January 24, 2024 • 38 mins
Tune into this episode of Math Teacher Lounge, featuring math teaching and learning specialist and friend of the podcast, Fawn Nguyen. Listen as Nguyen dives into the unique and powerful relationship between math fluency and problem-solving, and learn about how problem-solving activities can be leveraged to engage all students and fuel math fluency development.
Learn why math fluency and problem-solving have a symbiotic relationship and how to create problem-solving activities that generate individual and peer learning opportunities, including examples from the California framework. Plus, hear tips for supporting the relationship between problem-solving and math fluency that can implemented in the classroom, throughout the school building, and at home.
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Episode Transcript
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Speaker 1 (00:00):
People would just abruptly say to me, how does my
(00:03):
student be able to engage inthis problem if they don't know
their facts? And I, I think outof all the things that I hear,
that one breaks my heartbecause somehow we're equating
computational and memorizationwith be able to think, be able
to enjoy mathematics.
Speaker 2 (00:19):
Hi, and welcome back to another episode of Math
Teacher Lounge. I'm BethanyLockhart Johnson .
Speaker 3 (00:24):
And I'm Dan Meyer.
Great to see you folks here.
Hi, Bethany.
Speaker 2 (00:27):
Hi. We are on episode 10 of this season, all
about fluency. And you know, Igotta say, we started out this
whole season talking about howyou felt about fluency. I'm not
ready to yet, Dan, to hear howfar you've come, but I just
wanna say that like the guestwe have today, if no one else
(00:50):
can convince you about thepower of fluency, it , it's,
it's this guest.
Speaker 3 (00:55):
Yeah. I feel like me and this guest, and I will ,
uh, not be mysterious about it.
It's Fawn Win fan favoritesecond time appearance that I,
I feel like Fawn and I comefrom a bit of a similar
tradition in that we like bothappreciate problem solving,
mathematical modeling, thinkingthat isn't often easily
captured in like, let's justsay a , a math drill worksheet.
(01:18):
Um, so I, I feel like I, Iwe've recruited onto the show
today, someone with whom like,I'll , I'll be on the same
frequency with Fawn and I hopeto learn a little about how
Fawn sees fluency and problemsolving and how they relate.
Speaker 2 (01:33):
Well, that's exactly what I mean. Like, I feel like
there's something about the wayyou both talk about math and
problem solving that will justlike finally be like , oh, I'll
listen to you Fawn , notBethany
Speaker 3 (01:47):
. Right,
right. Not
Speaker 2 (01:48):
Our other fantastic experts and thinker. Fawn got
you
Speaker 3 (01:52):
Fond though. Yeah, fond has that but
Speaker 2 (01:54):
Fond though. Yeah.
Yeah . We're both fans. And ,um, hey, so before we dive in
though, how are you doing, howis the world of fluency in your
first graders lives?
Speaker 3 (02:04):
Right, right. Thank you for asking. Um, we are ,
uh, coming up on end of thefirst semester and I would say
my approach to fluency from dayone, the day whatever we're at
day 100 or so , um, has changeda lot over the course of this
season. You know, throughlearning with experts here on
the show, through learning withyou, I find myself much more
excited to inject elements intofluency, work with my kids that
(02:26):
are gamey, that involve choice,that involve , um, you know ,
thinking about structures thattry to help like properties of
math emerge that aren't justlike the numerical answers ,
um, you know, on the paper. Andso I would say I'm , uh, you
know, people are generallypretty math positive around
here, still in kinder andfirst. So we'll take that for a
(02:47):
win. People aren't freaked outby seeing numbers and we still
have some conversations hereand there that are, you know ,
kind of creative and fun. SoI'm, I'm, I'm loving it. Thank
you for your help with that.
Speaker 2 (02:55):
Hey, I gotta say, I was hoping I'd get far more
texts from you featuring, like,images of you all mid gameplay
, but, but I understand you'reso in it that you're not even
thinking about I , you knowwhat, I should share this magic
fluency moment with Bethany .
But yeah , I'm happy to hear,I'm happy to hear it's, it's
going well.
Speaker 3 (03:14):
Yeah, it has been careful with that. What , what
you wish for there. 'cause Imight just FaceTime you in and
have you teach my kids on everyafternoon or try to, anyway,
. So watch out withthat
Speaker 2 (03:24):
For having your children. I would be delighted
too . I would be delighted too. So we're bringing back fa now
. Fawn joined us season two ofthe show. Season two, Dan,
season two. That's when we wereon video.
Speaker 3 (03:37):
Yes. We had characters in our drama that
were still alive. Like there'ssome major plot points , um,
that hadn't yet been revealed ,uh, in our drama that is Math
Teacher lounge. Back in seasontwo, we were talking to Fawn
about was it problem solvingthen? I wanna say it was
problem solving. Yeah ,
Speaker 2 (03:52):
Yeah, yeah , for sure. And it was, you know, we
got such good feedback aboutit. Yeah.
Speaker 3 (03:56):
So if you were not around in season two, or for
some reason don't know aboutFawn , lemme just tell you a
bit about Fawn . Fawn iscurrently a specialist on the
math teaching and learning teamhere at Amplify. Um , but in
F's career, F's I , I thinkbetter known as, as a math
teacher, and certainly like a,a capital M, capital T math
teacher, a math teacher'steacher. Um, fawn loves the
(04:16):
work , uh, as much or more thananyone I've ever met. Teachers
will attend Fawn Sessionsaround the world and will just
vibe with her immediately forlots of reasons. But I think a
lot of it is just that she, shereally gets the work and
presents as someone who hasdone the work for a very long
time, knows what kids are like,knows what math teaching is
like, and , um, it's just very,I guess authentic is an
(04:37):
overused word. I think Fawn isone of the most authentic math
teacher types I know. Um, yeah.
