Latest / Rounding Up / Understanding Units Coordination with Dr. Amy Hackenberg

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Understanding Units Coordination with Dr. Amy Hackenberg · Rounding Up

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The term units coordination describes the way students understand the
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organization of units as they approach problem solving situations
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and how their understanding influences their problem solving strategies.
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In this episode,
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we're talking with Amy Hackenberg from the University of Indiana about how
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educators can recognize and support students at different stages of
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units coordination.
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Welcome to the podcast Amy.
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I'm excited to be chatting with you today about units coordination.
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Well, thank you for having me. I'm very excited to be here, Mike,
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and to talk with you.
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Fantastic.
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So we've had previous guests come on the podcast and they've talked about the
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importance of unitizing, but for guests who haven't heard those episodes,
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I'm wondering if we could start by offering a definition for Unitizing,
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but then follow that up with an explanation of what unit's coordination
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is.
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Yeah, sure.
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So unitizing basically means to take a segment of experience as one thing,
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which we do all the time in order to even just relate to each other and tell
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stories about our day.
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I think of my morning as a segment of experience and can tell someone else about
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it.
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And we also do it mathematically when we construct number and it's a very long
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process,
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but children began by compounding sensory experiences like sounds and rhythms
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as well as visual and tactical experiences of objects into experiential units,
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experiential segments of experience that they can think about,
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like hearing bells ringing could be an impetus to take a single bong as a unit
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and later people construct units from what they imagine and even later on
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abstract units that aren't tied to any particular sensory material. It's again,
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long process, but once we start to do that, we construct arithmatic units,
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which we can think of as discreet ones.
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It all starts with unitizing segments of experience to create arithmetical items
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that we might count with whole numbers.
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What's really interesting about that is this notion of unitizing grows out
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of our lived experiences in a way that I think I hadn't thought about
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this notion that a unit of experience might be something like a morning or
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lunchtime.
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That's a fascinating way to think about even before we get to say composing
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sets of 10 into a unit,
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that these notions of a unit exists in our daily lives.
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And we make them out of our daily lives. That's how we make units.
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And what you said about a 10 is also important because as we progress onward,
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we do take more than one, one as a unit,
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like thinking of four flowers in a row in a garden as a single unit,
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as both one unit and as four little flowers means it has a dual meaning,
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at least we call it a composite unit at that point.
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That's a common term for that.
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So that's another example unitizing that is interest to teachers.
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Well, I'm excited to shift and talk about units coordination.
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How would you describe that?
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Yeah,
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so units coordination is a way for teachers and researchers to understand how
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children create units and organize units to interpret problem
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situations and to solve problems.
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So it originated in understanding how children construct whole number
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multiplication and division,
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but it has since expanded from just that to be thinking more broadly
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about units and structuring units and organizing and creating more units and how
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people do that in solving problems.
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Before we dig into the fine grain details of students' thinking,
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I wonder if you can explain the role that units coordination plays in students'
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journey through elementary mathematics and maybe how that matters in middle
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school and beyond middle school.
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So that's where a lot of the research is right now,
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especially at the middle school level and starting to move into high school.
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But units coordination was originally about trying to understand how
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elementary school children construct whole number multiplication and division,
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but it's also found to greatly influence elementary school children's
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understanding of fractions, decimals,
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measurement and on into middle school students understanding of those
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same ideas and topics, fractions ratios and proportional reasoning,
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rational numbers, writing and transforming algebraic equations,
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even combinatorial reasoning.
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So there's a lot of ways in which units coordination influences
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different aspects of children's thinking and is relevant in lots of different
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domains in the curriculum.
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Part of what's interesting for me is that I don't think I'm alone in saying that
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this big idea around units coordination sounds really new to me.
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It's not language that I learned in my pre-service work and also in my
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practice.
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So I think what's coming together for me is there's a larger set of
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ideas that flow through elementary school and into middle
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school and high school mathematics,
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and it's helpful to hear you talk about that from the youngest children who are
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thinking about the notion of units in their daily lives to the way that
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this notion of units and units coordination continues to play through elementary
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school into middle school and high school.
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Yeah,
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it's nice that you're noticing that because I do think that's something that's a
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strength of units coordination in it can be this unifying idea,
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although there's lots of variation and lots of variation in what you see with
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elementary students versus middle school students versus high school students
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versus even college students.
