Season 4 | Episode 15 – Dr. DeAnn Huinker & Dr. Melissa Hedges, Math Trajectories for Young Learners, Part 2

DeAnn Huinker & Melissa Hedges, Math Trajectories for Young Learners, Part 2

ROUNDING UP: SEASON 4 | EPISODE 15

Research confirms that early mathematics experiences play a more significant role than we once imagined. Studies suggest that specific number competencies in 4-year-olds are strong predictors of fifth grade mathematics success. So what does it look like to provide meaningful mathematical experiences for our youngest learners? 

Today, we'll explore this question with DeAnn Huinker from UW-Milwaukee and Melissa Hedges from the Milwaukee Public Schools.

BIOGRAPHY

Dr. DeAnn Huinker is a professor of mathematics education in the Department of Teaching and Learning and directs the University of Wisconsin-Milwaukee Center for Mathematics and Science Education Research. Dr. Huinker teaches courses in mathematics education at the early childhood, elementary, and middle school levels.

Dr. Melissa Hedges is a curriculum specialist who supports K–5 and K–8 schools for the Milwaukee Public Schools. 

RESOURCES

Learning Trajectories website, featuring the work of Doug Clements and Julie Sarama 

Math Trajectories for Young Learners book by DeAnn Huinker and Melissa Hedges

TRANSCRIPT

Mike Wallus: A note to our listeners: This episode contains the second half of my conversation with DeAnn Huinker and Melissa Hedges about math trajectories for young learners. If you've not already listened to the first half of the conversation, I encourage you to go back and give it a listen. The second half of the conversation begins with DeAnn and Melissa discussing practices that educators can use to provide students a more meaningful experience with skip-counting. 

Melissa Hedges: One of the things, Mike, that I would add on that actually I just thought about is when you were talking about the importance of us letting the children figure out how they want to approach that task of organizing their count is it's coming from the child. And Clements and Sarama talk about the beautiful work about the trajectory, [which] is that we see that the mathematics comes from the child and we can nurture that along in developmentally appropriate ways. 

The other idea that popped into my mind is it's kind of a parallel to when our children get older and we want to teach them a way to add and a way to subtract, and I'm going to show you how to do it and you follow my procedure. I'm going to show it. You follow my procedure. We know that that's not best practice either. And so we're really looking at, how do we grab onto that idea of number sense and move forward with it in a way that's meaningful with children from as young as 1 and 2 all the way up? 

Mike: DeAnn, I was going to ask a question to follow up on something that you said just now when you said even something like skip-counting should be done with quantities. And you, I think, anticipated the question I was going to ask, which is: What are the implications of this idea of connecting number and quantity for processes that we have used in the past, like rote counting or skip-counting? And I think what you're saying is we need to attend to those things that, like the counting sequence, we should not create an artificial barrier between speaking the words in sequence and quantity. Am I reading you right or is there more nuance than I'm describing? 

DeAnn Huinker: I think you're right on target, Mike. (laughs) Connecting those things to quantity. And I mean, the one that's always salient for me is skip-counting. Skip-counting is such a rote skill for so many children that they don't realize when they go, “5, 10, 15” that they actually have seen, “Oh, there's five [items], there's five more items, there's five more items.” So it's making that connection to quantity for something like skip-counting, but also on the counting trajectory, then we start thinking about, “What's a ten? And what makes a ten?” And, “What is 30? And how many tens are composing or embedded in that number 30?” And again, it's not just to rotely say, “3 tens.” No. “Show me those objects. Can you make those tens?” Because sometimes we find disconnects. Kids will tell us things and then we say, “Can you show me?” And it doesn't match. (laughs) So we continually start thinking about quantities and putting [objects] with quantities. 

Let me add one more thing. In the counting trajectory—and this was very intentional for Melissa—is when we have kids count, we'd like to give them like 31 or 32 counters to see whether [...] they can actually bridge that decade and to go beyond. The other thing that we did, so getting like beyond a ten, also we find when kids get to the number 100, they stop. They just think that's the end. I got to 100, I'm going to stop. And then we say, "Oh, what would be the next number?" And some will say 110, some will say 200, some will give us something else that we find bridging 100 is on the trajectory. And that's actually a really critical point. And again, we want it with quantities with objects. 

