Episode 203: Number Sense with Christina Tondevold and Mona Iehl

Check out Mona’s podcast Math Chat

Mona’s website Mona Math

Mona Iehl’s book Word Problem Workshop: 5 Steps to Creating a Classroom of Problem Solvers (set to be published October 2025, but you can preorder now)

Welcome fellow Recovering Traditionalists to Episode 203: Number Sense with Christina Tondevold and Mona Iehl

This is the last episode I’ll be doing for this school year and hopefully the last one I’ll be doing without an office.  I’m hoping when I start the episodes again in August I will be in my new office with all the right recording equipment.

Recently I was a guest on Mona Iehl’s podcast called Math Chat.  She asked me to come on and chat about number sense and I asked her if I could use our chat as well.  So if you are a subscriber of the Math Chat podcast then you’ve probably heard this episode already.  If you aren’t following Math Chat, go check it out.  

This episode is definitely longer than my normal episodes but Mona and I had a great chat about number sense and I didn’t want to trim it down.  I could talk forever about number sense and if you want to dive deep into learning about the 8 number sense concepts I talk about with Mona I have full courses that you can take online this summer.  The Flexibility Formula is a completely online course that focuses on helping elementary teachers develop students’ flexibility with numbers to help them get truly fluent in math and not just memorize facts and procedures.

My core philosophy in the course is that flexibility with numbers is something we cannot directly teach to kids. I can’t tell kids that “5 is one more than 4, learn it. Know it. Use it.” Instead it comes from the experiences they have with numbers and the relationships they see between numbers.  So in the course I share what those relationships are that we need to help students build and give you math experiences that help your students build them.  If, after you listen to this interview I did with Mona about the 8 number sense concepts, you would like to learn more so that next school year you can help your students build these concepts, get enrolled in The Flexibility Formula.  There is a version for K-2 educators and one for 3rd-5th grade educators.  Go to BuildMathMinds.com/courses to get enrolled. 

On those show notes I’ll link up Mona’s podcast and ways you can learn more from Mona.

If you don’t know, Mona Iehl is an elementary math teacher and coach who advocates for a student-centered, engaging approach to teaching math. She focuses on helping teachers create classrooms where all students feel empowered to love and understand math through building a positive math identity and deep understanding. Along with being the host of the Math Chat podcast,  Mona is the author of “Word Problem Workshop: 5 Steps to Creating a Classroom of Problem Solvers”.  Okay, here’s the interview I did with Mona:

Mona: Hi Christina, welcome to Math Chat!

Christina: Hey Mona, thanks for having me. It’s excited to kind of talk to you again.

Mona: I know, I know, it’s really exciting to have you on the podcast. So let’s get right into it. I like to make these like super quick to the point for teachers, so I’ll see if I can be quick and to the point.

Christina: You get me talking about math and I can talk forever. You ask me—don’t talk a whole lot besides that.

Mona: Amazing! As long as you build your life around the thing you can talk about the most, why not?

Christina: There we go.

Mona: I also resonate with that, so that’s great. Let’s get into it then, shall we?

Mona: Okay. Your website’s called Build Math Minds. Can you talk to us about that? Like tell us about your work in building math minds.

Christina: Yeah, so my website actually has gone through a lot of transitions of names, and this one finally stuck. And I think it’s because that’s kind of what we all are doing. We’re trying to build math minds of our students, but we’re also building our own math mind.

I was very procedural with how I thought about mathematics. I was a good rule follower, and so that’s how I did math. I just kind of followed the rules, did the procedures, thought I was good in math because I got A’s in everything except for geometry where you had to actually think, prove things. I was not good at that. So geometry was not my favorite.

But it wasn’t until I started teaching mathematics that I realized I really didn’t have a math mind. I was good at arithmetic and rule following, and it just kind of advanced from there. I was a teacher and then kind of through a series of events became a math coach and started to share some of the things that I was doing because I was a math coach for an entire region of my state. And I live in Idaho, which is very rural, so it was like hundreds of miles between districts that I had to work with. And it was just difficult to kind of see everybody, and I felt really lonely because I was the only math coach for all of these districts.

Like, I know some places have like a math coach in their building. I was like a math coach for part of an entire state, and that was a very [lonely] place to be. So that’s how I got into kind of sharing what I was doing, just because I was like, “I wish I knew this stuff. I wish I had this information.” And that’s what I get to do now is just share information.

And I have courses, I have a membership site, and every year I do a free virtual math summit where I bring together all kinds of math minds because I’m not the only one who has information to share. And every year we have 20 to 30 presenters present around all things mathematics for elementary educators. And that’s what we do: we’re here to build our own minds so that we can build the math minds of others.