And Fawn has also done work asa math teacher coach in case
that wasn't , uh, you know,super obvious from the intro
there. So yeah, just reallyexcited , um, to have Fawn back
on the show and to help youfolks and all of us learn about
how fluency and problem solving, which I have had in my head
(04:58):
as somewhat, you know,antagonistic as somewhat like
unfriendly to one another tohelp us and especially me learn
how they relate. So pleasewelcome Fawn back on the show.
Hey, fawn , good to see youagain.
Speaker 1 (05:10):
Hey. Hey, Dan. Hi Bethany . My god , I finally
get to talk. Oh ,, mygoodness. 20 minutes in
Speaker 3 (05:21):
. That's a
great intro to Fawn. Yeah, fa
vibe right there. Thank you,fawn . Alright , yeah , I'll go
. We'll go shorter next time.
Yeah. Good to see
Speaker 1 (05:27):
You. Great to see you guys.
Speaker 2 (05:28):
Welcome to the chant . Thank
Speaker 1 (05:30):
You for having me.
Yeah, it's always a pleasure.
Speaker 2 (05:33):
So we have been asking all of our guests if
there is an area of your lifebeyond mathematics where you're
currently building fluency.
Speaker 1 (05:44):
I know I heard some others. I've been listening.
And , uh, it's baking, bake abaguette, not just any baking.
I love to cook. I love to cook.
And so on the stove top , I'm,I'm, I'm pretty good. But yeah,
baking is another beast. Andbut Baguette and , uh, someone
told me that that actually isthe measure, the quantitative
(06:06):
measure of , uh, the chef.
Right. The one who reallyknows. And , uh, well, no
wonder I've , I've been, yeah,it's tough. It's tough. It's
too many moving parts, too manyvariables. And , um, I don't
think I have the right water, Ifeel like, because I've tried
everything. So it's baking abaguette.
Speaker 3 (06:24):
We've had some chats about , uh, yeah. Making roast
chicken with Jason Ziba and nowa French baguettes with Fond
Wynn . We , we have like a, areal, we can pivot to a
different kind of podcast here,Bethany pretty easily, I think.
Yeah. Yeah .
Speaker 1 (06:37):
We , we have a channel for cooking.
Speaker 2 (06:39):
I love hearing about what folks are working on. Yeah
. And I think it just helps usbroaden our definition of
fluency. And I also think ithelps us remember that we're
all learning, it's all aboutlearning, and we all have
things that, you know, maybe weare fluent in our math facts,
and so it could be hard torelate to the topic, but then
(07:00):
when you remember, there'sthings that you're brand new at
too, it can help kind of groundyour work and your thinking and
remind you what it's like to bea beginner. So we have so much
we wanna talk to you about, butI have one more baguette
question. What would you say,like, thinking about one of the
first baguettes that you made,right? Right. And compared to
(07:22):
like what you are putting onthe table now, how has that
process changed or how hasthat, like your thinking about
the process changed?
Speaker 1 (07:31):
Right . I definitely have improved. For example,
I've got, you know, the , theproofing down. So yeah, there's
definitely, I , I can, youknow, I wish I did the before
picture right . Graduate ,because I'm definitely getting
better. So now, like Imentioned, maybe it's just the
water is that one part that'snot quite perfect. But
otherwise, yeah, I mean, youknow, made leaps and bounds. So
(07:53):
we , we get better at somethingwe spent , um, a lot of time
on.
Speaker 3 (07:57):
Um, we are so excited to talk to you because
one, you're fantastic, and two,you're an expert on problem
solving. Uh , before we delveinto the nitty gritty, I'm just
curious at a high level, howdoes an enthusiast on problem
solving think about fluency?
Speaker 1 (08:13):
Well, first what , anytime I hear the word fluency
, um, it always takes me to thecontext of what it means to be
fluent in a , a language,because I'm an English learner,
right? So to have flow, notstumbling, not being stuck. And
I finally can make a claim liketwo days ago that I can speak
English fluently. I mean , it'sjust right . So I'm, I'll
(08:35):
always be an English learner. Ifeel like that. And , um, now
sadly, however, I have to sayI'm no longer, I was just in
Vietnam last month and I am nolonger fluent in Vietnamese.
Right. 'cause I spent so muchtime I left when I was so
early. I spent so much time ,um, trying to learn English.
Oh, wow . And now that I haveit, yeah. So I , I can't say
I'm, I'm fluent in Vietnameseanymore. You know what's weird
(08:58):
though , uh, Bethany and Den isI woke up one morning and I
wish I'd written down the date.
I woke up one morning andrealized my thoughts were in
English, my thoughts inEnglish, right? It's a language
I'm trying to learn. I'm tryingto learn. I'm, I wanna get this
right. And then all of a suddenit's like, wow, I'm my thoughts
in English. Oh, wow. So that'swhen I knew I'd switched over.
(09:20):
It's weird. Right now, you knowwhat remains completely intact,
however, completely intact ismy Times table is in
Vietnamese, because I hadlearned that in Vietnamese,
right? So , oh . So even thoughI've, you know, yeah, I , um,
well maybe this had everythingto do with it, which was, it
(09:41):
was literally beaten into me.
And I've shared this storyhundreds of times, and you can
ask any Vietnamese , uh, whowent to Catholic school. So we
would stand and , uh, so we'dall stand up with our palm face
up like this one palm face upas , as we all stand. And the
teacher would walk around witha stick to hit us and just
point at us and say like, youknow, seven times eight. And if
(10:04):
the answer does not comepromptly, you know,
automatically, then we get hit.