Mike: I really appreciate this part of the conversation because I think for a teacher who's listening, it helps really get to the specific types of details that would allow them to create the kind of experiences that we think matter for children. 

I do want to take a step back though and talk about what's going on for students under the hood, so to speak. So as they're engaging in meaningful counting, what are the cognitive processes that they're learning to coordinate? 

Melissa: This is Melissa. So I'll start that one and then invite DeAnn to jump in as I work my way through my thinking. 

One of the pieces that, in addition to everything we talked about with all of the skills and ideas and understanding that comes to bear when little ones count, one of the big pieces that we're starting to talk and learn about a whole lot more is this idea of executive functioning. And so executive functioning are those skills that help us manage our attention, help us manage our behavior. They help us stay focused. They help us complete tasks, keep track of things. So hopefully as I'm saying this, what you have in your mind is a little one counting and you're thinking, “Oh my gosh, how do they know where to start?” “How do they know when to stop?” “How do they know when this has been counted with that hasn't been counted?” “What am I going to say next?” All of that tends to be couched very strongly in this idea of executive functions. So when we watch kids count, we know that they're really drawing on those executive functions. And it's actually a really beautiful marriage. So again, we're looking for kids to—are they able to stay on task? Can they keep track? Do they monitor themselves as they go? If someone—this happens a lot—if someone bumps into their collection and their collection gets a little shaky because their desk got moved or someone kicked a counter across the floor, do they remember where that goes and what that stood for in quantity? And for us, that really kind of comes down to some of those higher order skills and in particular, those ideas of the executive functions. 

So part of what we notice is that in particular with counting, though all of mathematics, much of what we do and ask kids to do, it takes planning, it takes self-monitoring, and it takes kind of a sense of control and agency over their work. We've talked a little bit about some of that other stuff in the way that it's the work of the child, and that's why we will always ask teachers to step back and just watch, just watch what they do, just watch what they do, because it gives us insight into so many skills, understandings, and kind of where they're at. 

DeAnn: Yeah. This is DeAnn. I was thinking of that same thing, Melissa, about this is the work of the child, right? As adults, we're kind of prone sometimes to say, “Let me show you how to do it.” But if we want to develop these executive function skills, these ideas and cognitive abilities under the hood, we have to give children opportunities. They need the time to think about how to organize that collection. That's always a great one to kind of think about. As adults, we're like, "Well, just line them up." And it's like, oh no, that's actually huge for a child to realize lining them up or organizing them in some way is a strategy, just like we do with larger numbers. It's a strategy for little kids. So again, that work needs to come from the child and they need to do some trial and error and adjustments in order to develop those things under the hood. And as adults, we can't take that opportunity away from children. We need to create the opportunities so they can explore more of their world and the quantitative world that we live in. 

Mike: Everything that we're talking about has some pretty major implications for instructional practice, but what I find myself thinking about is my own time teaching kindergarten. And when I reflect on that, I sometimes found myself falling into something that I would call a readiness trap. And what I mean by that is I had this notion that kids had to have a certain set of skills in place before they were ready to do something like counting a collection. And I think what you're going to tell me is that perhaps I had it backwards. Am I right? 

DeAnn: So this is DeAnn and I'm thinking, well, maybe it's not so much backwards, but it's a different perspective. So Melissa and I really struggle with this concept of readiness, and that's because we really frame our work from a developmental perspective. And as we think about learning trajectories, that's what they are. Learning trajectories is a developmental view of children's learning. So what really changes the question for us. We don't ask the question, “Are children ready?” What we ask is, “Oh, where are children currently in their learning?” And then we can start at that spot and then think about the experiences that would help support the next step in their learning. So from a learning trajectory perspective, we really view differences in children's understanding and abilities as just different starting points, OK? They're not deficits, nothing that needs to be remediated. Kids are ready to learn every single day. 