Mona: So I love it. That is exactly like it. And your previous name, and sometimes it comes up still, is Recovering Traditionalist, right?

Christina: And I like—I love this this whole thing because I always call myself a former math avoider because I avoided building my math mind through my whole schooling. Like I was just content to listen to the teacher, do what they said to do, and collect my A’s and move on, right? And never have to really think too hard about any of it.

Christina: Yep. I skipped the word problems at the end of the assignment because I did like all the standard problems. I knew I got all those right, but if I had to think through a problem, it was very difficult for me. I didn’t have a math mind at all.

And I do still have the Recovering Traditionalist. That was the blog that I started back in the day. So it is still there, and I like to call everybody—like and tell everybody else, “Hey, use it. We are all recovering traditionalists. We all learn this way. We’re all trying to do better.”

Sometimes we slip back into being very traditional style of teacher, but then we’ll, you know, be like, “Oh, that didn’t work quite right. What else can I try with my students?” Because not everybody learns the way that it was kind of dictated to us back in the day.

Mona: So yeah, and I think like if we’re honest about it, like we are—like it worked for some kids, but those—it anything was going to work for those kids, right? Like that traditional style. But if we’re honest, it didn’t really work for any of us, so we got to have a new way.

Christina: It was that illusion of understanding. I heard Mike Flynn say that once, and I loved it. I had that illusion of understanding for my own—like everybody thought I was good at math, but in reality, out in the world—like when I was—I remember this. I was teaching seventh and eighth grade in Las Vegas. That was my first teaching job ever. And going out to a restaurant, trying to figure out the tip, and just pulling out a calculator. Yet I’m a math teacher, and I couldn’t figure that.

Like, you know, I know how to plug it in. I know how to do the procedure. Sure. And it wasn’t—I mean, he wasn’t my husband at the time, but my husband had said, “Dude, just take 10% and then take like half of that or double it.” You know, like he was—he had this math mind.

Mona: Yeah!

Christina: And I was like, “What? Explain that.” Like, I could not do those things. I needed a calculator to do math in the real world. You know, I mean, I could do paper pencil, but calculator is right there too nowadays, so…

Mona: Yeah, it’s so true. And I’m always telling teachers like, that’s what we’re going for. We’re not teaching kids to like fill in worksheets. We’re teaching kids to be the person at the table that can calculate the discount or the tip, or calculate the discount, right? Like, what are we doing in the grocery store in our minds? That’s what we want our kids to do.

Christina: Yeah, they should be able to total things up so that they know—which is hard because real life has made that mathematical thinking not as necessary, right? Like nobody carries cash anymore to go pay for their groceries, so they don’t need to total it up to make sure that they have enough cash in their hand to pay for it. Right, that’s super embarrassing but—everybody if you don’t have enough money when you get to checkout. But nowadays everybody uses their card, and even if you don’t have enough in your bank account, it’s probably just going to give you an overdraft fee. And you’re not paying attention to that.

Like, there’s nothing set up that’s making a lot of people now have to use their math mind in the real world, which then does make it difficult for us to help people see like, “But we want them to be that person that knows how to manage their money, that knows how much to leave for a tip, that knows how to figure out the discount,” all of that stuff because it’s—it’s powerful. That knowledge is powerful when you’re out in the real world.

And you—you don’t even—even something simple as giving back change, right? I mean, my husband’s a baseball coach, and he—we run the concession stand at the high school baseball games. And so we have the kids—like when varsity is playing, JV runs concession stands, and counting back change—like that’s something I have to teach a kid when they’re in the concession stand with me. And so we—all those things really do matter, but it’s hard because life doesn’t really make them see that it matters. Totally, until you’re in the concession stand as a junior learning how to count quarters and nickels back to people.

Mona: I love it. And also, I think that is a whole different podcast episode on like the problem—sorry, I get off on tangents.

Christina: I love it. I love it.

Mona: No, because we are—we’re going to talk about number sense, which I know you can talk about too. But it’s making me think about the problems that we traditionally think of as math problems and real-world scenarios, and what we need to do is think about the scenarios that our kids are going to face as adults, right?

Christina: Right.

Mona: And like, what does that look like? And what are their jobs going to be? And what kinds of reasoning are they going to need in those jobs?

Christina: So I’m gonna think more about that. Everybody else should think about that too.

Mona: I love it. Okay, but you’re the expert in my eyes on number sense. So can we dive in? Tell us how you define it. How do you define number sense?

