May 9, 2024 • 24 mins
ROUNDING UP: SEASON 2 | EPISODE 17
Spatial reasoning can be a nebulous concept that is hard for many educators to define. In this episode, we’re talking about spatial reasoning with Dr. Robyn Pinilla from the University of Texas at El Paso. We’ll examine the connections between spatial reasoning and other mathematical concepts and explore different ways that educators can cultivate this type of reasoning with their students.
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Spatial Reasoning in the Early Years
TRANSCRIPT
Mike Wallus: Spatial reasoning can be a nebulous concept, and it's often hard for many educators to define. In this episode, we're talking about spatial reasoning with Dr. Robyn Pinilla from the University of Texas [at] El Paso. We'll examine the connections between spatial reasoning and other mathematical concepts and explore different ways that educators can cultivate this type of reasoning with their students.
Welcome to the podcast, Robyn. I'm really excited to be talking with you about spatial reasoning.
Robyn Pinilla: And I am excited to be here.
Mike: Well, let me start with a basic question. So when we're talking about spatial reasoning, is that just another way of saying that we're going to be talking about ideas that are associated with geometry? Or are we talking about something bigger?
Robyn: It's funny that you say it in that way, Mike, because geometry is definitely the closest mathematical content that we see in curricula, but it is something much bigger.
So I started with the misconception and then I used my own experiences to support that idea, that this was just geometry, because it was my favorite math course in high school because I could see the concepts modeled and I could make things more tangible. Drawing helped me to visualize some of those concepts that I was learning instead of just using a formula that I didn't necessarily understand. So, at that time, direct instruction really ruled, and I'm unsure what the conceptual understandings of my teachers even were because what I recall is doing numbers 3 through 47 odds in the back of the book and just plugging through these formulas. But spatial reasoning allows us to develop our concepts in a way that lead to deeper conceptual understanding. I liked geometry, and it gave me this vehicle for mathematizing the world. But geometry is really only one strand of spatial reasoning.
Mike: So you're already kind of poking around the question that I was going to ask next, which is the elevator description of, “What do we mean when we talk about spatial reasoning, and why does it matter? Why is it a big deal for students?”
Robyn: So spatial reasoning is a notoriously hard-to-define construct that deals with how things move in space. It's individually how they move in space, in relation to one another. A lot of my ideas come from a network analysis that [Cathy] Bruce and colleagues did back in 2017 that looked at the historical framing of what spati
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Episode Transcript
Available transcripts are automatically generated. Complete accuracy is not guaranteed.
(00:03):
Spatial reasoning canbe a nebulous concept,
and it's often hard for manyeducators to define. In this episode,
we're talking aboutspatial reasoning with Dr.
Robin Pania from theUniversity of Texas El Paso.
We'll examine the connections betweenspatial reasoning and other mathematical
concepts and explore different ways thateducators can cultivate this type of
(00:26):
reasoning with their students.Welcome to the podcast, Robin.
I'm really excited to be talkingwith you about spatial reasoning.
And I am excited to be here.
Well, let me start with a basic question.
So when we're talkingabout spatial reasoning,
is that just another way of saying thatwe're going to be talking about ideas
that are associated with geometry, orare we talking about something bigger?
(00:48):
It's funny that you sayit in that way, Mike,
because geometry is definitely the closestmathematical content that we see in
curricula, but it issomething much bigger.
So I started with the misconceptionand then I used my own experiences
to support that idea that this was justgeometry because it was my favorite math
course in high school because I couldsee the concepts modeled and I could make
(01:12):
things more tangible.
Drawing helped me to visualize someof those concepts that I was learning
instead of just using a formula thatI didn't necessarily understand.
So at that time,
direct instruction really ruledand I'm unsure what the conceptual
understandings of my teachers even werebecause what I recall is doing numbers
(01:33):
three through 47 odds in the back ofthe book and just plugging through these
formulas.
But spatial reasoning allows usto develop our concepts in a way
that lead to deeperconceptual understanding.
I liked geometry and it gave me thisvehicle for mathematizing the world,
but geometry is really onlystrand of spatial reasoning.
(01:54):
So you're already kind of poking aroundthe question that I was going to ask
next,
which is the elevator description of whatdo we mean when we talk about spatial
reasoning and why does it matter?Why is it a big deal for students?
So spatial reasoning is anotoriously hard to define construct
that deals with how things move in space.
(02:15):
It's individually how they move inspace in relation to one another.
A lot of my ideas come from anetwork analysis that Bruce and
colleagues did back in 2017that looked at the historical
framing of what spatial reasoning isand how we talk about it in different
fields because psychologists lookat spatial reasoning. Mathematics,
(02:37):
educators look at spatial reasoning.
There's also connectionsinto philosophy, the arts.
But when we start moving towardsmathematics more specifically,
it does deal with how things move inspace individually and in relation to one
another. So with geometry,
whether the objects are sliding andtransforming or we're composing and
decomposing to create new shapes,
(03:00):
those are the skills in two dimensionalgeometry that we do often see in
curricula.
But the underlying skills are alsocritical to everyday life and they can be
taught as well.