It's really us as adults. We have to reframe our preconceptions and train ourselves to really look at what children can do, not what they can't do. And that's where learning trajectories are so powerful because they help us identify those starting points and then they help us as educators know more clearly what is the next developmental milestone that we should be working on with that child. So it's our responsibility to be ready for the children that come to us, not the other way around. 

Mike: I really appreciate this idea of the progression as a series of starting points. I think that's a really helpful framing device, and it certainly puts the work that we do in the kind of light that you're advocating for. 

One of the other things I wanted to talk about is in the book [Math Trajectories for Young Learners], you all make reference to how important it is to develop a playful pedagogy. And I wonder if we could just try to talk about, “Well, what does that mean? What might that look like in a classroom?”

Melissa: So this is Melissa. I think in any district or agency that's supporting young children, this is a very hot topic, the idea of play or playful pedagogy. And what I like to do is to think that we can use play as a teaching platform and not just as a break from learning. Play actually can kind of lay the foundation for a lot of those learning experiences. I think it's powerful because playful learning, it nurtures important habits of mind that we can develop in some ways, but for our little ones, they develop very naturally through the idea of play. So we think about curiosity, creativity, problem solving, flexibility, persistence, all of that comes up as kids are playing. And so I think that both DeAnn and I would agree that the idea around playful pedagogy and mathematics learning trajectories really partner well because the trajectories help us see that mathematics develops over time based on experience and opportunity. So the trajectories don't replace play so much as [...] strengthen educators in recognizing during times when kids are playing or during those playful moments that an educator can have a stronger perspective or a more keen eye, I guess, on what they're noticing with their children. 

And when we think about playful pedagogy, where we're headed is not free play, but this idea of guided play. So in guided play, the teacher's going to set up the environment, they'll have a learning goal in mind. So for example, if I'm working and deepening my understanding as a classroom teacher around the counting trajectory, I'm going to have an idea of where my children are on the trajectory and what questions might I pose during play or ponderings might I provide to the children during play. So it's not me taking over that time or the teacher taking over that time, but it's really supporting or pushing the learning through some subtle prompts or some shared discoveries or maybe some purposeful questions. So, for example, if the kids are in the block area and they're building a tower or they just have blocks all over the floor, they're making a road, I might ask them, “How long is your road?” or “How tall is your tower?” and let them kind of ponder with that. And then, this is always a fun one, “What would happen if I put two more on?” or “What would happen if your tower grew by two more blocks? or “What would happen if three of them fell off?” And really just engaging in some of those playful conversations—not to take over the play, but to capitalize on the playful moment. 

Mike: I love that, particularly the notion of, “What if three fell off?” or “What if I had four more blocks and I wanted to make it bigger or longer?” It's a lovely way of organically injecting or assessing kids' thinking within the context as opposed to imposing a task in a way that it just has an entirely different feel to it. And yet at the same time, it's really informed by the trajectory in a way that helps it be like, “This is the right point for me to ask that particular question.”

Melissa: Exactly. So I can kind of give an example if, I'm thinking of maybe a 5-year-old and so, one of the levels of our counting trajectory is being able to do 1 more or 1 less, and it really sits around that idea of hierarchical inclusion. So if the kids are playing and I know that that's where this child might need or this group of children are ready to take that next step, those are questions I can pose in a very—you're right—in a very low-stress, not-high-stakes setting, and it's still very valuable information. 

Mike: Actually, that's a great segue because I wanted to ask the two of you about some of the ways that teachers are using the learning trajectories and the assessment protocols that are found in the book to monitor their students' growth. So I wonder if you could say a little bit more about that. 

DeAnn: This is DeAnn. I'll start and then I'll pass it back to Melissa. 