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Some of the research is on college students units coordination these days,
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but it is an interesting thread that can be helpful to think about in that way.
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Okay. With that in mind,
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let's introduce a context for units coordination and talk a little bit about the
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stages of student thinking.
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Yeah, so one way to understand some differences in how
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children up through say middle school students might coordinate units and
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engage in units.
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Coordination is to think about a problem and describe how solving it might
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happen. Here's a garden problem.
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Amaya is planting four pansies in a row. She plants 15 rows.
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How many pansies has she planted?
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There are three stages of units coordination. Broadly speaking,
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we've begun to understand more about the nuances there,
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but a stage refers to a set of ways of thinking that tend to fit
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together in how students understand and solve problems with whole numbers,
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fractions, quantities, and multiplicative relationships.
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It's sort of about a nexus of ideas and that we tend to see coming together and
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students don't usually think in a way that's characteristic of a different stage
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until they've made a significant change in their thinking like a big
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reorganization happens for them to move from one stage to the next. So students
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at stage one of units coordination are primarily in a one's world and their
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number sequence is not multiplicative. That's going to be hard to imagine,
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but they can take a group of ones as one thing,
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so they can make a composite unit and that means in the garden problem,
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they can take a row of pansies as one row as well as four little
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ones, and they can continue to do that over and over again.
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And so they can amass rows of four pansies and keep going.
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And what it usually looks like for them to solve the problem is they'll count by
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ones after any known skip counting patterns. So in this case they might be like,
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oh, I know four and eight, that's two rows,
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9, 10, 11, 12, that's three rows.
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Often using fingers or something to keep track or in some way to keep track and
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continuing to go up and get all the way barring counting errors to 60 pansies.
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And so for them the results 60 pansies is a composite unit.
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It's a unit of 60 units,
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but they don't maintain the structure that we see at all of the
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units of 60 as 15 fours. That's not something,
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even though they did track it and they're thinking they don't maintain that once
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they get to the 60,
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it's really just only a big composite unit of 60.
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So their view of the result is very different than an adult view might be.
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So the students at stage one can solve division problems,
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which means if they give some number of pansies and they're supposed to make
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rows of four, they can definitely do it, they can solve that,
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but they don't think of multiplication and division as inverses.
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So let me say what I mean by that. If they had this problem next,
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so Amaya's mom gave her 28 pansies,
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how many rows of four can she make? A student at stage one could solve that
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problem and they would be able to track fours over and over again and figure out
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that they got to seven fours once they get to 28.
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But then if immediately afterwards a teacher said, well,
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so how many pansies are there in seven rows of four,
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the student at stage one would start over and solve from the beginning.
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They wouldn't think that they had already solved it.
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And that's one telling sign of a student operating at stage one.
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And the reason is that the mental actions they engage in to do the segmenting or
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the tracking off of the fours and the 28 pansies are really different to them
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than what they use then the ways of thinking they use to create the seven rows
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of four and make the 28 that way. And so they don't recognize them as similar,
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so they feel like they have to engage in new problem solving to solve that
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problem. So to get back to the garden problem,
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students at stage two have a multiplicative number sequence,
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so they think of 60 as a one that they could repeat.
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Iterating is a term we often use.
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They could imagine it just being repeated over and over again.
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And this is a contrast to students at stage one who think of 60
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as like, oh,
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I got to have all 60 pansies there if I'm going to think about a number like 60.
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Whereas students at stage two do have a multiplicative number sequence and so
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they think, oh, I don't have to have all my 60 pansies.
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I can just think about one pansy and I just repeat it however many times I need
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to have however many pansies I want to imagine in my problem solving.
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So they anticipate 60 as one 60 times and that's obviously a
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great relief for kids who are dealing with big numbers. You can
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imagine it feels really onerous to think about a thousand if you feel like you
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have to have a thousand items in your mind, how could I possibly do that,
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but oh, I don't have to have a thousand,
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I can just have one and I can repeat it.
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That's a great economy efficiency in thinking that happens.
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So in the terms of the garden problem,
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students at stage two also have constructed a row as a thing to count,
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so a composite units one item as well.
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So four little items and they can amass fours just like I was talking about
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with students at stage one.