Christina: I’m honestly an expert because others have done the research before me.

Mona: Fair.

Christina: I am a curator of information that’s out there. So I never want to claim that any of this is mine at all. There are lots of definitions of number sense, and lots of people will say like different parts that make up number sense. And I feel like they’re all very similar, though. They all—it might be called something different, but it’s part of this piece that’s over here.

So the thing I use are eight number sense concepts. Four of them come from Clemens and Sarah. And if you’re not familiar with those, check out the learning trajectories. It’s just learningtrajectories.org, I believe. I think it’s .org. We’ll put it in the show notes.

Mona: We’ll put it in the show notes.

Christina: So the four that they—and they use different words for them. I kind of changed them to like teacher-friendly—

Mona: Love that.

Christina: —words, but one of the first ones is subitizing or subtizing, whichever way you say it. And that’s that instantly recognizing without needing to count.

And then the other three early numery ones are all dealing with counting. So it’s being able to—it’s—I call it verbal counting. They have different terms, more technical terms for them. But basically, it’s kids saying the numbers out loud in order, right? They need to do that.

I was just at a baseball game this weekend, and it was so funny because there was a girl off to the side, like one of the little sisters of somebody who was playing. And she’s counting, and it’s “27, 50, 42.” Like, she was just saying numbers, but they weren’t in order. I was like, “Oh, dude, I wish I could have grabbed a video camera of her because she was like prime example of someone who’s working on that verbal counting. She doesn’t have it in order, but she knows how to say numbers, right?”

But just because you could say numbers doesn’t mean you actually understand counting. So as they are developing verbal counting—so these are all kind of intermixed—you’re also working on counting objects, like attaching those numbers to something, because we should be counting for a purpose. It’s not just so that kids can say the numbers. We want them to count with a purpose. So getting them to object count is important. That’s another one.

And then the final one for early numeracy from Clemens and Sarama is called cardinality. And that is—the formal version of it, as is subitizing, those two are are two formal names. But cardinality is just knowing that the last number you say tells how many.

So that’s like when a kid is counting objects and you say, “How many goldfish do you have?” and they go, “One, two, three.” And you say, “So how many do you have?” “One, two, three.” They don’t understand that it’s a set of three, and they can say three to answer that question.

So that’s that’s the final stage of kind of counting. And they all work hand in hand. They’re all super important. But we in the early grades tend to spend a lot of time on counting, and then later, teachers are like, “How do I get my kids to stop counting when they’re adding and subtracting and multiplying and dividing?”

It’s like, well, because we’ve spent so much time—since they’ve been two to get them to be able to count. Now they’re 10, and you’re saying stop counting. Like, that’s hard because that’s the very first way that they ever learn to do anything. And it will always be their fallback.

There was a study done, and I honestly don’t even remember where it was. I just remember this information, and I think it was shared actually—so there’s an Iron Man competition that happens here in Idaho up in Coeur d’Alene. And one summer they were talking about some research about bicyclists and that when they are about to get in an accident, they pedal backwards.

Oh, well, on a professional bike, that does nothing. But when you first learn to ride a bike, pedaling backwards is your brakes. And so when you’re in trouble, your body instinctually goes back to the first way that you learn something.

And so our students, when they’re in trouble—basically mathematically—their first instinct will always be to go back to count, because they know it’s safe. They know it will give them the answer, even though it might take them 10 minutes to solve a problem. Counting is their foundation, and it will always be their backup plan.

Now, the way that we help them get kind of away from that—I mean, I still—I mean, I count on our fingers even as adults. So don’t diminish it. But we really don’t want them counting on their fingers when they’re doing like 8 plus 7. If they’re doing 8 plus 3, and they’re using [their fingers]—go ahead, whatever, sure, you know? But if they’re doing larger quantities, counting is not the most efficient.

So to move kids past that counting phase for most things—except occasionally, they will still go back to it, it’s okay—but moving them past that is the four other ones that come from Van de Walle and friends. Teaching Student-Centered Mathematics is like the math bible in my mind. And these four came from the PreK to 2 book.

But it is spatial relationship. So kids should have a visual that goes with a quantity. So a lot of early numeracy kids and and even older kids—even adults who are not teachers—if you stop somebody on the side of the road and say, “Close your eyes, picture seven,” they’re going to picture the digit seven. They don’t picture seven things, right?

We need kids to be able to close their eyes and see seven things. That’s where subitizing comes in because you can’t—even as adults, we can’t see seven individual things. It stops at about five unless they’re arranged in some way.

