Eric Sisofo, EdD, The Big Place Value Episode
ROUNDING UP: SEASON 2 | EPISODE 10
If you ask an educator to share some of the most important ideas in elementary mathematics, most would include place value on that list. But what does it mean to understand place value, really? And what types of language, practices, and tools support students as they build their understanding? On this week’s podcast, we discuss these questions and more with Dr. Eric Sisofo from the University of Delaware.
GUEST BIOGRAPHY
Dr. Eric Sisofo is an assistant professor specializing in mathematics education in the School of Education at the University of Delaware. He earned his EdD in educational leadership, curriculum, and instruction; his MEd in curriculum and instruction with a concentration in mathematics education; and his BS in mathematical sciences from the University of Delaware. With a focus on mathematics education, Dr. Sisofo’s education leadership portfolio, the capstone project for his doctoral degree, investigated the effects of lesson study on teacher candidates’ use of student thinking when planning and reflecting on their teaching.
RESOURCES
National Council of Teachers of Mathematics
TRANSCRIPT
Mike Wallus: If you ask an educator to share some of the most important ideas in elementary mathematics, I'm willing to bet that most would include place value on that list. But what does it mean to understand place value really? And what types of language practices and tools support students as they build their understanding? Today, we're digging deep into the topic of place value with Dr. Eric Sisofo from the University of Delaware.
Mike: Welcome to the podcast, Eric. We're glad to have you with us.
Eric Sisofo: Thanks for having me, Mike. Really excited to be here with you today.
Mike: I'm pretty excited to talk about place value. One of the things that's interesting is part of your work is preparing pre-service [teachers] to become classroom elementary teachers. And one of the things that I was thinking about is what do you want educators preparing to teach to understand about place value as they're getting ready to enter the field?
Eric: Yeah, that's a really great question.
In our math content courses at the University of Delaware, we focus on three big ideas about place value with our novice teachers. The first big idea is that place value is based on the idea of grouping a total amount of stuff or bundling a total amount of stuff into different size units. So, as you know, we use groups of ones, tens, hundreds, thousands, and so on—not just ones—in our base ten system to count or measure a total amount of stuff. And we write a numeral using the digits 0 through 9 to represent the amount of stuff that we measured. So interestingly, our novice teachers come to us with a really good understanding of this idea for whole numbers, but it's not as obvious to them for decimal quantities. So we spend a lot of time with our novice teachers helping them think conceptually about the different groupings, or bundlings, that they're using to measure a decimal amount of stuff. In particular, getting them used to using units of size: one-tenth, one-hundredth, one-thousandth, and so on. So, that's one big idea that really shines through whether you're dealing with whole numbers or decimal numbers, is that place value is all about grouping, or bundling, a total amount of stuff with very specific different-size units.
The second big idea we help our novice teachers make sense of at UD is that there's a relationship between different place value units. In particular, we want our novice teachers to realize that there's this 10 times relationship between place value units. And this relationship holds true for whole numbers and decimal numbers. So, 10 of one type of grouping will make one of the next larger-sized grouping in our decimal system. And that relationship holds true for all place value units in our place value system. So, there might be some kindergarten and first grade teachers listening who try to help their students realize that 10 ones are needed to make one 10. And some second- and third grade teachers who try to help their students see that 10 tens are needed to make 100. And 10 hundreds are needed to make 1,000, and so on. In fourth and fifth grade, we kind of extend that idea to decimal amounts. So helping our students realize that 10 of these one-tenths will create a one. Or 10 of the one-hundredths are needed to make one-tenth, and so on and so on for smaller and smaller place value units. So, that's the second big idea.
And the third big idea that we explicitly discuss with our pre-service teachers is that there's a big difference between the face value of a digit and the place value of a digit. So as you know, there are only 10 digits in our base ten place value system. And we can reuse those digits in different places, and they take on a different value. So, for example, for the number 444, the same digit, 4, shows up three different times in the numeral. So, the face value is four. It's the same each digit in the numeral, but each 4 represents a different place value or a different grouping or an amount of stuff. So, for 444, the 4 in the hundreds place means that you have four groupings of size 100, the 4 in the tens place means you have four groupings of size 10, and the 4 in the ones place means you have four groupings of size one.