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But what they are also able to do is break apart fours as they go
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along. They might say, well,
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I've got four and four is eight and one more is 12 and one more is 16 and
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one more is 20 and one more is 24 and one more is 28.
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Maybe at that point they say, oh, let's see, I don't know what one more four is,
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but two more is 30 and then two more is 32.
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So they can take the row apart. They don't all do this, but they,
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they have the mental capabilities to do that because they're not right in the
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midst of making the coordination happen.
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They're sort of a little bit able to stand above the coordination and take their
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rows apart if they need to.
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It sounds like part of what happens at stage one is you might have a kid who
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potentially could count by fours for lack of a better way of saying it,
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and they might say, well, four and four, so two sets of fours, eight.
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And then at some point it kind of breaks down where that memorized list of what
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happens when you count by four. And then kids are back to saying, okay,
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12, 13, 14, 15, 16, and if you were watching this listeners,
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you would see that I stuck out four fingers and then I'm like, okay,
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so that's three fours and so on.
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And so I would see a student who might appear to be thinking about units,
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but tell me if I'm correct in thinking that it's more a function of that they
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know a set of numbers in accounting sequence for counting by four is.
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So students at any stage may vary in the skip counting patterns.
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They know,
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I call it knowing a skip counting pattern to know automatically like 4, 8, 12,
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16 or whatever it is.
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So you could have a student at stage two who doesn't know their skip counting
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patterns very well,
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and you also could have a student at stage two who counts by one.
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So that's the issue there as you can't always tell just from what you see if you
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have to do more than the test of what I'm saying.
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It's just to give a sense of the stages.
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But the main thing is the outer boundary of what they can do at stage two is
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they don't have to count by ones.
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They can do other things because of the fact that their composite units have
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this special feature where they're multiplicative in nature.
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I mean the fancy term for it is they have iterable units of one,
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but let me say a little bit more about what happens when they get to 60. So
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let's say a student at stage two, they've gotten up to 60,
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they're 60 pansies and they're 15 rows of four.
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They will think of the 60 as 15 fours as they make
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it. So we call it a three levels of unit structure.
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60 is a unit of 15 units each containing four little
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ones. They'll think about that way as they solve the problem,
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but as they continue to work further and add more pansies on or do a further
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extension of the problem,
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they wouldn't maintain that three levels of units structure that we see.
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So that's important because it has implications for how they can build
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from what they've done.
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How would you know that they hadn't maintained it?
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What might they say or do that would give you that cue?
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Well, so you see it most if let's say I say, oh, guess what,
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we got 12 more pansies and you're going to put 'em in rows of four.
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Can you put those on? And then they put 'em on. Okay,
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they find out it's 72 now. Okay, so how many rows are we talking here?
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It would be a new problem for them to figure that out.
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It wouldn't be like they would be able to maintain that, oh,
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I had 15 rows and then I now have the three more added on.
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Got you. Okay.
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So you see they're having to remake stuff as adult learners. We would think,
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oh, you should already know that that's 15 fours, right?
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But they'll have to redo that in solving an extension of the problem like I was
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talking about there. So students at stage three,
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they also can definitely take four as a row of one and also four
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pansies. They can arrive at 60 and view it as a unit of units,
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but they also can view it as a unit of 15 units each containing four,
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and they maintain that. So if they were asked a further problem like,
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Hey, we're going to rearrange this garden,
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we're going to actually three rows together at a time,
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can you do that and how many rows would you have and how many pansies in each
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row? And what would be the total? They'd be able to say, oh yeah, I can,
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let's see, put my three rows together,
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that's going to be 12 and then I'm going to end up with five of them.
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And now they've created 60 as a unit of five rows each containing
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12,
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and they can still think of 60 as a unit of 15 units each containing four
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or 15 rows each containing four.
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So they can switch between different unit structures.
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It doesn't mean they automatically know it without thinking it through,
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but they can do it and they can go back and forth.
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And that has great implications for anticipating and for solving
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division problems and seeing them as inverses of multiplication and a whole lot
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of stuff, proportional reasoning, fractions, lots of things.
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I think what's really interesting about this is I really appreciate you walking
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through the mental processes or maybe even the mental scripts that the kids
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might engage in to help see behind the curtain, for lack of a better word.
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Because what strikes me is that there is a point probably early in my teaching
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career where I would've attended and focused mostly on did they get the answer?