So spatial relationships is is having those visuals, but also being able to see connectedness between those. So how does the visual of four connect to five? How does that visual of five help you see seven? How does 10 help you maybe see nine? Right, because our mind cannot see nine individual things. It has to tie to something else.

So spatial relationships is having that visual, but also typically those visuals are related to another visual that’s easier for our mind to hold on to. That’s why 10 frames have become so popular.

Mona: Yeah, I was like, “10 frames come to mind, five and 10.”

Christina: And that is one of the other ones. I’ll get to in just a second. There’s one in between, which is understanding one and two more or less. Kids need to instantly be able to tell you what’s one more, one less, two more, two less.

And then the next one is the benchmarks of five and 10, because our number system is a base 10. Two fives make a 10, and our mind can visually hold five things, right? It helps that our fingers stop at five. So anytime that they are thinking of things like seven, they should see it as five and two more, or they should see it as a 10, but three of my fingers are down, right? Like they should see how every number relates to the benchmarks of five and 10. And then those do progress up to like 50 and 100 and so on, but the base idea is understanding five and 10.

And then the last one, Van de Walle and friends called it “part-part-whole.” A lot of people now call it “part-part-total” because when you’re working with fractional amounts, it’s not always a whole. But so I still use part-part-whole because that’s what they have in their book. But it’s understanding that you have this total amount. It can be broken up into lots of different parts, but it always is that same total amount that you started with.

So helping kids understand those four number relationships from Van de Walle is really what helps start moving them past the counting phases and using more strategy-based things.

Because when they see 8 plus 7, they know—oh, they can visualize—let’s use the visual here, the spatial relationship. I can visualize eight in my mind, and I can see it’s just two away from having that 10 frame full. And I can see that seven is a five and two more, whether it’s with hands or in another 10 frame.

And they can visually—like, I mean, this takes a while in my brain, but they do it so quickly, you don’t even know they’re doing it—is that they will take two from that seven to make that 10. And then they’ve got 10 and five. And 10 and five, they know, is 15.

So those strategies that we’re trying to get kids to develop beyond the counting is all dependent upon spatial relationships, the benchmarks of five and 10, part-part-whole, knowing they can break apart the seven. They need to add two more to make the eight into a 10. All of those things are what’s dependent upon getting kids past those counting phases and into more strategy work, whether that’s with single-digit stuff, multi-digit, multiplication, all of that, fractions. All of those relationships help kids with fractions, decimals, any kind of number. Kids need to be developing these things.

Mona: Yes! Oh my gosh, that is such a good list for just like—I’m thinking about new teachers that I coach and just having that knowledge, right? Of like, “Oh yeah, we’re trying to get a benchmark of five and 10.” Like, they need to know that. So I wrote it all down, and I’ll put it in the show notes.

But sometimes it’s like—actually, I just had a conversation with a teacher who was like, “Well, I was just trying to show them the easiest way to get the answer.” And I was like, “I understand.” Like, I get it. Your heart is there. You want your kids to get the answer in first grade, right? Like, but showing them to directly model or directly draw all the quantities out and count them by ones isn’t necessarily how every kid is doing it, specifically mine, right?

Like, my kid, my first grader, puts eight in his head and counts up, even though I’m like, “What about a 10? Have we ever thought about—let’s look at these unifix cubes. Let’s look at our hands.” I’ve tried it all, but right now, he’s stuck in, “I want to count up.” Fine. But it also—when we understand those things and we understand what number sense is built on, we can better support kids in where they are, right?

Christina: And instead of believing that, “Well, if kids draw both and count them all by ones, that’s the best way”—that might not be the best way, right?

Mona: Yeah.

Christina: And even—even if you are count—even if you want to encourage that counting because the majority of your students are in the counting phase, just drawing them inside of a 10 frame, because then they can still count, but now kids have the opportunity—

Mona: Love it.

Christina: —to see ways that they might combine it to make it a little easier for them to then figure out the answer. Yes.

So it’s not that the counting is bad. It’s not that fingers are bad. I distinctly remember—I share a picture all the time in some stuff that I do with my youngest when he was in kindergarten, and he came up and asked me, “What’s 8 plus 8?” I go, “What do you think it is?” And the 8 plus 8 is not a kindergarten thing, right?

So I didn’t expect him to know the—he said, “13,” or some—I don’t remember what his actual answer was. I have it on video. But I go, “Okay, bud. So show me eight with your hands.” And you know, I asked him to do his fingers, and it was something like this. And then I’m—I did the same thing, and I put—he was standing in front of me, so he had a five against a five and a three against a three. And instantly, he goes, “Oh, it’s 16.” I didn’t say a thing. All I did was this, right?

