# Beth Hulbert, Making Sense of Unitizing: The Theme That Runs Through Elementary Mathematics

**ROUNDING UP: SEASON 3 | EPISODE 4**

During their elementary years, students grapple with many topics that involve relationships between different units. In fact, unitizing serves as a foundation for much of the mathematics students encounter during their elementary years. Today, we’re talking with Beth Hulbert from the Ongoing Assessment Project (OGAP) about the ways educators can encourage unitizing in their classrooms.

**BIOGRAPHY**

Beth Hulbert is an independent consultant focused on mathematics curriculum, instruction, and assessment at the K–8 level. She has been involved in all aspects of the Ongoing Assessment Project (OGAP) since its inception. Beth is coauthor of *A Focus on Multiplication and Division: Bringing Research to the Classroom*. The book was written to communicate how students develop their understanding of the concepts of multiplication and division.

**RESOURCES**

**TRANSCRIPT**

**Mike Wallus**: During their elementary years, students grapple with many topics that involve relationships between different units. This concept, called “unitizing,” serves as a foundation for much of the mathematics that students encounter during their elementary years. Today, we're talking with Beth Hulbert from the Ongoing Assessment Project (OGAP) about the ways educators can encourage unitizing in their classrooms.

Welcome to the podcast, Beth. We are really excited to talk with you today.

**Beth Hulbert**: Thanks. I'm really excited to be here.

**Mike**: I'm wondering if we can start with a fairly basic question: Can you explain OGAP and the mission of the organization?

**Beth**: Sure. So, OGAP stands for the Ongoing Assessment Project, and it started with a grant from the National Science Foundation to develop tools and resources for teachers to use in their classroom during math that were formative in nature. And we began with fractions. And the primary goal was to read, distill, and make the research accessible to classroom teachers, and at the same time develop tools and strategies that we could share with teachers that they could use to enhance whatever math program materials they were using.

Essentially, we started by developing materials, but it turned into professional development because we realized teachers didn't have a lot of opportunity to think deeply about the content at the level they teach. The more we dug into that content, the more it became clear to us that content was complicated. It was complicated to understand, it was complicated to teach, and it was complicated to learn.

So, we started with fractions, and we expanded to do work in multiplicative reasoning and then additive reasoning and proportional reasoning. And those cover the vast majority of the critical content in K–8. And our professional development is really focused on helping teachers understand how to use formative assessment effectively in their classroom. But also, our other goals are to give teachers a deep understanding of the content and an understanding of the math ed research, and then some support and strategies for using whatever program materials they want to use. And we say all the time that we're a program blind — we don't have any skin in the game about what program people are using. We are more interested in making people really effective users of their math program.

**Mike**: I want to ask a quick follow-up to that. When you think about the lived experience that educators have when they go through OGAP’s training, what are the features that you think have an impact on teachers when they go back into their classrooms?

**Beth**: Well, we have learning progressions in each of those four content strands. And learning progressions are maps of how students acquire the concepts related to, say, multiplicative reasoning or additive reasoning. And we use those to sort, analyze, and decide how we're going to respond to evidence in student work. They're really maps for equity and access, and they help teachers understand that there are multiple right ways to do some mathematics, but they're not all equal in efficiency and sophistication.

Another piece they take away of significant value is we have an item bank full of hundreds of short tasks that are meant to add value to, say, a lesson you taught in your math program. So, you teach a lesson, and you decide, “What is the primary goal of this lesson?” And we all know, no matter what the program is you're using, that every lesson has multiple goals, and they're all in varying degrees of importance. So partly, picking an item in our item bank is about helping yourself think about what was the most critical piece of that lesson that I want to know about that's critical for my students to understand for success tomorrow.

**Mike**: So, one big idea that runs through your work with teachers is this concept called “unitizing.” And it struck me that whether we're talking about addition, subtraction, multiplication, fractions, that this idea just keeps coming back and keeps coming up. I'm wondering if you could offer a brief definition of unitizing for folks who may not have heard that term before.

**Beth**: Sure. It became really clear as we read the research and thought about where the struggles kids have, that unitizing is at the core of a lot of struggles that students have. So, unitizing is the ability to call something “1,” say, but know it's worth maybe 1 or 100 or a 1,000, or even 1/10. So, think about your numbers in a place value system. In our base ten system, when a 1 is in the tenths place, it's not worth 1 anymore; it's worth 1 of 10. And so that idea that the 1 isn't the value of its face value, but it's the value of its place in that system. So, base ten is one of the first big ways that kids have to understand unitizing.