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And I think what you're helping remind me of is that it's the how,
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but there are particular ideas.
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And now I think I understand why the notion of units plural units coordination
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matters so much because a lot of what's happening is their ability to coordinate
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a unit made of units and then to be flexible with the
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units within that unit of units. Am I making proper sense of that, Amy?
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Yeah, for sure. That's great. That's exactly it.
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So the process and what units get created and how they get thought about and
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used is actually really,
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really important in trying to support kids' multiplicative thinking among other
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kinds of thinking too.
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I think this is a great segue because I suspected a lot of teachers are
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wondering about the kinds of tasks or practices or
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questions that they might use that could nudge students' thinking regarding
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units coordination.
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And I'm wondering what are some ideas you'd recommend for teachers as they're
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trying to think about how they assess but also advance their students' thinking
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when it comes to units coordination?
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That's a great question.
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And I mean the big response is have students engage in lots of reasoning with
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units, composite units, breaking apart numbers,
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strategically thinking about different solution pathways.
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So not just one solution pathway,
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but can you come up with multiple solutions for the problem,
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really sharing student solutions that involve breaking apart units.
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So if you're doing something like five sevens and finding out that kids are
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thinking of it as five fives and five twos, let's share that.
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How else could we break apart the five sevens, five fives and five twos?
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Why is that maybe helpful compared to other ways we might think about it?
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We might know five fives and five twos more easily than other ways of breaking
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it apart.
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And then even how are kids thinking about the five twos and the five fives and
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evaluating each of those.
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So basic things like that are super important. How many rows can we make with
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36 flowers with four per row? Thinking strategically about that.
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I know that five fours is 20 and I need 16 more flowers,
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so that's four fours because it's double two fours, so eight,
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so that means nine rows total.
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So I'm just kind of really briefly talking through,
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but posing these kinds of tasks and then asking for how students can break them
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up and think about them and presenting and making public that kind of
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thinking and reasoning. So valuing it in that way and sharing it.
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Same thing with lots of even more advanced multiplication problems.
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So for example, my daughter's in fourth grade right now,
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and so we've been working with her on 30 times 20 and doing something other than
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knowing three times two and then putting zeros on because she doesn't remember
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that. So to do 30 times 20, we asked her about 10 twenties, oh,
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she can figure that out. That's 200. And then can I iterate? Oh yeah,
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another 10 twenties, another 10 twenties.
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And then we did like 40 thirties, which was definitely harder.
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And so as part of the process of that,
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after she figured out 10 thirties when she was iterating her thirties,
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that was harder than iterating the twenties. She had to break apart numbers.
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When she got to 90 plus 30,
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she had to think about 90 plus 10 plus 20.
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So doing embedded,
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breaking apart of units with the prospect of trying to figure out a
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larger multiplication problem is super important. And interestingly,
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she could do 900 plus 300 and figure out that that was 900 and 100 to get a
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thousand and then 200 more. So that's additive reasoning,
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but it's the breaking apart of units and reconstituting them.
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That's what's really important in the process of solving multiplication and
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division problems. So that's my big thought about that.
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And the other thing is to not go to patterns too soon. I mean,
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this is related to what I just said about not thinking that I can just do
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three times two and then add zeros and count the zeros because that really
20:05
doesn't develop, it misses so much in what you can do with units.
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And so even if some kids do remember that and get the answer,
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they're really robbed of the experience that we're trying to give to my daughter
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of really thinking about, well,
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how can I figure out 40 thirties or 30 forties or 30 twenties?
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Right now I'm a big advocate of actually doing lots of counting by decade
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numbers because I feel like it's a way of really enhancing kids'
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work with larger multiplication.
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I've been sitting listening to you talk about this, Amy,
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and there are multiple things where I'm like, I need to ask her about this.
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I need to ask her about that. I need to ask about this other thing.
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So I'm going to ask you a couple follow ups.
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One of the things that is just an observation is the language you used when you
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were talking about your work with your daughter when the original task was 30
20:54
times 20 and you shifted the language to say 30 twenties,
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and then you step back even a little bit from there and you said, well,
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what's 30 tens? This language that you were using,
21:07
I wonder if you could be explicit about what you think that shift in
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language accomplishes.
21:13
Yeah, I've been also thinking a lot about this, so it's great. Yeah,
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one of the problems with multiplication notation is that it doesn't make clear
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anything about what the group is and what the number of groups you have are.