So this happens with decimal numbers too. With our novice teachers, we spend a lot of time trying to get them to name those units and not just say, for example, “3.4” miles when they're talking about a numeral. We wouldn't want them to say “3.4.” We instead want them to say “three and four-tenths,” or “three ones and four-tenths miles.” So saying the numeral “3.4” focuses mostly just on the face value of those digits and removes some of the mathematics that's embedded in the numeral. So instead saying the numerals “three ones and four-tenths” or “three and four-tenths” really requires you to think about the face value and the place value of each digit.
So those are the three big ideas that we discuss often with our novice teachers at the University of Delaware, and we hope that this helps them develop their conceptual understanding of those ideas so that they're better prepared to help their future students make sense of those same ideas.
Mike: You said a lot there, Eric. I'm really struck by point two, where you talk about the relationship between units. And I think what's hitting me is that I don't know that when I was a child learning mathematics—but even when I was an adult getting started teaching mathematics—that I really thought about relationships. I think about things like “add a zero” or even the language of “point-something.” And how in some ways some of the procedures or the tricks that we've used have actually obscured the relationship as opposed to shining a light on it. Does that make sense?
Eric: I think the same was true when I was growing up. That math was often taught to be a bunch of procedures or memorized kinds of things that my teacher taught me, that I didn't really understand the meaning behind what I was doing. And so, mathematics became more of just doing what I was told and memorizing things and not really understanding the reasoning why I was doing it.
Talking about relationships between things I think helps kids develop number sense. And so, when you talk about how 10 tenths are required to make one one, and knowing that that's how many of those one-tenths are needed to make one one, and that same pattern happens for every unit connected to the next larger unit. Seeing that in decimal numbers helps kids develop number sense about place value. And then when they start to need to operate on those numerals or on those numbers, if they need to add two decimal numbers together and they get more than 10 tenths when they add down the columns or something like that in a procedure—if you're doing it vertically. If they have more of a conceptual understanding of the relationship, maybe they'll say, “Oh, I have more than 10 tenths, so 10 of those tenths will allow me to get one one, and I'll leave the others in the tens place,” or something like that. So, it helps you to make sense of the regrouping that's going on and develop number sense so that when you operate and solve problems with these numbers, you actually understand the reasoning behind what you're doing as opposed to just memorizing a bunch of rules or steps.
Mike: Yeah. I will also say, just as an aside, I taught kindergarten and first grade for a long time, and just that idea of 10 ones and one ten, simultaneously, is such a big deal. And I think that idea of being able to say, “This unit is comprised of these equal-sized units,” how challenging that can be for educators to help build that understanding. But how rich and how worthwhile the payoff is when kids do understand that level of equivalence between different sets of units.
Eric: Absolutely, and it starts at a young age with children. And getting them to visualize those connections and that equivalence that a 10, one ten, can be broken up into these 10 ones or 10 ones can create one ten. And seeing that visually, multiple times, in lots of different situations really does pay off because that pattern will continue to show up throughout the grades. When you're going into second, third grade, like I said before, you’ve got to realize that 10 of these things we call “tens,” then we'll make a new unit called one “hundred.” Or 10 of these one hundreds will then make a unit that is called a “thousand.” And a thousand is equivalent to 10 hundreds. So these ideas are really critical pieces of students’ understanding about place value when they go ahead and try to add or subtract with these using different strategies or the standard algorithm. They're able to break numbers up, or decompose numbers into pieces that make sense to them. And their understanding of the mathematical relationships or ideas can just continue to grow and flourish.
Mike: I'm going to stay on this for one more question, Eric, and then I think you're already headed to the place where I want to go next. What you're making me think about is this work with kids not as, “How do I get an answer today?” but “What role is my helping kids understand these place value relationships going to play in their long-term success?”
Eric: Yeah, that's a great point. And learning mathematical ideas, it just doesn't happen in one lesson or in one week. When you have a complex idea like place value that, it spans over multiple years. And what kindergarten and first grade teachers are teaching them with respect to the relationship, or the equivalence, between 10 ones and one ten is setting the foundation, setting the stage for the students to start to make sense of a similar idea that happens in second grade. And then another similar idea that happens in third grade where they continue to think about this 10 times relationship between units, but just with larger and larger groupings. And then when you get to fourth, fifth, sixth, seventh grade, you're talking about smaller units, units smaller than one, and seeing that if we're using a decimal place value system, that there's still these relationships that occur. And that 10 times relationship holds true. And so, if we're going to help students make sense of those ideas in fourth and fifth grade with decimal units, we need to start laying that groundwork and helping them make sense of those relationships in the earlier grades as well.
