William Zahner, Understanding the Role of Language in Math Classrooms
ROUNDING UP: SEASON 3 | EPISODE 17
How can educators understand the relationship between language and the mathematical concepts and skills students engage with in their classrooms?
And how might educators think about the mathematical demands and the language demands of tasks when planning their instruction?
In this episode, we discuss these questions with Bill Zahner, director of the Center for Research in Mathematics and Science Education at San Diego State University.
BIOGRAPHY
Bill Zahner is a professor in the mathematics department at San Diego State University and the director of the Center for Research in Mathematics and Science Education. Zahner's research is focused on improving mathematics learning for all students, especially multilingual students who are classified as English Learners and students from historically marginalized communities that are underrepresented in STEM fields.
RESOURCES
National Council of Teachers of Mathematics
Mathematics Teacher: Learning and Teaching PK– 12
English Learners Success Forum
SDSU-ELSF Video Cases for Professional Development
The Math Learning Center materials
Bridges in Mathematics curriculum
Bridges in Mathematics Teachers Guides [BES login required]
TRANSCRIPT
Mike Wallus: How can educators understand the way that language interacts with the mathematical concepts and skills their students are learning? And how can educators focus on the mathematics of a task without losing sight of its language demands as their planning for instruction? We'll examine these topics with our guest, Bill Zahner, director of the Center for Research in Mathematics and Science Education at San Diego State University.
Welcome to the podcast, Bill. Thank you for joining us today.
Bill Zahner: Oh, thanks. I'm glad to be here.
Mike: So, I'd like to start by asking you to address a few ideas that often surface in conversations around multilingual learners and mathematics. The first is the notion that math is universal, and it's detached from language. What, if anything, is wrong with this idea and what impact might an idea like that have on the ways that we try to support multilingual learners?
Bill: Yeah, thanks for that. That's a great question because I think we have a common-sense and strongly held idea that math is math no matter where you are and who you are. And of course, the example that's always given is something like 2 plus 2 equals 4, no matter who you are or where you are. And that is true, I guess [in] the sense that 2 plus 2 is 4, unless you're in base 3 or something. But that is not necessarily what mathematics in its fullness is. And when we think about what mathematics broadly is, mathematics is a way of thinking and a way of reasoning and a way of using various tools to make sense of the world or to engage with those t
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(00:03):
How can educators understand theway that language interacts with the
mathematical concepts and skillstheir students are learning?
And how can educators focuson the mathematics of a task without losing sight of
its language demands as theirplanning for instruction?
We'll examine these topicswith our guest, bill Zaner,
director of the Center for Research inMathematics and Science Education at San
(00:27):
Diego State University.Welcome to the podcast, bill.
Thank you for joining us today.
Oh, thanks. I'm glad to be here.
So I'd like to start by asking you toaddress a few ideas that often surface in
conversations around multilinguallearners and mathematics.
The first is the notion that mathis universal and it's detached from
(00:49):
language. What, if anything,
is wrong with this idea and what impactmight an idea like that have on the
ways that we try to supportmultilingual learners?
Yeah, thanks for that.
That's a great question because Ithink we have a common sense and
strongly held idea that math ismath no matter where you are and who
(01:11):
you are. And of course,
the example that's always given issomething like two plus two equals four,
no matter who you are or whereyou are. And that is true,
I guess on the sense that two plus twois four unless you're in base three or
something.
But that is not necessarilywhat mathematics in
(01:31):
its fullness is.
And when we think aboutwhat mathematics broadly is,
mathematics is a way of thinkingand a way of reasoning and a way
of using various toolsto make sense of the
world or to engage withthose tools on their own
(01:52):
right.
And oftentimes that is deeplyembedded with language.
Probably the most straightforwardexample is anytime I ask someone to
justify or explain what they'rethinking in mathematics,
I'm immediately bringingin language into that case.
And we all know the old funny exampleswhere a kid is asked to show their
(02:16):
thinking and they draw a diagram ofthemselves with a thought bubble on a math
problem. And that's a really good casewhere I think a teacher can say, okay,
clearly that was not what I had inmind when I said show your thinking.
And instead,
the demand or the request was for astudent to show their reasoning or their
(02:36):
thought process typically in wordsor in a combination of words and
pictures and equations.
And so there's where I see thisidea that math is detached from
language is something of a myththat there's actually a lot of
mathematics.
And the interesting part of mathematicsis often deeply entwined with language.
(02:59):
So that's my first responseand thought about that.
And if you look at our common corestate standards for mathematics,
especially those standardsfor mathematical practice,
you see all sorts of connectionsto communication and to language
interspersed throughout those standards.
So create viable arguments,
that's a language practiceand even attend to precision,
(03:22):
which most of us tend to thinkof as round appropriately.
