Latest / Rounding Up / Math Trajectories for Young Learners *PART 1* with Dr. DeAnn Huinker & Dr. Melissa Hedges

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Math Trajectories for Young Learners *PART 1* 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.
• 0:17
  So what does it look like to provide meaningful mathematical experiences for our
• 0:21
  youngest learners? Today,
• 0:23
  we'll explore this question with Deanne Hinker from UW Milwaukee and Melissa
• 0:27
  Hedges from the Milwaukee Public Schools.
• 0:33
  Welcome back to the podcast, Deanne and Melissa.
• 0:35
  You have both been guests previously.
• 0:36
  It is a pleasure to have both of you back with us again to discuss your new
• 0:40
  book, Math Trajectories for Young Learners.
• 0:44
  Thank you for having us. We're both very excited to be here.
• 0:47
  Yes, I concur. Good to see you and be here again.
• 0:51
  So Deanne,
• 0:52
  I think what I'd like to do is just start with an important grounding question.
• 0:55
  What's a trajectory?
• 0:57
  That's exactly where we need to start, right?
• 1:00
  So as I think about what our learning trajectories,
• 1:03
  I always envision them as these roadmaps of children's mathematical development.
• 1:08
  And what makes them so compelling is that these learning pathways are
• 1:13
  highly predictable. We can see where children are in their learning,
• 1:18
  and then we can be more intentional in our teaching when we know where they are
• 1:22
  currently at.
• 1:24
  But if I kind of think about the development of learning trajectories,
• 1:27
  they really are based on weaving together insights from research and practice to
• 1:32
  give us this clear picture of the typical development of children's learning.
• 1:38
  And as we always think about these learning trajectories,
• 1:40
  there are three main components. The first component is a mathematical goal.
• 1:46
  This is the big ideas of math that children are learning. For example, counting,
• 1:51
  subatizing, decomposing shapes.
• 1:53
  The second component of a learning trajectory are developmental
• 1:58
  progressions. This is really the heart of a trajectory.
• 2:01
  And the progression lays out a sequence of distinct levels of
• 2:06
  thinking and reasoning that grow in mathematical sophistication.
• 2:11
  And then the third component are activities and tasks that align to
• 2:17
  and support children's movement along that particular trajectory.
• 2:22
  Now,
• 2:22
  it's really important that we point out the learning trajectories that we use in
• 2:26
  our work with teachers and children were developed by Doug Clements and Julie
• 2:31
  Sarama.
• 2:32
  So we have taken their trajectories and worked to make them more usable and
• 2:37
  applicable for teachers in our area.
• 2:41
  So what Doug and Julie did is they mapped out children's learning starting at
• 2:45
  birth when children are just borns, one-year-olds, two-year-olds,
• 2:51
  and they mapped it out up till about age eight. And right now,
• 2:56
  last count, they have about 20 learning trajectories,
• 3:01
  and they're in different topics like number, operations, geometry,
• 3:06
  and measurement. And we have to put in a plug. They have a wonderful website.
• 3:11
  It's learningtrajectories.org.
• 3:14
  We go there often to learn more about the trajectories and to get ideas for
• 3:18
  activities and tasks. Now,
• 3:21
  we're talking about this new book we have on math trajectories for young
• 3:25
  children. And in the book,
• 3:27
  we actually take a deep dive into just four of the trajectories.
• 3:32
  We look at counting, subatizing, composing numbers,
• 3:37
  and adding and subtracting. So back to your original question,
• 3:42
  what are they?
• 3:43
  Learning trajectories are highly predictable roadmaps of children's math
• 3:48
  learning that we can use to inform and support developmentally
• 3:52
  appropriate instruction.
• 3:55
  That's an incredibly helpful starting point.
• 3:58
  And I want to ask a follow-up just to get your thinking on the
• 4:03
  record.
• 4:04
  I wonder if you have thoughts about how you imagine educators could or should
• 4:09
  make use of the trajectories.
