March 6, 2025 • 30 mins
Assessment in the Early Years
Guest: Shelly Scheafer
ROUNDING UP: SEASON 3 | EPISODE 13
Mike (00:09.127)
Welcome to the podcast Shelley. Thank you so much for joining us today.
Shelly (00:12.956)
Thank you, Mike, for having me.
Mike (00:16.078)
So I'd like to start with this question. What makes the work of assessing younger children, particularly students in grades K through two, different from assessing students in upper elementary grades or even beyond?
Shelly (00:30.3)
There's a lot to that question, Mike. I think there's some obvious things. So effective assessment of our youngest learners is different because obviously our pre-K, first, even our second grade students are developmentally different from fourth and fifth graders. So when we think about assessing these early primary students,
we need to use appropriate assessment methods that match their stage of development. For example, when we think of typical paper pencil assessments and how we often ask students to show their thinking with pictures, numbers and words, our youngest learners are just starting to connect symbolic representations to mathematical ideas, let alone, you know, put letters together to make words. So
When we think of these assessments, we need to take into consideration that primary students are in the early stages of development with respect to their language, their reading, and their writing skills. And this in itself makes it challenging for them to fully articulate, write, sketch any of their mathematical thinking. So we often find that with young children in reviews,
you know, individual interviews can be really helpful. But even then, there's some drawbacks. Some children find it challenging, you know, to be put on the spot, to show in the moment, you know, on demand, you know, what they know. Others, you know, just aren't fully engaged or interested because you've called them over from something that they're busy doing. Or maybe, you know,
they're not yet comfortable with the setting or even the person doing the interview. So when we work with young children, we need to recognize all of these little peculiarities that come with working with that age. We also need to understand that their mathematical development is fluid, it's continually evolving. And this is why
Shelly (02:47.42)
they often or some may respond differently to the same proper question, especially if the setting or the context is changed. We may find that a kindergarten student who counts to 29 on Monday may count to 69 or even 100 later in the week, kind of depending on what's going on in their mind at the time. So this means that assessment with young children needs to be frequent.
informative and ongoing. So we're not necessarily waiting for the end of the unit to see, aha, did they get this? You know, what do we do? You know, we're looking at their work all of the time. And fortunately, some of the best assessments on young children are the observations in their natural setting, like times when maybe they're playing a math game or working with a center activity or even during just your classroom routines.
And it's these authentic situations that we can look at as assessments to help us capture a more accurate picture of their abilities because we not only get to hear what they say or see what they write on paper, we get to watch them in action. We get to see what they do when they're engaged in small group activities or playing games with friends.
Mike (04:11.832)
So I wanna go back to something you said and even in particular the way that you said it. You were talking about watching or noticing what students can do and you really emphasize the words do. Talk a little bit about what you were trying to convey with that, Shelley.
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Transcript
Episode Transcript
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(00:03):
How is the work of assessingyoung children different from assessing students in
upper elementary grades orin grades six through 12?
And what actions can we take to ensurewe understand our youngest learners'
thinking? Today we'retalking with Shelly Schafer,
senior manager of Content Developmentwith the math Learning Center about the
ways educators can understand andadvance the mathematical thinking of our
(00:27):
youngest learners. Welcometo the podcast, Shelly.
Thank you so much for joining us today.
Thank you, Mike, for having me.
So I'd like to start with this question.
What makes the work ofassessing younger children,
particularly studentsin grades K through two,
different from assessing students inupper elementary grades or even beyond?
(00:49):
Wow, there's a lot to that question, Mike.
I think there's some obvious things.
So effective assessment of ouryoungest learners is different
because obviously our pre-K, our first,
even our second grade students aredevelopmentally different from fourth and
(01:10):
fifth graders.
So when we think about assessingthese early primary students,
we need to use appropriate assessmentmethods that match their stage of
development. For example,
when we think of typical paper pencilassessments and how we often ask
students to show their thinkingwith pictures, numbers and words.
(01:31):
Our youngest learners,
our just starting to connect symbolicrepresentations to mathematical
ideas, let alone put letterstogether to make words.
So we need to take intoconsideration that primary students
are in the early stages of developmentwith their language, their reading,
(01:52):
and their writing skills.
And this makes it challengingfor them to fully articulate,
write, sketch,
any of their mathematical thinking.So we often find that with young
children, interviewscan be really helpful,
but even then there's some drawbacks.
Some children find it challengingto show in the moment what they
(02:15):
know.
Others just aren't fully engaged orinterested because you've called them over
from something that they're busy doing,
or maybe they're not yet comfortablewith the setting or even the person doing
the interview. So whenwe work within children,
we need to recognize all ofthese little peculiarities that
(02:35):
come with working with that age.
We also need to understand thattheir mathematical development is
fluid, it's continually evolving,
and this is why they often orsome may respond differently
to the same prompter question,
especially if the settingof the context has changed.
(02:56):
We may find that a kindergarten studentwho counts to nine on Monday may
count to 69 or even ahundred later in the week,
depending on what's going onin their mind at the time. So
this means that assessment withyoung children needs to be frequent,
formative, and ongoing.
So we're not necessarily waitingfor the end of the unit to see, aha,
(03:19):
did they get this? What do we do?
We're looking at theirwork all of the time.
And fortunately,
some of the best assessments on youngchildren are the observations in their
natural setting,
like times when maybe they're playinga math game or working with a center
activity or even during justyour classroom routines.
(03:41):
And it's these authenticsituations that we can look at
as assessments help us capturea more accurate picture of their
abilities because we not onlyget to hear what they say or see
what they write on paper, weget to watch them in action.
We get to see what they do whenthey're engaged in small group
(04:04):
activities or playing games with friends.
So I want to go back to something yousaid and the way that you said it.
You were talking aboutwatching or noticing what students can do and you really
emphasize the words.
Do talk a little bit about what youwere trying to convey with that Shelly.