But when you actuallyread the standard itself,
it's really aboutmathematical communication and definitions and using those
definitions with precision.Again, that's an example,
bringing it right back into the schoolmathematics domain where language and
mathematics are somewhat inseparablefrom my perspective here.
(03:45):
That's really helpful.
So the second idea that I often hearis the best way to support multilingual
learners is by focusingon facts or procedures.
And that language comes later, forlack of a better way of saying it.
And it seems like this isconnected to that first notion,
but I wanted to ask the question again,
what if anything is wrong with thisidea that a focus on facts or procedures
(04:09):
with language coming after the fact,
what impact do you suspect that thatwould have on the way that we support
multilingual learners?
So that's a great question too, becausethere's a grain of truth, right?
Both of these questions havesimultaneously a grain of truth and simultaneously
a fundamental problem in them.
So the grain of truth and an experiencethat I've heard from many folks who
(04:32):
learned mathematics in a secondlanguage was that they felt more
competent in mathematicsthan they did in say,
a literature class where the onlyactivity was engaging with texts
or engaging with words becausethere was a connection to the
numbers and to symbolsthat we're familiar.
(04:52):
So on one level,
I think that this idea of focusing onfacts or procedures comes out of this
observation that sometimes an emergentmultilingual student feels most
comfortable in thatcontext, in that setting.
But then the second part of the answergoes back to this first idea that really
what we're trying to teach students inschool mathematics now is not simply
(05:15):
or only how to applyprocedures to really big
numbers or to know your timestables fast.I think we have a much more
ambitious goal when it comes toteaching and learning mathematics.
That includes explaining,justifying, modeling,
using mathematics toanalyze the world and so on.
(05:37):
And so those practices aredeeply tied with language and
deeply tied with using communication.
And so if we want to develop those, well,
the best way to do that is to developthem to think about what are the
scaffolds,
what are the supports that we need tointegrate into our lessons or into our
(05:58):
designs to make that possible?
And so that might be the takeaway there,
is that if you simplylook at mathematics as
calculations, then this could be true.
But I think our vision of mathematicsis much broader than that,
and that's where I see this potential.
That's really clarifying.
(06:18):
I think the way that you unpack that isif you view mathematics as simply a set
of procedures or calculationsmaybe. But I would agree with you.
What we want for students isactually so much more than that.
One of the things that I heard you saywhen we were preparing for this interview
is that at the elementary level,
learning mathematics isa deeply social endeavor.
(06:41):
Tell us a little bit aboutwhat you mean by that bill.
Sure. So mathematics itself,
maybe as a premise is a social activity.
It's created by humans as a way ofengaging with the world and a way of
reasoning.
So the learning ofmathematics is also social
in the sense that we're givingstudents an introduction to this way of
(07:05):
engaging in the world usingnumbers and quantities and shapes
in order to make sense of our environment.
And when I think aboutlearning mathematics,
I think that we are not simply downloadingknowledge and sticking it into our
heads.
And in the modern day where artificialintelligence and computers can
(07:26):
do almost every calculationthat we can imagine,
although your AI may do itincorrectly just as a fair warning,
but in the modern day,
the actual answer is notwhat we're so focused on.
It's actually the processand the reasoning and the
modeling and justification ofthose choices. And so when I think
(07:49):
about learning mathematics aslearning to use these language tools,
learning to use these waysof communication, how do we learn to communicate?
We learn to communicate byengaging with other people,
by engaging with the ideas and theminds and the feelings and so on
of the folks around us, whetherit's the teacher and the student,
(08:11):
the student and the student,the whole class and the teacher.
That's where I really see the power.And most of us who have learned,
I think can attest to the fact thateven when we're engaging with a text,
really fundamentally we're engaging withsomething that was created by somebody
else. So fundamentally,
even when you're sitting by yourselfdoing a math word problem or doing
(08:33):
calculations,
someone has given that to you and youthink that that's important enough to do,
right? So from that stance,
I see all of teaching andlearning. Mathematics is social,
and maybe one of our goals inmathematics classrooms is beyond
memorizing the timestables, is learningto communicate with other people,
(08:53):
learning to be in thisactivity with other folks.
One of the things that strikes meabout what you were saying, bill,
is there's this kind ofvirtuous cycle, right?
That by engaging with language andhaving the social aspect of it,
you're actually alsodeepening the opportunity for students to make sense of the
math.
(09:14):
You're building the scaffolds thathelp kids communicate their ideas as
opposed to removing or stripping outthe language that's the context in some
ways that helps themfilter and make sense.
You could either be in a vicious cycle,
which comes from removing thelanguage or a virtuous cycle.
And it seems a little counterintuitivebecause I think people perceive language
(09:35):
as the thing that is holding kids backas opposed to the thing that might
actually help them moveforward and make sense.
And actually that's one of the reallyinteresting pieces that we've looked at in
my research and the broaderresearch is this question of
what makes mathematicslinguistically complex
(09:57):
is a complicated question.