• 4:11
  This is Melissa. I'll pick up with that question.
• 4:14
  So I'll piggyback on Deanne's response and thinking around this highly
• 4:19
  predictable nature of a trajectory as a way to ground my first
• 4:23
  comment and that we want to always look at a trajectory as a tool.
• 4:28
  So it's really meant as an important tool to help us understand where a child is
• 4:33
  and their thinking right now,
• 4:35
  and then what those next steps might be to push for some deeper
• 4:40
  mathematical understanding.
• 4:42
  So the first thing that when we work with teachers that we like to keep in mind,
• 4:46
  and one of the things that actually draw teachers to the trajectories is that
• 4:50
  they're strength-based. So it's not what a child can't do.
• 4:54
  It's what a child can do right now based off of experience and
• 4:59
  opportunity that they've had.
• 5:01
  We also really caution against using our trajectories as a way to kind of
• 5:06
  pigeonhole kids or rank kids or label kids,
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  because what we know is that as children have more experience and opportunity,
• 5:13
  they grow and they learn and they advance along that trajectory. So really it's
• 5:18
  a tool that's incredibly powerful when in the hands of a teacher that
• 5:22
  understands how they work to be able to think about where are the children right
• 5:27
  now in their classroom and what can they do to advance them.
• 5:30
  And I think the other point that I would emphasize other than what moves
• 5:35
  children along is experience and opportunity.
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  Children are going to be all over on the trajectory that's been our experience
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  and they're in the same classroom.
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  And it's not that some can't and some won't and some can,
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  it's just some need more experience and some need more opportunity.
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  So it's really opened up the door many ways to view a more equitable approach to
• 5:55
  mathematics instruction. The other thing that I would say is,
• 5:59
  and Deanne and I had big conversations about this when we were first using the
• 6:03
  trajectories,
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  is do we look at the ages? So the trajectories that Clemens and Serama develop
• 6:09
  do have age markers on them. And we were a bit back and forth on do we use them,
• 6:13
  do we not,
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  knowing that mathematical growth is meant to be viewed through a developmental
• 6:19
  lens.
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  So we had them on and then we had them off and then we shared them with teachers
• 6:25
  and many of our projects and the teachers were like, "No, no, no,
• 6:27
  put the ages back on. Trust us.
• 6:29
  We'll use them well." And so the ages are back onto the
• 6:34
  trajectories.
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  And what we've noticed is that they really do help us understand how to take
• 6:39
  either intentional steps forward or intentional steps back,
• 6:43
  depending on what kids are showing us on that trajectory.
• 6:47
  The other spot that I would maybe put a plugin for on where we could use a
• 6:51
  trajectory and what would be an appropriate use for it would be for our special
• 6:55
  educators out there and to really start to use them to support clear,
• 6:59
  measurable IEP goals grounded in a developmental progress. So that's
• 7:04
  kind of what our rule of thumb would be around a should and shouldn't with the
• 7:08
  trajectories.
• 7:10
  That's really helpful.
• 7:11
  You mentioned the notion of experiences and opportunities being critical.
• 7:16
  So I wanted to take perhaps a bit of a detour and talk about what
• 7:20
  research tells us about the impact of early mathematics experiences,
• 7:25
  what impact that has on children.
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  I wonder if you could share some of the research that you cite in the book with
• 7:30
  our listeners.
• 7:31
  Sure. This is Deanne and in the book we cite research throughout
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  all of the chapters and aligned to all of the different trajectories.
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  But as we think about our work,
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  there really are a few studies that we anchor in always as we think about
• 7:47
  children's learning.
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  And their research evidence is really clear that early mathematics
• 7:54
  matters.
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  The math that children learn in these early years in pre-kindergarten,
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  kindergarten, first grade, I mean, we're talking four, five, six year olds,
• 8:04
  seven year olds,
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  that their math learning is really more important than a lot of people think it
• 8:10
  is. Okay?