So he showed—he showed eight. I showed eight. And he knew this was eight because of some subitizing, yeah? And those who aren’t—I forget we’re doing a podcast. So those those who can’t see this, I’ve got the one whole hand and then three on the other, right? I’ve got five fingers and three fingers. And our palms are joined together, and he sees the fives coming together to make a 10, and then three and three is six.

But I could have just had him count, you know, and count out eight items and eight items. But we, you know, it’s also a rush of, you know, in the moment, just trying to cook dinner with a kid, right? I got my fingers are right here. But—but a lot of times, teachers think, “Fingers are bad. Fingers are bad.” They’re a foundational thing, and they’re always with your students. But it’s—it’s helping them see groups within their fingers that they can use to be a bit more efficient when they’re adding, subtracting, even multiplying, right?

Using their fingers is not a bad thing. If I want to think about seven groups of six, well, I might know five groups of six. And I can put my hand up with all five—five groups of six, I do know that one. That’s 30. And then I could go six more—30, and I’m putting my palm up, and then 36, and then 42 as I put each finger up. I’m doing another six, right?

And this might look like a kid is counting on their fingers to do multiplication. But what they did with their fingers is just so cool, right? That’s such an awesome strategy. And they’re starting to become more efficient and have a different strategy beyond counting. But they’re encouraging grouping using their fingers and the groups that they already do know around multiplication.

Which is why it’s so important to not just observe kids doing things in math, but asked them, “What are you up to?” I always say that, right? Like, “What are you up to?” Because I never want to assume that they’re doing math, you know? Like, if I walk up and I’m like, “Tell me how you’re solving.” They’re like, “Uh, I’m not. I’m just saying a song in my head,” you know? And then they don’t want to answer at all.

But if I’m like, “Tell me what you’re up to,” at first they get really confused on what that even means. But—but then if a student describes that to you, that’s something then you can have them share with the rest of the class.

Mona: That maybe other kids are doing that, and they think it’s like not okay, or they don’t even know what they’re doing, right? So putting that that student’s thinking out there for others to learn from or with.

Christina: Yep, yep. It just provides an opportunity. That’s what we’re doing, is providing opportunities for kids to build their math mind. Because when we do—when we dictate and say, “Hey, you have to solve it this way,” it’s—that’s the only way they’re going to think about it. And so we want to provide the opportunity that they can think about it differently.

Mona: Yeah, and using those number sense concepts is a huge way to do that.

Christina: Yeah, yeah. And exactly. And when we tell kids to solve it a certain way, like that teacher I was talking to that’s like, “But I just wanted to help them get to the answer.” Great, but now my kid is the one who’s like, “Okay, I’ll do it her way” and does it. And I said, “Why did you draw those circles?” And he was like, “Um,”—what did he say? He was like, “Yeah, that’s what she told us to do.”

I was like, “Oh, okay.” Like, you know, like I ask my kid about their math test all the time, and then I never talk to their teachers about it because that’s not fair. But I ask him all the time, and that’s what he’ll tell me. “Oh, well, that’s what she told us to do.”

I said, “Okay, but if she didn’t tell you what to do, what would you have done?” And he always tells me, “Oh, I would count on. I would do like this.” Okay, but then this first grade teacher told me that when I was having a conversation with her, and she was like, “Well, I just told him to do that so it got him to the answer faster.” And I thought, “That’s probably what my kid’s teacher is doing. She just thinks that she is helping them,” right?

Just like this teacher is. But instead, that shift of—yes, you can—we’re not saying don’t help them. We’re saying don’t provide a cycle that we just talked about at the beginning, which is where kids just do what you say to do to make you happy.

Christina: Yeah, yep. And—and it’s hard because we do—our—we went into teaching because we want to help kids.

Mona: Yep, that that’s why we did it, right? It’s not because we get summers off. That’s what everybody thinks.

Christina: But teaching, because we want to help kids and we want to provide them a good experience in school. So it’s never—teachers are never going into it like, “Oh, I’m doing this to kind of destroy a kid.”

Mona: 100% right, no.

Christina: It is always of, “I want to help this kid or this group of kids,” right? And so that—that’s difficult because the way that we see to help them is to give them the steps when they don’t know the steps, right?

And so a lot of times, it is just backtracking a bit and moving into more of the modeling of the mathematics, which is difficult a lot of times because it takes a lot of extra time and all of, you know, getting manipulatives out and all of that kind of stuff.