And so sometimes we think of things likelooking at the word count as a way to
say, if there are fewer words, it's lesscomplex. And if there are more words,
it's more complex. Butthat's not totally true.
And similarly, if there's no context,it's easier or more accessible.
(10:18):
And if there is a context,then it's less accessible.
And I don't see these as binary choices.
I see these as happeningon a somewhat complicated
terrain where we want tothink about how do these words
or these contexts addto student understanding
(10:40):
or potentially impede?
And that's where I think this socialaspect of learning mathematics,
as you described,
it could be a virtuous cycle sothat we can use language in order to
engage in the processof learning language.
Or the vicious cycle is youwithhold all language and then
get frustrated when studentscan't apply their mathematics.
(11:03):
Maybe the most stereotypicalanswer, my kids can do this,
but as soon as they get a word problem,they can't do it. And it's like, well,
did you give them opportunitiesto learn how to do this?
Or is this the first time?That would explain a lot.
Well, it's an interesting question too,
because I think what sits behind that insome ways is the idea that you're kind
of going to reach a point,
(11:23):
or students might reach a point wherethey're ready for word problems.
And I think what we're really saying isit's actually through engaging with word
problems that you build your proficiency,
your skillset that actually allows youto become a stronger mathematician.
Right? Exactly. And it'sa daily practice, right?
(11:44):
It's not something that you justhold off to the end of the unit,
and then you have the word problems,
but it's part of the process of learningand thinking about how you integrate
and support that. That's the keyquestion that I really wrestle with.
Not trivial, but I think that's the keyand the most important part of this.
Well,
I think that's actually a really goodsegue because I wanted to shift and talk
(12:04):
about some of the concrete or productiveways that educators can support
multilingual learners. And inpreparing for this conversation,
one of the things that I've heard youstress is this notion of a consistent
context.
So can you just talk a little bit moreabout what you mean by that and how
educators can use that when they'relooking at their lessons or when they're
(12:25):
writing lessons or looking at thecurriculum that they're using?
Absolutely. In our past work,
we engaged in some cyclesof design research with
teachers looking at theirmathematics curriculum and
opportunities to engage multilinguallearners in communication and reasoning in
the classroom.
(12:46):
And one of the surprising thingsthat we found just by looking
at a couple of standard textbookswas a surprising number of
contexts were introduced thatare all related to the same
concept.
So the concept would be somethinglike rate of change or ratio,
and then the contexts,
(13:08):
there would be a half dozen of themin the same section of the book. Now,
this was, I should sayat a secondary level,
so not quite where most ofthe bridges work is happening,
but I think it's an interesting lessonfor us that we took away from this.
Actually at the elementary level,
Catherine Cheval has madethe same observation.
What we realized was thatcontexts are not good or bad by
(13:31):
themselves.In fact,
they can be highly supportive ofstudent reasoning or they can get in the
way, and it's how theyare used and introduced.
And so the other way we thoughtabout this was when you introduce a
context,
you want to make sure that that contextis one that you give sufficient time for
the students to understand andto engage with that is relatable,
(13:54):
that everyone has access to it,
not something that's just completelyunrelated to students' experiences.
And then you can reallyleverage that relatable,
understandable contextfor multiple problems and
iterations and opportunitiesto go deeper and deeper.
To give a concrete example of that,
when we were looking at thisratio and rate of change,
(14:17):
we went all the way back to one ofthe fundamental contexts that's been
studied for a long time,
which is motion and speed and distanceand time. And that seemed like a
really important topic because we knowthat that starts all the way back in
elementary school and continues throughcollege level physics and beyond.
So it was a rich context.
(14:38):
It was also something that was accessiblein the sense that we could do things
like act out story problemsor reenact a race that's
described in a story problem.
And so the students themselves hadaccess to the context in a deep way.
And then last,
that context was one that we couldcome back to again and again,
(15:01):
so we could do variations onthat context on that story.
And I think there's lots of examplesof materials out there that start off
with a core context and build it out.
And thinking of some ofthe bridges materials,
even on the countingand the multiplication,
I think there's stories of the insectsand their legs and wings and counting and
multiplying, and that's a reallynice example of it's accessible.
(15:25):
You can go find insects almostanywhere you are. Kids like it,
they enjoy thinking about insectsand other icky, creepy crawly things.
And then you can take that and run withit in lots of different ways, right?
Counting multiplication,division ratio, and so on.
This last bit of our conversation has methinking about what it might look like
(15:46):
to plan a lesson for a class ora group of multilingual learners.
And I know that it's important that Ithink about mathematical demands as well
as the language demands of a given task.
Can you unpack why it's importantto set math and language development
learning goals for atask or a set of tasks,
and what are the opportunitiesthat come along with that?
(16:10):
If I'm thinking about both ofthose things during my planning?
Yeah, that's a great question, andI want to mark the shift, right?
