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Building a Meaningful Understanding of Properties Through Fact Fluency Tasks with Kyndall Thomas · Rounding Up

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  Building fluency with multiplication and division is essential for students in
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  the upper elementary grades.
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  This work also presents opportunities to build students' understanding of the
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  algebraic properties that become increasingly important in secondary
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  mathematics. In this episode,
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  we're talking with Kendall Thomas about practical ways educators can support
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  fluency development and build students understanding of algebraic properties.
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  Hi, Kendall. Welcome to the podcast.
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  I'm really excited to be talking with you today.
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  Hi, Mike. Thanks for having me.
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  I'm excited to dive into some math talk with you also.
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  Kendall, tell us a little bit about your background.
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  What brought you to this work?
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  Yeah. I started in the classroom. I was in upper elementary.
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  I served fifth grade students and I taught specifically math and science.
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  And then I moved into a more interventionist role where I was a specialist that
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  worked with teachers and also worked with small groups intervention students.
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  And through that I was able for the first time to really develop an
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  understanding of that mathematical progression that happens at each grade level
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  and the formative things that are introduced at the lower elementary
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  and then kind of fade out,
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  but still need to be brought back at the upper elementary.
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  So I've heard other folks talk about the ways students can learn about the
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  algebraic properties as they're building fluency,
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  but I feel like you've taken this a step further.
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  You have some ideas around how we can use visual models to make those properties
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  visible,
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  and I wonder if you could talk a little bit about what you mean by making
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  properties visible and maybe why you think this is an opportunity that's too
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  good to pass up.
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  My thought is bringing visual models back into the classroom with our higher
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  upper elementary students so that they can use those models to build a natural
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  immersion of some of the algebraic properties so that they can emerge rather
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  than just be rules that we are teaching by supporting students learning through
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  building models, with manipulatives,
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  we're able to build a bridge in a student's mind between their experience with
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  those models and then their mental capacity to visualize those models.
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  This is where the opportunity to bring properties to life is too good to pass
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  up.
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  Okay, so let's get specific. Where would you start?
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  Which of the properties do you see as an opportunity to help students understand
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  as they're building an understanding of fluency?
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  So when I begin laying the foundation for understanding of the operations and
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  multiplication and division,
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  I intentionally layer in two other major algebraic properties for discovery,
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  the commutative property and the distributive property.
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  We're not setting our students up for success when we simply introduce these
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  properties as abstract rules to memorize.
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  Strong visual models allow students to discover the why behind the rules.
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  They're able to see these properties in action before I even spend any time
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  naming them. For example,
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  they get to witness or discover factors can switch order without changing the
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  product,
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  how grouping affects computation and how numbers can be broken apart and
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  recombined for efficient counting and solving strategies by teaching basic facts
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  in this structured and intentional way through the behavior of numbers and the
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  authentic discovery of properties, we're not only building fluency,
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  but we're also developing deep conceptual understanding.
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  Students begin to recognize patterns, understand rules, make connections,
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  and rely on reasoning instead of rote memorization.
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  That approach supports long-term mathematical flexibility,
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  which is exactly what we want our students to be able to do.
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  I want to ask you about two particular tools, the number rack and the 10 frame.
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  Tell me a little bit about what's powerful about the way the TENS frame is set
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  up that helps students make sense of multiplication.
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  What is it about the way it's designed that you love?
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  The tens frame is so powerful because it's set up in our
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  Base 10 system already.
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  It introduces the tens in a way that is two rows of
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  five, which is going to lead into properties being identified.
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  So let me break that up into each individual thing that I love about it. First,
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  the 10 frame being broken up into the two rows of five that's going to allow
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  students to be able to see that distributive property happening where we're
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  counting our fives first and then adding some more into each group.
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  So when we're seeing a factor like eight times two,
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  we're seeing that as two groups of five and two groups
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  of three.
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  I think what you're making me remember is how it's difficult to
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  help kids visualize that, right? It's a challenge.
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  You can say four times four is the same as four times two
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  plus four times two, but that's still an abstraction of what's happening, right?
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  The visual really brings it to life in a way that even if you're representing
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  that with an equation and doing a true false equation where it's four times four
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  is the same as four times two plus four times two,
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  that's still at a level of abstraction that's not necessarily accessible for
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  children.
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  And as we're talking through this,
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  if I see students and they're working on four groups of three and
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  they're seeing those threes as a double fact plus one more group,
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  I'm on the board writing out the equation and I'm using the parentheses
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  as that introduction to what this looks like abstractly.
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  They're building it and they're building those visuals both with their hands and
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  with their minds, and then I'm bringing it to life in the equation on the board.
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  So I think what I see in my mind as I hear you describe that is you have kids
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  with a set of materials you're doing, for lack of a better word,
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  a translation into a more abstract version of that,
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  and you're helping kids connect the physical materials that they have in front
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  of them to that abstraction and really kind of drawing the connection between
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  the two. Am I getting that right?
