The five different representations we created and connected last time, based on those five red counters and four yellow counters, highlighted ALL of the two-way pathways between representations that I just mentioned. We connected each representation to all four of the other representations! That’s pretty impressive in and of itself.
Connecting the Five Representations to Themselves
But today, we’re going to take it a step further and bring student learning even deeper.
What It Means to Connect Within a Representation
If you take an even closer look at Lesh’s Translation Model, you’ll notice that each of the five bubbles has its own two-way arrow to the side of it. What does this pathway indicate? It is letting us know that not only can we connect one representation to another, like a concrete representation to a visual one, but we connect two different concrete representations to each other.
What could this look like? If students are successful with the counter representation, we can introduce a second manipulative to help deepen their understanding. Perhaps we use different colored connecting cubes.
Again students show two parts, where there are 5 of one color and 4 of the other. Students understand that there are 9 cubes in total.
Then, we can connect this concrete representation to their previous concrete representation. If we go back to the original two-sided counters, we can help students compare and connect the two different ways they concretely showed 5+4=9.
These are the connections that will deepen student understanding! Being able to translate between these two models is where the magic is going to happen!
Connecting Within Visual Representations
Comparing Concrete Visuals and Abstract Visuals
So what can this look like with other representations? Last time we talked about creating a basic visual representation of the counters by drawing five red circles and four yellow circles.
Now, in addition to their sketch, which is a more concrete visual, the student could use a more abstract visual like a number bond to show the idea of 5+4=9. Both are considered a visual representation, although one representation leans towards concrete, and one leans towards abstract.
When a student is able to connect the two different visuals expressing the same idea, they have connected this representation to itself and further deepened their understanding!
Some other visual representations the student could have used include a bar model, a number line showing jumps, or drawing the images on a ten frame. Just like there are countless ways to show this concept with manipulatives, there are countless visual representations our students could create. Our job is to encourage the creation of representations and facilitate connection-making between them.
Translating Between Symbolic Representations
Supporting Flexible Mathematical Thinking
So what about symbolic representations (in the CRA model this is called the Abstract)? It’s a fairly straightforward representation, but how do symbolic representations connect to themselves in Lesh’s translation model?
Well, let’s say that the student was showing how they counted their five red counters and four yellow counters. This student actually wanted to write 4 + 4 + 1 because that is how they solved the problem. They knew 4 + 4 = 8, so 4 + 5 = 9 because it’s 4 + 4 + 1 more. Understanding that both expressions can describe the same concept is just one example of how to translate within the symbolic representation.
Making Verbal to Verbal Connections
From Oral Explanations to Written Descriptions
In our last episode, we asked a student about their drawing, and they shared “I had nine counters, five of them were red and four of them were yellow and that makes nine.” This was an example of the student connecting their concrete representation to a verbal representation.
Now, if wanted, we could take it a step further by asking the student to write their thoughts down. This will require the student to revisit their thoughts communicated orally and condense them into a written description, like “Four counters and five counters make nine counters.” This extra step of condensing their language into a second form, allowed students to connect two verbal representations.
Recognizing Contextual Patterns Across Situations
Helping Students See the Structure Beneath the Story
Phew! Are you still with me? It’s a lot, I know, but we only have ONE more representation to go before we dive into how to easily plan with these connections in mind.
Last time, we had our student give context to their five red circles and four yellow circles. They came up with the following situation: “D’Andre wrote 4 thank you cards on Monday, and then wrote 5 more on Tuesday. He has written 9 thank you cards.” This showed a translation between the visual drawing and a contextual representation.
This same context could be connected to their concrete counters, their symbolic number sentence, or their verbal explanation.
Additionally, the student could connect their previous context of D’Andre’s thank you cards to other contexts.
For example, “Liana read five books last week and this week she read four. She has read nine books in total.” Or “Yuki has 5 beads, and Cora has 4 beads. They made a necklace together with 9 beads.” By understanding that these situations give context to the same idea, the student has translated between different contextual representations.
Making Representation Planning Easy and Intentional
Becoming More Intentional With What’s Already Working
So, now we know how to connect the five representations to one another, and to themselves. But what does this look like in the classroom?
The truth is that much of the groundwork is likely already happening in your classroom. Think about your lessons from this past week. Did they include a manipulative? Did they include numerals or number sentences? Were students using drawings to share their work? Were students talking about their thinking, or creating story problems?
I’d bet at least two of those things happened in your classroom this past week, and probably more! So, the various representations are already happening. What we get to do now is be more intentional with how we incorporate them, and look for ways to highlight the connections between them so the magic can happen.
A Practical Strategy Using Post-It Notes
I am going to share a personal strategy that I used, even though it might sound silly to you. I used to print off the image of Lesh’s translation model onto post-its.
So, I print a copy of Lesh’s Translation Model onto the post-its, and for a few math lessons each week, I grab one of my post-its and quickly highlight 2-3 representations that I have planned to include. Sometimes I realize that I’m focusing too heavily on one representation and need to reconsider my plans to include a second or third. However, typically (for you too I’m sure) the representations are already there.
So why go through this exercise? Well, by highlighting the specific representations I am holding myself accountable. I’m ensuring that I’m planning with multiple representations in mind.
But then I take it a step further. I skim through my plans and look for places where I need to focus on making that connection between the different representations.
If I am using counters for one problem, and having them sketch their thinking on a different problem, I’m actually using these representations in isolation of each other. I am not creating an opportunity to connect the two. This is where I need to be more intentional.
What I’ve found to be most helpful (and realistic if I’m being honest) is actually cutting down the amount of problems my students complete. This way we can work deeper and foster those connections and heightened understanding.
Yes, following this method, you will cover fewer problems, but you will go deeper with them – you will prioritize depth over breath! Again, I want to stress that I did NOT do this with every lesson every week. That is not realistic. Instead, I gave myself grace, and focused on a few lessons throughout the week.
Meaning Makers, I’m so happy we took more time to dive into Lesh’s Translation Model. What were your big takeaways? What are some ways you’d like to incorporate connections between the representations into your classroom? Share your thoughts and let’s keep this conversation going! Have a great one!