Another kind of unitizing would be money. Money's a really nice example of unitizing. So, I can see one thing, it's called a nickel, but it's worth 5. And I can see one thing that's smaller, and it's called a dime, and it's worth 10. And so, the idea that 1 would be worth 5 and 1 would be worth 10, that's unitizing. And it's an abstract idea, but it provides the foundation for pretty much everything kids are going to learn from first grade on. And when you hear that kids are struggling, say, in third and fourth grade, I promise you that one of their fundamental struggles is a unitizing struggle.

**Mike**: Well, let's start where you all started when you began this work in OGAP. Let's start with multiplication. Can you talk a little bit about how this notion of unitizing plays out in the context of multiplication?

**Beth**: Sure. In multiplication, one of the first ways you think about unitizing is, say, in the example of 3 times 4. One of those numbers is a unit or a composite unit, and the other number is how many times you copy or iterate that unit. So, your composite unit in that case could be 3, and you're going to repeat or iterate it 4 times. Or your composite unit could be 4, and you're going to repeat or iterate it 3 times.

When I was in school, the teacher wrote “3 times 4” up on the board and she said, “3 tells you how many groups you have, and 4 tells you how many you put in each group.” But if you think about the process you go through when you draw that in that definition, you draw 1, 2, 3 circles, then you go 1, 2, 3, 4; 1, 2, 3, 4; 1, 2, 3, 4; 1, 2, 3, 4. And in creating that model, you never once thought about a unit, you thought about single items in a group. So, you counted 1, 2, 3, 4 three times, and there was never really any thought about the unit.

In a composite unit way of thinking about it, you would say, “I have a composite unit of 3, and I'm going to replicate it 4 times.” And in that case, every time, say, you stamped that — you had this stamp that was 3 — every time you stamped it, that one action would mean 3, right? 1 to 3, 1 to 3, 1 to 3, 1 to 3. So, in really early number work, kids think 1 to 1. When little kids are counting a small quantity, they'll count 1, 2, 3, 4. But what we want them to think about in multiplication is a many-to-1 action. When each of those quantities happens, it's not one thing. Even though you make one action, it's 4 things or 3 things, depending upon what your unit is. If you needed 3 times 8, you could take your 3-times-4 and add four more 3-times-4s to that. So, you have your four 3s and now you need four more 3s. And that allows you to use a fact to get a fact you don't know because you've got that unit and that understanding that it's not by 1, but by a unit.

When it gets to larger multiplication, we don't really want to be working by drawing by 1s, and we don't even want to be stamping 27 nineteen times, right? But it's a first step into multiplication, this idea that you have a composite unit, and in the case of 3 times 4 and 3 times 7, seeing that 3 is common. So, there's your common composite unit. You [need] four of them for 3 times 4, and you need seven of them for 3 times 7.

So, it allows you to see those relationships, which if you look at the standards, the relationships are the glue. So, it's not enough to memorize your multiplication facts. If you don't have a strong relationship understanding there, it does fall short of a depth of understanding.

**Mike**: I think it was interesting to hear you talk about that, Beth, because one of the things that struck me is some of the language that you used, and I was comparing it in my head to some of the language that I've used in the past. So, I know I've talked about 3 times 4, but I thought it was really interesting how you used “iterations of” or “duplicated” …

**Beth**: “Copies.”

**Mike**: … or “copies,” right? What you make me think is that those language choices are a little bit clearer. I can visualize them in a way that “3 times 4” is a little bit more abstract or obscure. I may be thinking of that wrong, but I'm curious how you think the language that you use when you're trying to get kids to think about composite numbers matters.

**Beth**: Well, I'll say this, that when you draw your 3 circles and count 4 dots in each circle, the result is the same model than if you thought of it as a unit of 3 stamped 4 times. In the end, the model looks the same, but the physical and mental process you went through is significantly different. So, you thought when you drew every dot, you were thinking about, “1, 1, 1, 1, 1, 1, 1.” When you thought about your composite unit copied or iterated, you thought about this unit being repeated over and over. And that changes the way you're even thinking about what those numbers mean.