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  Yeah. As the students are doing the physical work of math,
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  I'm translating it into its own language up on the board. Absolutely.
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  I think what's clear to me from this conversation is the way that the tools can
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  illuminate the property,
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  and I think this also helps me think about what my role is as a teacher in terms
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  of building a bridge to an abstraction.
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  Do you actually feel like there's a point where you do introduce the formal
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  language of it, and if you do, how do you decide when.
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  The vocabulary kind of comes after the concept has been discovered,
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  but I don't like to introduce the vocabulary first as a rote memorization tool
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  because that has no meaning to it.
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  I think if I were to summarize this,
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  you're giving them a physical experience with the properties.
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  You're translating that into an abstraction,
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  and then once they've got an experience that they can hang those ideas on top
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  of,
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  then you might decide to introduce the formal language to them at some point.
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  Yeah, absolutely.
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  So just as a refresher,
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  for folks who might teach upper elementary and don't have a lot of lived
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  experiences with the number rack, be it the 10 or 20 or the a hundred,
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  can you describe a little bit about the structure and maybe what about the
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  structure in particular is important?
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  The structure of a number rack has rows,
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  and each row has 10 beads in it,
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  and typically those beads are divided into two sets of five,
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  five red beads and five white beads.
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  Then we typically move into a number rack that has two rows so that we're
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  working within 20. Now.
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  My thought is to take that into our third, fourth, and fifth grade,
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  our upper elementary students and use the hundreds rack and rack where now we
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  have 10 rows and we have 10 beads in each row,
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  still split up into five red and five white so that we can use
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  that to teach things. If we're looking at the zero property,
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  students are starting to notice that the rows represent the groups,
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  then rows with the beads on it. That's one group,
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  and so if we're building zero groups of three,
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  we don't have a group that we can access to put three beads in.
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  If we're looking at it with the commutative property,
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  students are able to say one group of three,
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  we have one row and we're putting three beads in it.
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  But what happens when we switch those factors?
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  Now we're utilizing three of our rows,
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  but we're only sliding over one bead.
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  The number rack is also so important to when we get to the distributive
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  property because of the way that they have separated those colors.
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  So when we're looking at a factor like seven times six,
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  seven groups of six,
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  then we're to be accessing seven rows with six beads in
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  each.
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  That is already set up in the structure of the tool to have five red
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  beads and one white bead showing seven groups of
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  five and seven groups of one put together.
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  That is super powerful.
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  One of the things that really jumped out that I want to mark is if I treat
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  the rows like the groups and then I treat the beads like the number
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  of things in each group,
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  I can model one group with three inside of it,
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  or I can model three groups with one inside of it,
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  and I can really make the difference between those things clear,
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  but also the way that the product is still the same clear,
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  so I've got an actual physical model that helps kids understand what
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  was often a rule that was just like one times three is the same as three times
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  one because it is,
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  but you're actually saying this is a tool that helps us make meaning of that.
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  The other thing that jumps out from what you said is if I'm doing six
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  times five or six times seven and I push over six and six looks like
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  five red, one white,
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  I'm automatically set up to make sense of the distributive property because
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  the visual helps me see it. Am I getting that right?
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  Yes, except let me correct you on that last one. You said six times five,
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  and you said, if I slide over six now six is our group number.
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  We have to be deliberate that six groups of five,
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  so we're grabbing our groups first, but absolutely, yes.
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  That is the key structure there. That's the idea.
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  This is why this would've been very helpful for a young Mike Wallace.
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  Well, before we go,
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  are there any resources that you'd recommend to a listener that have either
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  informed your thinking or that might help someone take what you've been talking
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  about and put these ideas into action?
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  Yeah.
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  I've been putting this practice into play here at my own district and tracking
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  its progress for a while now.
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  After seeing the success in my own halls here in Sandy,
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  I've started to reach out and work with other educators on purposeful tool use
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  and mathematical progression. If it resonates with you,
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  whether you're in the classroom or in a leadership role,
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  I would genuinely love to connect and learn alongside you.
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  You're always welcome to reach out to me directly at Kindle Thomas
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  fifty6@yahoo.com. I anticipate more conversations in collaboration,
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  and I'd love to bring them to life through trainings moving forward.
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  I believe that when teachers are confident in their own understanding,
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  they build that same confidence in students.
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  I think that's a great place to stop. Kendall, thank you so much.
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  It has really been a pleasure talking with you and learning from you.
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  Thank you so much for having me. It's been fun.
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  This podcast is brought to you by the Math Learning Center and the Meyer Math
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  Foundation dedicated to inspiring and enabling all individuals to discover
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  and develop their mathematical confidence and ability.