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And so just saying 30 times 20, I mean,
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you can think of that as 30 twenties or I can think of that as 2030s,
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but the language doesn't contain it,
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so it doesn't refer to the action I might do in thinking about how to
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actually figure it out.
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And kids have to bring a lot to the table then to really read that into that
21:43
multiplication notation. It's even more so with fractions.
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I can say more about that in a second.
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So I really am advocating with my pre-service teachers is that we speak in
21:52
iterative language with the multiplication. So we try to always say,
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I'm talking about five sevens, or I'm talking about seven fives, 30 forties,
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40 thirties,
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and then of course with the decade numbers knowing that we can go down to 10 of
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something and that that's easier to figure out, and then we can build on that.
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So like 10 twenties and then, oh,
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I'm going to need three of those 10 twenties to get to 30 twenties.
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Which really to some degree is helping them make meaningful sense of the
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associative property as well.
22:23
Right? Yeah, exactly. It's very mathematically rich. Unfortunately,
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it's not necessarily worked on a lot I am finding,
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and I think it's a real missed opportunity.
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I think there's a lot that kids could do with that that would really build
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strong meanings for multiplication and strong ideas of base 10 as well.
22:43
Yeah, absolutely.
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I think one of the things that I've been obsessed with lately is this notion of
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a nudge or small sized shifts in my practice that I can make.
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Part of what I'd like to mark for the audience is the shift in the language,
22:57
as you described, 30 twenties or five sevens,
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those are moves that a teacher could make to help clarify the fact that units
23:06
are involved and help students visualize with a bit more clarity what's going
23:10
on.
23:10
That feels like something that a teacher could take up and really have an impact
23:14
on students' understanding.
23:16
Yeah, I think so. I think it is something that is reasonable,
23:19
and what's nice is it also can flow right into fractions because then instead of
23:24
saying just three fifths, we say three, one fifths, four, one fifths, five,
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one fifths, six, one fifths, seven,
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it allows for fractions larger than one to have maybe more of an iterative
23:35
meaning. Not that that's a simple thing at all.
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That's a whole nother podcast we could do,
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but I've done a lot of research on that, but.
23:43
Well, I think you're hitting on something important though, Amy,
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because this notion of what is a unit fraction?
23:48
It's really four fifths is a group of four one fifths,
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and that's a critical understanding that I think often floats underneath
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students' understanding in ways that if we could make that clearer or help build
24:02
that understanding,
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that also has huge ramifications for what comes later in their mathematics
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learning experience.
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So I'm a big proponent of iterative language there as well.
24:14
You have me thinking about something else too,
24:16
which is the importance of context and having students deal with
24:21
measurement division problems specifically as a way to build their
24:25
understanding.
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And I know I'm using language right now for the audience that might not be super
24:29
clear,
24:30
but I'm wondering if you could talk a little bit about what measurement division
24:34
means in context and maybe why that would be valuable for students.
24:39
So in multiplication and division structures,
24:42
if we're talking about equal groups, there's always some number of equal groups,
24:45
some number in the equal group, so a size of the group,
24:49
and then a total number of items. With measurement division,
24:53
we know the total number of items, and we know the number of items in a group,
24:57
but we don't know the number of groups.
24:58
So my example of you've got 36 flowers and you want to put them in rows of four
25:03
would be a measurement division problem,
25:05
because we know that there are four in each row, and we know we have 36,
25:08
but we don't know how many rows we're going to make.
25:11
And so those are really nice to pair with work on equal groups
25:16
multiplication problems because they are very closely related. And for kids,
25:20
they can become closely related as they solve them and realize like, oh,
25:23
I can use my multiplication strategies to build up my fours and find out when I
25:28
get to 36 and oh, then I do, I know how many rows I've made.
25:32
So it's highly linked to what we're talking about here.
25:37
What I found myself thinking about is that in solving that problem,
25:40
one of the ways that a kid could do that is they're iterating a set, right?
25:44
So potentially they're iterating a set of fours multiple times,
25:47
and then they're finding out how many of those sets of four they have.
25:51
So I think part of what you're helping me think about is the way that the
25:55
structure of a measurement division problem maybe shines a
26:00
flashlight on this notion of groups and the number in each group,
26:04
and also some of the ideas you were talking about earlier with units
26:08
coordination.