And when we're thinking about theeveryday constructs that are being built
through our interactions with the world,
I like to think about the GPS on our car.
(03:21):
So spatial reasoning has a lot of spatialtemporal processes that are going on.
It's not just thinking about the waysthat things move in relation to one
another or the connections to mathematics,
but also the way that wemove through this world,
the way that we navigate throughit. So I'll give a little example.
Spatial temporal processes have todo with us running errands perhaps.
(03:43):
How long does it take you to getfrom work to the store to home,
and how many things can you purchase inthe store knowing how full your fridge
currently is?
What pots and pans are you going touse to cook the food that you purchase,
and what volume of that food are youand your family going to consume?
So all of those daily tasks involveconceptions of how much space things
(04:06):
take, and we could call it capacity,
which situates nicely withinthe measurement domain of mathematics education,
but it's also spatial reasoningand it extends further than that.
That is helpful.
I think you opened up my understandingof what we're actually talking about,
and I think the piece that was reallyinteresting is how in that example of I'm
(04:28):
going to the grocery store, how longwill it take? How full is my fridge?
What are the different tools that I'lluse to prepare what capacity do they
have?
I think that really helped me broadenout my own thinking about what spatial
reasoning actually is.
I wonder if we could shift a bit and youcould help unpack for educators who are
listening,
(04:48):
a few examples of tasks that kidsmight encounter that could support the
development of spatial reasoning.
Sure.
My research and work is primarily focusedon early childhood and elementary,
so I'm going to focus therebut then kind of expand up.
Number one, let's play.
That's the first thing that I wantto walk into a classroom and see.
(05:12):
I want to see the kids engagingwith blocks, legos, duplos,
and building with andwithout specific intentions.
Not everything has to havea preconceived lesson.
One of the activities I've been doingactually with teachers and professional
development sessions lately is apresentation called Who's Its and what
sits.
I have the teacherscreate what sits that do.
(05:36):
That sits meaning they createsomething that does something.
I don't give them a prompt of whatproblem they're going to be solving or
anything specific for themto build, but rather say,
here are materials. We givethem large dulo blocks,
magnet tiles and magforms,
(05:56):
different types of woodencardboard and foam blocks,
PVC pipes,
which are really interesting in the waysthat teachers use them and have them
start thinking as thoughthey're the children in the class and they're trying to
build something that takes spaceand can be used in different ways.
So the session we dida couple of weeks ago,
(06:17):
some teachers came up with Firstthere was a swing that they had put a
little frog in that theycontrolled with magnets.
So they had used the PVC pipe atthe top to be the top of the swing
connected over,
and then were using the magnets to guideit back and forth without ever having
to touch the swing,
and I just thought that was the coolestway for them to be using these materials
(06:39):
in really playful creative ways thatcould also engender them taking those
lessons back into their classroom.
I have also recently been remindedof the importance of modeling with
fractions.
So are you familiar with thewhich one doesn't belong Tasks?
Absolutely love them.
Yes.
There's also a website for fractiontalks that children can look at visual
(07:03):
representations of fractions and determinewhich one doesn't belong for some
reason,
that helps us to see the ways thatchildren are thinking about the fractional
spaces and then justifying theirreason around them. With that,
we can talk about the spatialpositioning of the fractional pieces that
are colored in or the waysthat they're separated.
(07:26):
If those colored pieces are in differentplaces on the figure that's being
shown,
they open up some nice spaces for us totalk about different concepts and use
that language of spatial reasoning thatis critical for teachers to engage in to
show the ways that studentscan think about those things.
So I want to go backto this notion of play,
(07:48):
and what I'm curious about is why issituating this in play going to help
these ideas around spatial reasoningcome out as opposed to say,
situating it in a morecontrolled structure?
Well, I think by situatingspatial reasoning within play,
we do allow teachers to respond inthe moment rather than having these
lesson plans that they are requiredto plan out from the beginning.
(08:12):
A lot of the ideaswithin spatial reasoning,
because it's a nebulous constructand it's learned through our everyday
experiences and interactions withthe world, they are harder to plan.
And so when children are engagedin play in the classroom,
teachers can respond very naturallyso that they're incorporating
(08:33):
the mathematizing of the world intowhat the students are already doing.
If you take, for example,
one of my old teachersused to do a treasure hunt,
great way to incorporate spatialreasoning with early childhood elementary
classrooms where she would set up amapping task is really what it was,
(08:53):
but it was introducing the children tothe school itself and navigating that
environment, which is criticalfor spatial reasoning skills.
And they would play this gingerbreadman type game of she would read the book
and then everybody would be involvedwith this treasure hunt where the kiddos
would start out in the classroom and theywould get a clue to help them navigate
(09:15):
toward the cafeteria. Whenthey got to the cafeteria,
the gingerbread man would already begone. He would've already run off,
so they would get their next clue tohelp them navigate to the playground,
so on and so forth. They would go tothe nurse's office, the principal,
the library,
all of the critical places that theywould be going through on a daily basis or
when they needed to within the school.
(09:37):
And it reminds me that there was alsoa teacher I once interviewed who used
orienteering skills with her students.Have you ever heard of orienteering?