The problem is, well, it , ithurts a lot. But the problem
is, what normally happens isone of us in the room, and it's
like, who is gonna be is, youknow, we're humans, right? We,
we , we naturally just kind ofpull back, so we flinch and
pull back. If you pull yourhand back and the teacher
(10:26):
misses your hand, then you'regonna get twice. Then you get
two . Yeah. So that happened afew times. So what we do is a
lot of us do, is we take theother hand and we grab our
wrist just to hold it in placeand then look away. But then if
we look away, we don't know ifthe teacher's pointing at us.
So it's like, oh, so that's,that's traumatic, right? That,
that is, that's how I learnedthe times table by just, it was
(10:50):
literally beat into to us. Uh ,yeah , I don't joke about that.
That's just, yeah. So it isterrifying. I remember missing
it a few times or, you know,I'm just taking my time just
trying to figure out what itis. And it didn't come out fast
enough for the teachers andthen Yep . Got hit.
Speaker 2 (11:06):
What an interesting, like, connection to fluency.
Like the , the , you know, whatan interesting connection to,
to thinking about how you learnsomething and to thinking about
how your relationship with mathdeveloped, right? Like , right.
Speaker 1 (11:21):
Yeah. And so I , I don't know if that's why I'm,
so, I'm literally scared. Imean, I freeze up anything.
There's time when something istimed. Even games like Kahoot,
which is very popular amongteachers, right? Kahoot games
when I'm playing a , a game ,um, I don't know all those
apps. And if there's a timefactor, I I just lose it as I
(11:42):
give up. I don't even tryanymore. So that's how , um,
yeah, I don't know if that tiesback to being hit, but
something, right. It's , it'spretty good evidence.
Speaker 2 (11:51):
I would , I would say there's probably a
Speaker 1 (11:52):
Link. Yeah . Right ?
Yeah. Uh , alright , well, so Ithink math , um, back to the
math fluency, it's , it's quitesimilar to language fluency,
right? It's doing mathematicswith a flow, not fast, I didn't
say fast, but flow, it's likefluid. So it's having
confidence in knowing efficientand accurate ways to compute.
(12:12):
It's being flexible in one'sthinking and applying
strategies. And school math, Iwanna say seems to tie fluency
more to basic facts andcomputation. And this is
evidenced , um, by the explicitlanguage that I see in the
grade level standards, right? Imean, I don't recall, I don't
know if you , correct me if I'mwrong, but I don't recall
seeing fluency standards inhigh school. So, which further
(12:37):
sends a message that, hey,you're in high school now, you
should have these proceduresdown, right? So it , it puts a
stop to fluency to me that,that this seems, when it's not,
it's not mentioned as often,you know, you should be fluent
in high school and in uppergrades. And I think that's just
something, I don't want it tobe that way. I don't want that
to be the case. I think you canalways learn new strategies and
(12:58):
learn more efficient ones. Sothe , the intersection for
problem solving that youbrought up, I think, you know,
the three words that we alwaysuse to describe fluency, right?
It's , um, efficient. We wantto be efficient with problem
solving so we don't get boggeddown in too many steps and lose
track of our reasoning. And ,uh, we certainly wanna be
(13:21):
flexible in flexible in problemsolving. Just like in fluency
is knowing more than onestrategy, knowing more than one
approach. That way you canchoose in order to pick the
right one, we have to have morethan one to choose from. And
also we might use one strategyto problem solve and a
different strategy to check theresult. And then the third word
(13:42):
we use to describe fluency thatyou can also describe problem
solving is to be accurate,right? So of course we wanna be
accurate during problemsolving. Uh , we wanna do
careful. That means carefulrecording, mindfully systematic
in our recording and alwaysdouble checking our work. I
mean, all of those habits ofmind or just , um, those
(14:03):
quantitative qualitativemeasures apply if I take all
the problem solving strategiesor the common problem solving
strategies and put it side byside with fluency strategies,
right? If side by side theyhave quite similar, for
example, or very powerfulproblem solving strategy is to
do a simpler problem andfluency. That's what we're , we
(14:26):
are doing when we aredecomposing and breaking up
numbers, right? To do a simplerproblem. Another one is working
backwards in problem solving .
It's helpful to sometimes thinkabout applicable to use working
backwards. And that's inverseoperations. You know, I, my ad
in a subtraction problem, I'madding, I'm moving forward, you
know, all these inverseoperations , um, make a table
(14:49):
in problem solving. That'sliterally the organization of
the Times table , right? Andthen this one I love, this is
from , uh, Jocelyn Zucker. I ,I love this strategy for
problem solving. It's calledwishful thinking, right? The
problem's difficult, we're notquite sure. So we, we wish it
were this part or we, we wishthis constraint wasn't there.
(15:10):
We wish this, we wish that, sothat , um, we can control some
stuff, make it easier. Andthat's just when we do in
fluency, right? I for example,don't multiply by seven ever. I
don't, I I , there actually,there are many, honestly, if
you ask me to multiply a threedigit or more , um, I stop at
(15:32):
two. I stop at two. Uh , if I,I take a three digit number,
multiply by. If I had tomultiply by three, I will
multiply by two and then addthat whole , you want me
multiply by four? I do double,double , right? I tell my
students this , I don'tmultiply by five, I multiply by
10 and I divide by two. 'causethere's , there's just easier
operations. So that's what Imean by wishful thinking is ,
(15:56):
um, we do a lot of rounding. IfI had to multiply by eight and
nine-tenths, eight andnine-tenths, okay, that's a
little taxing on my brain. I'mgonna try multiplying by nine
instead or by 10 and take awaythat kind of thing. So just a
ton of strategies. And for meit's more, it's fun because I ,
you know, it , it , it's fun toplay. Why not? Why not play
(16:18):
with the numbers in that way,
Speaker 2 (16:19):
The language that you're using to describe
problem solving ? You talkabout those elements of fun. I
love the, I haven't heard yousaid Josh Zucker's wishful
thinking. I hadn't heard thatidea of, well, imagine it in
another way, right? And I'mimagining the strategies that
students are using to solveproblems and using derived
(16:41):
facts, using known facts tofigure out facts that they
don't know. And when I'mthinking about students in my
class or when I'm thinkingabout games that they're
playing, students are actively, like, I want them actively
playing around with thenumbers, right? And trying out,
well, let's see, let's see ifthat works. Or how did you
(17:01):
think about that in sharingtheir strategy and those
conversations that arehappening in fluency practice.