• 8:12
  So as we think about these kind of anchor studies that we look at,
• 8:16
  one of the major studies in this area is from Greg Duncan and his colleagues,
• 8:21
  and there was a study published in 2007.
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  And what they did is they examined data from thousands of children
• 8:30
  drawing information from six large scale studies,
• 8:34
  and they found that the math knowledge and abilities of four and five year olds
• 8:40
  was the strongest predictor of later achievement. I mean,
• 8:44
  four and five year olds, that's just as they're starting school.
• 8:47
  Wow.
• 8:48
  Yeah.
• 8:49
  One of the surprising findings was that they found early math
• 8:54
  knowledge and abilities was a stronger predictor than social
• 8:58
  emotional skills, stronger than family background,
• 9:03
  and stronger than family income,
• 9:05
  that it was the math knowledge that was predictive.
• 9:08
  That's incredible.
• 9:10
  Yes.
• 9:11
  A couple other surprising things from this study was that early math
• 9:16
  was a stronger predictor than early reading. Now,
• 9:20
  we know reading is really important and we know reading gets a lot of emphasis
• 9:25
  in the early grades,
• 9:27
  but math is a stronger predictor than reading.
• 9:32
  And then one last thing I'll say about this study is that early math not only
• 9:37
  predicts later math achievement,
• 9:39
  it also predicts later reading achievement.
• 9:44
  So that is always a surprise as we share that information with teachers,
• 9:49
  that early math seems to matter as much and perhaps more than early
• 9:54
  reading abilities. There's a couple other studies I'll share with you as well.
• 9:59
  So there's this body of research that talks about early math is very predictive
• 10:04
  of later learning, but we're teachers, we're educators. We like to know,
• 10:09
  well, what math seems to be most important.
• 10:14
  So there was a study in 2016 that looked at children's math learning
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  in pre-kindergarten, four-year-olds,
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  and then looked at their learning again back in fifth grade. And what was unique
• 10:26
  about this study is they looked closely at what specific math
• 10:31
  topics seemed to matter the most.
• 10:34
  And what they found was that advanced number competencies were the
• 10:39
  strongest predictors of later achievement. Now,
• 10:43
  what are advanced number competencies?
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  So these are the three that really stood out as being
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  important.
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  One was being able to count a set of objects with cardinality.
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  So in other words, counting things,
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  not just being able to recite a count sequence, no.
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  So not verbal, route counting, but actually counting things,
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  putting those numbers to objects.
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  Another thing that they found was really important was being able to count
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  forward from any number.
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  So if I said start at seven and keep counting,
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  start at 23 and keep counting,
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  that that was predictive of later learning.
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  And the reason for that is when kids can count forward from a number,
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  it helps them understand the structure of the number system,
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  something we're always working on. And then the third thing that they found as
• 11:41
  part of advanced number competencies was conceptual subatizing.
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  Now, what that is,
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  is being able to see a number such as five as composed of subgroups,
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  like five being composed of four and one or three and two.
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  So subatizing is being able to see the parts of a number,
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  and that was really important for these four year olds to begin working on for
• 12:06
  later learning. All right. One more, Mike, that I can share?
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  Fire away. Yes.
• 12:13
  Okay.
• 12:14
  So this last area of research that I want to share is actually really important
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  as we think about the work of teachers in kindergarten and first grade in
• 12:23
  particular.
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  So what these researchers did is they looked at children's learning at the
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  beginning of kindergarten and then at the end of first grade. So wow,
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  think of the math kids learn from five, six years old,
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  and they found that these gains in what children can do was more predictive
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  of later achievement than just what knowledge they had coming in.
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  So learning gains,
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  what children do and learn in math in kindergarten and first grade is
• 12:51
  predictive of their mathematical success up through third grade.
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  And then another study hooked it even further and said, wait a minute,
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  what they learn in kindergarten and first grade even predicts children's math
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  achievement into high school.
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  So there's just a growing body of research and evidence that early math
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  is really important. The math learning of four year olds, five year olds,
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  six year olds, and seven year olds really builds this foundation that
• 13:22
  determines children's mathematical success many years later.