The connection I'm making isto something like Geocashing,
but I think you shouldhelp me understand it.
Yeah, that's really similar.
So it's this idea thatchildren would find their way,
places path finding and way finding arealso spatial reasoning skills that are
(10:01):
applied within our real world.
And so while it maynot be as scientific or
sophisticated as doing geocaching,
it has children with theidea of navigating in our real world helps them start
to learn cardinality and the differentways of thinking about traversing to a
different location,
which these are all things that mightbetter relate to social studies or
(10:25):
technology, other STEMdomains specifically,
but that are undergirdedby the spatial reasoning,
which does have thosemathematics connections.
I think the first thing that occurredis all of the directional language that
could emerge from something liketrying to find the J of red boy.
And then the other piece that youmade me think about just now is this
(10:47):
opportunity to quantifydistance in different ways,
and I'm sure there are otherthings that you could draw out,
especially in a play setting where thestructure is a little bit looser and it
gives you a little bitmore space, as you said,
to respond to kids rather than feelinglike you have to impose the structure.
Yeah, absolutely.
There's an ability when teachers areengaging in authentic ways with the
(11:10):
students, that they're able tosupport language development, support,
ideation and creation withoutnecessarily having kids sit down
and fill out a worksheetthat says, where is the ball?
The ball is sitting ontop of the shelf. Instead,
we can be on the floor working withstudents and providing those directions
(11:31):
of, oh, hey,
I need you to get me those materialsfrom the shelf on the other side of the
room,
but thinking about how can I say thatin a way that better supports children
understanding the spatial reasoningthat's occurring in our room.
So maybe it's find the pencilinside the blue cup on top of
the shelf that's behindthe pencil sharpener,
(11:53):
getting really specific in the waysthat we talk about things so that we're
ingraining those ideas in such a waythat it becomes part of the way that the
kids communicate as well.
You have me thinking that there's anintentionality in language choice that can
create that, but then Iwould imagine as a teacher,
I could also revoice what students aresaying and perhaps introduce language in
(12:15):
that way as well.
Yeah,
and now you have me thinking abouta really fun routine number talks,
of course.
And if we do the idea of a.talkinstead of a number talk,
thinking about the spatial structuringof the dots that we're seeing and the
different ways that you cansee those arrangements and
describe the quantificationof the arrangement.
(12:39):
It's a nice way to introduce educatorsto spatial reasoning because it might be
something that they're already doing inthe classroom while also providing an
avenue for children to see spatialstructuring in a way that they're already
accustomed to as well based on theroutines that they're receiving from the
teacher.
I think what's really exciting aboutthis, Robin, is the more that we talk,
(13:00):
the more two things jump out.
I think one is my language choicesallow me to introduce these
ideas in a way that I don't know thatI'd thought about as a practitioner.
Part two is that we can'treally necessarily draw a distinction between work
we're doing around numbers and quantityand spatial reasoning that there are
opportunities within our workaround number quantity and within
(13:23):
math content to inject the language ofspatial reasoning and have it become a
part of the experience for students.
Yeah,
and that's important that I haveconveyed that without explicitly
saying it because that's the very workthat I'm doing with teachers in their
classrooms at this time. One,
(13:44):
as you're talking aboutlanguage, and I hate to do this,
but I'm going to take us a littlebit off topic for a moment.
I keep seeing this idea on Twitteror whatever we call it at this point,
that some people actually don'thear music in their heads.
This idea is wild to me because I havesongs playing in my head all the time,
(14:06):
but at the same time,
what if we think about the ideathat some people don't also
visualize things,
they don't imagine thosemovements continuously that I just
see. And so as teachers,
we really need to focus on that sameidea that children need opportunities
to practice what we thinkthey should be able to hear,
(14:29):
but also practice what we thinkthey should be able to see.
I'm not a cognitive scientist. Ican't see inside someone's head,
but I am a teacher by trade,
so I want to emphasize that teachers cando what's within their locus of control
so that children can have opportunitiesto talk about those tasks.
One that I recently sawwas a lesson on clocks.
(14:51):
So while I was sitting there watchingher teach, she was using a Judy clock.
She was having fun games with the kidsto do a little competition where they
could read the clock andtell her what time it was.
But I was just starting to think aboutall of the ways that we could talk about
the shorter and longer hands,the minute and our hands,
the ways that we could talk about themrotating around that center point.
(15:13):
What shape does the hand make as itgoes around that center point and what
happens if it doesn't rotate fully?
Now I'm going back to those fractionalideas from earlier with the which one
doesn't belong tasks of havingfull shapes versus half shapes,
and how we see those shapes inour real lives that we can then
relate with visualized shapes thatsome children may or may not be able to
(15:36):
see.
You have me thinking aboutsomething. First of all,
I'm so glad that you mentionedthe role of visualization.
You had me thinking about a conversationI was having with a colleague a while
ago,
and we had read a text that we werediscussing and the point of conversation
came up. I read this and there's acertain image that popped into my head,
and the joke we were making is I'm prettycertain that the image that I saw in
(15:58):
my head having read this textis not the same as what you saw.