Young children are doers whenthey work on a math pass,
(04:25):
they show their thinking and their actionswith finger formations and objects.
And we can see if a student hasone-to-one correspondence when they're
counting, if they group their objects,how they line 'em up, do they tag them?
Do they move them as they count them?
They may not always have the verbalskills to articulate their thinking,
(04:46):
but we can also attend to thingslike head nodding, finger counting,
and even how they cluster or matchobjects. I'm going to give you an example.
Let's say that I'm watching some earlyfirst graders and they're solving the
expression six plus seven,
and the first studentpicks up a number rack,
(05:07):
and if you're not familiarwith a number rack,
it's a tool with two rows of beads.
And on the first row there are five redbeads and five white beads. And on the
second row there's five redbeads and five white beads.
And the students solving six plusseven begins by pushing over five red
beads in one push and thenone more bead on the top row.
(05:30):
And then they do thesame thing for the seven.
They push over five redbeads and two white beads,
and they haven't said a word to me,
I'm just watching their actionsand I'm already able to tell, hmm,
that student could subitize a group offive because I saw 'em push over all
five beads in one push.
(05:50):
And that they know that sixis composed of five in one
and seven is composed of five and two,
and they haven't said a word. I'mjust watching what they're doing.
And then I might watch thestudent and I see 'em pause,
nothing's being said,
but I start to notice this slightlittle head nodding. And then they say
(06:13):
13, and they give me the answerand they're really pleased.
I didn't get a lot of language from them,
but boy did I get a lot fromwatching how they solve that problem.
And I want to contrast that observationwith a student who might be solving the
same expression, six plusseven, and they might go, Hmm,
(06:34):
six.
And then they start popping up onefinger at a time while counting
7,
8,
9,
10,
11,
12,
13.
And when they get seven fingersheld up, they say 13. Again,
they've approached that problemquite differently, but again,
(06:54):
I get that information that theyunderstood the equation they were able to
count on starting with six,
and they kept track of their count withtheir fingers and they knew to stop
when seven fingers were raised.
And I might even have a different studentthat might start talking to me and
(07:15):
they say, well, six plus six is 12,
and seven is one more than six.
So the answer is 13.
And if this were being done ona paper pencil as an assessment
item or they were answeringon some kind of a device,
all I would know about my students isthat they were able to get the correct
(07:37):
answer.
I wouldn't really know a lotabout how they got the answer,
what skills do they have?What was their thinking?
And there's not a lot that I canwork with to plan my instruction.
Does that kind of make sense?
Absolutely. I think the waythat you described this,
attending to behaviors, to gestures,
(07:59):
to the way that kids areinteracting with manipulatives,
the self-talk that's happening,it makes a ton of sense.
And I think for me, when Ithink back to my own practice,
I wish I could wind the clock back becauseI think I was attending a lot to what
kids were saying and sometimestheir written communication.
(08:19):
And there was a lot that I could havealso taken in if I was attending to
those things in a little bit more depth.
It also strikes me that this might feela bit overwhelming for an educator.
How could an educator knowwhat they're looking for?
I do think it can feeloverwhelming at first,
but as teachers begin tomake informal observations,
(08:41):
really listening and watching studentsactions as part of just their daily
practice, something thatthey're doing on a normal basis,
they start to develop these kindof intuitive understandings of how
children learn, what to expect them to do,
what they might say next if they see acertain action. And after several years,
(09:02):
let's say, of teaching kindergarten,
if you've been a kindergartenteacher for four or 5, 6,
20 plus years,
you start to notice thesepatterns of behavior,
things that five and six year oldsseem to say and think and do on a
fairly consistent basis. And that kindof helps you know what you're looking at.
(09:22):
And fortunately, we have severalresearchers that have been, let's say,
kid watching for 40 years,
and they have identified stages throughwhich most children pass as they develop
their counting skills or maybe strategiesfor solving addition and subtraction
problems.And these stages are
(09:43):
laid out as progressions of thinkingor actions that students exhibit
as they develop understandingover periods of time.
And listeners might know these aslearning progressions or learning
trajectories.
And these are ways to convey anidea of concept in little bits of
(10:03):
understanding.
So when I was sharing the thinkingand actions of three students
solving six plus seven listenersfamiliar with cognitively guided
instruction, CGI,
they might've recognized the sequenceof strategies that children go
through when they're solvingaddition and subtraction problems.
(10:24):
So in my first student,
they didn't say anything but gave mean answer was using direct modeling.
We saw them push over five and one beadsfor six and then five and two beads for
seven, and then kind ofpause at their model.
And I could tell withtheir head nodding that
they were counting quietly in theirhead counting all the beads to get the
(10:48):
answer. And that's one of those firststages that we see and recognize with
direct modeling.
And that gives me information onwhat I might do with a student
coming next time.
I might work on the second strategythat I conveyed with my second
student where they were able to count on,
(11:08):
they started with that six,
and then they counted seven more usingtheir fingers to keep track of their
count and got the answer.
And then that third kind of level inthat progression as we're moving of
understanding was shown with my thirdstudent when they were able to use a
derived fact strategy.The student said, oh,
well I know that six plus sixis 12. I knew my double fact,
(11:33):
and then I use that relationship ofknowing that seven is one more than six.
And so that's kind of how we move kidsthrough. And so when I'm watching them,
I can kind of pinpoint where theyare and where they might go next,
and I can also thinkabout what I might do.
And so it's this knowledge of developmentand progressions and how children,
(11:53):
number concepts that can helpteachers recognize the skills
as they emerge, as they beginto see them with their students,
and they can use those to guide theirinstruction for that student or look at
the class overall and plan theirinstruction or think about more open-ended
kinds of questions that they can ask thatrecognize these different levels that
(12:15):
students are working with.
As a K one teacher,
I remember that I spent a lot of mytime tracking students with things like
checklists.