I so see that connection toproblem solving. Because in
problem solving, I want mystudents to think about, well,
what did you use to solve that?
Or what do you understand aboutthe problem? And how did you
get that part? Or where didyour thinking come from? And
(17:21):
those conversations and thatmeaning making it, it, the
playfulness with which you talkabout problem solving, I think
a hundred percent, it's sooften missing from fluency
practice. And so I love thoselinks that you're building.
Speaker 3 (17:38):
I think it often feels like that fluency is
there to support problemsolving is how it's often
positioned where kids need tobe drilled on their basic facts
and had to have to , uh, haveeverything down to automaticity
and then they can engage insome problem solving
experiences. But like Bethanysaid, it's really nice to see
the bidirectionality here whereyou can inject elements of
(18:02):
problem solving into fluency.
How they can support oneanother. If you had, you know,
any kind of way other ideas toinject that energy of problem
solving into fluency work, I'msure we'd love to hear it.
Speaker 1 (18:14):
Well, first, I, I wanna mention what you , you
said earlier about , um, youknow, that you have to be
fluent in order to problemsolve. And I unfortunately, I ,
I hear that a lot. And actuallypeople would just abruptly say
to me , um, how does my studentbe able to engage in this
problem if they don't knowtheir facts? Right ? And I, I
think out of all the thingsthat I hear, that one breaks my
(18:36):
heart. It really breaks myheart because somehow we're
equating computational andmemorization with be able to
think, be able to enjoymathematics, right? And then
the intersection betweenproblem solving fluency is
symbiotic. There's, you know,if I can think of it that way,
right? One helps the other, itreally, you don't need, it's
(18:57):
not a prerequisite. And ifanything , um, when you have a
kid engaged in the problem,problem based , that, that
engagement, right? We want kidsto be interested in the
problem, then that would justbring out, now they're
interested, now they wannaspend time working on
something. And then so it , it, you know, one should be
(19:17):
playing off the other . And Iremember , um, in Dr. Staley's
podcast, I really appreciatehearing you say, Dan, about the
binary, right? That when we saythat the kids can do this or
not do able to do something,and I think, you know, where's
the progression when we saythey can't do something that ,
that's just, that just makes,you know , me just wanna die
(19:41):
because Yeah. Well , it , it'ssuch a heavy label that you
can't do something. I mean, Iknow we say that as adults
often, but when we're, whenwe're labeling children that
they can't do something and,and something such as fluency,
which is, is supposedly take ,um, this lifespan, right? This
K through 12 at least. Andagain, I keep mentioning Dr.
(20:01):
Staley just because I, that'sthe last episode I had listened
to, and I so much appreciateeverything he said. He
mentioned the story and math asa story, and how is this
vertical, how mathematics ismore vertical. And then we
have, we owe it to thechildren, we owe it to the kids
that they get the time toprocess. And it just seems,
(20:22):
yeah. It's like what? Theycan't be exposed to this
interesting problem justbecause they don't have their
number facts.
Speaker 3 (20:30):
Yeah. My feeling is that when, when people say that
students can't do a particularproblem solving experience,
it's often because of poordesign of that experience, not
because of any kind oflimitation on the kids
themselves, like students ,right ? Right . Kids, people
are all sense making mammals,like they look at things and
make sense of them without evenhaving to, in lots of ways. And
(20:50):
when, you know, I've worked onsome of the problems that, that
you've produced for people.
Mm-Hmm . , like,they're visual. They propose a
question without even reallyneeding to state it out loud,
right ? For instance, theyinvite me to experiment and to,
to push things around and seewhat happens next , um, to talk
to people. These are all justlike basic fundamental, you
(21:12):
know, attributes of being aliveand as a human. These are very,
like, natural competencies.
We're not talking aboutknowing, you know, the long
division algorithm here, whichis different of course. Um , so
I'm just wondering, I guesslike when we think about the ,
the kinds of drill sheets thatpeople often produce and other
kinds of fluency experiencesthat students could be having,
(21:32):
that involved everything I justmentioned, that's in your
problem solving work, you know,experimentation, visuals, et
cetera, communication. Likewhat are ways that those
attributes can be transferredonto fluency exercises? Have
you seen any, have you producedany that you think capture
elements of problem solvingbetter than the fluency?
(21:53):
Exercise Worksheets do.
Speaker 1 (21:55):
I mean, right. There are really smart worksheets,
really smart worksheets, and I,I don't get to see them often,
but they're smart because they,they focus, they have the kids
focused on, it's calledvariation I might be familiar
with, right ? It's a term,right? It's variation, whereas
it's not just a bunch ofvariety of problems, but it's
(22:15):
just, it's smart practicebecause it's having kids honing
onto one thing that'sdifferent, right? So those are
wonderful. I mean, worksheetsare not bad. Tests are not bad.