What you said that really struck homefor me is I might be making some real
assumptions about the pictures thatkids see in their head and helping build
those internal images, those mentalmovies. That's a part of our work as well.
Absolutely. Because I'm thinking aboutthe way that we have prototypical shapes.
(16:20):
So a few years ago I was workingwith some assessments and
the children were supposed to be ableto recognize an equilateral triangle,
whether it was gravity basedor facing another orientation,
and there were some childrenwho automatically could see that the triangle was a
triangle no matter whichdirection it was pointing,
(16:44):
whereas others only recognize it ifa triangle, if it were gravity based.
And so we need to be teaching theproperties of the shapes beyond just that
image recognition that oftentimesour younger students come out with.
I tend to think of visualizationand language as supporting one another with the
(17:04):
idea that when we are talking, we'realso writing a descriptive essay.
Our words are what createthe intended picture.
Can't say that it's alwaysthe picture that comes out,
but the intended picture for the audience.
What we're hopeful for in classroomsis that because we're sharing physical
spaces and tangible experiences that
(17:27):
the language used around thoseexperiences could create shared meaning.
That's one of the most difficult piecesin talking about spatial reason or quite
frankly, anything else,
is that oftentimes our words may havedifferent meanings depending on who the
speaker and who the listener are.
And so navigating what those differencesare can be quite challenging,
(17:52):
which is why spatial reasoningis still so hard to define.
Absolutely.
My other follow up is if you were tooffer people a way to get started,
particularly on visualization,
is there a kind of task that you imaginemight move them along that pathway?
I think the first thing to do isreally grasp an approximation.
(18:12):
I'm not going to say figureout what spatial reasoning is,
but just an approximation or acouple of the skills therein that
you feel comfortable with.
So spatial reasoning is really the setof skills that undergirds almost all of
our daily actions,
but it also can be insertedinto the lessons that teachers are already teaching.
(18:33):
I think that we do have to acknowledgethat spatial reasoning is hard to define,
but the good news is that we doreason spatially all day every day.
If I am in a classroom,
I want to look first at theteaching that's happening,
the routines that are already there,
and see where some spatial reasoningmight actually fit in with our young
(18:54):
classes. I like to thinkabout calendar math.
Every single kindergarten, firstgrade classroom that you walk into,
they're going to havethat calendar on the wall,
so how can you work into theroutines that are occurring,
that spatial language to describe thedifferent components of the routine?
So as a kiddo is counting on thathundreds chart talking about the ways
(19:18):
in which they're moving thepointer along the numbers.
When they're counting by tens,
talk about the ways that they're movingdown when they're finding the patterns
that are on the calendar,
because all of those littlecalendar numbers for the day,
they wind up having a pattern withinthem in most of the curricular kits.
So thinking about just the ways thatwe can use language they're in now with
(19:39):
older students,
I think that offering that variety ofmodels or manipulatives for them to use
and then encourage them to translatefrom having a concrete manipulative
into those more representationalideas is great regardless of age or
grade.
So students benefit from the modelingwhen they do diagramming of their
(19:59):
models.
That is translating the 3D model to 2D,
which is another componentof spatial reasoning.
And that gets me to thissticky point of I'm not
arguing against automaticity or beingable to solve equations without physical
or visual models,
but I'm just acknowledging this idea thatoffering alternative ways for students
(20:23):
to engage with content is reallycritical because we're no longer at
a phase that we need ourchildren to become computers.
We have programs for that.
We need children who are able tothink and solve problems in novel
ways because that's the directionthat we're moving in problem solving.
That's fantastic. My finalquestion before we close things up.
(20:47):
If you were to make a recommendationfor someone who's listening and they're
intrigued and they want to keep learning,
are there any particular resources thatyou'd offer people that they might be
able to go to?
Yeah, absolutely.
So the first one that I like isthe Learning Trajectories website.
It's a learning trajectories.org. It'sproduced by Doug Clements and Julie Sama.
(21:08):
There are wonderful tasksthat are associated with
spatial reasoning skills fromvery young children in the
infants and toddler stages all theway up until seven or eight years old.
So that's a great place to go thatwill allow you to see how children are
performing in differentareas of spatial reasoning.
(21:29):
There is also a book calledTaking Shape by Kathy Bruin,
colleagues that I believewas produced in 2016,
and the grade levels might be a littlebit different because it is on the
Canadian school system, butit's for K through two students,
and that offers both the tasks andthe spatial reasoning skills that are
associated with them. Formore of the research side,
(21:53):
there's a book by Brent Davis andthe Spatial Reasoning study group
called Spatial Reasoningin the early Years,
and that volume has been one of my go-tos
in understanding both the history ofspatial reasoning in our schools and also
ways to start thinking aboutspatializing school mathematics.
One of the things that I really appreciateabout this conversation is you've
(22:16):
helped me make a lot moresense of spatial reasoning,
but the other thing thatyou've done for me at least,
is see that there are ways that Ican make choices with my planning,
with my language that I couldpick up and do tomorrow.
There's not a discreet separate bitthat is about spatial reasoning.