Mike: That's a great segue because I suspect there are probably educators who are listening who are curious about the types of learning activities that they could put into place that would help build that deeper understanding of place value. And I'm curious, when you think about learning activities that you think really do help build that understanding, what are some of the things that come to mind for you?
Eric: Well, I'll talk about some specific activities in response to this, and thankfully there are some really high-quality instructional materials and math curricula out there that suggest some specific activities for teachers to use to help students make sense of place value. I personally think there are lots of cool instructional routines nowadays that teachers can use to help students make sense of place value ideas too. Actually, some of the math curricula embed these instructional routines within their lesson plans. But what I love about the instructional routines is that they're fairly easy to implement. They usually don't take that much time, and as long as you do them fairly consistently with your students, they can have real benefits for the children's thinking over time.
So one of the instructional routines that could really help students develop place value ideas in the younger grades is something called “counting collections.” And with counting collections, students are asked to just count a collection of objects. It could be beans or paper clips or straws or Unifix cubes, whatever you have available in your classroom. And when counting, students are encouraged to make different bundles that help them keep track of the total more efficiently than if they were just counting by ones. So, let's say we asked our first- or second grade class to count a collection of 36 Unifix cubes or something like that. And when counting, students can put every group of 10 cubes into a cup or make stacks of 10 cubes by connecting them together to represent every grouping of 10. And so, if they continue to make stacks of 10 Unifix cubes as they count the total of 36, they'll get three stacks of 10 cubes or three cups of 10 cubes and six singletons. And then teachers can have students represent their count in a place value table where the columns are labeled with tens and ones. So, they would put a 3 in the tens column and a 6 in the ones column to show why the numeral 36 represents the total. So giving students multiple opportunities to make the connection between counting an amount of stuff and using groupings of tens and ones, writing that numeral that corresponds to that quantity in a place value table, let's say, and using words like 3 tens and 6 ones will hopefully help students over time to make sense of that idea.
Mike: You're bringing me back to that language you used at the beginning, Eric, where you talked about face value versus place value. What strikes me is that counting collections task, where kids are literally counting physical objects, grouping them into, in the case you used, tens, you actually have a physical representation that they've created themself that helps them think about, “OK, here's the face value. Where do you see this particular chunk of that and what place value does it hold?” That's a lovely, super simple, as you said, but really powerful way to kind of take all those big ideas—like 10 times as many, grouping, place value versus face value—and really touch all of those big ideas for kids in a short amount of time.
Eric: Absolutely. What's nice is that this instructional routine, counting collections, can be used with older students too. So, when you're discussing decimal quantities let's say, you just have to make it very clear what represents one. So, suppose we were in a fourth- or fifth grade class, and we still wanted students to count 36 Unifix cubes, but we make it very clear that every cup of 10 cubes, or every stack of 10 cubes, represents, let's say, 1 pound. Then every stack of 10 cubes represents 1 pound. So every cube would represent just one-tenth of a pound. Then as the students count the 36 Unifix cubes, they would still get three stacks of 10 cubes, but this time each stack represents one. And they would get six singleton cubes where each singleton cube represents one-tenth of a pound. So if you have students represent this quantity in a place value table labeled ones and tenths, they still get 3 in the ones place this time and 6 in the tenths place. So over time, students will learn that the face value of a digit tells you how many of a particular-size grouping you need, and the place value tells you the size of the grouping needed to make the total quantity.
Mike: That totally makes sense.
Eric: I guess another instructional routine that I really like is called “choral counting.” And with choral counting, teachers ask students to count together as a class starting from a particular number and jumping either forward or backward by a particular amount. So for example, suppose we ask students to start at 5 and count by tens together. The teacher would record their counting on the board in several rows. And so as the students count together, saying “5, 15, 25, 35,” and so on, the teacher's writing these numerals across the board. He or she puts 10 numbers in a row. That means that when the students get to 105, the teacher starts a new row beginning at 105 and records all the way to 195, and then the third row would start at 205 and go all the way to 295. And after a few rows are recorded on the board, teachers could ask students to look for any patterns that they see in the numerals on the board and to see if those patterns can help them predict what number might come in the next row.
So students might notice that 10 is being added across from one number to the next going across, or 100 is being added down the columns. Or 10 tens are needed to make a hundred. And having students notice those patterns and discuss how they see those patterns and then share their reasoning for how they can use that pattern to predict what's going to happen further down in the rows could be really helpful for them too. Again, this can be used with decimal numbers and even fractional numbers. So this is something that I think can also be really helpful, and it's done in a fun and engaging way. It seems like a puzzle. And I know patterns are a big part of mathematics, and choral counting is just a neat way to incorporate those ideas.