We've gone from thinking about the demandsto thinking about the goals and where
we're going to go next.
And so when I think aboutintegrating mathematical goals,
mathematical learning goalsand language learning goals,
(16:30):
I often go back to these ideasthat we call like the practices or
these standards that are abouthow you engage in mathematics.
And then I think about linkingthose back to the content itself.
And so there's kind of atwo piece element to that.
And so when we're setting ourgoals and lesson planning,
at least here in thegreat state of California,
(16:52):
sometimes we'll have these templates thathave what standard are you addressing?
What language standard are you addressing?What ELD standard are you addressing?
What SEL standard are you addressing?
And I've seen sometimes teachers approachthat as a checkbox, right? Tick, tick,
tick, tick, tick. But I seethat as a missed opportunity.
If you just look at this,
you're plugging things in because as westarted with talking about how learning
(17:15):
mathematics is deeply social andintegrated with language that we can
integrate the mathematical goalsand the language goals in a lesson.
And I think really good materials shouldbe suggesting that to the teacher,
you shouldn't be doing thisyourself every day from scratch.
But I think really highquality materials will say,
(17:36):
here's the mathematical goal andhere's an associated language goal,
whether it's productive orreceptive functions of language.
And here's how the language goalconnects the mathematical goal.
Now, just to get really concrete,
if we're talking about anexample of reasoning with ratios,
so I was going back to that,then it might be generalized,
(17:59):
the relationship between distance andtime and that the ratio of distance
and time gives you thisquantity called speed,
and that different combinationsof distance and time can lead to the same speed.
And so explain and justifyand show using words,
pictures, diagrams. So thatwould be a language goal,
(18:20):
but it's also very mucha mathematical goal.
And I guess I see the mathematicalcontent, the practices,
and the language really braidedtogether in these goals.
And that I think is the ideal,and at least from our work,
has been most powerful andproductive for students.
This is off script, but I'm going toask it and you can pass if you want to.
(18:42):
I wonder if you could just share alittle bit about what the impact of those
kind of practices that you described,have you seen what that impact looks like?
Either for an educator who has made thestep and is doing that integration or
for students who are in a classroomwhere an educator is purposely thinking
about that level of integration?
Yeah, I can talk a little bitabout that. In our research,
(19:04):
we have tried to measure theeffects of some of these efforts.
It is a difficult thing to measurebecause it's not just a simple true false
test question type of thing that youcan give a multiple choice test for.
But one of the ways that we've lookedfor the impact that these types of
intentional designs is bylooking at patterns of student
participation in classroomdiscussions and seeing who is
(19:29):
accessing the floor ofthe discussion and then
looking at other resultslike giving an assessment,
but deeper than looking at the outcome,
the binary correct versus incorrect,
also looking at the quality ofthe explanation that's provided.
So how you justify an answer.
(19:51):
Does the student provide adeeper or a more mathematically
complete explanation?
That is an area where I thinkmore investigation is needed,
and it's also very hardto vary systematically. So from a research perspective,
you may not want to putthis into the final version,
but from a research perspective,
it's very hard to fix and isolatethese things because they are
(20:14):
integrated,
because language and mathematicsare so deeply integrated
that trying to fix everything and do this,
what caused this water to taste likewater? Was it the hydrogen or the oxygen?
Well, you can't reallypull those apart, right?
The water molecule ishydrogen and oxygen together.
(20:36):
I think that's a lovely analogy for whatwe were talking about with mathematical
goals and language goals.
That I think is really a helpful way tothink about the extent to which they're
intertwined one another.
Yeah, I need to give fullcredit to Vygotsky, I think,
who said that something might beVygotsky. I'll need to check my notes.
I think you're in good company if you'requoting Vygotsky. Before we close,
(20:59):
I'd love to just ask you abit about resources. I say this often on the podcast.
We have 20 to 25 minutesto dig deeply into an idea,
and I know people who are listening oftenthink about, where do I go from here?
Are there any particular resources thatyou would suggest for someone who wanted
to continue learning about what it is tosupport multilingual learners in a math
(21:22):
classroom?
Sure. Happy to share that.
So I think on the individualand collective level,
so say a group of teachers,
there's a beautiful book by CatherineCheval and her colleagues about supporting
multilingual learners and mathematics,
and I really see thatas a valuable resource.
I've used that in reading groups withteachers and use that in book studies,
(21:45):
and it's been very productiveand powerful for us. Beyond that,
of course, I think the NCTM providesa number of really useful resources,
and there are articles, for example,
in the mathematics teacherlearning and teaching PK to 12,
that could make for a really wonderfulstudy or opportunity to engage more
(22:05):
deeply. And then I would sayon a broader perspective,
I've worked with organizations likethe English Learner Success Forum and
others.
We've done some case studies and littlekind of classroom studies that are
accessible on my website,
so you can go to that.But there's also from that organization,
some really valuable insights ifyou're looking at adopting new
(22:27):
materials or evaluating things,
that gives you a principledset of guidelines to follow.