It's really an integrated set of ideasand concepts and skills that I can
(22:39):
start to build upon right awaywhatever curriculum I have.
And that's the point.Often in mathematics,
we think more explicitlyabout algebraic or numeric
reasoning, but less frequently inclassrooms about spatial reasoning.
But spatial reasoning supportsnot only mathematics development,
but other stem domains as well,
(22:59):
and even skills that crossover intosocial studies and language arts as we're
talking about mapping, aswe're talking about language.
So as students have these experiences,
they too can start tomathematize the world,
see spatial connectionsas they go out to recess,
as they go home from school,
as they're walking through theirneighborhoods or just around the house.
(23:22):
And it's ingrained ideas of measurementthat we are looking at on a daily
basis, the ways that we plan outour days and plan out our movements,
whether it's really a plan or justour reactions to the world that
support building these skills over time.
And so there are those reallypractical applications,
(23:44):
but it also comes to supporting overallmathematics development and then later
STEM career interests,
which is why I get excited about thework and want to be able to share it with
more and more people.
I think that's a greatplace to stop for listeners.
We're going to link all of the contentthat Robin shared to our show notes.
And Robin, I'll just say again,thank you so much for joining us.
(24:05):
It's really been apleasure talking with you.
Yes, absolutely. Thanks so much.
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.
Spatial reasoning canbe a nebulous concept,
and it's often hard for manyeducators to define. In this episode,
we're talking aboutspatial reasoning with Dr.
Robin Pania from theUniversity of Texas El Paso.
We'll examine the connections betweenspatial reasoning and other mathematical
concepts and explore different ways thateducators can cultivate this type of
(00:26):
reasoning with their students.Welcome to the podcast, Robin.
I'm really excited to be talkingwith you about spatial reasoning.
And I am excited to be here.
Well, let me start with a basic question.
So when we're talkingabout spatial reasoning,
is that just another way of saying thatwe're going to be talking about ideas
that are associated with geometry, orare we talking about something bigger?
(00:48):
It's funny that you sayit in that way, Mike,
because geometry is definitely the closestmathematical content that we see in
curricula, but it issomething much bigger.
So I started with the misconceptionand then I used my own experiences
to support that idea that this was justgeometry because it was my favorite math
course in high school because I couldsee the concepts modeled and I could make
(01:12):
things more tangible.
Drawing helped me to visualize someof those concepts that I was learning
instead of just using a formula thatI didn't necessarily understand.
So at that time,
direct instruction really ruledand I'm unsure what the conceptual
understandings of my teachers even werebecause what I recall is doing numbers
(01:33):
three through 47 odds in the back ofthe book and just plugging through these
formulas.
But spatial reasoning allows usto develop our concepts in a way
that lead to deeperconceptual understanding.
I liked geometry and it gave me thisvehicle for mathematizing the world,
but geometry is really onlystrand of spatial reasoning.
(01:54):
So you're already kind of poking aroundthe question that I was going to ask
next,
which is the elevator description of whatdo we mean when we talk about spatial
reasoning and why does it matter?Why is it a big deal for students?
So spatial reasoning is anotoriously hard to define construct
that deals with how things move in space.
(02:15):
It's individually how they move inspace in relation to one another.
A lot of my ideas come from anetwork analysis that Bruce and
colleagues did back in 2017that looked at the historical
framing of what spatial reasoning isand how we talk about it in different
fields because psychologists lookat spatial reasoning. Mathematics,
(02:37):
educators look at spatial reasoning.
There's also connectionsinto philosophy, the arts.
But when we start moving towardsmathematics more specifically,
it does deal with how things move inspace individually and in relation to one
another. So with geometry,
whether the objects are sliding andtransforming or we're composing and
decomposing to create new shapes,
(03:00):
those are the skills in two dimensionalgeometry that we do often see in
curricula.
But the underlying skills are alsocritical to everyday life and they can be
taught as well.
And when we're thinking about theeveryday constructs that are being built
through our interactions with the world,
I like to think about the GPS on our car.
(03:21):
So spatial reasoning has a lot of spatialtemporal processes that are going on.
It's not just thinking about the waysthat things move in relation to one
another or the connections to mathematics,
but also the way that wemove through this world,
the way that we navigate throughit. So I'll give a little example.
Spatial temporal processes have todo with us running errands perhaps.
(03:43):
How long does it take you to getfrom work to the store to home,
and how many things can you purchase inthe store knowing how full your fridge
currently is?
What pots and pans are you going touse to cook the food that you purchase,
and what volume of that food are youand your family going to consume?
So all of those daily tasks involveconceptions of how much space things
(04:06):
take, and we could call it capacity,
which situates nicely withinthe measurement domain of mathematics education,
but it's also spatial reasoningand it extends further than that.
That is helpful.
I think you opened up my understandingof what we're actually talking about,
and I think the piece that was reallyinteresting is how in that example of I'm
(04:28):
going to the grocery store, how longwill it take? How full is my fridge?
What are the different tools that I'lluse to prepare what capacity do they
have?
I think that really helped me broadenout my own thinking about what spatial
reasoning actually is.