So I note if students hador didn't have a skill,
and as I hear you talk,
that feels fairly oversimplified when wethink about this idea of developmental
(12:37):
progressions.
How do you suggest that teachersapproach capturing evidence of student
learning, Shelly?
Well,
we have to really think about assessmentand children's learning is something
that is ongoing andinvolving, and if we do,
it becomes part of whatwe can do every day.
We can look for opportunities to observestudents' skills in authentic settings.
(13:01):
Maybe it's something that we're havingthem write down on their whiteboard,
or maybe it's something where they'reshowing the answer with finger formations
or we're giving a thumbs up or athumbs down to check in on their
understanding. We might notbe checking on every student,
but we're capturing a few and we cantake note because we're doing this on a
(13:22):
daily basis of who we want to checkin with, what do we want to see?
We can also do a littlemore formal planning when we draw from what we're going
to do already in our lesson.
Let's say for example thatour lesson today includes
a.talk or a number talk,
something that we're going to write down.We're going to record student
(13:43):
thinking. And during the lesson,
the teacher's going to be busyfacilitating the discussion,
recording the student's thinkingand making all of those notes.
But if we write the child's name,
honor their thinking and give it thatcaption on that public record at the end
of the lesson, we can capture apicture, just use an iPad quickly,
(14:05):
take a picture of that student's thinking,
and then we can record that wherewe're keeping track of our students.
So we have, okay, another moment in time.
And it's this collection ofevidence that we keep growing.
We can also by capturingthese public records note,
whose voice and thinking we'reelevating in the classroom.
(14:27):
So it gives us how are they thinking andwho are we listening to and making sure
that we're spreading thatout. And hearing everyone,
I think like you mentionedchecklists that you use.
I did.
Yeah.
And even checklists can play arole in observation and assessments
when they have a focus and a wayto capture students' thinking.
(14:49):
One of the things we did in thirdedition is we designed additional
tools for gathering and recordinginformation during workplaces.
That's a routine where students areplaying games and or engaged with partners
doing some sort of a math activity.
And we designed thesebased on what we might see
(15:10):
students do at these differentgames and activities.
And we didn't necessarily think aboutthis is something you're going to do with
every student or even in one daybecause these are spanned out
over a period of four to six weekswhere that they can go to these games
and we might see the students goto these activities multiple times.
(15:33):
And so let's say thatkindergarten students are playing
something like the game beat you to10 where they're spinning a spinner,
they're counting cubes,
and they're trying to race theirpartner to collect 10 cubes. And with an
activity like that,
I might want to focus onstudents who I still want to see,
do they have one-to-one correspondence?Are they developing cardinality?
(15:57):
Are they able to count out a set?
And those might be kinds of skillsthat you might've had typically on a
checklist, right, Mike for kindergarten.
But I could use this activity togather that note and make any comment.
So just for those kids I'm looking at,
or maybe first graders are playing agame like sort the sum where they're
(16:19):
drawing two different dominoes andthey're supposed to find how many they
have in all. So with a game like that,
I might focus on what aretheir strategies? Are they counting all the dots?
Are they counting on from one set of thedots on one side and then counting on
the other?
Are they starting with the greater numberor the most dots? Are they starting
(16:40):
with the one always on the left?
Or I might even see they mightinstantly recognize some of those.
So I might know the skills that I wantto look for with those games and be
making notes, whichkind of feels checklist,
but I can target thattime to do it on students.
I want that informationby thinking ahead of time,
(17:01):
what can I get by watching observingthese students at these games? I mean,
as you know, young children love it.
Older children love it when the teachergoes over and wants to watch them play
or even better wants to engagein the gameplay with them.
But I can still use that as an assessment.
That's helpful, Shelly,for a couple of reasons.
(17:22):
One of the things that you said wasreally powerful is thinking about not just
the assessment tools that mightbe within your curriculum,
but looking at the task itself thatyou're going to have students engage with,
be it a game or a project or some kind of
activity and really thinking,
what can I get from this as a personwho's trying to make sense of students'
(17:43):
thinking?
And I think my checklist suddenly feelsreally different when I've got a clear
vision of what can I get from this taskor this game that students are playing
and looking for evidence of that versusfeeling like I was pulling kids over
one-on-one,
which I think I would still do becausethere's some depth that I might want to
capture,
but it changes the way that I think aboutwhat I might do and also what I might
(18:07):
get out of a task that really resonatesfor me. The other thing that you made me
think about is the extentto which I remember thinking is I need to make sure if
a student has got it or not. Got it.
What you're making me think can reallycome out of this experience of observing
students when they're working on a taskor with a partner is that I can gather
(18:28):
more evidence about theapplication of that idea.
I can see the extent to which studentsare doing something like counting on in
the context of a game or a task,
and that adds to the evidencethat I gather in a one-on-one interview with them.
But it gives me a chance to see,
is this way of thinking something thatstudents are applying in different
contexts or did it just happen at thatone particular moment in time when I was
(18:51):
with 'em?
So that really helps me think about howthose two different ways of assessing
students be it one-on-one or observingthem and seeing what's happening,
support one another.
And I think you also made me thinkit really hit on this idea that
students, like I said,
their learning is evolving overtime and it might change with the
(19:15):
context so that they show us that theyknow something in one context with these
numbers or this scenario,
but they don't necessarily alwayssee that it applies across the
board. They don't make generalizations.
That's something that we really haveto work with students to develop.
And they're also young children.
Think about how quickly a three-year-oldand a four-year-old change
(19:40):
the same five to six, six to seven, Imean, they're evolving all the time.
And so we want to get this informationfor them on a regular basis.
A unit of instruction maybe a month or more long,
and a lot can happen in that time.
So we want to make sure that we continueto check in with them and help them to
develop if needed or that we advance them,
(20:03):
we nudge them along, we challengethem with maybe a question,
will that apply to every number?