Mike: Yeah, I've seen people do things like counting by unit fractions too, and in this case counting by tenths, right? One-tenths, two-tenths, three-tenths, and so on. And then there's a point where the teacher might start a new column, and you could make a strategic choice to say, “I'm going to start a new column when we get to ten-tenths.” Or you could do it at five-tenths. But regardless, one of the things that's lovely is choral counting can really help kids see structure in a way that counting out loud, if it doesn't have the, kind of, written component of building it along rows and columns, it's harder to discern that. You might hear it in the language, but choral accounting really helps kids see that structure in a way that, from my experience at least, is really powerful for them.
Eric: And like you said, the teacher, strategically, chooses when to make the new row happen to help students, kind of, see particular patterns or groupings. And like you said, you could do it with fractions too. So even unit fractions: zero, one-seventh, two-sevenths, three-sevenths, four-sevenths all the way to six-sevenths. And then you might start a new row at seven-sevenths, which is the same as one. And so, [students] kind of realize that, “Oh, I get a new one when I regroup seven of these sevenths together. And so with decimal numbers, I need 10 of the one-tenths to get to one.” And so if you help kids, kind of, realize that these numerals that we write down correspond with units and smaller amounts of stuff, and you need a certain amount of those units to make the next-sized unit or something like that, like I said, it can go a long way even into fractional or decimal kinds of quantities.
Mike: I think you're taking this conversation in a place I was hoping it would go, Eric, because to be autobiographical, one thing that I think is an advance in the field from the time when I was learning mathematics as a child is, rather than having just a procedure with no visual or manipulative support, we have made progress using a set of manipulative tools. And at the same time, there's definitely nuance to how manipulatives might support kids' understanding of place value and also ways where, if we're not careful, it might actually just replace the algorithm that we had with a different algorithm that just happens to be shaped like cubes.
What I wanted to unpack with you is: What's the best-case use for manipulatives? What can manipulatives do to help kids think about place value? And is there any place where you would imagine asking teachers to kind of approach with caution?
Eric: Well, yeah. To start off, I'll just begin by saying that I really believe manipulatives can play a critical role in developing an understanding of a lot of mathematical ideas, including place value. And there's been a lot of research about how concrete materials can help students visualize amounts of stuff and visualize relationships among different amounts of stuff. And in particular, research has suggested that the CRA progression, have you heard of CRA before?
Mike: Let me check. Concrete, Representational and Abstract. Am I right?
Eric: That's right. So because “C,” the concrete representation, is first in this progression, this means that we should first give students opportunities to represent an amount of stuff with concrete manipulatives before having them draw pictures or write the amount with a numeral. To help kindergarten and first grade students begin to develop understandings of our base ten place value system, I think it's super important to maybe use Unifix cubes to make stacks of 10 cubes. We could use bundles of 10 straws wrapped up with a rubber band and singleton straws. We could use cups of 10 beans and singleton beans—basically use any concrete manipulative that allows us to easily group stuff into tens and ones and give students multiple opportunities to understand that grouping of tens and ones are important to count by. And I think at the same time, making connections between the concrete representation—the “C” in CRA—and the abstract representation—the “A,” which is the symbol or the numeral we write down—is so important.
So using place value tables, like I was saying before, and writing the symbols in the place value table that corresponds with the grouping that children used with the actual stuff that they counted will help them over time make sense that we use these groupings of tens and ones to count or measure stuff. And then in second grade, you can start using base ten blocks to do the same type of thing, but for maybe groupings of hundreds, thousands, and beyond. And then in fourth and fifth grade, base ten blocks are really good for tenths and hundredths and ones, and so on, like that. But for each of these, making connections between the concrete stuff and the abstract symbols that we use to represent that stuff. So one of the main values that concrete manipulatives bring to the table, I think, is that they allow students to represent some fairly abstract mathematical ideas with actual stuff that you can see and manipulate with your hands. And it allows students to get visual images in their heads of what the numerals and the symbols mean. And so it brings meaning to the mathematics.