But it’s—it’s having options for kids all the time. And unfortunately, our textbooks are one of the hardest things to get out of because the textbook will always tell you exactly how to solve the problem. It will say, “Use blah, blah, blah—counting on, use the make 10 strategy, use bridge over 10,” whatever it might be. It says, “Have your students use this strategy.”

Well, if you just take white out and white that out and give them the problem and say, “Show me how you think about this problem,” that’s all it takes. That’s it. And—and giving them stuff there because if—if all they have is just the problem in front of them and you say, “Solve this,” you get the deer in headlights like, “I don’t know what to—” and you will have kids who know what to do. They’re those kids that already have some strategies, have some ways that they’re thinking about it.

But if—if you spend a lot of time helping kids understand that mathematics is about making sense, and we work through problems, and we model them, we use manipulatives, we draw, we do use symbols, that all of those options are available, and you aren’t constantly saying, “Use this one way,” then it does open the door for it. But a lot of times, the textbooks are really prescribed.

Mona: Yeah, and you don’t have to have them solve it the make 10 way. Kids just need to solve it, right? That’s what our standards say. It says solve addition—you know, within 10 or whatever your your standard says—using strategies. And some of them do say the counting on, you know, in the early grades because that’s where a lot of kids are, developmentally.

But you also are going to have a lot of kids who have developed thinking strategies, and we—we don’t want to hinder them from showing their thinking either. And you’re going to have some kids who are just counting one by one by one as well. So you’ve got counter, counting on, you got direct modelers, you got direct fact kids, and you got some kids who just know the facts, and they just know it. Like, are—do we make them have to draw circles every single time?

Christina: Because they’re going to soon hate mathematics.

Mona: That’s right. Let them do that. So it’s—it’s about feeling okay letting kids solve it the way that they want, but laying a foundation of counting, yes, but also number sense ideas that when they see 8 + 7, they might stop and think, “Oh, that’s just two away from 10, and I could use this strategy.”

And even—even I might be counting on, but I might jump a little. I might use some jumps, right? Like, there’s—is still in that counting on, but when he sees 8 plus 7, he might say, “Oh, if I jump two or count on two, it gets me into 10.” And then from 10, he might still count one by one by one, right? But that’s—that’s moving towards a strategy, right, of that make a 10 idea. But he’s not quite wanting to use a make a 10.

Mona: So, right, that—

Christina: Quite yet, right? They—I think what—Oh, sorry.

Mona: Nope, nope. We’re both so excited. I was going to say—so we started by talking about number sense, which I think this is perfect because here’s the deal. Like, I run into a lot of teachers that are like, “Well, these kids are missing a lot of number sense.” That’s one of the things that teachers say often, right?

Like, “My middle schoolers don’t have number sense. My fifth or fourth graders don’t have number sense.” But what we’re actually talking about is problem-solving, right? And all of these, like approaches to thinking—

Christina: Yeah, thinking through a problem.

Mona: Yeah, exactly. And so I just—I love this. Let’s tie this back to like, okay, so you are seeing that kids are not flexibly thinking. They are not thinking. They’re just relying on these strategies that are inefficient. I think that’s what people mean, right, when they say their older students are lacking number sense, or even their first graders, maybe second.

Christina: That’s a very good point. They—they say that because they see kids still counting typically.

Mona: Yeah. And so what I hear us saying is that we need to provide these opportunities for students to solve in ways that make sense to them. And then we can build their efficiency and effectiveness in their ways that they think and solve. You want to say more about that?

Christina: Yeah, yeah. And I think the—the very first place to start is to see if your students have a visual of numbers. Because if they—if they can’t visualize, like that—the research on predictors of future success in mathematics with preschool kids is if they have spatial reasoning, like those things where you have like a picture of an F, right, and then you have to figure out which one below shows the F—that same style of F, but rotated, or, you know, that kind of a thing.

It—and and maybe it wasn’t that type, but it—that’s an example of a spatial reasoning task. But kids who—who do well with those kinds of spatial reasoning tasks do well in mathematics later on. And it’s because they can visualize things, and they aren’t tied to just that—like, it’s really hard to manipulate a 27 if all you see is a two and a seven, and you’re adding that to 38. Like, what do you—what do you do?

But if you can visualize 27 in a 10 frame, and you could see 38, and you can visualize that 10 frame or on a rekenrek or whatever visual it might be, or with base 10 blocks, right—they need visuals of numbers, whatever those are—multi-digit, fractions, decimals. Kids need those visuals because the visuals then help them be able to manipulate and move those parts around to make it easier to use and to add, subtract, multiply, or divide with.