And I think that's really helpful foreducators because we don't have to do
this all on our own.
This is not a reinvent the wheel atevery single site kind of situation.
And so I always encourage people tolook for those resources. And of course,
(22:49):
I will say that the MLC materials,the bridges and mathematics,
I think have been reallybeautifully designed with a lot of
these principles rightbehind them. So for example,
if you look through the teacher'sguides on the bridges and mathematics,
those integrated math and language andpractice goals are a part of the design.
(23:11):
Well, I think that's a great place tostop. Thank you so much for joining us,
bill. This has been insightful and it'sreally been a pleasure talking with you.
Oh, well thank you. I appreciate it.
And that's a wrap forseason three of Rounding Up.
I want to thank all of our guests andthe MLC staff who make these podcasts
possible, as well as all ofour listeners for tuning in.
(23:33):
Have a great summer and we'll beback in September for season four.
This podcast is brought to you by theMath Learning Center and the Meyer Math
Foundation dedicated to inspiring andenabling all individuals to discover
and develop their mathematicalconfidence and ability.
How can educators understand theway that language interacts with the
mathematical concepts and skillstheir students are learning?
And how can educators focuson the mathematics of a task without losing sight of
its language demands as theirplanning for instruction?
We'll examine these topicswith our guest, bill Zaner,
director of the Center for Research inMathematics and Science Education at San
(00:27):
Diego State University.Welcome to the podcast, bill.
Thank you for joining us today.
Oh, thanks. I'm glad to be here.
So I'd like to start by asking you toaddress a few ideas that often surface in
conversations around multilinguallearners and mathematics.
The first is the notion that mathis universal and it's detached from
(00:49):
language. What, if anything,
is wrong with this idea and what impactmight an idea like that have on the
ways that we try to supportmultilingual learners?
Yeah, thanks for that.
That's a great question because Ithink we have a common sense and
strongly held idea that math ismath no matter where you are and who
(01:11):
you are. And of course,
the example that's always given issomething like two plus two equals four,
no matter who you are or whereyou are. And that is true,
I guess on the sense that two plus twois four unless you're in base three or
something.
But that is not necessarilywhat mathematics in
(01:31):
its fullness is.
And when we think aboutwhat mathematics broadly is,
mathematics is a way of thinkingand a way of reasoning and a way
of using various toolsto make sense of the
world or to engage withthose tools on their own
(01:52):
right.
And oftentimes that is deeplyembedded with language.
Probably the most straightforwardexample is anytime I ask someone to
justify or explain what they'rethinking in mathematics,
I'm immediately bringingin language into that case.
And we all know the old funny exampleswhere a kid is asked to show their
(02:16):
thinking and they draw a diagram ofthemselves with a thought bubble on a math
problem. And that's a really good casewhere I think a teacher can say, okay,
clearly that was not what I had inmind when I said show your thinking.
And instead,
the demand or the request was for astudent to show their reasoning or their
(02:36):
thought process typically in wordsor in a combination of words and
pictures and equations.
And so there's where I see thisidea that math is detached from
language is something of a myththat there's actually a lot of
mathematics.
And the interesting part of mathematicsis often deeply entwined with language.
(02:59):
So that's my first responseand thought about that.
And if you look at our common corestate standards for mathematics,
especially those standardsfor mathematical practice,
you see all sorts of connectionsto communication and to language
interspersed throughout those standards.
So create viable arguments,
that's a language practiceand even attend to precision,
(03:22):
which most of us tend to thinkof as round appropriately.
But when you actuallyread the standard itself,
it's really aboutmathematical communication and definitions and using those
definitions with precision.Again, that's an example,
bringing it right back into the schoolmathematics domain where language and
mathematics are somewhat inseparablefrom my perspective here.
(03:45):
That's really helpful.
So the second idea that I often hearis the best way to support multilingual
learners is by focusingon facts or procedures.
And that language comes later, forlack of a better way of saying it.
And it seems like this isconnected to that first notion,
but I wanted to ask the question again,
what if anything is wrong with thisidea that a focus on facts or procedures
(04:09):
with language coming after the fact,
what impact do you suspect that thatwould have on the way that we support
multilingual learners?
So that's a great question too, becausethere's a grain of truth, right?
Both of these questions havesimultaneously a grain of truth and simultaneously
a fundamental problem in them.
So the grain of truth and an experiencethat I've heard from many folks who
(04:32):
learned mathematics in a secondlanguage was that they felt more
competent in mathematicsthan they did in say,
a literature class where the onlyactivity was engaging with texts
or engaging with words becausethere was a connection to the
numbers and to symbolsthat we're familiar.
(04:52):
So on one level,
I think that this idea of focusing onfacts or procedures comes out of this
observation that sometimes an emergentmultilingual student feels most
comfortable in thatcontext, in that setting.
