Latest / Rounding Up / Math Trajectories for Young Learners *PART 2* with Dr. DeAnn Huinker & Dr. Melissa Hedges

Transcript

Math Trajectories for Young Learners *PART 2* with Dr. DeAnn Huinker & Dr. Melissa Hedges · Rounding Up

0:03
Research confirms that early mathematics experiences play a more significant
0:07
role than we once imagined.
0:09
Studies suggest that specific number competencies in four-year-olds are strong
0:14
predictors of fifth grade mathematics success.
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So what does it look like to provide meaningful mathematical experiences for our
0:21
youngest learners? Today,
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we'll explore this question with DeAnn Hinker from UW Milwaukee and Melissa
0:27
Hedges from the Milwaukee Public Schools.
0:32
A note to our listeners.
0:33
This episode contains the second half of my conversation with DeAnn Hinker and
0:37
Melissa Hedges about math trajectories for young learners.
0:41
If you've not already listened to the first half of the conversation,
0:44
I encourage you to go back and give it a listen.
0:46
The second half of the conversation begins with DeAnn and Melissa discussing
0:50
practices that educators can use to provide students a more meaningful
0:54
experience with skip counting.
0:57
One of the things, Mike,
0:57
that I would add on that actually I just thought about is when you were talking
1:02
about the importance of us letting the children figure out how they want to
1:06
approach that task of organizing their count is it's coming from the child.
1:11
And Clements and Serama talk about the beautiful work about the trajectory is
1:15
that we see that the mathematics comes from the child and we can nurture that
1:19
along in developmentally appropriate ways.
1:22
The other idea that popped into my mind is it's kind of a parallel to when our
1:26
children get older and we want to teach them a way to add and a way to subtract,
1:31
and I'm going to show you how to do it and you follow my procedure.
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I'm going to show it. You follow my procedure.
1:37
We know that that's not best practice either.
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And so we're really looking at how do we grab onto that idea of number sense and
1:44
move forward with it in a way that's meaningful with children from as young
1:49
as one and two all the way up.
1:52
DeAnn,
1:53
I was going to ask a question to follow up on something that you said just now
1:57
when you said even something like skip counting should be done with quantities.
2:01
And you, I think anticipated the question I was going to ask,
2:05
which is what are the implications of this idea of connecting number and
2:10
quantity for processes that we have used in the past like route counting or skip
2:14
counting?
2:15
And I think what you're saying is we need to attend to those things that like
2:20
the counting sequence,
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we should not create an artificial barrier between speaking the words in
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sequence and quantity.
2:28
Am I reading you right or is there more nuance than I'm describing?
2:33
I think you're right on target, Mike. Connecting those things to quantity.
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And I mean, the one that's always salient for me is skip counting.
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Skip counting is such a route skill for so many children that they don't
2:47
realize when they go five, 10, 15 that they've actually have seen,
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oh, there's five, there's five more items, there's five more items.
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So it's making that connection to quantity for something like skip counting,
3:00
but also on the counting trajectory,
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then we start thinking about what's a 10 and what makes a 10 and what is
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30 and how many 10s are composing or embedded in
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that number 30? And again, it's not just rotely say three 10s.
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No, show me those objects. Can you make those 10s?
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Because sometimes we find disconnects. Kids will tell us things and then we say,
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"Can you show me?
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" And it doesn't match. So we continually start thinking about
3:31
quantities and putting with quantities. Let me add one more thing.
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In the counting trajectory, and this was very intentional for Melissa,
3:39
is when we have kids count,
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we'd like to give them like 31 or 32 counters to see
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whether that they can actually bridge that decade and to go
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beyond. The other thing that we did, so getting like beyond a 10,
3:56
also we find when kids get to the number 100,
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they stop. They just think that's the end. I got to a hundred,
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I'm going to stop. And then we say, "Oh,
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what would be the next number?" And some will say 110,
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some will say 200,
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some will give us something else that we find bridging 100 is
4:20
on the trajectory. And that's actually a really critical point. And again,
4:24
we want it with quantities with objects.
4:29
I really appreciate this part of the conversation because I think for a teacher
4:33
who's listening,
4:34
it helps really get to the specific types of details that would allow
4:39
them to create the kind of experiences that we think matter for children.
4:44
I do want to take a step back though and talk about what's going on for students
4:50
under the hood, so to speak. So as they're engaging in meaningful counting,
4:54
what are the cognitive processes that they're learning to coordinate?