Additionally, I think concrete manipulatives can be used to help students really make sense of the meaning of the four operations too, by performing actions on the concrete stuff. So for example, if we're modeling the meaning of addition, we can use concrete manipulatives to represent the two or more numerals as amounts of stuff and show the addition by actually combining all the stuff together and then figuring out, “Well, how much is this stuff altogether?” And then if we're going to represent this with a base ten numeral, we’ve got to break all the stuff into groupings that base ten numerals use. So, ones, tens, hundreds—if needed, tenths, hundredths, thousandths. And one thing that you said that maybe we need to be cautious about is we don't want those manipulatives to always be a crutch for students, I don't think. So we need to help students make the transition between those concrete manipulatives and abstract symbols by making connections, looking at similarities, looking at differences.
I guess another concern that educators should be aware of is that you want to be strategic—again, which manipulatives you think would match the students’ development in terms of their mathematical thinking. So for example, I probably wouldn't use base ten blocks in kindergarten or first grade, to be honest. When students are just learning about tens and ones, because the long in a base ten block is already put together for them. The 10-unit cubes are already formed into a long. So some of the cognitive work is already done for them in the base ten blocks, and so you're kind of removing some of the thinking. And so that's why I would choose Unifix cubes over base ten blocks, or I would choose straws to, kind of, represent this relationship between ones and tens in those early grades before I start using base ten blocks. So those are two things that I think we have to be thoughtful about when we're using manipulatives.
Mike: My wife and I have this conversation very often, and it's fascinating to me. I think about what happens in my head when a multi-addition problem gets posed. So say it was 13 plus 46, right? In my head, I start to decompose those numbers into place value chunks, and in some cases I'll round them to compensate. Or in some cases I'll almost visualize a number line, and I'll add those chunks to get to landmarks. And she'll say to me, “I see the standard algorithm with those two things lined up.” And I just think to myself how big of a gift we're actually giving kids, giving them these tools that can then transfer eventually. They become these representations that happen in their heads, and how much more they have in their toolbox when it comes to thinking about operating than many of us did who grew up learning just a set of algorithms.
Eric: Yeah, and like you said, decomposing numerals or numbers into place value parts is huge because the standard algorithm does the same thing. When you're doing the standard addition algorithm in vertical form, you're still adding things up, and you're breaking the two numbers up by place value. It's just that you're doing it in a very specific way. You're starting with the smallest unit first, and you add those up, and if you get more than 10 of that particular unit, then you put a little 1 at the top to represent, “Oh, I get one of the next-sized unit because 10 of one unit makes one of the next size.” And so it's interesting how the standard algorithm kind of flows from some of these more informal strategies that you were talking about—decomposing or compensating or rounding these numbers and other strategies that you were talking about—really, I think help students understand, and manipulatives too, help students understand that you can break these numbers up into pieces where you can figure out how close this amount of stuff is to another amount of stuff and round it up or round it down and then compensate based off of that. And that helps prepare students to make sense of those standard algorithms when we go ahead and teach those.
Mike: And I think you put your finger on the thing. I suspect that some people would be listening to this, and they might think, “Boy, Mike really doesn't like the standard algorithm.” What I would say is, “The concern I have is that oftentimes the way that we've introduced the algorithm obscures the place value ideas that we really want kids to have so that they're actually making sense of it.” So, I think we need to give kids options as opposed to giving them one way to do it, and perhaps doing it in a way that obscures the mathematics.
Eric: And I'm not against the standard algorithm at all. We teach the standard algorithms at the University of Delaware to our novice teachers and try to help them make sense of those standard algorithms in ways that talk about those big ideas that we've been discussing throughout the podcast. And talking about the place values of the units, talking about how when you get 10 of a particular unit, it makes one of the next-sized unit. And thinking about how the standard algorithm can be taught in a more conceptual way as opposed to a procedural, memorized kind of set of steps. And I think that's how it sounds like you were taught the standard algorithm, and I know I was taught that too. But giving them the foundation with making sense of the mathematical relationships between place value units in the early grades and continuing that throughout, will help students make sense of those standard algorithms much more efficiently and soundly.
Mike: Yeah, absolutely. One of the pieces that you started to talk about earlier is how do you help bring meaning to both place value and, ultimately, things like standard algorithms. I'm thinking about the role of language, meaning the language that we use when we talk in our classrooms, when we talk about numbers and quantities. And I'm wondering if you have any thoughts about the ways that educators can use language to support students’ understanding of place value?
Eric: Oh, yeah. That's a huge part of our teaching. How we as teachers talk about mathematics and how we ask our students to communicate their thinking, I think is a critical piece of their learning. As I was saying earlier, instead of saying “3.4,” but expecting students to say “three and four-tenths,” can help them make sense of the meaning of each digit and the total value of the numeral as opposed to just saying “3.4.”