So a student discovers whenwe add one to every number,
it's like saying the nextnumber, so six and one more,
seven and eight and one more is nine.And you can challenge them. Ooh,
does that always work?What if the number was 22?
(20:25):
What if it was 132? Would it always work?
So when you're checking in with kids,
you have those opportunities to keepthem thinking, to help them grow.
I want to pick up on something thatwe haven't necessarily said aloud,
but I'd like to explore it.
Looking at young students' workfrom an asset-based perspective,
particularly with younger students,
(20:46):
I've had points in time where therefelt like so much that I needed to teach
them, and sometimes I felt myselffocusing on what they couldn't do.
Looking back, I wish I had thoughtabout my work as noticing the assets,
the strategies, the ways of thinkingthat they were accumulating.
Are there practices you think support anasset-based approach to assessment with
(21:08):
young learners?
I think probably the biggest thing wecan do is broaden our thinking about
assessment.
The National Council of Teachers ofMathematics wrote in catalyzing change in
early childhood and elementarymathematics that the primary purpose of
assessment is to gatherevidence of children's thinking,
(21:30):
understanding and reasoningto inform both instructional
decisions and studentin teaching learning.
If we consider assessments andobservations as tools to inform our
instruction,
we need to pay attention to thedetails of the child's thinking.
And when we're payingattention to the details,
(21:53):
what the child is bringing tothe table, what they can do,
that's where our focus goes.So the question becomes,
what is the student understanding?
What assets do they bring to the task?
It's no longer can they doit or can they not do it?
And when we know,
(22:14):
when we focusing on just what thatstudent can do and we have some
understanding of understandinglearning progressions,
how students learn,
then we can place what they'redoing kind of on that trajectory
in that progression,
and that becomes knowledge.And with that knowledge then we
(22:36):
can help students move alongthe progression to more
developed understanding.For example, again,
if I go back to my six plus seven andwe notice that a student is direct
modeling, they're counting outeach of the sets and counting all,
we can start to nudgethem toward counting on.
We might cover, they'reusing that number rack.
(22:58):
We might cover the first row and say, Ooh,
you just really showed me a good physicalrepresentation of six plus seven,
and I noticed that you were countingthe beats to see how many were there.
I'm wondering if I cover this firstrow, how many beats am I covering?
Six. I wonder, could youstart your counting at six?
(23:18):
We can work with what they know,
and I can do that because I'vefocused on where they are in
that progression and wherethat development is going.
And I have a goal of where I wantstudents to go to further their thinking,
not that being any one place is rightor wrong, or yes, they can do it. No,
(23:38):
they can't. It's my understanding of whatassets they bring that I can build on.
Is that kind of what you were after?
It is. And I think you also addressedsomething that again has gone unsaid,
but I think you unpacked it there,
which is assessment is reallydesigned to inform my instruction.
And I think the example you offeredis a really lovely one where we have a
(24:01):
student who's direct modeling and they'remaking sense of number in a certain
way, and their strategy reflects that.
And that helps us think about thekinds of nudges we can offer that might
shift that thinking or press them tomake sense of numbers in a different way.
That really the assessmentis it is a moment in time,
but it also informs the way that you thinkabout what you're going to do next to
(24:25):
keep nudging that student's thinking.
Exactly. And we have to knowthat if we have 20 students,
they all have 20 little plans,
that they're on 20 littlepathways of their learning.
And so we need to think about everybody.
So we're going to askquestions that help 'em do 'em,
and we're going to honortheir thinking. So again,
I'm going to go back to doingthat.talk with those students.
(24:48):
And so I'm honoring all these differentways that students are finding the total
number of dots,
and then I'm asking them to look forwhat's the same within their thinking so
that other studentsalso can serve to nudge
kids, have them let them tryand explore a different idea or,
(25:08):
Ooh, can we try that Mike'sway and see if we can do that?
Or what do you notice about how Mikesolve the problem and how Shelly solve the
problem? Where is their thinkingthe same? Where is it different?
And so we're honoring everybody'splace of where they're at,
but they're stilllearning from each other.
You have made multiple mentions to thisidea of progressions or trajectories,
(25:32):
and I'm wondering,
what are some of the resources thathelped you build an understanding of
children's developmentalprogression? Shelly?
Honestly,
I can say that I learned a lot fromthe students I taught in my classroom.
My roots run deep in earlychildhood. With that said,
I think I stand on the back of giants,
teacher practitioner researchersfor early childhood who have spent
(25:57):
decades observing childrenand recording their thinking.
I mentioned cognitivelyguided instruction,
which features the research of ThomasCarpenter and his team and their book.
Children's Mathematics is agreat guide for K five teachers.
Another teacher researcheris Kathy Richardson,
(26:18):
and some listeners mayknow her from her books,
the Developing Number concept seriesor number talks in the primary
classroom.
She also wrote a book called HowChildren Learn Number Concepts,
A Guide to the Critical Learning Phases,
which Targets Pre-Kindergartenthrough grade four.
And then I think also the workof Julie Sama and Doug Clement.
(26:42):
They have a website thatlooks at learning progressions
starting at birth all theway through grade three.
And this website islearning trajectories.org,
and it's one of thosethat is always evolving,
so it not only explainslearning trajectories for all
early childhood math concepts,
(27:05):
but there are literally thousands ofvideos and lessons for teaching math
and new content is always being added.
So any of those would reallygive teachers some good ideas
on how children learn theprogressions that they go for and
really help them notice and put areference to what they're seeing kids do.
(27:28):
You mentioned Giants,
and those are some gigantic folks inthe world of mathematics education.
I had a really similar experiencewith both CGI and Kathy Richardson in
that a lot of what they're describingare the things that I was seeing in
classrooms.
What it really helped me do isunderstand how to place that behavior
(27:49):
and what the meaning of it was interms of students' understanding of
mathematics.
And it also helped me think about thatas an asset that then I could build on.
Are there resources you would invite ourlisteners to engage with if they want
to continue learning?