4:58
This is Melissa.
4:59
So I'll start that one and then invite DeAnn to jump in as I work my way
5:03
through my thinking. One of the pieces that,
5:06
in addition to everything we talked about with all of the skills and ideas and
5:10
understanding that comes to bear when little ones count,
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one of the big pieces that we're starting to talk and learn about a whole lot
5:18
more is this idea of executive functioning.
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And so executive functioning are those skills that help us manage our attention,
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help us manage our behavior. They help us stay focused.
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They help us complete tasks, keep track of things.
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So hopefully as I'm saying this,
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what you have in your mind is a little one counting and you're thinking,
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oh my gosh, how do they know where to start? How do they know when to stop?
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How do they know when this has been counted with that hasn't been counted? What
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am I going to say next?
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All of that tends to be couched very strongly in this idea of executive
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functions. So when we watch kids count,
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we know that they're really drawing on those executive functions.
5:56
And it's actually a really beautiful marriage. So again,
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we're looking for kids to ... Are they able to stay on task?
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Can they keep track? Do they monitor themselves as they go?
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This happens a lot.
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If someone bumps into their collection and their collection gets a little shaky
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because their desk got moved or someone kicked a counter across the floor,
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do they remember where that goes and what that stood for in quantity?
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And for us,
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that really kind of comes down to some of those higher order skills and in
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particular, those ideas of the executive functions.
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So part of what we notice is that in particular with counting,
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though all of mathematics, much of what we do and ask kids to do,
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it takes planning, it takes self-monitoring,
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and it takes kind of a sense of control and agency over their work. We've talked
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a little bit about some of that other stuff in the way that it's the work of the
6:50
child, and that's why we will always ask teachers to step back and just watch,
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just watch what they do, just watch what they do,
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because it gives us insight into so many skills, understandings,
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and kind of where they're at.
7:03
Yeah. This is DeAnn. I was thinking of that same thing, Melissa,
7:06
about this is the work of the child, right? As adults,
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we're kind of prone sometimes to say, let me show you how to do it.
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But if we want to develop these executive function skills,
7:17
these ideas and cognitive abilities under the hood,
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we have to give children opportunities.
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They need the time to think about how to organize that collection.
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That's always a great one to kind of think about. As adults, we're like, "Well,
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just line them up." And it's like, oh no,
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that's actually huge for a child to realize lining them up or
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organizing them in some way is a strategy,
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just like we do with larger numbers. It's a strategy for little kids. So again,
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that work needs to come from the child and they need to do some trial
7:50
and error and adjustments in order to develop those things under the hood. And
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as adults, we can't take that opportunity away from children.
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We need to create the opportunities so they can explore more of their world and
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the quantitative world that we live in.
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Everything that we're talking about has some pretty major implications for
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instructional practice,
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but what I find myself thinking about is my own time teaching kindergarten.
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And when I reflect on that,
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I sometimes found myself falling into something that I would call a readiness
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trap.
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And what I mean by that is I had this notion that kids had to have a certain set
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of skills in place before they were ready to do something like counting a
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collection.
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And I think what you're going to tell me is that perhaps I had it backwards.
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Am I right?
8:39
So this is DeAnn and I'm thinking, well, maybe it's not so much backwards,
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but it's a different perspective.
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So Melissa and I really struggle with this concept of readiness,
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and that's because we really frame our work from a developmental perspective.
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And as we think about learning trajectories, that's what they are.
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Learning trajectories is a developmental view of children's learning.
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So what really changes the question for us. We don't ask the question,
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"Are children ready?" What we ask is, "Oh,
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where are children currently in their learning?" And then we can
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start at that spot and then think about the experiences that would
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help support the next step in their learning.
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So from a learning trajectory perspective,
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we really view differences in children's understanding and abilities as just
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different starting points. They're not deficits,
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nothing that needs to be remediated.
9:40
Kids are ready to learn every single day. It's really us as
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adults.
9:46
We have to reframe our preconceptions and train ourselves to
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really look at what children can do, not what they can't do.
9:56
And that's where learning trajectories are so powerful because they help
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us identify those starting points and then they help us as educators know
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more clearly what is the next developmental milestone that we should
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be working on with that child.
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So it's our responsibility to be ready for the children that come
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to us, not the other way around.
10:21
I really appreciate this idea of the progression as a series of starting points.
10:26
I think that's a really helpful framing device,
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and it certainly puts the work that we do in the kind of light that you're
10:32
advocating for. One of the other things I wanted to talk about is in the book,
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you all make reference to how important it is to develop a playful pedagogy.