So, you ask us about the assessment protocols. So maybe we should explain what an assessment protocol is. One thing that we've done with the trajectories that were developed by Doug Clements and Julie Sarama, we've taken those trajectories, but as we're thinking about making them useful for teachers, we actually have developed some structured assessment protocols that are aligned to the trajectory with [tasks] and prompts that we can use with children to help find those starting points. As I mention in the book, we have five assessment protocols in there, like one for counting, one for subitizing, one for adding and subtracting and so on. And then teachers can take these and use them to [say], “Let me ask this question. Oh, they did great there. Let me jump up a couple levels. Let me ask a question there.” Or maybe I want to back up to a previous level and ask so that we can kind of get a sense of those starting points for then building instruction.

All right, and then Melissa, you can share how else teachers are using them in and out in the district. 

Melissa: I think one of the important aspects that I firmly believe in when a teacher approaches their teaching of mathematics through the lens of a learning trajectory, a mathematics learning trajectory, is that it really does lay the foundation for equitable teaching and learning opportunities. So not only does it lay the path for a developmental approach, it's also incredibly equitable in that we've looked at trajectories as identifying children's strengths. And in that way, it's not what they don't know, it's, “Where are they, and what are those [experiences] that they need?” So it's not that somebody is never going to learn it. Again, they need more experience and opportunity. And that's, I think, probably been one of the biggest takeaways as we've looked at how we are using trajectories here in the Milwaukee Public Schools, in particular the counting trajectory. So to get a really nice handle on where children are developmentally, if we have, for example, in a first grade classroom where they're moving into composing that unit of 10, and we know that we've got kids that are struggling with cardinality, even counting collections of one, two, three, four, five [objects], we know that that's going to be a struggle. So what is it that we can do to accelerate some of those learning opportunities and give more learning opportunities for children so when they get to those big key milestones, we have an idea of why they may be struggling? And it's not that they can't; it's not that they won't; it's not that they don't understand. They just need more experience and more opportunity and more guidance with that work. 

So that's one of the ways I think that has really allowed us to support our teachers and have our teachers feel a great sense of autonomy in making instructional decisions for their students. That it's not, “The book is telling me to do this or this is telling me to do that.” It’s, “Here is something that's really honored a developmental approach to what kids know, and how can I take that then and apply that in my classroom with my students?” 

The other thing that it really has helped us do on a big broad level is think about, “Where do we want children to work towards by the end of 3-year-old kindergarten or 4-year-old kindergarten or 5-year-old kindergarten or first grade or second grade in a way that, again, matches the developmental nature of children's mathematical growth?”

Mike: What I really appreciate about what you shared is there's certainly the systems level way of thinking about using this as a tool, but I appreciate the fact that as an educator who might be reading the book, I can also see directly into my own classroom practice and think about moves that I can make to support students and also to understand where they are and what comes next for them. That's super helpful. 

Melissa: Yeah. It's those small little moments. It's really as, just staying keyed in and tuned to those small moments. 

Mike: I'm going to ask a question at this point in the interview that I suspect is difficult to narrow down an answer, but I want to give it a try just because there's so much from my reading of the book that was powerful. And at the same time, I'm hoping that we can give people a chance to think about how they might start to take action.

So here's the question: If you were to, say, recommend one or two small-scale practices for listeners who want to take the ideas we're talking about and put them into action in their classrooms, what might you recommend? 

DeAnn: This is DeAnn. I'll get us started. 

First of all, [...] developing this book really came out of our own work with teachers. We have spent many, many hours with teachers of the young grades and helping them to improve their practice. And then with them, we started learning about the trajectories and learning with them as they started thinking about and applying these to the classrooms. So a place to start for one's own professional learning and to deepen your understanding is just to pick a trajectory and just read through it. “Ooh, what's happening with children that are 1-, 2-years-old, all the way up through children who are 6, 7, and 8.” And just reading through this progression of levels and then starting to deepen your knowledge of what are these kind of steps that we're taking them through. 

And I'll use an example. I think one of the biggest surprises I had for myself in this work is I never really understood before studying the trajectories that counting then leads to unitizing, which then leads to looking at groups, which then takes us to place value. And we talk about counting as being the on-ramp to place value. And I didn't really think about that connection until I just started reading and studying the counting learning trajectory myself and thinking about, “How do children go across all these levels?”