But then the second part of the answergoes back to this first idea that really
what we're trying to teach students inschool mathematics now is not simply
(05:15):
or only how to applyprocedures to really big
numbers or to know your timestables fast.I think we have a much more
ambitious goal when it comes toteaching and learning mathematics.
That includes explaining,justifying, modeling,
using mathematics toanalyze the world and so on.
(05:37):
And so those practices aredeeply tied with language and
deeply tied with using communication.
And so if we want to develop those, well,
the best way to do that is to developthem to think about what are the
scaffolds,
what are the supports that we need tointegrate into our lessons or into our
(05:58):
designs to make that possible?
And so that might be the takeaway there,
is that if you simplylook at mathematics as
calculations, then this could be true.
But I think our vision of mathematicsis much broader than that,
and that's where I see this potential.
That's really clarifying.
(06:18):
I think the way that you unpack that isif you view mathematics as simply a set
of procedures or calculationsmaybe. But I would agree with you.
What we want for students isactually so much more than that.
One of the things that I heard you saywhen we were preparing for this interview
is that at the elementary level,
learning mathematics isa deeply social endeavor.
(06:41):
Tell us a little bit aboutwhat you mean by that bill.
Sure. So mathematics itself,
maybe as a premise is a social activity.
It's created by humans as a way ofengaging with the world and a way of
reasoning.
So the learning ofmathematics is also social
in the sense that we're givingstudents an introduction to this way of
(07:05):
engaging in the world usingnumbers and quantities and shapes
in order to make sense of our environment.
And when I think aboutlearning mathematics,
I think that we are not simply downloadingknowledge and sticking it into our
heads.
And in the modern day where artificialintelligence and computers can
(07:26):
do almost every calculationthat we can imagine,
although your AI may do itincorrectly just as a fair warning,
but in the modern day,
the actual answer is notwhat we're so focused on.
It's actually the processand the reasoning and the
modeling and justification ofthose choices. And so when I think
(07:49):
about learning mathematics aslearning to use these language tools,
learning to use these waysof communication, how do we learn to communicate?
We learn to communicate byengaging with other people,
by engaging with the ideas and theminds and the feelings and so on
of the folks around us, whetherit's the teacher and the student,
(08:11):
the student and the student,the whole class and the teacher.
That's where I really see the power.And most of us who have learned,
I think can attest to the fact thateven when we're engaging with a text,
really fundamentally we're engaging withsomething that was created by somebody
else. So fundamentally,
even when you're sitting by yourselfdoing a math word problem or doing
(08:33):
calculations,
someone has given that to you and youthink that that's important enough to do,
right? So from that stance,
I see all of teaching andlearning. Mathematics is social,
and maybe one of our goals inmathematics classrooms is beyond
memorizing the timestables, is learningto communicate with other people,
(08:53):
learning to be in thisactivity with other folks.
One of the things that strikes meabout what you were saying, bill,
is there's this kind ofvirtuous cycle, right?
That by engaging with language andhaving the social aspect of it,
you're actually alsodeepening the opportunity for students to make sense of the
math.
(09:14):
You're building the scaffolds thathelp kids communicate their ideas as
opposed to removing or stripping outthe language that's the context in some
ways that helps themfilter and make sense.
You could either be in a vicious cycle,
which comes from removing thelanguage or a virtuous cycle.
And it seems a little counterintuitivebecause I think people perceive language
(09:35):
as the thing that is holding kids backas opposed to the thing that might
actually help them moveforward and make sense.
And actually that's one of the reallyinteresting pieces that we've looked at in
my research and the broaderresearch is this question of
what makes mathematicslinguistically complex
(09:57):
is a complicated question.
And so sometimes we think of things likelooking at the word count as a way to
say, if there are fewer words, it's lesscomplex. And if there are more words,
it's more complex. Butthat's not totally true.
And similarly, if there's no context,it's easier or more accessible.
(10:18):
And if there is a context,then it's less accessible.
And I don't see these as binary choices.
I see these as happeningon a somewhat complicated
terrain where we want tothink about how do these words
or these contexts addto student understanding
(10:40):
or potentially impede?
And that's where I think this socialaspect of learning mathematics,
as you described,
it could be a virtuous cycle sothat we can use language in order to
engage in the processof learning language.
Or the vicious cycle is youwithhold all language and then
get frustrated when studentscan't apply their mathematics.
(11:03):
Maybe the most stereotypicalanswer, my kids can do this,
but as soon as they get a word problem,they can't do it. And it's like, well,
did you give them opportunitiesto learn how to do this?
Or is this the first time?That would explain a lot.
Well, it's an interesting question too,
because I think what sits behind that insome ways is the idea that you're kind
of going to reach a point,
(11:23):
or students might reach a point wherethey're ready for word problems.