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And I wonder if we could just try to talk about, what does that mean?
10:46
What might that look like in a classroom?
10:48
So this is Melissa.
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I think in any district or agency that's supporting young children,
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this is a very hot topic, the idea of play or playful pedagogy.
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And what I like to do is to think that we can use play as a teaching platform
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and not just as a break from learning.
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Play actually can kind of lay the foundation for a lot of those learning
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experiences.
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I think it's powerful because playful learning it nurtures important habits of
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mind that we can develop in some ways, but for our little ones,
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they develop very naturally through the idea of play.
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So we think about curiosity, creativity, problem solving, flexibility,
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persistence, all of that comes up as kids are playing.
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And so I think that both DeAnn and I would agree that the idea around
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playful pedagogy and mathematics learning trajectories really
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partner well because the trajectories help us see that mathematics develops over
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time based on experience and opportunity. So the trajectories don't replace
11:50
play so much as it does is it strengthens educators in
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recognizing during times when kids are playing or during those playful
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moments that an educator can have a stronger perspective or a more keen eye,
12:05
I guess, on what they're noticing with their children.
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And when we think about playful pedagogy, where we're headed is not free play,
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but this idea of guided play. So in guided play,
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the teacher's going to set up the environment,
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they'll have a learning goal in mind. So for example,
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if I'm working and deepening my understanding as a classroom teacher around the
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counting trajectory,
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I'm going to have an idea of where my children are on the trajectory and
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what questions might I pose during play or
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ponderings might I provide to the children during play.
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So it's not me taking over that time or the teacher taking over that time,
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but it's really supporting or pushing the learning through some subtle prompts
12:48
or some shared discoveries or maybe some purposeful questions. So for
12:52
example,
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if the kids are in the block area and they're building a tower or they just
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have blocks all over the floor, they're making a road, I might ask them,
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"How long is your road or how tall is your tower?" And let them
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kind of ponder with that. And then this is always a fun one,
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"What would happen if I put two more on or what would happen if your tower grew
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by two more blocks?
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Or what would happen if three of them fell off?" And really just engaging in
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some of those playful conversations not to take over the play,
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but to capitalize on the playful moment.
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I love that,
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particularly the notion of what if three fell off or what if I had four more
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blocks and I wanted to make it bigger or longer.
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It's a lovely way of organically injecting or assessing kids'
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thinking within the context as opposed to imposing a task in a way
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that it just has an entirely different feel to it. And yet at the same time,
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it's really informed by the trajectory in a way that helps it be like,
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"This is the right point for me to ask that particular question.".
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Exactly.
13:58
So I can kind of give an example if I'm thinking of maybe a five-year-old
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and so one of the levels of our counting trajectory is being able to do one more
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or one lesson and it really sits around that idea of hierarchical inclusion.
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So if the kids are playing and I know that that's where this child might need or
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this group of children are ready to take that next step,
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those are questions I can pose in a very, you're right, in a very low stress,
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not high stakes setting, and it's still very valuable information.
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Actually,
14:31
that's a great segue because I wanted to ask the two of you about some of the
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ways that teachers are using the learning trajectories and the assessment
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protocols that are found in the book to monitor their students' growth.
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So I wonder if you could say a little bit more about that.
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This is DeAnn. I'll start and then I'll pass it back to Melissa.
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So you ask us about the assessment protocols.
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So maybe we should explain what an assessment protocol is.
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One thing that we've done with the trajectories that were developed by
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Doug Clements and Julie Sarama, we've taken those trajectories,
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but as we're thinking about making them useful for teachers,
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we actually have developed some structured assessment protocols that are
15:11
aligned to the trajectory with task and prompts that we can
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use with children to help find those starting points.
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As I mentioned in the book, we have five assessment protocols in there,
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like one for counting, one for subatizing,
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one for adding and subtracting and so on.
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And then teachers can take these and use them to, "Let me ask this question.
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Oh, they did great there. Let me jump up a couple levels.
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Let me ask a question there,
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or maybe I want to back up to a previous level and ask so that we can kind of
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get a sense of those starting points for then building instruction." All right.
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And then Melissa,
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you can share how else teachers are using them in and out in the district.
15:52
I think one of the important aspects that I firmly believe in when a teacher
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approaches their teaching of mathematics through the lens of a learning
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trajectory,
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a mathematics learning trajectory is that it really does lay the foundation for
16:08
equitable teaching and learning opportunities.