It, it's what's on there, , I always say , you
know, right. It depends on thecontent and, and the context of
it. But , um, yeah, I mean, youcan turn a , a , um, a
(22:36):
numerical test into problemsolving. So I'm, I'm less of ,
um, you know, thosenon-routine. I used to be
really love those non-routinetasks. But I think you can take
anything from the textbook,from, from text and turn that
into problem solving simply bywhen and how you introduce it,
when, if you just do it earlierprior to teaching it, there you
(22:58):
go. It becomes, right. So if Igave you a multiplication
problem, solve it , it's a ,it's an exercise for us. But if
we gave a multiplicationproblem to a child who has not
learned it, that's problemsolving. So there's that,
there's that basic, just, youknow, the definite problem
solving simply means, I don'tquite know how to do this, but
I understand multiplication, orI understand repeat addition .
(23:21):
I understand something aboutit. Right? And, and , um, yeah,
and, and you can put it incontext, but it , it's just all
around us. You don't have tosearch for, I think the degree
to which , um, somethingbecomes problem solving is
simply how familiar we are withit. Whether we have a
prescribed set of steps.
Speaker 3 (23:40):
I'd love to push a little bit more on this right
here, where, like, if I gave a, a calculus question to a
third grader, like they, thatwould be earlier than they'd
been taught it , obviously. And, um, so , but like, it , it
wouldn't feel like a problemsolving exper experience, I
would imagine so much as anexperience of , of frustration
and futility, right ? So Ithink that there's like maybe
something extra, I'd love tohear from you about what makes
(24:01):
the experience a problemsolving experience.
Speaker 1 (24:04):
I'm gonna steal what Terrence Tao , uh, I heard him
say once, and I love that hesaid just out of reach, just
like , so it's right there,just right there, right? So for
example, I've, I've been inelementary classroom where the
teacher says they haven'tlearned , um, multiplication
yet. So I thought, okay, that'sperfect. That's what I'll do,
right? That's , or they'll,they'll learn , um, about area
(24:27):
in the spring, like perfect.
That's then that's , I'll do,that's in the fall. Just , just
, they're gonna get to it soonenough. But I just know they
have, you know, have enough,have enough so that it becomes
problem solving.
Speaker 2 (24:40):
I, I think about in, you know, particularly like
thinking about work with Megan, Frankie and cognitively
guided instruction. Yeah . I II've seen so many examples of
students who, like, forinstance, in conversation, not
in like a specific roteassessment, but in conversation
saying to a student, like Ieven my kindergarten student
(25:02):
saying, you know , uh, you havethree packs of gum and in each
pack is four pieces. Like, howmany pieces of gum do I have?
Right? Mm-Hmm . .
Yeah . And looking at the, I'mnot expecting the student to
use multiplication, right? Butthinking about how does the
students solve it and what arethey using to solve that
problem, right? Right. And sosometimes students are saying,
(25:25):
well, I know four and four iseight, or they're , they're ,
you know, we have tools on thetable. And so students are
sometimes pulling out thetools, or, or another example
that I do in my fluency work aproblem. We have Dr. Garino on
the , uh, podcast, and we hadworked with my students on this
problem where Dr. Garino hasfive bunnies in the hutch.
(25:47):
Sometimes the bunnies areinside the hutch and sometimes
the bunnies are outside of thehutch. What are all the
different ways the bunnies canbe, you know, she could come
out right and see the bunnies,right? And there are ways that
students could solve it that, Imean, there's, they could make
up, you know, equations, all ,all sorts of things. I'm sure
there are very advancedmathematical, right ? You know,
if that you could extend fromfive bunnies to a million
(26:08):
bunnies, right? Like , andBethany,
Speaker 1 (26:10):
That's a kind of lesson that I lose sleep. I
mean, I am so excited. I, Iloosely yes . Can't wait
because what , what you'resaying, I can't wait to see
their strategies. I mean, yes,I think that's, you know, my
blog will will , you know, if,if anybody reads my blog, it's
just those types of lessonsthat it's just, I can't wait to
share this because it blows, itgoes in a direction. I'm super
(26:31):
proud. It's, you know, I don'twant class to end kind of
thing. You know, I've done, youknow, Dan Dan's, the taco card
brings to mind just pennypyramid. Just those things
where, you know, I, I , youknow, there's some calculus
problems, but no, we can dowhat we can do, right? We can
do, and they can take it as, asfar as they can, but just so,
so right there, it's, it'sproblem solving when it's just
(26:54):
a little outreach.
Speaker 2 (26:54):
But I give those examples too, because what I
am, as you're talking about thelink between problem solving
and fluency, I'm feeling like,let's say I'm doing that
problem in my class. What Ithink I hear you saying is then
you're attending to thestrategies that students are
using or the meaning makingthat students are using. And
are you saying that then thatcan be transferred to the way
(27:18):
that they're building fluency?
Or are you saying fluency ishappening as they're
Speaker 1 (27:23):
Both, I, I want to say both.
Speaker 2 (27:25):
Yeah . Yeah. I'm , I'm imagining that the teacher
that's listening to this andthinking about their fluency
work is solved, right? They're,they're hoping to develop
fluency in a way that willsupport problem solving. And
I'm wondering, you know, I lovehearing the way you think about
those links.
Speaker 1 (27:42):
Yeah. So for example, if by engaging that
task, and with classmates,right? They're always doing
these in groups. When they see,they're motivated by when the ,
the group's strategies is , oh,we're building this table. And
they can hear, I mean, when ,when we, especially when we
randomly group kids, they willknow who has what facts, right?
(28:04):
And then when it , it's helpingeach other out, and they say,
okay, you know, I'm, I'm, I'mdoing this my way, I'm doing
this my way. But somebody elsenext to them is like, oh, I'm
doing this way. So hopefully,you know, they're seeing the
efficiency, right? They'reseeing , oh, that's more
efficient. And that's where thenumber talks come in also is so
(28:25):
powerful. The number talks. Ithink the most powerful thing
about the number talks, whichis mental math for me, when I
say number talks, is the sharedstrategies, the share
strategies is my favorite part.