26:09
Yeah, for sure.
26:10
And in terms of continuing the theme of using iterative language,
26:14
then when you get the result of that problem, nine rows, oh,
26:18
what does that nine mean? Oh, it means nine fours make 36.
26:21
So that's a meaning both for four times nine equals 36,
26:24
as well as 36 divided by four equals nine. So it's nice to emphasize that,
26:28
and as students build those meanings and have repeated work with
26:33
that kind of thing, they usually, often, we don't know all the mechanisms here,
26:38
but they usually come to be able to at least make that coordination in their
26:43
problem solving activity and ultimately make it so they can anticipate it,
26:46
like we're talking about with stage three.
26:49
One of the things that is really helpful is in the course of this interview,
26:54
we've talked a lot about what might the behavior of a student at stage one or
26:58
stage two or stage three not only look like,
27:02
but what might it mean for how they're thinking?
27:05
And I think what I'm really appreciating about this, Amy,
27:08
is there are a few practical things that an educator could do to support
27:13
students. One is iterative language as we've been talking about,
27:16
and the other is measurement division using a particular problem structure like
27:21
measurement division to shine a light on these parts that we think are really
27:25
important for kids to attend to if they're in fact going to make some of the
27:29
shifts that we're hoping for.
27:31
Yeah, for sure.
27:32
And then also exploring the boundaries of what the kids' strategies are and
27:36
asking for multiple solutions because you might see kids,
27:39
even students at stage three that might be counting by ones, and so you want to,
27:43
oh, can you solve that another way? Is there another way you can do it?
27:47
And so seeing what they see as possible,
27:49
what they're able to think about is also really important to support units
27:52
coordination.
27:54
Absolutely. Before we close,
27:56
I typically ask a question about resources or training or learning experiences
28:01
that would help someone who's listening continue learning or continue to think
28:05
about how they could take up these ideas in their practice. You particularly,
28:09
I know have written some work around this and I also suspect that you might have
28:13
some recommendations in terms of organizations that can help educators
28:17
really dig into these ideas if they saw that as something that was important for
28:22
their growth. Would you be willing to talk a little bit about resources,
28:25
organizations or even the types of experience you think support teachers as
28:30
they're making sense of all of this?
28:32
Yeah. Well, yes.
28:34
I was planning to talk about Integral at this point because Integral Numeracy
28:38
Solutions has a lot of great supportive materials for all this kind of work.
28:43
And everything that I'm talking about is something that is sort of built into
28:47
much of what they do for people who are unfamiliar. It's a bit,
28:50
council used to be a call to council of people who got together and have really
28:55
developed materials that are supportive of teachers working one-on-one to
29:00
support students who might be struggling as well as whole group instruction all
29:04
around developing strong number sense.
29:06
And it's very well developed set of materials both for classroom use as well as
29:11
for teacher development. And we,
29:14
meaning me and my two co-authors,
29:16
Andy Norton and Bob Wright wrote a book in the series for teachers on
29:21
fractions called Developing Fractions Knowledge. And that was published,
29:24
oh my gosh, nine years ago now.
29:26
So Andy and I are working on a second edition right now,
29:29
and in that book we address units coordination and talk about its usefulness for
29:34
teachers.
29:35
It's mostly though a book about fractions and about how units coordination is
29:39
relevant in trying to support students' fractions knowledge and to help assess
29:43
students' thinking and also promote their learning.
29:46
So that is one resource I can recommend on units coordination with a revision
29:51
coming in the next year.
29:53
That's fantastic. So I'll say for listeners,
29:55
we'll include a link to Integr Grow Numeracy Solutions if you want to check out
29:59
the organization.
30:00
And Amy will also add a link directly to the book so that if someone wanted to
30:05
dig in and explore that way they had the option.
30:08
I think that's probably a great place to stop,
30:09
although I certainly would love to continue.
30:12
I want to thank you so much for joining us.
30:13
It's really been a pleasure talking with you.
30:15
Yeah, likewise, Mike.
30:17
I've really enjoyed it and I look forward to further conversations.
30:25
This podcast is brought to you by the Math Learning Center and the Meyer Math
30:28
Foundation dedicated to inspiring and enabling all individuals to discover
30:33
and develop their mathematical confidence and ability.