• 13:27
  This feels like a really great segue to a conversation about what it means
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  to provide students opportunities for meaningful counting.
• 13:35
  That feels particularly significant when I heard all of the ideas that you were
• 13:40
  sharing in the research.
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  I'm wondering if you could talk about the features of a meaningful counting
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  experience.
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  If we were to try to break that down and think about what does that mean?
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  What does that look like?
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  What types of experiences count as meaningful when it comes to counting?
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  Could you all talk about that a little bit?
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  Yeah, that's a great question, Mike. This is Melissa.
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  So I think what's interesting about the idea of meaningful counting is the more
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  Deanna and I studied the trajectory and spent time working with teachers and
• 14:10
  students,
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  we came to the conclusion that the counting trajectory in particular
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  is anchored or a cornerstone of that counting trajectory is really meaningful
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  counting. That once a skill is acquired,
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  and we'll talk a little bit more about meaningful counting,
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  but once that skill is acquired,
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  it just builds and develops as kids grow and have more experience with number
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  and quantity. So when we think about meaningful counting,
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  the phrase that we like to use is that numbers represent quantity.
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  And it's just not that kids are saying numbers out loud,
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  is that when they say five, they know what five means.
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  They know how many that is.
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  They can connect it to a context that they can go grab five of
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  something. They might know that five is bigger than two or that 10
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  is bigger than five. So they start to really play with this idea of quantity.
• 15:01
  And specifically when we're talking about kids engaging in meaningful counting,
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  there's really key skills and understandings that we're looking and watching for
• 15:10
  as children count.
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  The first one Deanne already alluded to is this idea of cardinality.
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  So when I count how many I have, one, two, three, four, five,
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  if that's the size of my set, when someone asks me how many is it,
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  I can say five without needing to go back and count so I can hold that quantity.
• 15:27
  Another one is stable count sequence. So we used to call it rote count sequence.
• 15:32
  And again,
• 15:32
  Deanne referenced the idea that really when we're asking kids to count,
• 15:36
  we're asking more than just saying numbers.
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  So we think about the stability and the confidence in their counting. One of the
• 15:42
  pieces that we've started to really watch very carefully and think carefully
• 15:46
  about with our children as we're watching many of them count is their ability to
• 15:51
  organize. So it's not the job of the teacher to organize the counter,
• 15:55
  to tell the child how to lay out the counters.
• 15:57
  It really is the work of the child because it brings to bear counting,
• 16:02
  saying the numbers, maintaining cardinality,
• 16:04
  as well as sets them up and sets us up to see where they at with that one-to-one
• 16:08
  correspondence.
• 16:09
  So can they organize a set of counters in such a way that allows them to say one
• 16:13
  number, one touch, one object?
• 16:16
  And then as they continue to coordinate those skills,
• 16:19
  are they able to say back and hold onto the idea of quantity?
• 16:23
  So the other ideas that we like to consider,
• 16:26
  mostly because they're embedded in the trajectory and we've seen them become
• 16:29
  incredibly important as we work with children,
• 16:32
  is the idea of producing a set. So when I ask a child,
• 16:36
  "Can you give me five?" They give me five,
• 16:40
  or are they able to stop when they get to five? Do they keep counting?
• 16:44
  Do they pick up a handful of counters and dump it in my hand?
• 16:47
  So all of those things are what we're looking for as we're thinking about the
• 16:50
  idea of producing a set. And then finally, even for our youngest ones,
• 16:55
  we really place a fair importance on the idea of representing a count.
• 17:00
  So can they demonstrate,
• 17:02
  can they show on paper what they did or how many they have?
• 17:05
  So we leave with a very rudimentary math sketch.
• 17:09
  So if they've counted a collection of five,
• 17:11
  how would they represent five on that paper?