So the first place to start is to see if your students have a visual. And that can be as easy as, you know, “Close your eyes, picture seven. Now draw for me on your paper what you did,” or a sticky note—just have a sticky note. That’s one of my favorite tasks, actually, is use a sticky note, and they have to give you the sticky note before they leave for lunch or wherever they’re headed out, right? Just do it real quick.

And then when they’re out at recess, you flip through those sticky notes and see how many of them just have the digit seven, how many of them are drawing individual seven things, how many of them have grouped them into like a five and a two or into a 10 frame or whatever, right?

They need something that they can—because if it’s still individual items, seven individual items, they honestly did not see seven. They—they see some stuff in their mind, and then when they drew it on paper, they counted to make sure it was seven. “There you go. Seven.”

So—so you need to see how many of them can group and show seven items in a way that’s easily recognizable. And that’s subitizing. If your students aren’t doing that, start with subitizing. Subitizing is kind of the—the catchall spot that you can start doing lots of subitizing activities with your students.

And if you’re not familiar with subitizing, just Google it. It will—it will come up. But doing quick images is is a great place where you just flash some kind—some image up there with an amount and then have it disappear after a few seconds and see if they can describe what they saw.

So we want it there, but then we want it to disappear so that they have to hold it in their head. If we leave it up there, they’re still able to see it, and they’re able to count. And I’m totally pointing at the screen for those of you that can’t see.

So doing lots of subitizing activities, and then if they’ve got lots of visuals, if it shows that they are seeing seven and it’s grouped in a way, then start doing spatial relationship ones where they’re—where you are showing seven and then you’re showing five right after it. And—and they disappear, but then you ask, “What was the same and what’s different?” So they’re starting to see connections between visuals is super important.

So that the visual side of it and then the benchmarks of five and 10 because our system is a five and 10—I mean, it’s a 10 system, but two fives help us make the—any kind of 10. So that’s—that’s another really important one.

I’m not saying the others aren’t important, but if you need a place to start, the visuals and then helping kids see relationships between the five and 10. And those go together because if you notice the—when I said show a visual of seven and then after it show a visual of five, I’m helping them relate to a benchmark of five. If I showed seven and then I show 10, I’m helping them relate to a benchmark of 10, right?

So even as I’m working on subitizing and I’m working on spatial relationships, I’m still building the 10 and five, 10 and five relationship. So when I’m doing subitizing activities and I’m showing seven, I don’t show seven in a row. I show—I show five pattern and then a two pattern with it because kids can see the five and two, and then the five and two make the seven.

So anytime that you are doing subitizing and spatial relationships, I’m still pulling in the five and ten-ness of a number because it’s—it’s really, really powerful.

Mona: Yes, and having this understanding as a teacher that like the subitizing matters, that the benchmark of five and 10 matters, that, you know, all of the things—like all the things you described about number sense—all of having that knowledge. But then when you’re in front of kids, you have to like not just tell them, right? Like, you can’t just be like, “Hey, seven is five and two, remember that,” right?

Christina: Like and holding up your hands, “There is five in seven.”

Mona: Like, if—if teaching was just—if parenting—teaching was just telling kids something and they know it and do it forever—

Christina: Yeah, it’d be so easy, right?

Mona: It’d be so easy. It’d be—our life, our life would be so easy. We wouldn’t have this podcast. But what we need to do is say, “What do you see here?” Right? Like another one I feel like is good for this is like “which one doesn’t belong” in with five and—or with 10 frames and getting kids to make those comparison—

Christina: Routine that’s out there. If you just Google “number routines,” there is so much out there about building number sense. It’s—it’s so amazing. But number routines is part of it. But even—even if you’re doing story problems, right?

I—number routines, story problems, and games are the three things I believe every teacher should be doing. If you’re working on story problems, check the numbers that are in those story problems. Have them be things that are relating to fives and tens. Right?

If—if Johnny has $7, and he gets some more for his birthday, now he has—guess what?—$10, right? You—you choose those numbers. You can change those numbers that are in your textbook so that they are working on developing how a number relates to five and 10, or one and two more or less, wherever you’re at.

And if you’re older grades, it’s one or two tens more or less. You’re adding a 10 or a 20, or you’re subtracting a 10 or a 20. And then as with fractions, it’s one or two of whatever unit. So if you’re—if it’s three-fourths, they had three-fourths of the cake left, and they ate one-fourth of it, how much is left now? That’s one less—one-fourth more or less, whichever way you’re doing it.