Mike: I want to just jump in and say thank you for saying that because that's something that I've been pondering as I've been listening to you all the way back, DeAnn, to when you talked about connecting skip-counting to physical quantities. What struck me about that is that it allows me to start to imagine a unit that's not just 1, right? If I'm skip-counting by 2s, and I have 2, that's kind of the starting point for unitizing, which—I think the other thing that jumps out is, that's actually eventually going to lead to a deeper understanding of, say, multiplication. So there's a lot in this that really when you understand what's going on across that trajectory, it really helps you understand what's critical about what kids are learning and what also is critical about the kind of experience that I as an educator want to make sure that I'm offering to students. 

DeAnn: I'll just build on that a little bit. (laughs) Melissa might too, is, “Wow, counting is amazing.” And I think Melissa would say, “Counting is the foundation of everything.” But that counting is much more than I think most of us as adults realized. That counting does take us through this idea of making these groups and then thinking about units and units of 10, understanding the place value or our base ten system and understanding place value. It's just amazing when you really start to dig into a little deeper about the math, and the math learning, but how it goes across so many years.

Mike: Melissa, how about you? Do you have a recommendation or do you want to build on something that DeAnn shared?

Melissa: I think what I'll recommend might be a build on. One of the best ways that I would encourage folks to get into understanding how a trajectory could be a really powerful tool in the classroom is pick a child, or one or two children, and give that trajectory a try. Just do it as written, don't stray from it. Just kind of give it a feel, and see what it is that you're learning about your children or child as they work through that trajectory. Because again, it's those small moments when we're looking for those small transitions. Like, if a child—one of the tasks in the counting trajectory assessment is counting a collection of 31. And what do we notice? Do they try to count by 2s? Do they just count by 1s? Do they begin to do some of that grouping of 10 and 10 and 10 and 1 more? One of the most fascinating things we found out as we've watched kids work through the trajectory is when they get a collection, a little bit of a larger collection, let's say 43, they might begin to do some of that grouping and they'll go, “10, 20, 30, 40.” And then they hit what we would say as “41” and they say “50, 60, 70”. (laughs) 

So I would encourage folks just to probably start with where DeAnn started us, which is understand the mathematics a little bit that you're looking for, read through that trajectory, get a feel for what that progression is looking like. Maybe you start to very naturally think of a child that you know, maybe they're kind of sitting here, maybe they're kind of sitting there, and then give that trajectory a try, and see what you learn about your kids. 

The other thing that I would say is that we've got a lovely set of videos in the book. There's over 50 videos of many of us doing little tasks with children that will help illustrate what some of those assessment tasks look like or what kids' thinking sounds like. 

The other lovely part is that we've provided some activities as well. So if you're thinking, “Oh, somebody is at this level or I'm looking to expand my teaching of number and quantity in my classroom,” there's lots of really lovely tested, tried-and-true tasks in there that a teacher could pick up and use tomorrow. 

Mike: I think one of the things I'd like to do before we close is just give you all an opportunity to share with listeners where they could go if they wanted to buy the book and learn a little bit more. And then I'll also offer, is there anything else that you think might be a reference point for teachers who want to continue learning about the things you've shared today? 

DeAnn: So the book we're talking about, Math Trajectories for Young Learners is published by NCTM, so the National Council of Teachers of Mathematics. It can be purchased from them. They even have a nice little website, nctm.org/trajectories, and that will take you right to a website that can give you access for ordering the book. I will also point out that it's available certainly in paperback, but it can also be purchased in digital formats. So you can download a PDF or something to read on your Kindle or some other reading device.

Mike: I think that's a great place to stop. DeAnn and Melissa, thank you so much for your time. It's really been a pleasure talking with you today. 

DeAnn: Always a pleasure talking with you and thinking with you, Mike, about children's learning. 

Melissa: Completely agree. Thank you again for having us, Mike. 

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.

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