I think if listeners are interestedin learning more about developmental
progressions in math, the resourcesI mentioned, children's Mathematics,
(28:13):
cognitive guided instructionby Thomas Carpenter Al,
the How Children Learn Number Conceptsby Kathy Richardson or SAMA in Clements
learning directories.org are goodplaces to start. But honestly, Mike,
it's about teachers makingpurposeful observations,
understanding what they'reseeing and hearing,
and then knowing what todo with that information.
(28:37):
The Latin root of the wordassessment means to sit beside.
I would like to invite our listenersto sit beside their students. Listen,
watch, question, take note,
because developing the capacityto observe children in action,
listening to their thinking and thenacting upon what they see in hear
(29:00):
takes practice, takes effort,
and once teachers become fascinatedwith children sitting in front of us,
we can become students of ourstudents. As Alan Fisher would say.
That's when teachersreally see the benefits.
They'll recognize that all theirstudents have math abilities,
(29:21):
and that these math abilitiesare specific and actionable.
And when we nurture our students withwhat they know and what they need to know,
they will grow.
I think that's a great placeto stop. Shelly Schaeffer,
thank you so much for joining us.
Thank you so much, Mike, for having me.It's been a pleasure talking with you.
(29:47):
This podcast is brought to you by theMath Learning Center and the Meyer Math
Foundation dedicated to inspiringand enabling all individuals to
discover and develop theirmathematical confidence and ability.
How is the work of assessingyoung children different from assessing students in
upper elementary grades orin grades six through 12?
And what actions can we take to ensurewe understand our youngest learners'
thinking? Today we'retalking with Shelly Schafer,
senior manager of Content Developmentwith the math Learning Center about the
ways educators can understand andadvance the mathematical thinking of our
(00:27):
youngest learners. Welcometo the podcast, Shelly.
Thank you so much for joining us today.
Thank you, Mike, for having me.
So I'd like to start with this question.
What makes the work ofassessing younger children,
particularly studentsin grades K through two,
different from assessing students inupper elementary grades or even beyond?
(00:49):
Wow, there's a lot to that question, Mike.
I think there's some obvious things.
So effective assessment of ouryoungest learners is different
because obviously our pre-K, our first,
even our second grade students aredevelopmentally different from fourth and
(01:10):
fifth graders.
So when we think about assessingthese early primary students,
we need to use appropriate assessmentmethods that match their stage of
development. For example,
when we think of typical paper pencilassessments and how we often ask
students to show their thinkingwith pictures, numbers and words.
(01:31):
Our youngest learners,
our just starting to connect symbolicrepresentations to mathematical
ideas, let alone put letterstogether to make words.
So we need to take intoconsideration that primary students
are in the early stages of developmentwith their language, their reading,
(01:52):
and their writing skills.
And this makes it challengingfor them to fully articulate,
write, sketch,
any of their mathematical thinking.So we often find that with young
children, interviewscan be really helpful,
but even then there's some drawbacks.
Some children find it challengingto show in the moment what they
(02:15):
know.
Others just aren't fully engaged orinterested because you've called them over
from something that they're busy doing,
or maybe they're not yet comfortablewith the setting or even the person doing
the interview. So whenwe work within children,
we need to recognize all ofthese little peculiarities that
(02:35):
come with working with that age.
We also need to understand thattheir mathematical development is
fluid, it's continually evolving,
and this is why they often orsome may respond differently
to the same prompter question,
especially if the settingof the context has changed.
(02:56):
We may find that a kindergarten studentwho counts to nine on Monday may
count to 69 or even ahundred later in the week,
depending on what's going onin their mind at the time. So
this means that assessment withyoung children needs to be frequent,
formative, and ongoing.
So we're not necessarily waitingfor the end of the unit to see, aha,
(03:19):
did they get this? What do we do?
We're looking at theirwork all of the time.
And fortunately,
some of the best assessments on youngchildren are the observations in their
natural setting,
like times when maybe they're playinga math game or working with a center
activity or even during justyour classroom routines.
(03:41):
And it's these authenticsituations that we can look at
as assessments help us capturea more accurate picture of their
abilities because we not onlyget to hear what they say or see
what they write on paper, weget to watch them in action.
We get to see what they do whenthey're engaged in small group
(04:04):
activities or playing games with friends.
So I want to go back to something yousaid and the way that you said it.
You were talking aboutwatching or noticing what students can do and you really
emphasize the words.
Do talk a little bit about what youwere trying to convey with that Shelly.
Young children are doers whenthey work on a math pass,
(04:25):
they show their thinking and their actionswith finger formations and objects.
And we can see if a student hasone-to-one correspondence when they're
counting, if they group their objects,how they line 'em up, do they tag them?
Do they move them as they count them?
They may not always have the verbalskills to articulate their thinking,
(04:46):
but we can also attend to thingslike head nodding, finger counting,
and even how they cluster or matchobjects. I'm going to give you an example.
Let's say that I'm watching some earlyfirst graders and they're solving the
expression six plus seven,
and the first studentpicks up a number rack,
(05:07):
and if you're not familiarwith a number rack,
it's a tool with two rows of beads.
And on the first row there are five redbeads and five white beads. And on the
second row there's five redbeads and five white beads.
And the students solving six plusseven begins by pushing over five red
beads in one push and thenone more bead on the top row.
(05:30):
And then they do thesame thing for the seven.
They push over five redbeads and two white beads,
and they haven't said a word to me,
I'm just watching their actionsand I'm already able to tell, hmm,
that student could subitize a group offive because I saw 'em push over all
five beads in one push.
(05:50):
And that they know that sixis composed of five in one
and seven is composed of five and two,
and they haven't said a word. I'mjust watching what they're doing.
And then I might watch thestudent and I see 'em pause,
nothing's being said,
but I start to notice this slightlittle head nodding. And then they say
(06:13):
13, and they give me the answerand they're really pleased.