And one of those big, significant things that makes addition different than multiplication when you look at equations is, in addition, those numbers mean the same thing. You have three things, and you have four things, and you're going to put them together. If you had 3 plus 4, and you changed that 4 to a 5, you're going to change one of your quantities by 1, impacting your answer by 1. In multiplication, if you have 3 times 4, and you change that 4 to a 5, your factor increases by 1, but your product increases by the value of your composite unit. So, it's a change of the other factor. And that is [a] significant change in how you think about multiplication, and it allows you to pave the way, essentially, to proportional reasoning, which is that replicating [of] your unit.

**Mike**: One of the things I'd appreciated about what you said was it's a change in how you're thinking. Because when I think back to Mike Wallus, classroom teacher, I don't know that I understood that as my work. What I thought of my work at that point in time was, “I need to teach kids how to use an algorithm or how to get an answer.” But I think where you're really leading is we really need to be attending to, “What's the thinking that underlies whatever is happening?”

**Beth**: Yes. And that's what our work is all about, is, “How do you give teachers a sort of lens into, or a look into, how kids are thinking and how that impacts whether they can employ more efficient and sophisticated relationships and strategies in their thinking?”

And it's not enough to know your multiplication facts. And the research is pretty clear on the fact that memorizing is difficult. If you're memorizing 100 single facts just by memory, the likelihood you're not going to remember some is high. But if you understand the *relationship* between those numbers, then you can use your 3 times 4 to get your 3 times 5 or your 3 times 8. So, the language that you use is important, and the way you leave kids thinking about something is important.

And this idea of the composite unit, it's thematic, right? It goes through fractions and additive and proportional, but it's not the only definition of multiplication. So, you've got to also think of multiplication as scaling — that comes later, but you also have to think of multiplication as area and as dimensions. But that first experience with multiplication has to be that composite-unit experience.

**Mike**: You've got me thinking already about how these ideas around unitizing that students can start to make sense of when they're multiplying whole numbers, that that would have a significant impact when they [start] to think about fractions or rational numbers. Can you talk a little bit about unitizing in the context of fractions, Beth?

**Beth**: Sure. The fraction standards have been most difficult for teachers to get their heads around because the way that the standards promote thinking about fractions is significantly different than the way most of us were taught fractions. So, in the standards and in the research, you come across the term “unit fraction,” and you can probably recognize the unitizing piece in the unit fraction. So, a unit fraction is a fraction where 1 is in the numerator, it's one unit of a fraction. So, in the case of 3/4, you have three of the one-fourths. Now, this is a bit of a shift in how we were taught. Most of us were taught, “Oh, we have 3/4. It means you have 4 things, but you only keep 3 of them,” right? We learned about the name “numerator” and the name “denominator.”

And, of course, we know in fractions, in particular, kids really struggle. Adults really struggle. Fractions are difficult because they seem to be a set of numbers that don't have anything in common with any other numbers. But once you start to think about unitizing and that composite unit, there's a standard in third grade that talks about “decompose any fraction into the sum of unit fractions.” So, in the case of 5/6, you would identify the unit fraction as 1/6, and you would have five of those 1/6. So, your unit fraction is 1/6, and you're going to iterate it or copy it or repeat it 5 times.

**Mike**: I can hear the parallels between the way you described this work with whole numbers, right? I have 1/4, and I've duplicated or copied that 5 times, and that's what 5/4 is. It feels really helpful to see the through line between how we think about helping kids think about composite numbers and multiplying with whole numbers to what you just described with unit fractions.

**Beth**: Yeah, and even the language, that language in fractions, is similar too. So, you talk about that 5 one-fourths. You decompose the five-fourths into 5 of the one-fourths, or you recompose those 5 one-fourths. This is a fourth-grade standard. You recompose those 5 one-fourths into 3 one-fourths (or three-fourths) and 2 one-fourths (or two-fourths). So, even reading a fraction like seven-eighths as “7 one-eighths,” helps to really understand what that seven-eighths means, and it keeps you from reading it as “7 out of 8.” Because when you read a fraction as “7 out of 8,” it sounds like you're talking about a whole number over another whole number.

And so again, that connection to the composite unit in multiplication extends to that composite unit or that unit fraction or unitizing in multiplication. And really, even when we talk about multiplying fractions, we talk about multiplying, say, a whole number times a fraction, “5 times one-fourth,” that would be the same as saying, “I'm going to repeat one-fourth 5 times,” as opposed to, we were told, “Put a 1 under the 5 and multiply across the numerator and multiply across the denominator.” But that didn't help kids really understand what was happening.