Another area of mathematics where we tend to focus on the face value of digits, like I was saying before, rather than the place value, is when we teach the standard algorithms. So it kind of connects again. I believe it's really important that students and teachers alike should think about and use the place value words of the digits when they communicate their reasoning. So if we're adding 36 plus 48 using the standard addition algorithm and vertical format, we start at the right and say, “Well, 6 plus 8 equals 14, put the 4 carry the 1. But what does that little 1 represent, is what we want to talk about or have our students make sense of. And it's actually the 10 ones that we regrouped into 1 ten.
So we need to say that that equivalence happened or that regrouping or that exchange happened, and talk about how that little 1 that's carried over is actually the one ten that we got and not just call it a 1 that we carry over. So continuing with the standard algorithm for 36 plus 48, going over to the tens column, we usually often just say, “3 plus 4 plus the 1 gives us 8,” and we put down the 8 and get the answer of 84. But what does the 3 and the 4 and the 1 really represent? “Oh, they're all tens.” So, we might say that we're combining 3 tens, or 30, with 4 tens, or 40. And the other 10 that we got from the regrouping to get 8 tens, or 80, as opposed to just calling it “8.”
So talking about the digits in this way and using the place value meaning, and talking about the regrouping, all of this is really bringing meaning to what's actually happening mathematically. That's a big part of it. I guess to add onto that, when I was talking about the standard algorithm, I didn't use the words “add” or “plus;” I was saying “put together,” “combine,” to talk about the actual action of what we're doing with those two amounts of stuff. Even that language is, I think, really important. That kind of emphasizes the action that we're taking when we're using the plus symbol to put two things together. And also, I didn't say “carry.” Instead, I said, we want to “regroup” or “exchange” these 10 ones for one ten. So I'm a big believer in using language that tries to precisely describe the mathematical ideas accurately because I just have seen over and over again how this language can benefit students' understanding of the ideas too.
Mike: I think what strikes me too is that the kinds of suggestions you're talking about in terms of describing the units, the quantities, the actions, these are things that I hope folks feel like they could turn around and use tomorrow and have an immediate impact on their kids.
Eric: I hope so too. That would be fantastic.
Mike: Well, before we close the interview, I wanted to ask you, for many teachers thinking about things like place value or any big idea that they're teaching, often is kind of on the job learning and you're learning along with your kids, at least initially. So, I wanted to step back and ask if you had any recommendations for an educator who's listening to the podcast. If there are articles, books, things online, particular resources that you think would help an educator build that understanding or think about how to build that understanding with their students?
Eric: Yeah. One is to listen to podcasts about mathematics teaching and learning like this one. There's a little plug for you, Mike.
Both: [laugh]
Eric: I guess …
Mike: I'll take it.
Eric: Yeah! Another way that comes to mind is if your school uses a math curriculum that aims to help students make sense of ideas, often the curriculum materials have some mathematical background pages that teachers can read to really deepen their understanding of the mathematics. There's some really good math curricula out there now that can be really educative for teachers.
I think teachers also can learn from each other. I believe teachers should collaborate with each other, talk about teaching specific lessons with each other, and through their discussions, teachers can learn from one another about the mathematics that they teach and different ways that they can try to help their students make sense of some of those ideas.
Another thing that I would suggest is to become a member of an organization like NCTM, the National Council of Teachers of Mathematics. I know NCTM has some awesome resources for practitioners to help teachers continue to learn about mathematical ideas and different ways to teach particular ideas to kids. And you can attend a regional or national conference with some of these organizations. I know I've been to several of them, and I always learn some really great ideas about teaching place value or fractions or early algebraic thinking. Whatever it is, there's so many neat ideas that you can learn from others. I've been teaching math for so many years. What's cool is that I'm still learning, you know, about math and how to teach math in effective ways, and I keep learning every day, which is really one of the fun things about teaching as a profession. You just keep learning. So, I guess one thing I would suggest is to keep plugging away. Stay positive as you work through any struggles you might experience, and just know that we all wrestle with parts of teaching mathematics especially. So stay curious and keep working to make sense of those concepts that you want your students to make sense of so that they can be problem solvers and thinkers and sensemakers.
Mike: I think that’s a great place to leave it. Eric, thank you so much for joining us. It's really been a pleasure talking to you.
Eric: Thanks, Mike. It's been a pleasure.
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.