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ROUNDING UP: Building a Meaningful Understanding of Properties Through Fact Fluency Tasks with Kyndall Thomas
Building fluency with multiplication and division is essential for students in the upper elementary grades. This work also presents opportunities to build students' understanding of the algebraic properties that become increasingly important in secondary mathematics.
In this episode, we're talking with Kyndall Thomas about practical ways educators can support fluency development and build students' understanding of algebraic properties.
BIOGRAPHY Kyndall Thomas serves as a math interventionist and resource teacher with the Oregon Trail School District, focusing on data-driven support and empowering teachers to spark a love of numbers in their students.
TRANSCRIPT Mike Wallus: Hi, Kyndall. Welcome to the podcast. I'm really excited to be talking with you today.
Kyndall Thomas: Hi, Mike. Thanks for having me. I'm excited to dive into some math talk with you also.
Mike: Kyndall, tell us a little bit about your background. What brought you to this work?
Kyndall: Yeah. I started in the classroom. I was in upper elementary. I served fifth grade students, and I taught specifically math and science. And then I moved into a more interventionist role where I was a specialist that worked with teachers and also worked with small groups, intervention students. And through that I was able for the first time to really develop an understanding of that mathematical progression that happens at each grade level and the formative things that are introduced at the lower elementary [grades] and then kind of fade out, but still need to be brought back at the upper elementary.
Mike: So I've heard other folks talk about the ways students can learn about the algebraic properties as they're building fluency, but I feel like you've taken this a step further. You have some ideas around how we can use visual models to make those properties visible. And I wonder if you could talk a little bit about what you mean by making properties visible and maybe why you think this is an opportunity that's too good to pass up?
Kyndall: My thought is bringing visual models back into the classroom with our higher upper elementary students so that they can use those models to build a natural immersion of some of the algebraic properties so that they can emerge rather than just be rules that we are teaching. By supporting students' learning through building models with manipulatives, we're able to build a bridge in a student's mind between their experience with those models and then their mental capacity to visualize those models. This is where the opportunity to bring properties to life is too good to pass up.
Mike: OK, so let's get specific. Where would you start? Which of the properties do you see as an opportunity to help students understand as they're building an understanding of fluency?
Kyndall: So, when I begin laying the foundation for understanding of the operations and multiplication and division, I intentionally layer in two other major algebraic properties for discovery: the commutative property and the distributive property. We're not setting our students up for success when we simply introduce these properties as abstract rules to memorize. Strong visual models allow students to discover the why behind the rules. They're able to see these properties in action before I even spend any time naming them.
For example, they get to witness or discover how factors can switch order without changing the product, how grouping affects computation, and how numbers can be broken apart and recombined for efficient counting and solving strategies. By teaching basic facts in this structured and intentional way through the behavior of numbers and the authentic discovery of properties, we're not only building fluency, but we're also developing deep conceptual understanding. Students begin to recognize patterns, understand rules, make connections, and rely on reasoning instead of rote memorization. That approach supports long-term mathematical flexibility, which is exactly what we want our students to be able to do.
Mike: I want to ask you about two particular tools: the number rack and the 10-frame. Tell me a little bit about what's powerful about the way the [10-frame] is set up that helps students make sense of multiplication. What is it about the way it's designed that you love?
Kyndall: The [10-frame] is so powerful because it's set up in our base ten system already. It introduces the tens in a way that is two rows of 5, which is going to lead into properties being identified. So, let me break that up into each individual thing that I love about it.
First, the [10-frame] being broken up into the two rows of 5. That's going to allow students to be able to see that distributive property happening, where we're counting our 5s first and then adding some more into each group. So, when we're seeing a factor like 8 times 2, we're seeing that as two groups of 5 and two groups of 3.
Mike: I think what you're making me remember is how it's difficult to help kids visualize that, right? It's a challenge. You can say "'4 times 4' is the same as '4 times 2 plus 4 times 2,'" but that's still an abstraction of what's happening, right? The visual really brings it to life in a way that—even if you're representing that with an equation and doing a true-false equation where it's 4 times 4 is the same as 4 times 2 plus 4 times 2—that's still at a level of abstraction that's not necessarily accessible for children.
Kyndall: And as we're talking through this, if I see students and they're working on four groups of 3 and they're seeing those 3s as a double fact plus one more group, I'm on the board writing out the equation, and I'm using the parentheses as that introduction to what this looks like abstractly. They're building it, and they're building those visuals both with their hands and with their minds, and then I'm bringing it to life in the equation on the board.