Not just , um, not justbecause, okay, I'm hearing
different things, but the bestpart is they're saying, I just
learned a new way. Bethany didit this way. Oh my gosh, I just
(28:46):
learned it. And then the nexttime they name it after you,
Bethany Mm-Hmm ,. So I started Bethany's way,
or I really like how Danstarted the problem, and I
started the same way. So it ,it's, we have mathematicians ,
um, right, have names, andthey're sitting right in front
of us, all of them. And, andit's empowering. We talk about
wanting to , um, empower thekids. How, how do they know
(29:07):
that they have a place inmathematics? It's, it's only
because they have a voice. Andso the number talks not just
in, in, in the fluency skills,but building those skills,
build confidence, but also justthe fact knowing that they have
a voice is, is , um, one feedsoff the other, right? When ,
when you're, you're this good,and then when you're told, yay
(29:29):
, you know, you applauded, you,you , um, you , you respected
for your thing, and that just,it just feeds on, its its own.
And then now you wanna workmore on that. So, so there's
that balance. Whereas aworksheet doesn't do the same
thing for you. It doesn't giveyou that feedback that other
classmates can give feedback to. In addition to number talks,
(29:50):
which I do as warmups , um,every other day, by the way, we
do number talks every otherday. And , um, you know, I , I
want teachers, I alwaysencourage teachers to always
ask for estimates all the time.
'cause that builds number senseand play games. And I know you
had , um, Jennifer on , um, thelive session that was, that's
fun, right? And , um, so playgames , uh, strategic games
(30:13):
like she had. Um, and then Iwant to also mention that
numeracy routines, becausenumeracy is part of fluency
that's important. I thinkfluency is, that's why fluency
is such a bigger category forme. And , um, one of my
favorite activities withfluency also is , um, different
bases working other than Base10 and James Tanton exploding
(30:36):
dots come to mind when kidslearn, you know, to work in
base five, base three, all of asudden they really pay
attention, especially in thedecals unit, right? And now,
now we get to the decals andit's like, yeah, it's, it's on
our Base 10 system. What if wewere, you know, an alien
dropped in and they onlyfunction in base seven, that
kind of thing. So , um, andthen , um, school-wide, nothing
(30:58):
should happen in isolation,right ? I don't want math to
just happen within the, themath classroom. We wanted that
to , um, be far reaching asmuch as we can. So I was
thinking, you know, I wouldlove to have, you know, schools
adopt where over the paannouncement that happens every
morning that they maybe justend with a mental math problem.
Just, just toss it out thereand, and whatever class wants
(31:19):
to pick it up, pick it up. Butit has that, that school kind
of theme that , um, communitysense and , um, newsletters,
home, newsletters home,encourage parents to play with
mathematics with their kids.
And , um, yeah, justschool-wide things. And, you
know, I, I would love to starta staff meeting with a number
talk. Really just, just drop iteverywhere, everywhere you can
(31:43):
because , uh, once you promotesomething so that it , it's
natural and , um, you know, itbecomes embedded. And it's not,
oh, I'm going to math. I onlydo this in a math class. It
really should be borderless
Speaker 3 (31:56):
Fa . That was , I really appreciate you adding
those bo those , uh, agendaitems for, you know, the fa
Winn School of the Future atthe one at some point.
Speaker 1 (32:04):
Just a little bit more.
Speaker 2 (32:05):
Well, I love that the way you're talking about
those ideas, fawn, are , you'renot talking about something
that requires massive overhaulof like school systems or
massive budget, reallocation, . These are things that
you could start incorporating.
Um, and you know, I feel likethe work that, and the , what
I've heard you talk about, I'veheard you talk about work like
(32:26):
not only in the school, but atdistrict levels. And, you know,
we, here we're three matheducators all based in
California, and you know,California has a brand new math
framework, which it's goingbeyond just fluency, but it
places a greater emphasis onreal world application, right?
(32:46):
And so I'm curious what standsout to you in, in the new
framework as you start to divein
Speaker 1 (32:53):
There? There are several things. One of the key
things, and I really applaudthe framework, is it's the most
equity-based focusedrecommendations, right? And we
are in a state, we have 6million students in a state .
So I'm glad equity is at theforefront. Also, the advocates
for collaborative inquiry-basedlearning. And I think , um,
(33:16):
back to the binary, I I, I'mhearing it around Twitter in
terms of, you know, it's, it'sdirect instruction versus where
it's, it's a blend. And so I'mglad to hear that. No, we're
not saying it's one or theother, but it's, it's problem
based . But that doesn't meanit's all just, you know, give a
kid a problem and let 'em go atit. But it has directed
(33:36):
instruction also. And , um, thepush, because of the push for
collaborative inquiry-basedlearning, and along with the
equity-based, it just, it makesthis very wonderful promise
that we will support allstudents from diverse
backgrounds, include Englishlearners, like I am and
(33:56):
neurodivergent learners to knowthat we all belong in a math
class. Another thing that , um,the framework does is the big
ideas, right? The , the bigideas is a big thing. And we
have data science along withthe fluency, but the data
science and math content itselfsupports the equity based in a
way that when the students canexamine the important issues
(34:19):
and address the inequities thatthey may see in their own
communities. So that part is,is wonderful. I think what we
all need now is , uh, whatwe've always needed is the
teachers providing that supportbecause , um, we, it took a
bunch of years, right? Three,four years for that to get
finalized, get approved, and ,um, we need to focus on
(34:41):
providing teachers with thesupport, not just in the
pedagogy, but the, the mathcontent to bring that framework
to life.
