ROUNDING UP: Math Trajectories for Young Learners *PART 2* with Dr. DeAnn Huinker & Dr. Melissa Hedges 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 learnin