Mike: So, I think what I see in my mind as I hear you describe that is, you have kids with a set of materials. You're doing, for lack of a better word, a translation into a more abstract version of that, and you're helping kids connect the physical materials that they have in front of them to that abstraction and really kind of drawing the connection between the two. Am I getting that right?
Kyndall: Yeah. As the students are doing the physical work of math, I'm translating it into its own language up on the board. Absolutely.
Mike: I think what's clear to me from this conversation is the way that the tools can illuminate the property, and I think this also helps me think about what my role is as a teacher in terms of building a bridge to an abstraction. Do you actually feel like there's a point where you do introduce the formal language of it? And if you do, how do you decide when?
Kyndall: So, the vocabulary kind of comes after the concept has been discovered. But I don't like to introduce the vocabulary first as a rote memorization tool because that has no meaning to it.
Mike: I think if I were to summarize this, you're giving them a physical experience with the properties. You're translating that into an abstraction. And then once they've got an experience that they can hang those ideas on top of, then you might decide to introduce the formal language to them at some point.
Kyndall: Yeah, absolutely.
Mike: So, just as a refresher, for folks who might teach upper elementary and don't have a lot of lived experiences with the number rack—be it the ten or twenty or the hundred—can you describe a little bit about the structure, and maybe what about the structure in particular is important?
Kyndall: The structure of a number rack has rows, and each row has 10 beads in it. And typically those beads are divided into two sets of 5: five red beads and five white beads. Then we typically move into a number rack that has two rows so that we're working within 20.
Now, my thought is to take that [to] our third, fourth, and fifth grade, our upper elementary students, and use the hundreds rekenrek [i.e., number rack], where now we have 10 rows and we have 10 beads in each row—still split up into five red [beads] and five white—so that we can use that to teach things. If we're looking at the zero property, students are starting to notice that the rows represent the groups—the rows with the beads on it, that's one group. And so, if we're building zero groups of 3, we don't have a group that we can access to put three beads in. If we're looking at it with the commutative property, students are able to say, "One group of 3. We have one row and we're putting three beads in it."
But what happens when we switch those factors? Now we're utilizing three of our rows, but we're only sliding over one bead. The number rack is also so important when we get to the distributive property because of the way that they have separated those colors. So when we're looking at a factor like 7 times 6—seven groups of 6—then we're gonna be accessing seven rows with six beads in each. That is already set up in the structure of the tool to have five red beads and one white bead showing seven groups of 5 and seven groups of 1 put together.
Mike: That is super powerful. One of the things that really jumped out that I want to mark is: If I treat the rows like the groups and then I treat the beads like the number of things in each group, I can model one group with three inside of it, or I can model three groups with one inside of it, and I can really make the difference between those things clear, but also [I can make] the way that the product is still the same clear, right? So, I've got an actual physical model that helps kids understand what was often a rule that was just like 1 times 3 is the same as 3 times 1, because it is. But you're actually saying this is a tool that helps us make meaning of that.
The other thing that jumps out from what you said is: If I'm doing 6 times 5 or 6 times 7 and I push over six [beads], and six looks like five red, one white, I'm automatically set up to make sense of the distributive property because the visual helps me see it. Am I getting that right?
Kyndall: Yes, except let me correct you on that last one. You said "6 times 5," and you said, "If I slide over six," Now, six is our group number. We have to be deliberate; that's six groups of 5. So, we're grabbing our groups first, but absolutely, yes. That is the key structure there. [laughs] That's the idea.
Mike: This is why this would've been very helpful for a young Mike Wallus.
Kyndall: [laughs]
Mike: Well, before we go, are there any resources that you'd recommend to a listener that have either informed your thinking or that might help someone take what you've been talking about and put these ideas into action?
Kyndall: Yeah. I've been putting this practice into play here at my own district and tracking its progress for a while now. After seeing the success in my own halls here in Sandy, [Oregon,] I've started to reach out and work with other educators on purposeful tool use and mathematical progression.
If it resonates with you, whether you're in the classroom or in a leadership role, I would genuinely love to connect and learn alongside you. You're always welcome to reach out to me directly at KyndallThomas56@yahoo.com. I anticipate more conversations in collaboration, and I'd love to bring them to life through trainings moving forward. I believe that when teachers are confident in their own understanding, they build that same confidence in students.
Mike: I think that's a great place to stop. Kyndall, thank you so much. It has really been a pleasure talking with you and learning from you.
Kyndall: Thank you so much for having me. It's been fun.
Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling all individuals to discover and develop their mathematical confidence and ability.
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