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So not only does it lay the path for a developmental approach,
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it's also incredibly equitable in that we've looked at trajectories as
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identifying children's strengths. And in that way,
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it's not what they don't know,
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it's where are they and what are those experience that they need.
16:29
So it's not that somebody is never going to learn it. Again,
16:31
they need more experience and opportunity. And that's, I think,
16:35
probably been one of the biggest takeaways as we've looked at how we are using
16:39
trajectories here in the Milwaukee Public Schools,
16:42
in particular the counting trajectory.
16:44
So to get a really nice handle on where children are
16:48
developmentally, if we have, for example,
16:51
in a first grade classroom where they're moving into composing that unit of 10,
16:57
and we know that we've got kids that are struggling with cardinality,
17:00
even counting collections of one, two, three, four, five,
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we know that that's going to be a struggle. So what is it that we can do to
17:07
accelerate some of those learning opportunities and give more learning
17:11
opportunities for children so when they get to those big key
17:16
milestones, we have an idea of why they may be struggling.
17:20
And it's not that they can't, it's not that they won't,
17:22
it's not that they don't understand.
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They just need more experience and more opportunity and more guidance with that
17:27
work.
17:28
So that's one of the ways I think that has really allowed us to support our
17:32
teachers and have our teachers feel a great sense of autonomy in making
17:36
instructional decisions for their students,
17:39
that it's not the book is telling me to do this or this is telling me to do
17:42
that,
17:42
is here is something that's really honored a developmental approach to what
17:47
kids know and how can I take that then and apply that in my classroom with my
17:52
students? The other thing that it really has helped us do on a big broad level
17:57
is think about where do we want children to work towards by the end of
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three-year-old kindergarten or four-year-old kindergarten or five-year-old
18:05
kindergarten or first grade or second grade in a way that, again,
18:09
matches the developmental nature of children's mathematical
18:12
growth.What.
18:15
I really appreciate about what you shared is there's certainly the systems level
18:18
way of thinking about using this as a tool,
18:20
but I appreciate the fact that as an educator who might be reading the book,
18:25
I can also see directly into my own classroom practice and think about moves
18:29
that I can make to support students and also to understand where they are
18:34
and what comes next for them. That's super helpful.
18:38
Yeah. It's those small little moments.
18:40
It's really as just staying keyed in and tuned to those small moments.
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I'm going to ask a question at this point in the interview that I suspect is
18:48
difficult to narrow down an answer,
18:51
but I want to give it a try just because there's so much from my reading of the
18:55
book that was powerful. And at the same time,
18:58
I'm hoping that we can give people a chance to think about how they might start
19:02
to take action. So here's the question. If you were to say,
19:06
recommend one or two small scale practices for listeners who want to take the
19:11
ideas we're talking about and put them into action in their classrooms,
19:14
what might you recommend?
19:17
This is DeAnn. I'll get us started. First of all,
19:20
in developing this book really came out of our own work with teachers.
19:25
We have spent many,
19:27
many hours with teachers of the young grades and helping them to
19:31
improve their practice. And then with them,
19:34
we started learning about the trajectories and learning with them as they
19:39
started thinking about and applying these to the classrooms.
19:42
So a place to start for one's own professional learning and to deepen
19:47
your understanding is just to pick a trajectory and just read through it.
19:52
Ooh, what's happening with children that are one, two years old,
19:57
all the way up through children who are six, seven, and eight,
20:00
and just reading through this progression of levels and then
20:05
starting to deepen your knowledge of what are these kind of steps that we're
20:09
taking them through. And I'll use an example.
20:13
I think one of the biggest surprises I had for myself in this work
20:18
is I never really understood before studying the trajectories that
20:23
counting then leads to unitizing,
20:26
which then leads to looking at groups,
20:29
which then takes us to place value. And we talk about
20:34
counting as being the on- ramp to place value.
20:38
And I didn't really think about that connection until I just started
20:43
reading and studying the counting learning trajectory myself and thinking about
20:48
how do children go across all these levels.
20:52
I want to just jump in and say thank you for saying that because that's
20:56
something that I've been pondering as I've been listening to you all the way
20:59
back to when you talked about connecting skip counting to physical
21:04
quantities.
21:05
What struck me about that is that it allows me to start to imagine a unit that's
21:10
not just one, right? If I'm skipped counting by twos and I have two,
21:15
that's kind of the starting point for unitizing,
21:19
which I think the other thing that jumps out is that's actually eventually going
21:24
to lead to a deeper understanding of say multiplication.