I wonder if we could shift a bit and youcould help unpack for educators who are
listening,
(04:48):
a few examples of tasks that kidsmight encounter that could support the
development of spatial reasoning.
Sure.
My research and work is primarily focusedon early childhood and elementary,
so I'm going to focus therebut then kind of expand up.
Number one, let's play.
That's the first thing that I wantto walk into a classroom and see.
(05:12):
I want to see the kids engagingwith blocks, legos, duplos,
and building with andwithout specific intentions.
Not everything has to havea preconceived lesson.
One of the activities I've been doingactually with teachers and professional
development sessions lately is apresentation called Who's Its and what
sits.
I have the teacherscreate what sits that do.
(05:36):
That sits meaning they createsomething that does something.
I don't give them a prompt of whatproblem they're going to be solving or
anything specific for themto build, but rather say,
here are materials. We givethem large dulo blocks,
magnet tiles and magforms,
(05:56):
different types of woodencardboard and foam blocks,
PVC pipes,
which are really interesting in the waysthat teachers use them and have them
start thinking as thoughthey're the children in the class and they're trying to
build something that takes spaceand can be used in different ways.
So the session we dida couple of weeks ago,
(06:17):
some teachers came up with Firstthere was a swing that they had put a
little frog in that theycontrolled with magnets.
So they had used the PVC pipe atthe top to be the top of the swing
connected over,
and then were using the magnets to guideit back and forth without ever having
to touch the swing,
and I just thought that was the coolestway for them to be using these materials
(06:39):
in really playful creative ways thatcould also engender them taking those
lessons back into their classroom.
I have also recently been remindedof the importance of modeling with
fractions.
So are you familiar with thewhich one doesn't belong Tasks?
Absolutely love them.
Yes.
There's also a website for fractiontalks that children can look at visual
(07:03):
representations of fractions and determinewhich one doesn't belong for some
reason,
that helps us to see the ways thatchildren are thinking about the fractional
spaces and then justifying theirreason around them. With that,
we can talk about the spatialpositioning of the fractional pieces that
are colored in or the waysthat they're separated.
(07:26):
If those colored pieces are in differentplaces on the figure that's being
shown,
they open up some nice spaces for us totalk about different concepts and use
that language of spatial reasoning thatis critical for teachers to engage in to
show the ways that studentscan think about those things.
So I want to go backto this notion of play,
(07:48):
and what I'm curious about is why issituating this in play going to help
these ideas around spatial reasoningcome out as opposed to say,
situating it in a morecontrolled structure?
Well, I think by situatingspatial reasoning within play,
we do allow teachers to respond inthe moment rather than having these
lesson plans that they are requiredto plan out from the beginning.
(08:12):
A lot of the ideaswithin spatial reasoning,
because it's a nebulous constructand it's learned through our everyday
experiences and interactions withthe world, they are harder to plan.
And so when children are engagedin play in the classroom,
teachers can respond very naturallyso that they're incorporating
(08:33):
the mathematizing of the world intowhat the students are already doing.
If you take, for example,
one of my old teachersused to do a treasure hunt,
great way to incorporate spatialreasoning with early childhood elementary
classrooms where she would set up amapping task is really what it was,
(08:53):
but it was introducing the children tothe school itself and navigating that
environment, which is criticalfor spatial reasoning skills.
And they would play this gingerbreadman type game of she would read the book
and then everybody would be involvedwith this treasure hunt where the kiddos
would start out in the classroom and theywould get a clue to help them navigate
(09:15):
toward the cafeteria. Whenthey got to the cafeteria,
the gingerbread man would already begone. He would've already run off,
so they would get their next clue tohelp them navigate to the playground,
so on and so forth. They would go tothe nurse's office, the principal,
the library,
all of the critical places that theywould be going through on a daily basis or
when they needed to within the school.
(09:37):
And it reminds me that there was alsoa teacher I once interviewed who used
orienteering skills with her students.Have you ever heard of orienteering?
The connection I'm making isto something like Geocashing,
but I think you shouldhelp me understand it.
Yeah, that's really similar.
So it's this idea thatchildren would find their way,
places path finding and way finding arealso spatial reasoning skills that are
(10:01):
applied within our real world.
And so while it maynot be as scientific or
sophisticated as doing geocaching,
it has children with theidea of navigating in our real world helps them start
to learn cardinality and the differentways of thinking about traversing to a
different location,
which these are all things that mightbetter relate to social studies or
(10:25):
technology, other STEMdomains specifically,
but that are undergirdedby the spatial reasoning,
which does have thosemathematics connections.
I think the first thing that occurredis all of the directional language that
could emerge from something liketrying to find the J of red boy.
And then the other piece that youmade me think about just now is this
(10:47):
opportunity to quantifydistance in different ways,
and I'm sure there are otherthings that you could draw out,
especially in a play setting where thestructure is a little bit looser and it
gives you a little bitmore space, as you said,
to respond to kids rather than feelinglike you have to impose the structure.
Yeah, absolutely.
There's an ability when teachers areengaging in authentic ways with the
(11:10):
students, that they're able tosupport language development, support,
ideation and creation withoutnecessarily having kids sit down
and fill out a worksheetthat says, where is the ball?