**Mike**: So, this progression of ideas that we've talked about from multiplication to fractions, what you've got me thinking about is, “What does it mean to think about unitizing with younger kids?” Particularly perhaps, kids in kindergarten, first, or second grade. I'm wondering how, or what you think educators could do to draw out the big ideas about units and unitizing with students in those grade levels?

**Beth**: Well, really we don't expect kindergartners to strictly unitize because it's a relatively abstract idea. The big focus in kindergarten is for a student to understand “four” means “4” — “4 ones” — and “seven” means “7 ones.”

But where we do unitize is in the use of our models in early grades. In kindergarten, the use of a 5-frame or a [10-frame]. So, let's use the [10-frame] to count by 10s: 10, 20, 30. And then, how many [10-frames] did it take us to count to 30? It took three. There's the beginning of your unitizing idea. The idea that we would say, “It took three of the [10-frames] to make 30” is really starting to plant that idea of unitizing: 3 can mean 30.

And in first grade, when we start to expose kids to coin values, time, telling time, you know. One of the examples we use is, “[When] was 1 minus 1 [equal to] 59?” And that was, “When you read for 1 hour and your friend read for 1 minute less than you, how long did they read?” So, all time is really a unitizing idea. So, all measures — measure conversion, time, money, and the big one in first grade is base ten.

And first grade and second grade have the opportunity to solidify strong base ten so that when kids enter third grade, they've already developed a concept of unitizing within the base ten system. In first grade, the idea that in a number like 78, the 7 is actually worth more than the 8, even though at face value, the 7 seems less than the 8. The idea that 7 is greater than the 8 in a number like 78 is unitizing. In second grade, when we have a number like 378, we can unitize that into 3 hundreds, 7 tens, and 8 ones — or 37 tens and 8 ones — and there's your re-unitizing. And that's actually a standard in second grade. Or 378 ones.

So, in first and second grade, really what teachers have to commit to is developing really strong, flexible base ten understanding. Because that's the first place kids have to struggle with, this idea of the face value of a number isn't the same as the place value of a number.

**Mike**: Yeah, yeah. So, my question is, would you describe that as the seeds of unitizing? Like, conserving? That's the thing that popped into my head, is, “Maybe that's what I'm actually starting to do when I'm trying to get kids to go from counting each individual 1 and naming the total when they say the last 1.”

**Beth**: So, there are some early number concepts that need to be solidified for kids to be able to unitize, right? So, conservation is certainly one of them. And we work on conservation all throughout elementary school. As numbers get larger, as they have different features to them, they're more complex. Conservation doesn't get fixed in kindergarten; it's just pre-K and K are the places where we start to build that really early understanding with small quantities. There's cardinality, hierarchical inclusion, those are all concepts that we focus on and develop in the earliest grades that feed into a child's ability to unitize.

So, the thing about unitizing that happens in the earliest grades is it's pretty informal. In pre-K and K, you might make piles of 10, you might count quantities. Counting collections is something we talk a lot about, and we talk a lot about the importance of counting in early math instruction — actually, all the way up through, but particularly in early math. And let's say you had a group of kids, and they were counting out piles of, say, 45 things, and they put them in piles of 10 and then a pile of 5, and they were able to go back and say, “10, 20, 30, 40, and 5.”

So, there's a lot that's happening there. So, one is, they're able to make those piles of 10, so they could count to 10. But the other one is, they have conservation. And the other one is, they have a rote-count sequence that got developed outside of this use of that rote-count sequence, and now they're applying that. So, [there are] *so many* balls in the air when a student can do something like that. The unitizing question would be, “You counted 45 things. How many piles of 10 did you have?” There's your unitizing question. And in kindergarten, there are students — even though we say it's not something we work on in kindergarten — there are certainly students who could look at that and say, “45 is four piles of 10 and 5 extra.” So, when I say we don't really do it in kindergarten, we have exposure, but it's very relaxed. It becomes a lot more significant in first and second grade.

**Mike**: You said earlier that teachers in first and second grade really have to commit to building a flexible understanding of base ten. What I wanted to ask you is, how would you describe that? And the reason I ask is, I also think it's possible to build an *inflexible* understanding of base ten. So, I wonder how you would differentiate between the kind of practices that might lead to a relatively inflexible understanding of base ten versus the kind of practices that lead to a more flexible understanding.