I didn't get a lot of language from them,
but boy did I get a lot fromwatching how they solve that problem.
And I want to contrast that observationwith a student who might be solving the
same expression, six plusseven, and they might go, Hmm,
(06:34):
six.
And then they start popping up onefinger at a time while counting
7,
8,
9,
10,
11,
12,
13.
And when they get seven fingersheld up, they say 13. Again,
they've approached that problemquite differently, but again,
(06:54):
I get that information that theyunderstood the equation they were able to
count on starting with six,
and they kept track of their count withtheir fingers and they knew to stop
when seven fingers were raised.
And I might even have a different studentthat might start talking to me and
(07:15):
they say, well, six plus six is 12,
and seven is one more than six.
So the answer is 13.
And if this were being done ona paper pencil as an assessment
item or they were answeringon some kind of a device,
all I would know about my students isthat they were able to get the correct
(07:37):
answer.
I wouldn't really know a lotabout how they got the answer,
what skills do they have?What was their thinking?
And there's not a lot that I canwork with to plan my instruction.
Does that kind of make sense?
Absolutely. I think the waythat you described this,
attending to behaviors, to gestures,
(07:59):
to the way that kids areinteracting with manipulatives,
the self-talk that's happening,it makes a ton of sense.
And I think for me, when Ithink back to my own practice,
I wish I could wind the clock back becauseI think I was attending a lot to what
kids were saying and sometimestheir written communication.
(08:19):
And there was a lot that I could havealso taken in if I was attending to
those things in a little bit more depth.
It also strikes me that this might feela bit overwhelming for an educator.
How could an educator knowwhat they're looking for?
I do think it can feeloverwhelming at first,
but as teachers begin tomake informal observations,
(08:41):
really listening and watching studentsactions as part of just their daily
practice, something thatthey're doing on a normal basis,
they start to develop these kindof intuitive understandings of how
children learn, what to expect them to do,
what they might say next if they see acertain action. And after several years,
(09:02):
let's say, of teaching kindergarten,
if you've been a kindergartenteacher for four or 5, 6,
20 plus years,
you start to notice thesepatterns of behavior,
things that five and six year oldsseem to say and think and do on a
fairly consistent basis. And that kindof helps you know what you're looking at.
(09:22):
And fortunately, we have severalresearchers that have been, let's say,
kid watching for 40 years,
and they have identified stages throughwhich most children pass as they develop
their counting skills or maybe strategiesfor solving addition and subtraction
problems.And these stages are
(09:43):
laid out as progressions of thinkingor actions that students exhibit
as they develop understandingover periods of time.
And listeners might know these aslearning progressions or learning
trajectories.
And these are ways to convey anidea of concept in little bits of
(10:03):
understanding.
So when I was sharing the thinkingand actions of three students
solving six plus seven listenersfamiliar with cognitively guided
instruction, CGI,
they might've recognized the sequenceof strategies that children go
through when they're solvingaddition and subtraction problems.
(10:24):
So in my first student,
they didn't say anything but gave mean answer was using direct modeling.
We saw them push over five and one beadsfor six and then five and two beads for
seven, and then kind ofpause at their model.
And I could tell withtheir head nodding that
they were counting quietly in theirhead counting all the beads to get the
(10:48):
answer. And that's one of those firststages that we see and recognize with
direct modeling.
And that gives me information onwhat I might do with a student
coming next time.
I might work on the second strategythat I conveyed with my second
student where they were able to count on,
(11:08):
they started with that six,
and then they counted seven more usingtheir fingers to keep track of their
count and got the answer.
And then that third kind of level inthat progression as we're moving of
understanding was shown with my thirdstudent when they were able to use a
derived fact strategy.The student said, oh,
well I know that six plus sixis 12. I knew my double fact,
(11:33):
and then I use that relationship ofknowing that seven is one more than six.
And so that's kind of how we move kidsthrough. And so when I'm watching them,
I can kind of pinpoint where theyare and where they might go next,
and I can also thinkabout what I might do.
And so it's this knowledge of developmentand progressions and how children,
(11:53):
number concepts that can helpteachers recognize the skills
as they emerge, as they beginto see them with their students,
and they can use those to guide theirinstruction for that student or look at
the class overall and plan theirinstruction or think about more open-ended
kinds of questions that they can ask thatrecognize these different levels that
(12:15):
students are working with.
As a K one teacher,
I remember that I spent a lot of mytime tracking students with things like
checklists.
So I note if students hador didn't have a skill,
and as I hear you talk,
that feels fairly oversimplified when wethink about this idea of developmental
(12:37):
progressions.
How do you suggest that teachersapproach capturing evidence of student
learning, Shelly?
Well,
we have to really think about assessmentand children's learning is something
that is ongoing andinvolving, and if we do,
it becomes part of whatwe can do every day.
We can look for opportunities to observestudents' skills in authentic settings.
(13:01):
Maybe it's something that we're havingthem write down on their whiteboard,
or maybe it's something where they'reshowing the answer with finger formations
or we're giving a thumbs up or athumbs down to check in on their
understanding. We might notbe checking on every student,
but we're capturing a few and we cantake note because we're doing this on a
(13:22):
daily basis of who we want to checkin with, what do we want to see?
We can also do a littlemore formal planning when we draw from what we're going
to do already in our lesson.
Let's say for example thatour lesson today includes
a.talk or a number talk,
something that we're going to write down.We're going to record student
(13:43):
thinking. And during the lesson,
the teacher's going to be busyfacilitating the discussion,
recording the student's thinkingand making all of those notes.
But if we write the child's name,
honor their thinking and give it thatcaption on that public record at the end
of the lesson, we can capture apicture, just use an iPad quickly,
(14:05):
take a picture of that student's thinking,
and then we can record that wherewe're keeping track of our students.
So we have, okay, another moment in time.
And it's this collection ofevidence that we keep growing.