21:28
So there's a lot in this that really when you understand what's going
21:33
on across that trajectory,
21:37
it really helps you understand what's critical about what kids are
21:42
learning and what also is critical about the kind of experience that I
21:47
as an educator want to make sure that I'm offering to students.
21:52
I'll just build on that a little bit, Melissa might too, is, wow,
21:57
counting is amazing.
21:59
And I think Melissa would say counting is the foundation of everything,
22:03
but that counting is much more than I think most of us as adults
22:09
realized that counting does take us through this idea
22:14
of making these groups and then thinking about units and units of 10,
22:18
understanding the place value or our base 10 system and understanding place
22:22
value.
22:23
It's just amazing when you really start to dig into a little deeper about
22:28
the math and the math learning, but how it goes across so many years.
22:35
Melissa, how about you?
22:36
Do you have a recommendation or do you want to build on something that DeAnn
22:39
shared?
22:40
I think what I'll recommend might be a build on.
22:43
One of the best ways that I would encourage folks to get into
22:48
understanding how a trajectory could be a really powerful tool in the classroom
22:53
is pick a child or one or two children and
22:58
give that trajectory a try. Just do it as written,
23:02
don't stray from it.
23:03
Just kind of give it a feel and see what it is that you're learning about your
23:07
children or child as they work through that trajectory. Because again,
23:11
it's those small moments when we're looking for those small transitions.
23:15
Like if a child,
23:16
one of the tasks in the counting trajectory assessment is counting a collection
23:20
of 31, and what do we notice? Do they try to count by twos?
23:24
Do they just count by ones?
23:26
Do they begin to do some of that grouping of 10 and 10 and 10 and
23:31
one more? One of the most fascinating things we found out as we've watched kids
23:35
work through the trajectory is when they get a collection,
23:38
a little bit of a larger collection, let's say 43,
23:42
they might begin to do some of that grouping and they'll go 10, 20, 30, 40,
23:47
and then they hit what we would say as 41 and they say 50, 67.
23:53
So I would encourage folks just to probably start with where DeAnn
23:58
started us,
23:59
which is understand the mathematics a little bit that you're looking for,
24:03
read through that trajectory,
24:04
get a feel for what that progression is looking like.
24:07
Maybe you start to very naturally think of a child that you know,
24:10
maybe they're kind of sitting here, maybe they're kind of sitting there,
24:14
and then give that trajectory a try and see what you learn about your kids.
24:18
The other thing that I would say is that we've got a lovely set of videos in the
24:23
book. There's over 50 videos of many of us doing little
24:28
tasks with children that will help illustrate what some of those assessment
24:32
tasks look like or what kids' thinking sounds like.
24:36
The other lovely part is that we've provided some activities as well.
24:40
So if you're thinking, "Oh,
24:41
somebody is at this level or I'm looking to expand my teaching of
24:46
number and quantity in my classroom," there's lots of really lovely tested,
24:50
tried and true tasks in there that a teacher could pick up and use tomorrow.
24:56
I think one of the things I'd like to do before we close is just give you all an
24:59
opportunity to share with listeners where they could go if they wanted to buy
25:03
the book and learn a little bit more. And then I'll also offer,
25:06
is there anything else that you think might be a reference point for teachers
25:10
who want to continue learning about the things you've shared today?
25:14
So the book we're talking about,
25:15
Math Trajectories for Young Learners is published by NCTM,
25:20
so the National Council of Teachers of Mathematics.
25:23
It can be purchased from them. They even have a nice little website,
25:29
nctm.org/trajjectories,
25:31
and that will take you right to a website that can give you access for ordering
25:36
the book. I will also point out that it's available certainly in paperback,
25:41
but it can also be purchased in digital formats.
25:43
So you can download a PDF or something to read on your Kindle or some other
25:48
reading device.
25:50
I think that's a great place to stop. DeAnn and Melissa,
25:53
thank you so much for your time.
25:54
It's really been a pleasure talking with you today.
25:56
Always a pleasure talking with you and thinking with you, Mike,
26:00
about children's learning.
26:01
Completely agree. Thank you again for having us, Mike.
26:07
This podcast is brought to you by the Math Learning Center and the Meyer Math
26:11
Foundation,
26:12
dedicated to inspiring and enabling all individuals to discover and develop
26:16
their mathematical confidence and ability.