The ball is sitting ontop of the shelf. Instead,
we can be on the floor working withstudents and providing those directions
(11:31):
of, oh, hey,
I need you to get me those materialsfrom the shelf on the other side of the
room,
but thinking about how can I say thatin a way that better supports children
understanding the spatial reasoningthat's occurring in our room.
So maybe it's find the pencilinside the blue cup on top of
the shelf that's behindthe pencil sharpener,
(11:53):
getting really specific in the waysthat we talk about things so that we're
ingraining those ideas in such a waythat it becomes part of the way that the
kids communicate as well.
You have me thinking that there's anintentionality in language choice that can
create that, but then Iwould imagine as a teacher,
I could also revoice what students aresaying and perhaps introduce language in
(12:15):
that way as well.
Yeah,
and now you have me thinking abouta really fun routine number talks,
of course.
And if we do the idea of a.talkinstead of a number talk,
thinking about the spatial structuringof the dots that we're seeing and the
different ways that you cansee those arrangements and
describe the quantificationof the arrangement.
(12:39):
It's a nice way to introduce educatorsto spatial reasoning because it might be
something that they're already doing inthe classroom while also providing an
avenue for children to see spatialstructuring in a way that they're already
accustomed to as well based on theroutines that they're receiving from the
teacher.
I think what's really exciting aboutthis, Robin, is the more that we talk,
(13:00):
the more two things jump out.
I think one is my language choicesallow me to introduce these
ideas in a way that I don't know thatI'd thought about as a practitioner.
Part two is that we can'treally necessarily draw a distinction between work
we're doing around numbers and quantityand spatial reasoning that there are
opportunities within our workaround number quantity and within
(13:23):
math content to inject the language ofspatial reasoning and have it become a
part of the experience for students.
Yeah,
and that's important that I haveconveyed that without explicitly
saying it because that's the very workthat I'm doing with teachers in their
classrooms at this time. One,
(13:44):
as you're talking aboutlanguage, and I hate to do this,
but I'm going to take us a littlebit off topic for a moment.
I keep seeing this idea on Twitteror whatever we call it at this point,
that some people actually don'thear music in their heads.
This idea is wild to me because I havesongs playing in my head all the time,
(14:06):
but at the same time,
what if we think about the ideathat some people don't also
visualize things,
they don't imagine thosemovements continuously that I just
see. And so as teachers,
we really need to focus on that sameidea that children need opportunities
to practice what we thinkthey should be able to hear,
(14:29):
but also practice what we thinkthey should be able to see.
I'm not a cognitive scientist. Ican't see inside someone's head,
but I am a teacher by trade,
so I want to emphasize that teachers cando what's within their locus of control
so that children can have opportunitiesto talk about those tasks.
One that I recently sawwas a lesson on clocks.
(14:51):
So while I was sitting there watchingher teach, she was using a Judy clock.
She was having fun games with the kidsto do a little competition where they
could read the clock andtell her what time it was.
But I was just starting to think aboutall of the ways that we could talk about
the shorter and longer hands,the minute and our hands,
the ways that we could talk about themrotating around that center point.
(15:13):
What shape does the hand make as itgoes around that center point and what
happens if it doesn't rotate fully?
Now I'm going back to those fractionalideas from earlier with the which one
doesn't belong tasks of havingfull shapes versus half shapes,
and how we see those shapes inour real lives that we can then
relate with visualized shapes thatsome children may or may not be able to
(15:36):
see.
You have me thinking aboutsomething. First of all,
I'm so glad that you mentionedthe role of visualization.
You had me thinking about a conversationI was having with a colleague a while
ago,
and we had read a text that we werediscussing and the point of conversation
came up. I read this and there's acertain image that popped into my head,
and the joke we were making is I'm prettycertain that the image that I saw in
(15:58):
my head having read this textis not the same as what you saw.
What you said that really struck homefor me is I might be making some real
assumptions about the pictures thatkids see in their head and helping build
those internal images, those mentalmovies. That's a part of our work as well.
Absolutely. Because I'm thinking aboutthe way that we have prototypical shapes.
(16:20):
So a few years ago I was workingwith some assessments and
the children were supposed to be ableto recognize an equilateral triangle,
whether it was gravity basedor facing another orientation,
and there were some childrenwho automatically could see that the triangle was a
triangle no matter whichdirection it was pointing,
(16:44):
whereas others only recognize it ifa triangle, if it were gravity based.
And so we need to be teaching theproperties of the shapes beyond just that
image recognition that oftentimesour younger students come out with.
I tend to think of visualizationand language as supporting one another with the
(17:04):
idea that when we are talking, we'realso writing a descriptive essay.
Our words are what createthe intended picture.
Can't say that it's alwaysthe picture that comes out,
but the intended picture for the audience.
What we're hopeful for in classroomsis that because we're sharing physical
spaces and tangible experiences that
(17:27):
the language used around thoseexperiences could create shared meaning.
That's one of the most difficult piecesin talking about spatial reason or quite
frankly, anything else,
is that oftentimes our words may havedifferent meanings depending on who the
speaker and who the listener are.