We can also by capturingthese public records note,
whose voice and thinking we'reelevating in the classroom.
(14:27):
So it gives us how are they thinking andwho are we listening to and making sure
that we're spreading thatout. And hearing everyone,
I think like you mentionedchecklists that you use.
I did.
Yeah.
And even checklists can play arole in observation and assessments
when they have a focus and a wayto capture students' thinking.
(14:49):
One of the things we did in thirdedition is we designed additional
tools for gathering and recordinginformation during workplaces.
That's a routine where students areplaying games and or engaged with partners
doing some sort of a math activity.
And we designed thesebased on what we might see
(15:10):
students do at these differentgames and activities.
And we didn't necessarily think aboutthis is something you're going to do with
every student or even in one daybecause these are spanned out
over a period of four to six weekswhere that they can go to these games
and we might see the students goto these activities multiple times.
(15:33):
And so let's say thatkindergarten students are playing
something like the game beat you to10 where they're spinning a spinner,
they're counting cubes,
and they're trying to race theirpartner to collect 10 cubes. And with an
activity like that,
I might want to focus onstudents who I still want to see,
do they have one-to-one correspondence?Are they developing cardinality?
(15:57):
Are they able to count out a set?
And those might be kinds of skillsthat you might've had typically on a
checklist, right, Mike for kindergarten.
But I could use this activity togather that note and make any comment.
So just for those kids I'm looking at,
or maybe first graders are playing agame like sort the sum where they're
(16:19):
drawing two different dominoes andthey're supposed to find how many they
have in all. So with a game like that,
I might focus on what aretheir strategies? Are they counting all the dots?
Are they counting on from one set of thedots on one side and then counting on
the other?
Are they starting with the greater numberor the most dots? Are they starting
(16:40):
with the one always on the left?
Or I might even see they mightinstantly recognize some of those.
So I might know the skills that I wantto look for with those games and be
making notes, whichkind of feels checklist,
but I can target thattime to do it on students.
I want that informationby thinking ahead of time,
(17:01):
what can I get by watching observingthese students at these games? I mean,
as you know, young children love it.
Older children love it when the teachergoes over and wants to watch them play
or even better wants to engagein the gameplay with them.
But I can still use that as an assessment.
That's helpful, Shelly,for a couple of reasons.
(17:22):
One of the things that you said wasreally powerful is thinking about not just
the assessment tools that mightbe within your curriculum,
but looking at the task itself thatyou're going to have students engage with,
be it a game or a project or some kind of
activity and really thinking,
what can I get from this as a personwho's trying to make sense of students'
(17:43):
thinking?
And I think my checklist suddenly feelsreally different when I've got a clear
vision of what can I get from this taskor this game that students are playing
and looking for evidence of that versusfeeling like I was pulling kids over
one-on-one,
which I think I would still do becausethere's some depth that I might want to
capture,
but it changes the way that I think aboutwhat I might do and also what I might
(18:07):
get out of a task that really resonatesfor me. The other thing that you made me
think about is the extentto which I remember thinking is I need to make sure if
a student has got it or not. Got it.
What you're making me think can reallycome out of this experience of observing
students when they're working on a taskor with a partner is that I can gather
(18:28):
more evidence about theapplication of that idea.
I can see the extent to which studentsare doing something like counting on in
the context of a game or a task,
and that adds to the evidencethat I gather in a one-on-one interview with them.
But it gives me a chance to see,
is this way of thinking something thatstudents are applying in different
contexts or did it just happen at thatone particular moment in time when I was
(18:51):
with 'em?
So that really helps me think about howthose two different ways of assessing
students be it one-on-one or observingthem and seeing what's happening,
support one another.
And I think you also made me thinkit really hit on this idea that
students, like I said,
their learning is evolving overtime and it might change with the
(19:15):
context so that they show us that theyknow something in one context with these
numbers or this scenario,
but they don't necessarily alwayssee that it applies across the
board. They don't make generalizations.
That's something that we really haveto work with students to develop.
And they're also young children.
Think about how quickly a three-year-oldand a four-year-old change
(19:40):
the same five to six, six to seven, Imean, they're evolving all the time.
And so we want to get this informationfor them on a regular basis.
A unit of instruction maybe a month or more long,
and a lot can happen in that time.
So we want to make sure that we continueto check in with them and help them to
develop if needed or that we advance them,
(20:03):
we nudge them along, we challengethem with maybe a question,
will that apply to every number?
So a student discovers whenwe add one to every number,
it's like saying the nextnumber, so six and one more,
seven and eight and one more is nine.And you can challenge them. Ooh,
does that always work?What if the number was 22?
(20:25):
What if it was 132? Would it always work?
So when you're checking in with kids,
you have those opportunities to keepthem thinking, to help them grow.
I want to pick up on something thatwe haven't necessarily said aloud,
but I'd like to explore it.
Looking at young students' workfrom an asset-based perspective,
particularly with younger students,
(20:46):
I've had points in time where therefelt like so much that I needed to teach
them, and sometimes I felt myselffocusing on what they couldn't do.
Looking back, I wish I had thoughtabout my work as noticing the assets,
the strategies, the ways of thinkingthat they were accumulating.
Are there practices you think support anasset-based approach to assessment with
(21:08):
young learners?
I think probably the biggest thing wecan do is broaden our thinking about
assessment.
The National Council of Teachers ofMathematics wrote in catalyzing change in
early childhood and elementarymathematics that the primary purpose of
assessment is to gatherevidence of children's thinking,
(21:30):
understanding and reasoningto inform both instructional
decisions and studentin teaching learning.
If we consider assessments andobservations as tools to inform our
instruction,
we need to pay attention to thedetails of the child's thinking.
And when we're payingattention to the details,
(21:53):
what the child is bringing tothe table, what they can do,
that's where our focus goes.So the question becomes,
what is the student understanding?
What assets do they bring to the task?
It's no longer can they doit or can they not do it?