**Beth**: So, I think counting collections. I already said we talk a lot about counting collections and the primary training. Having kids count things and make groups of 10, focus on your 10 and your 5. We tell kindergarten teachers that the first month or two of school, the most important number you learn is 5. It's not 10 because our brain likes 5, and we can manage 5 easily. Our hand is very helpful. So, building that unit of 5 toward putting two 5s together to make a 10. I mean, I have a 3-year-old granddaughter, and she knows 5, and she knows that she can hold up both her hands and show me 10. But if she had to show me 7, she would actually start back at 1 and count up to 7. So, taking advantage of those units that are baked in already and focusing on them helps in the earliest grades.

And then really, I like materials to go into kids’ hands where they're doing the building. I feel like second grade is a great time to hand kids base ten blocks, but first grade is not. In first grade, kids should be snapping cubes together and building their own units because the more they build their own units of 5 or 10, the more it's meaningful and useful for them.

The other thing I'm going to say, and Bridges has this as a tool, which I really like, is they have dark lines at their 5s and 10s on their base ten blocks. And that helps. Even though people are going to say, “Kids can tell you it's 100,” they didn't build it. And so, there's a leap of faith there that is an abstraction that we take for granted. So, what we want is kids using those manipulatives in ways that they constructed those groupings, and that helps a lot. Also, no operations for addition and subtraction. You shouldn't be adding and subtracting without using base ten. So, adding and subtracting on a number line helps you practice not just addition and subtraction, but also base ten. So, because base ten’s so important, it could be taught all year long in second grade with *everything* you do. We call second grade the sweet spot of math because all the most important math can be taught together in second grade.

**Mike**: One of the things that you made me think about is something that a colleague said, which is this idea that 10 is simultaneously 10 ones and one unit of 10. And I really connect that with what you said about the need for kids to actually, physically *build* the units in first grade.

**Beth**: What you just said, that's unitizing. I can call this 10 ones, and I can call this 1 worth 10. And it's more in your face in the earliest grades because we often are very comfortable having kids make piles of 10 things or seeing the marks on a base ten block, say. Or snapping 10 Unifix cubes together, 5 red and 5 yellow Unifix cubes or something to see those two 5s inside that unit of 10. And then also there's your math hand, your 5s and your 10s and your [10-frames] are your 5s and your 10s. So, we take full advantage of that.

But as kids get older, the math that's going to happen is going to rely on kids already coming wired with that concept. And if we don't push it in those early grades by putting your hands on things and building them and sketching what you've just built and transferring it to the pictorial and the abstract in very strategic ways, then you could go a long way and look like you know what you're doing — but [you] don't really. And base ten is one of those ways, we think because kids can tell you the 7 is in the tens place, they really understand. But the reality is that's a low bar, and it probably isn't an indication a student really understands. There's a lot more to ask.

**Mike**: Well, I think that's a good place for my next question, which is to ask you what resources OGAP has available, either for someone who might participate in the training, other kinds of resources. Could you just unpack the resources, the training, the other things that OGAP has available, and perhaps how people could learn more about it or be in touch if they were interested in training?

**Beth**: Sure. Well, if they want to be in touch, they can go to ogapmathllc.com, and that's our website. And there's a link there to send us a message, and we are really good at getting back to people. We've written books on each of our four content strands. The titles of all those books are “A Focus on … .” So, we have *A Focus on Addition and Subtraction*, *A Focus on Multiplication and Division*, *A Focus on Fractions*, [and] *A Focus on Ratios and Proportions*, and you can buy them on Amazon. Our progressions are readily available on our website. You can look around on our website, and all our progressions are there so people can have access to those.

We do training all over. We don't do any open training. In other words, we only do training with districts who want to do the work with more than just one person. So, we contract with districts and work with them directly. We help districts use their math program. Some of the follow-up work we've done is [to] help them see the possibilities within their program, help them look at their program and see how they might need to add more. And once people come to training, they have access to all our resources, the item bank, the progressions, the training, the book, all that stuff.

**Mike**: So, listeners, know that we're going to add links to the resources that Beth is referencing to the show notes for this particular episode.

And, Beth, I want to just say thank you so much for this really interesting conversation. I'm so glad we had a chance to talk with you today.

**Beth**: Well, I'm really happy to talk to you, so it was a good time.

**Mike**: Fantastic.

**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.