• 17:14
  What that allows then the teacher to do is to continue to leverage where the
• 17:19
  trajectory goes as well as what they know about young children to bring in
• 17:23
  meaningful experiences tied to writing numbers,
• 17:27
  tied to having conversations about numbers so the kids aren't doing worksheets,
• 17:32
  they're actually documenting something very important to them,
• 17:36
  which is this collection of whatever it is that they just counted in a way that
• 17:40
  makes sense to them. And so I think the other part that I like to talk about
• 17:45
  when we think about meaningful counting is this idea of hierarchical inclusion.
• 17:49
  It's that idea that children understand that numbers are nested one within each
• 17:54
  other and that each number in the count sequence is exactly one higher
• 18:00
  than what they said before.
• 18:02
  So many times our reference with that is with our teachers are those little
• 18:07
  nesting dolls.
• 18:08
  So we think about one and then we wrap two around it and then we wrap three
• 18:13
  around it. So when we think about the number three, we're thinking, well,
• 18:16
  it's actually the quantity of two and one more.
• 18:19
  And we see that as a really powerful understanding in particular as our
• 18:24
  children get older and we ask them not just what is one more or one less,
• 18:29
  but what is 10 more or 10 less that they take that and they
• 18:33
  extend that in meaningful ways. So again, the idea of meaningful counting,
• 18:38
  regardless of where we are on the trajectory,
• 18:40
  it's the idea that numbers represent quantities.
• 18:44
  And the neat thing about the trajectory is the counting trajectory in particular
• 18:48
  is that they give us really beautiful markers as to when to watch for these.
• 18:52
  So we tend to talk about the trajectories as levels.
• 18:55
  So we'll say at level six on our counting trajectory is where we see
• 19:00
  cardinality first start to kind of show up, where we're starting to look for it.
• 19:05
  And then we watch that idea of cardinality grow as children get older,
• 19:09
  as they have more experience and opportunity,
• 19:11
  and as they work with larger numbers.
• 19:14
  That's incredibly helpful. So I think one of the things that really jumped out,
• 19:18
  and I want to mark this and give you all an opportunity to be a little bit more
• 19:22
  explicit than you already were,
• 19:24
  this importance of linking numbers and quantities.
• 19:27
  And I wonder if you could say a bit more about what you mean,
• 19:30
  just to make sure that our listeners have a full understanding of why that is so
• 19:35
  significant.
• 19:37
  All right, this is Deanne. I'll jump in and get started, and Melissa can add on.
• 19:42
  As we first started to study the learning trajectory,
• 19:46
  the one thing we noticed was the importance of connecting things to quantity.
• 19:50
  Even some of the original levels didn't necessarily say quantity,
• 19:55
  but we anchor our work to developing meaning for our
• 20:00
  work. And we always think about, even when we're skip counting,
• 20:04
  it should be done with objects that we should be able to see skip counting as
• 20:09
  quantities, not just as words that I'm reciting.
• 20:13
  So across the trajectory,
• 20:16
  we put this huge emphasis on always connecting them to items,
• 20:21
  to things,
• 20:22
  or to actions and to movements so that it's not just a word,
• 20:27
  but that word has some meaning and significance for the child.
• 20:31
  I think that takes me to the other bit of language, Melissa,
• 20:35
  that you said that I want to come back to.
• 20:38
  You said at one point when you were describing meaningful counting experiences,
• 20:41
  you said one number, one touch, one object.
• 20:46
  And I wonder if you could unpack that,
• 20:48
  particularly one touch for young children and why that feels significant.
• 20:53
  That's a great question. And I'll come at this through a lens of watching many,
• 20:58
  many children count and working with lots and lots of teachers.
• 21:01
  When children are counting a set, many times they'll look and they'll go one,
• 21:06
  two, three, four, five, six, seven, igh, nine,
• 21:08
  and then however many are in the collection,
• 21:09
  they'll just say nine by just looking.
• 21:12
  And one of the things that we've noticed is that sometimes we need to
• 21:17
  explicitly give permission to children to do what they need to do with that
• 21:22
  collection to find out how many. Sometimes they're afraid to touch the items.