Speaker 3 (34:48):
Yeah. Right? The framework. I mean, it is just
always so interesting whenthese documents come out, the
common Core math framework andothers like, like teaching is
the single largest professionin the us . There's just so
many teachers of so manydifferent capacities, and from
novice to veteran, from, youknow , people who have math
degrees and who don't, who aregeneralists and specialists and
so on. And it really, it's avery lovely marker to put out
(35:10):
there a flag on the ground ismy feeling about it. And now
I'm left to wonder, yeah . Sohow do we, how do we back those
words up with deeds with, likeyou said, fawn , um, with ,
with, you know, the pedagogicalsupport for teachers and what
good is pedagogical supportthat teachers don't have useful
materials to work with thatbring out, right . From
students, that invite fromstudents. So it's exactly, the
(35:32):
work is just beginning here isis what it seems. But I love
how you highlighted what's so,so good, good about this
document.
Speaker 1 (35:38):
Thank you. And , um, this is something I read and I
just want , um, and it stickswith me that I love. And , um,
I don't know who said it, but ,um, I love it. It says, it's
not that students don't needmath, it's that math needs
students, right? All of themmath needs students. 'cause we
always say, you know, you needmath to do this and that. I
(35:59):
said , no, mathematics needskids, needs humans. All of them
Speaker 3 (36:05):
Shout out to, I think , uh, Rochelle Gutierrez
has made that a part of her,her messaging about
mathematics. I love that.
Speaker 1 (36:11):
I could've guessed that Yeah .
Speaker 3 (36:13):
Matches, right? Uh , well th thank you so much for
coming on the show and , andtalk to us again about problem
solving fan favorite, but alsohow to link that up with
fluency and help the two ofthose nourish one another
rather than being kinda thesecompeting priorities in a
class. It's been wonderful tochat, so thank you so much. Fun
. Thank
Speaker 2 (36:28):
You, fa
Speaker 1 (36:30):
Thank you so much.
Appreciate you guys always.
Thank you. Thank you. Pleasesee you in real life soon.
Speaker 3 (36:35):
Soon enough.
Speaker 2 (36:38):
Thanks so much for listening to our conversation
with Von Nguyen specialist onthe math teaching and learning
team at Amplify.
Speaker 3 (36:45):
If you folks are interested in delving deeper
into the future of math inCalifornia, we're excited to
announce that Bethany and Iwill both be a part of a brand
new webinar series exploringjust that beginning of
February. You can catch the twoof us hosting conversations as
part of a free series all aboutmath in California.
Speaker 2 (37:03):
That's right. We'll be breaking down the new
California Math framework ingreater detail. We're talking
about building equity andengagement in California's math
classrooms. And we'll answerall the questions you've ever
had about authentictasks and problem-based
learning. Actually, Dan, wecan't promise all , all of the
questions will be answered, butwe're gonna do our best. I
think together. We can at leasttackle a few of 'em , right,
(37:25):
Dan? Yeah , let's
Speaker 3 (37:26):
Do it.
Speaker 2 (37:27):
For more information and to register for the
webinars, go to amplify.com/ca.
Math webinars, that'samplify.com/ca. Math webinars,
all one word. And we're alsogonna put a link into the show
notes. And speaking of shownotes, that's where you can
find links to follow Fawn andwatch her first appearance in
(37:48):
the Math Teacher Lounge wayback in season two. Next time
on the show, we are wrapping upour fluency focus season. We're
gonna be talking about whatwe've learned, what questions
do we still have, and try tosynthesize some of all these
amazing thought leaders andwhat we've learned together.
(38:10):
Find that episode andeverything else we've done this
season in the Math Teacherlounge, feed . Wherever you get
podcasts, you could find moreinformation on all of Amplify
shows at our podcast hub. Go toamplify.com/hub.
Speaker 3 (38:24):
Thanks for listening.
Speaker 2 (38:25):
Bye.
People would just abruptly say to me, how does my
(00:03):
student be able to engage inthis problem if they don't know
their facts? And I, I think outof all the things that I hear,
that one breaks my heartbecause somehow we're equating
computational and memorizationwith be able to think, be able
to enjoy mathematics.
Speaker 2 (00:19):
Hi, and welcome back to another episode of Math
Teacher Lounge. I'm BethanyLockhart Johnson .
Speaker 3 (00:24):
And I'm Dan Meyer.
Great to see you folks here.
Hi, Bethany.
Speaker 2 (00:27):
Hi. We are on episode 10 of this season, all
about fluency. And you know, Igotta say, we started out this
whole season talking about howyou felt about fluency. I'm not
ready to yet, Dan, to hear howfar you've come, but I just
wanna say that like the guestwe have today, if no one else
(00:50):
can convince you about thepower of fluency, it , it's,
it's this guest.
Speaker 3 (00:55):
Yeah. I feel like me and this guest, and I will ,
uh, not be mysterious about it.
It's Fawn Win fan favoritesecond time appearance that I,
I feel like Fawn and I comefrom a bit of a similar
tradition in that we like bothappreciate problem solving,
mathematical modeling, thinkingthat isn't often easily
captured in like, let's justsay a , a math drill worksheet.
(01:18):
Um, so I, I feel like I, Iwe've recruited onto the show
today, someone with whom like,I'll , I'll be on the same
frequency with Fawn and I hopeto learn a little about how
Fawn sees fluency and problemsolving and how they relate.
Speaker 2 (01:33):
Well, that's exactly what I mean. Like, I feel like
there's something about the wayyou both talk about math and
problem solving that will justlike finally be like , oh, I'll
listen to you Fawn , notBethany
Speaker 3 (01:47):
Speaker 2 (01:48):
Our other fantastic experts and thinker. Fawn got
you
Speaker 3 (01:52):
Fond though. Yeah, fond has that but
Speaker 2 (01:54):
Fond though. Yeah.