So you can—you can change the numbers within your story problems to focus on whatever number sense concept you’re looking at. And then when you’re playing games, if you’re playing Bump, right, have it be something that is relating to the making a 10. So if five is the number on—on the board, and four and six and seven—well, you’ve got to roll to make a 10.

Right? So if I roll a six, I could put my—my space on the four because six and four make a 10, right? So whatever games you’re playing, make them really focused on five-ness, 10-ness, one and two more or less, having lots of visuals in there.

It’s—it’s everything that you’re doing. Once you understand those eight number sense concepts, you’ll see opportunities to pull them in, right? If—if your word problems are not helping kids build a sense of five and 10-ness, then maybe those need to—it’s not all the time you need to do that, right? But if your kids are struggling with understanding the benchmarks of five and 10, change it up. Let’s focus on that.

Mona: Yes, yes, yes, yes. Okay, this was so good, and I feel like this last little bit gave us a lot of things to do. So let me recap. Number one, if you’re like, “My kids don’t have number sense,” we have to see if they have a visual. Like, close your eyes—

Christina: Sticky notes are like the—the pro-tier exit ticket.

Mona: I agree, yeah. Sticky notes and index cards, ‘cuz I bet you have a whole bin of index cards ‘cuz it was on the supply list, and somebody brought in so many. I always had so many index cards.

Christina: So anyway, and—and let me just say too, it doesn’t need to be seven. That’s an example for early grades, right? But if—if you’re a fifth grade teacher, have them close their eyes and picture—picture three-fourths because three-fourths should be something that they can visualize quickly, right?

I’m not going to say visualize seven-twelfths. Who—yeah, when do we ever see that? Well, and that’s more than five, right? Like, that’s what we—

Mona: We intend to talk about, like we—we want to see that they have been able to group something. And I think that’s another point here, right? Like seven is a little bit to 10 but a little bit to five. And so when you’re thinking about numbers, we want kids to—it’s—

Christina: Yeah, yeah. I love it. It’s such a—it’s such a quick, easy way to check in on number sense if you’re feeling like, “Is this a number sense thing?” or whatever. So see if they have that visual. Love it.

Mona: And then do some subitizing, routines, games, and think about all of those eight things that Christina shared—all the number sense definitions, which I know are on her website, and we’ll link to those in the show notes.

And think—I think that what you just said was like, these eight things you’re gonna—once you know them, you’re going to see where they show up all the time. And if they’re not—if there’s not opportunities for kids to practice one more or less or two more or two less, then make opportunities, like strategically choose the numbers in your word problems, strategically order your “which one doesn’t belong” or your “alike and different” or your whatever game like you said you’re playing. Make it about that.

Christina: Love it. Anything else you want to say to the teachers listening?

Mona: No, that’s it. It’s—you don’t have to add more. It’s just modifying what’s there to really focus in on those key number sense components that will help build your—your students’ foundation beyond counting. Because counting is wonderful. Keep counting. Yes, we have to count. Let’s start—start building number sense with it.

Christina: Yeah, and giving—and you said also like giving those opportunities for students to show their understanding and not falling into the trap that we’re helping students by giving like an illusion of their understanding. Instead, we want to like give them an opportunity to show us if they really do understand.

Mona: Love it.

Christina: Yeah, anybody can follow and parrot back what a teacher said. I mean, not anybody—I shouldn’t say anybody—but the majority of your students will follow and just procedurally give you what you’re asking for because they want to be a good student, right?

Just like we want to be a good teacher, they want to be a good student. And they want to give you what you’re asking for. But if you open it up and just say, “Solve this in a way that makes sense to you,” you really will see what they’re understanding and what they’re not understanding.

Mona: Yes, yeah. Okay, let’s end on that. That’s so good. All right, tell us how we can connect with you further, Christina.

Christina: So everything’s at buildmathminds.com. We’ve got YouTube videos. The Recovering Traditionalist has a ton too, but all of those are on YouTube also. And we have the podcast, Build Math Minds. We’re going to try to joint—do this with you as well through that. But yeah, everything’s through there. And Instagram, Build Math Minds.

Mona: Great, all those places. Awesome. Thank you so much, Christina.

Christina: Thank you.

Normally at this point in the podcast I say “Until next week”, but this is my last episode for a few months while I take a break over the summer.  So instead I’ll end by saying I hope you have an amazing end of the school year, an even better summer break, and I’ll see you back here in August so we can chat more about how to build math minds next school year.

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