• 21:26
  Sometimes they don't know that they can. And we don't come right out and say,
• 21:29
  "Go ahead and touch them." But we just say, "Gosh,
• 21:32
  is there another way that you could find out how many?" And what we notice are
• 21:36
  some amazing and interesting ways kids organize their collections. So
• 21:41
  sometimes to be able to get to that one touch, one, number one object,
• 21:45
  they'll lay them out in a row.
• 21:47
  Sometimes they'll lay them out in a circle and they'll mark the one that they
• 21:50
  started with. Sometimes with our little guys in particular,
• 21:54
  we like to give them collections where they have to sit things up,
• 21:58
  so like the little counting bears. So if the bears are lying down,
• 22:02
  the kids will be very intentional in, "I set it up and I count it.
• 22:06
  I set it up and I count it.
• 22:07
  " And they all many times have to be facing the same direction as well.
• 22:12
  So the kids are very particular about how does this fit into
• 22:17
  the counting experience.
• 22:18
  And I would say that's one thing that's been really significant for us in
• 22:23
  understanding that it really is the work of the child to do that one touch,
• 22:27
  one object,
• 22:28
  one count in a way that matters to them. And that a teacher can very easily
• 22:33
  lay it out and say, "Find out how many,
• 22:35
  remember to touch one and tell me the number." Then it's not coming from the
• 22:39
  child. Then we don't know what they know. So that's been a really,
• 22:43
  really interesting aspect for us to watch in kids is how are they choosing to go
• 22:47
  into and enter into counting that.
• 22:50
  And we look at that as problem solving from our youngest,
• 22:54
  from our three-year-olds all the way up is what are you going to do with that
• 22:56
  pile of stuff in front of you?
• 22:58
  And that's an authentic problem for them and it's meaningful.
• 23:01
  I think what jumps out about that from me is the structure of what you just
• 23:05
  described is actually an experience and it's an opportunity to make
• 23:10
  sense of counting versus what perhaps has typically happened,
• 23:15
  which is a procedure for counting that we're asking kids to replicate and show
• 23:20
  us again.
• 23:21
  And what strikes me is you're advocating for a sense-making opportunity because
• 23:26
  that's the work of the child as opposed to, let me show you how to do it,
• 23:31
  you do it again and show it back to me,
• 23:34
  but what might be missing is meaning or connection to something
• 23:39
  that's real and that sets up what we think might be a house of cards or
• 23:44
  at the very least it has significant implications as you described in the
• 23:47
  research.
• 23:48
  One of the things, Mike,
• 23:49
  that I would add on that actually I just thought about is when you were talking
• 23:53
  about the importance of us letting the children figure out how they want to
• 23:58
  approach that task of organizing their count is it's coming from the child and
• 24:03
  Clements and Serama talk about the beautiful work about the trajectory is that
• 24:07
  we see that the mathematics comes from the child and we can nurture that along
• 24:11
  in developmentally appropriate ways.
• 24:14
  The other idea that popped into my mind is it's kind of a parallel to when our
• 24:18
  children get older and we want to teach them a way to add and a way to subtract.
• 24:22
  And I'm going to show you how to do it and you follow my procedure.
• 24:26
  I'm going to show it, you follow my procedure.
• 24:28
  We know that that's not best practice either.
• 24:31
  And so we're really looking at how do we grab onto that idea of number sense and
• 24:36
  move forward with it in a way that's meaningful with children from as young
• 24:41
  as one and two all the way up.
• 24:44
  I hope you've enjoyed the first half of our conversation with Deanne and Melissa
• 24:48
  as much as I have.
• 24:49
  We'll release the second half of our conversation on April 9th.
• 24:56
  This podcast is brought to you by the Math Learning Center and the Meyer Math
• 25:00
  Foundation,
• 25:01
  dedicated to inspiring and enabling all individuals to discover and develop
• 25:05
  their mathematical confidence and ability.