And I think what we're really saying isit's actually through engaging with word
problems that you build your proficiency,
your skillset that actually allows youto become a stronger mathematician.
Right? Exactly. And it'sa daily practice, right?
(11:44):
It's not something that you justhold off to the end of the unit,
and then you have the word problems,
but it's part of the process of learningand thinking about how you integrate
and support that. That's the keyquestion that I really wrestle with.
Not trivial, but I think that's the keyand the most important part of this.
Well,
I think that's actually a really goodsegue because I wanted to shift and talk
(12:04):
about some of the concrete or productiveways that educators can support
multilingual learners. And inpreparing for this conversation,
one of the things that I've heard youstress is this notion of a consistent
context.
So can you just talk a little bit moreabout what you mean by that and how
educators can use that when they'relooking at their lessons or when they're
(12:25):
writing lessons or looking at thecurriculum that they're using?
Absolutely. In our past work,
we engaged in some cyclesof design research with
teachers looking at theirmathematics curriculum and
opportunities to engage multilinguallearners in communication and reasoning in
the classroom.
(12:46):
And one of the surprising thingsthat we found just by looking
at a couple of standard textbookswas a surprising number of
contexts were introduced thatare all related to the same
concept.
So the concept would be somethinglike rate of change or ratio,
and then the contexts,
(13:08):
there would be a half dozen of themin the same section of the book. Now,
this was, I should sayat a secondary level,
so not quite where most ofthe bridges work is happening,
but I think it's an interesting lessonfor us that we took away from this.
Actually at the elementary level,
Catherine Cheval has madethe same observation.
What we realized was thatcontexts are not good or bad by
(13:31):
themselves.In fact,
they can be highly supportive ofstudent reasoning or they can get in the
way, and it's how theyare used and introduced.
And so the other way we thoughtabout this was when you introduce a
context,
you want to make sure that that contextis one that you give sufficient time for
the students to understand andto engage with that is relatable,
(13:54):
that everyone has access to it,
not something that's just completelyunrelated to students' experiences.
And then you can reallyleverage that relatable,
understandable contextfor multiple problems and
iterations and opportunitiesto go deeper and deeper.
To give a concrete example of that,
when we were looking at thisratio and rate of change,
(14:17):
we went all the way back to one ofthe fundamental contexts that's been
studied for a long time,
which is motion and speed and distanceand time. And that seemed like a
really important topic because we knowthat that starts all the way back in
elementary school and continues throughcollege level physics and beyond.
So it was a rich context.
(14:38):
It was also something that was accessiblein the sense that we could do things
like act out story problemsor reenact a race that's
described in a story problem.
And so the students themselves hadaccess to the context in a deep way.
And then last,
that context was one that we couldcome back to again and again,
(15:01):
so we could do variations onthat context on that story.
And I think there's lots of examplesof materials out there that start off
with a core context and build it out.
And thinking of some ofthe bridges materials,
even on the countingand the multiplication,
I think there's stories of the insectsand their legs and wings and counting and
multiplying, and that's a reallynice example of it's accessible.
(15:25):
You can go find insects almostanywhere you are. Kids like it,
they enjoy thinking about insectsand other icky, creepy crawly things.
And then you can take that and run withit in lots of different ways, right?
Counting multiplication,division ratio, and so on.
This last bit of our conversation has methinking about what it might look like
(15:46):
to plan a lesson for a class ora group of multilingual learners.
And I know that it's important that Ithink about mathematical demands as well
as the language demands of a given task.
Can you unpack why it's importantto set math and language development
learning goals for atask or a set of tasks,
and what are the opportunitiesthat come along with that?
(16:10):
If I'm thinking about both ofthose things during my planning?
Yeah, that's a great question, andI want to mark the shift, right?
We've gone from thinking about the demandsto thinking about the goals and where
we're going to go next.
And so when I think aboutintegrating mathematical goals,
mathematical learning goalsand language learning goals,
(16:30):
I often go back to these ideasthat we call like the practices or
these standards that are abouthow you engage in mathematics.
And then I think about linkingthose back to the content itself.
And so there's kind of atwo piece element to that.
And so when we're setting ourgoals and lesson planning,
at least here in thegreat state of California,
(16:52):
sometimes we'll have these templates thathave what standard are you addressing?
What language standard are you addressing?What ELD standard are you addressing?
What SEL standard are you addressing?
And I've seen sometimes teachers approachthat as a checkbox, right? Tick, tick,
tick, tick, tick. But I seethat as a missed opportunity.
If you just look at this,
you're plugging things in because as westarted with talking about how learning
(17:15):
mathematics is deeply social andintegrated with language that we can
integrate the mathematical goalsand the language goals in a lesson.
And I think really good materials shouldbe suggesting that to the teacher,
you shouldn't be doing thisyourself every day from scratch.
But I think really highquality materials will say,
(17:36):
here's the mathematical goal andhere's an associated language goal,
whether it's productive orreceptive functions of language.