And so navigating what those differencesare can be quite challenging,
(17:52):
which is why spatial reasoningis still so hard to define.
Absolutely.
My other follow up is if you were tooffer people a way to get started,
particularly on visualization,
is there a kind of task that you imaginemight move them along that pathway?
I think the first thing to do isreally grasp an approximation.
(18:12):
I'm not going to say figureout what spatial reasoning is,
but just an approximation or acouple of the skills therein that
you feel comfortable with.
So spatial reasoning is really the setof skills that undergirds almost all of
our daily actions,
but it also can be insertedinto the lessons that teachers are already teaching.
(18:33):
I think that we do have to acknowledgethat spatial reasoning is hard to define,
but the good news is that we doreason spatially all day every day.
If I am in a classroom,
I want to look first at theteaching that's happening,
the routines that are already there,
and see where some spatial reasoningmight actually fit in with our young
(18:54):
classes. I like to thinkabout calendar math.
Every single kindergarten, firstgrade classroom that you walk into,
they're going to havethat calendar on the wall,
so how can you work into theroutines that are occurring,
that spatial language to describe thedifferent components of the routine?
So as a kiddo is counting on thathundreds chart talking about the ways
(19:18):
in which they're moving thepointer along the numbers.
When they're counting by tens,
talk about the ways that they're movingdown when they're finding the patterns
that are on the calendar,
because all of those littlecalendar numbers for the day,
they wind up having a pattern withinthem in most of the curricular kits.
So thinking about just the ways thatwe can use language they're in now with
(19:39):
older students,
I think that offering that variety ofmodels or manipulatives for them to use
and then encourage them to translatefrom having a concrete manipulative
into those more representationalideas is great regardless of age or
grade.
So students benefit from the modelingwhen they do diagramming of their
(19:59):
models.
That is translating the 3D model to 2D,
which is another componentof spatial reasoning.
And that gets me to thissticky point of I'm not
arguing against automaticity or beingable to solve equations without physical
or visual models,
but I'm just acknowledging this idea thatoffering alternative ways for students
(20:23):
to engage with content is reallycritical because we're no longer at
a phase that we need ourchildren to become computers.
We have programs for that.
We need children who are able tothink and solve problems in novel
ways because that's the directionthat we're moving in problem solving.
That's fantastic. My finalquestion before we close things up.
(20:47):
If you were to make a recommendationfor someone who's listening and they're
intrigued and they want to keep learning,
are there any particular resources thatyou'd offer people that they might be
able to go to?
Yeah, absolutely.
So the first one that I like isthe Learning Trajectories website.
It's a learning trajectories.org. It'sproduced by Doug Clements and Julie Sama.
(21:08):
There are wonderful tasksthat are associated with
spatial reasoning skills fromvery young children in the
infants and toddler stages all theway up until seven or eight years old.
So that's a great place to go thatwill allow you to see how children are
performing in differentareas of spatial reasoning.
(21:29):
There is also a book calledTaking Shape by Kathy Bruin,
colleagues that I believewas produced in 2016,
and the grade levels might be a littlebit different because it is on the
Canadian school system, butit's for K through two students,
and that offers both the tasks andthe spatial reasoning skills that are
associated with them. Formore of the research side,
(21:53):
there's a book by Brent Davis andthe Spatial Reasoning study group
called Spatial Reasoningin the early Years,
and that volume has been one of my go-tos
in understanding both the history ofspatial reasoning in our schools and also
ways to start thinking aboutspatializing school mathematics.
One of the things that I really appreciateabout this conversation is you've
(22:16):
helped me make a lot moresense of spatial reasoning,
but the other thing thatyou've done for me at least,
is see that there are ways that Ican make choices with my planning,
with my language that I couldpick up and do tomorrow.
There's not a discreet separate bitthat is about spatial reasoning.
It's really an integrated set of ideasand concepts and skills that I can
(22:39):
start to build upon right awaywhatever curriculum I have.
And that's the point.Often in mathematics,
we think more explicitlyabout algebraic or numeric
reasoning, but less frequently inclassrooms about spatial reasoning.
But spatial reasoning supportsnot only mathematics development,
but other stem domains as well,
(22:59):
and even skills that crossover intosocial studies and language arts as we're
talking about mapping, aswe're talking about language.
So as students have these experiences,
they too can start tomathematize the world,
see spatial connectionsas they go out to recess,
as they go home from school,
as they're walking through theirneighborhoods or just around the house.
(23:22):
And it's ingrained ideas of measurementthat we are looking at on a daily
basis, the ways that we plan outour days and plan out our movements,
whether it's really a plan or justour reactions to the world that
support building these skills over time.
And so there are those reallypractical applications,
(23:44):
but it also comes to supporting overallmathematics development and then later
STEM career interests,
which is why I get excited about thework and want to be able to share it with
more and more people.
I think that's a greatplace to stop for listeners.
We're going to link all of the contentthat Robin shared to our show notes.
And Robin, I'll just say again,thank you so much for joining us.
(24:05):
It's really been apleasure talking with you.
Yes, absolutely. Thanks so much.
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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