And when we know,
(22:14):
when we focusing on just what thatstudent can do and we have some
understanding of understandinglearning progressions,
how students learn,
then we can place what they'redoing kind of on that trajectory
in that progression,
and that becomes knowledge.And with that knowledge then we
(22:36):
can help students move alongthe progression to more
developed understanding.For example, again,
if I go back to my six plus seven andwe notice that a student is direct
modeling, they're counting outeach of the sets and counting all,
we can start to nudgethem toward counting on.
We might cover, they'reusing that number rack.
(22:58):
We might cover the first row and say, Ooh,
you just really showed me a good physicalrepresentation of six plus seven,
and I noticed that you were countingthe beats to see how many were there.
I'm wondering if I cover this firstrow, how many beats am I covering?
Six. I wonder, could youstart your counting at six?
(23:18):
We can work with what they know,
and I can do that because I'vefocused on where they are in
that progression and wherethat development is going.
And I have a goal of where I wantstudents to go to further their thinking,
not that being any one place is rightor wrong, or yes, they can do it. No,
(23:38):
they can't. It's my understanding of whatassets they bring that I can build on.
Is that kind of what you were after?
It is. And I think you also addressedsomething that again has gone unsaid,
but I think you unpacked it there,
which is assessment is reallydesigned to inform my instruction.
And I think the example you offeredis a really lovely one where we have a
(24:01):
student who's direct modeling and they'remaking sense of number in a certain
way, and their strategy reflects that.
And that helps us think about thekinds of nudges we can offer that might
shift that thinking or press them tomake sense of numbers in a different way.
That really the assessmentis it is a moment in time,
but it also informs the way that you thinkabout what you're going to do next to
(24:25):
keep nudging that student's thinking.
Exactly. And we have to knowthat if we have 20 students,
they all have 20 little plans,
that they're on 20 littlepathways of their learning.
And so we need to think about everybody.
So we're going to askquestions that help 'em do 'em,
and we're going to honortheir thinking. So again,
I'm going to go back to doingthat.talk with those students.
(24:48):
And so I'm honoring all these differentways that students are finding the total
number of dots,
and then I'm asking them to look forwhat's the same within their thinking so
that other studentsalso can serve to nudge
kids, have them let them tryand explore a different idea or,
(25:08):
Ooh, can we try that Mike'sway and see if we can do that?
Or what do you notice about how Mikesolve the problem and how Shelly solve the
problem? Where is their thinkingthe same? Where is it different?
And so we're honoring everybody'splace of where they're at,
but they're stilllearning from each other.
You have made multiple mentions to thisidea of progressions or trajectories,
(25:32):
and I'm wondering,
what are some of the resources thathelped you build an understanding of
children's developmentalprogression? Shelly?
Honestly,
I can say that I learned a lot fromthe students I taught in my classroom.
My roots run deep in earlychildhood. With that said,
I think I stand on the back of giants,
teacher practitioner researchersfor early childhood who have spent
(25:57):
decades observing childrenand recording their thinking.
I mentioned cognitivelyguided instruction,
which features the research of ThomasCarpenter and his team and their book.
Children's Mathematics is agreat guide for K five teachers.
Another teacher researcheris Kathy Richardson,
(26:18):
and some listeners mayknow her from her books,
the Developing Number concept seriesor number talks in the primary
classroom.
She also wrote a book called HowChildren Learn Number Concepts,
A Guide to the Critical Learning Phases,
which Targets Pre-Kindergartenthrough grade four.
And then I think also the workof Julie Sama and Doug Clement.
(26:42):
They have a website thatlooks at learning progressions
starting at birth all theway through grade three.
And this website islearning trajectories.org,
and it's one of thosethat is always evolving,
so it not only explainslearning trajectories for all
early childhood math concepts,
(27:05):
but there are literally thousands ofvideos and lessons for teaching math
and new content is always being added.
So any of those would reallygive teachers some good ideas
on how children learn theprogressions that they go for and
really help them notice and put areference to what they're seeing kids do.
(27:28):
You mentioned Giants,
and those are some gigantic folks inthe world of mathematics education.
I had a really similar experiencewith both CGI and Kathy Richardson in
that a lot of what they're describingare the things that I was seeing in
classrooms.
What it really helped me do isunderstand how to place that behavior
(27:49):
and what the meaning of it was interms of students' understanding of
mathematics.
And it also helped me think about thatas an asset that then I could build on.
Are there resources you would invite ourlisteners to engage with if they want
to continue learning?
I think if listeners are interestedin learning more about developmental
progressions in math, the resourcesI mentioned, children's Mathematics,
(28:13):
cognitive guided instructionby Thomas Carpenter Al,
the How Children Learn Number Conceptsby Kathy Richardson or SAMA in Clements
learning directories.org are goodplaces to start. But honestly, Mike,
it's about teachers makingpurposeful observations,
understanding what they'reseeing and hearing,
and then knowing what todo with that information.
(28:37):
The Latin root of the wordassessment means to sit beside.
I would like to invite our listenersto sit beside their students. Listen,
watch, question, take note,
because developing the capacityto observe children in action,
listening to their thinking and thenacting upon what they see in hear
(29:00):
takes practice, takes effort,
and once teachers become fascinatedwith children sitting in front of us,
we can become students of ourstudents. As Alan Fisher would say.
That's when teachersreally see the benefits.
They'll recognize that all theirstudents have math abilities,
(29:21):
and that these math abilitiesare specific and actionable.
And when we nurture our students withwhat they know and what they need to know,
they will grow.
I think that's a great placeto stop. Shelly Schaeffer,
thank you so much for joining us.
Thank you so much, Mike, for having me.It's been a pleasure talking with you.
(29:47):
This podcast is brought to you by theMath Learning Center and the Meyer Math
Foundation dedicated to inspiringand enabling all individuals to
discover and develop theirmathematical confidence and ability.
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