And here's how the language goalconnects the mathematical goal.
Now, just to get really concrete,
if we're talking about anexample of reasoning with ratios,
so I was going back to that,then it might be generalized,
(17:59):
the relationship between distance andtime and that the ratio of distance
and time gives you thisquantity called speed,
and that different combinationsof distance and time can lead to the same speed.
And so explain and justifyand show using words,
pictures, diagrams. So thatwould be a language goal,
(18:20):
but it's also very mucha mathematical goal.
And I guess I see the mathematicalcontent, the practices,
and the language really braidedtogether in these goals.
And that I think is the ideal,and at least from our work,
has been most powerful andproductive for students.
This is off script, but I'm going toask it and you can pass if you want to.
(18:42):
I wonder if you could just share alittle bit about what the impact of those
kind of practices that you described,have you seen what that impact looks like?
Either for an educator who has made thestep and is doing that integration or
for students who are in a classroomwhere an educator is purposely thinking
about that level of integration?
Yeah, I can talk a little bitabout that. In our research,
(19:04):
we have tried to measure theeffects of some of these efforts.
It is a difficult thing to measurebecause it's not just a simple true false
test question type of thing that youcan give a multiple choice test for.
But one of the ways that we've lookedfor the impact that these types of
intentional designs is bylooking at patterns of student
participation in classroomdiscussions and seeing who is
(19:29):
accessing the floor ofthe discussion and then
looking at other resultslike giving an assessment,
but deeper than looking at the outcome,
the binary correct versus incorrect,
also looking at the quality ofthe explanation that's provided.
So how you justify an answer.
(19:51):
Does the student provide adeeper or a more mathematically
complete explanation?
That is an area where I thinkmore investigation is needed,
and it's also very hardto vary systematically. So from a research perspective,
you may not want to putthis into the final version,
but from a research perspective,
it's very hard to fix and isolatethese things because they are
(20:14):
integrated,
because language and mathematicsare so deeply integrated
that trying to fix everything and do this,
what caused this water to taste likewater? Was it the hydrogen or the oxygen?
Well, you can't reallypull those apart, right?
The water molecule ishydrogen and oxygen together.
(20:36):
I think that's a lovely analogy for whatwe were talking about with mathematical
goals and language goals.
That I think is really a helpful way tothink about the extent to which they're
intertwined one another.
Yeah, I need to give fullcredit to Vygotsky, I think,
who said that something might beVygotsky. I'll need to check my notes.
I think you're in good company if you'requoting Vygotsky. Before we close,
(20:59):
I'd love to just ask you abit about resources. I say this often on the podcast.
We have 20 to 25 minutesto dig deeply into an idea,
and I know people who are listening oftenthink about, where do I go from here?
Are there any particular resources thatyou would suggest for someone who wanted
to continue learning about what it is tosupport multilingual learners in a math
(21:22):
classroom?
Sure. Happy to share that.
So I think on the individualand collective level,
so say a group of teachers,
there's a beautiful book by CatherineCheval and her colleagues about supporting
multilingual learners and mathematics,
and I really see thatas a valuable resource.
I've used that in reading groups withteachers and use that in book studies,
(21:45):
and it's been very productiveand powerful for us. Beyond that,
of course, I think the NCTM providesa number of really useful resources,
and there are articles, for example,
in the mathematics teacherlearning and teaching PK to 12,
that could make for a really wonderfulstudy or opportunity to engage more
(22:05):
deeply. And then I would sayon a broader perspective,
I've worked with organizations likethe English Learner Success Forum and
others.
We've done some case studies and littlekind of classroom studies that are
accessible on my website,
so you can go to that.But there's also from that organization,
some really valuable insights ifyou're looking at adopting new
(22:27):
materials or evaluating things,
that gives you a principledset of guidelines to follow.
And I think that's really helpful foreducators because we don't have to do
this all on our own.
This is not a reinvent the wheel atevery single site kind of situation.
And so I always encourage people tolook for those resources. And of course,
(22:49):
I will say that the MLC materials,the bridges and mathematics,
I think have been reallybeautifully designed with a lot of
these principles rightbehind them. So for example,
if you look through the teacher'sguides on the bridges and mathematics,
those integrated math and language andpractice goals are a part of the design.
(23:11):
Well, I think that's a great place tostop. Thank you so much for joining us,
bill. This has been insightful and it'sreally been a pleasure talking with you.
Oh, well thank you. I appreciate it.
And that's a wrap forseason three of Rounding Up.
I want to thank all of our guests andthe MLC staff who make these podcasts
possible, as well as all ofour listeners for tuning in.
(23:33):
Have a great summer and we'll beback in September for season four.
This podcast is brought to you by theMath Learning Center and the Meyer Math
Foundation dedicated to inspiring andenabling all individuals to discover
and develop their mathematicalconfidence and ability.
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