Yeah . We're both fans. And ,um, hey, so before we dive in
though, how are you doing, howis the world of fluency in your
first graders lives?
Speaker 3 (02:04):
Right, right. Thank you for asking. Um, we are ,
uh, coming up on end of thefirst semester and I would say
my approach to fluency from dayone, the day whatever we're at
day 100 or so , um, has changeda lot over the course of this
season. You know, throughlearning with experts here on
the show, through learning withyou, I find myself much more
excited to inject elements intofluency, work with my kids that
(02:26):
are gamey, that involve choice,that involve , um, you know ,
thinking about structures thattry to help like properties of
math emerge that aren't justlike the numerical answers ,
um, you know, on the paper. Andso I would say I'm , uh, you
know, people are generallypretty math positive around
here, still in kinder andfirst. So we'll take that for a
(02:47):
win. People aren't freaked outby seeing numbers and we still
have some conversations hereand there that are, you know ,
kind of creative and fun. SoI'm, I'm, I'm loving it. Thank
you for your help with that.
Speaker 2 (02:55):
Hey, I gotta say, I was hoping I'd get far more
texts from you featuring, like,images of you all mid gameplay
, but, but I understand you'reso in it that you're not even
thinking about I , you knowwhat, I should share this magic
fluency moment with Bethany .
But yeah , I'm happy to hear,I'm happy to hear it's, it's
going well.
Speaker 3 (03:14):
Yeah, it has been careful with that. What , what
you wish for there. 'cause Imight just FaceTime you in and
have you teach my kids on everyafternoon or try to, anyway,
Speaker 2 (03:24):
For having your children. I would be delighted
too . I would be delighted too. So we're bringing back fa now
. Fawn joined us season two ofthe show. Season two, Dan,
season two. That's when we wereon video.
Speaker 3 (03:37):
Yes. We had characters in our drama that
were still alive. Like there'ssome major plot points , um,
that hadn't yet been revealed ,uh, in our drama that is Math
Teacher lounge. Back in seasontwo, we were talking to Fawn
about was it problem solvingthen? I wanna say it was
problem solving. Yeah ,
Speaker 2 (03:52):
Yeah, yeah , for sure. And it was, you know, we
got such good feedback aboutit. Yeah.
Speaker 3 (03:56):
So if you were not around in season two, or for
some reason don't know aboutFawn , lemme just tell you a
bit about Fawn . Fawn iscurrently a specialist on the
math teaching and learning teamhere at Amplify. Um , but in
F's career, F's I , I thinkbetter known as, as a math
teacher, and certainly like a,a capital M, capital T math
teacher, a math teacher'steacher. Um, fawn loves the
(04:16):
work , uh, as much or more thananyone I've ever met. Teachers
will attend Fawn Sessionsaround the world and will just
vibe with her immediately forlots of reasons. But I think a
lot of it is just that she, shereally gets the work and
presents as someone who hasdone the work for a very long
time, knows what kids are like,knows what math teaching is
like, and , um, it's just very,I guess authentic is an
(04:37):
overused word. I think Fawn isone of the most authentic math
teacher types I know. Um, yeah.
And Fawn has also done work asa math teacher coach in case
that wasn't , uh, you know,super obvious from the intro
there. So yeah, just reallyexcited , um, to have Fawn back
on the show and to help youfolks and all of us learn about
how fluency and problem solving, which I have had in my head
(04:58):
as somewhat, you know,antagonistic as somewhat like
unfriendly to one another tohelp us and especially me learn
how they relate. So pleasewelcome Fawn back on the show.
Hey, fawn , good to see youagain.
Speaker 1 (05:10):
Hey. Hey, Dan. Hi Bethany . My god , I finally
get to talk. Oh ,
Speaker 3 (05:21):
vibe right there. Thank you,fawn . Alright , yeah , I'll go
. We'll go shorter next time.
Yeah. Good to see
Speaker 1 (05:27):
You. Great to see you guys.
Speaker 2 (05:28):
Welcome to the chant . Thank
Speaker 1 (05:30):
You for having me.
Yeah, it's always a pleasure.
Speaker 2 (05:33):
So we have been asking all of our guests if
there is an area of your lifebeyond mathematics where you're
currently building fluency.
Speaker 1 (05:44):
I know I heard some others. I've been listening.
And , uh, it's baking, bake abaguette, not just any baking.
I love to cook. I love to cook.
And so on the stove top , I'm,I'm, I'm pretty good. But yeah,
baking is another beast. Andbut Baguette and , uh, someone
told me that that actually isthe measure, the quantitative
(06:06):
measure of , uh, the chef.
Right. The one who reallyknows. And , uh, well, no
wonder I've , I've been, yeah,it's tough. It's tough. It's
too many moving parts, too manyvariables. And , um, I don't
think I have the right water, Ifeel like, because I've tried
everything. So it's baking abaguette.
Speaker 3 (06:24):
We've had some chats about , uh, yeah. Making roast
chicken with Jason Ziba and nowa French baguettes with Fond
Wynn . We , we have like a, areal, we can pivot to a
different kind of podcast here,Bethany pretty easily, I think.
Yeah. Yeah .
Speaker 1 (06:37):
We , we have a channel for cooking.
Speaker 2 (06:39):
I love hearing about what folks are working on. Yeah
. And I think it just helps usbroaden our definition of
fluency. And I also think ithelps us remember that we're
all learning, it's all aboutlearning, and we all have
things that, you know, maybe weare fluent in our math facts,
and so it could be hard torelate to the topic, but then
(07:00):
when you remember, there'sthings that you're brand new at
too, it can help kind of groundyour work and your thinking and
remind you what it's like to bea beginner. So we have so much
we wanna talk to you about, butI have one more baguette
question. What would you say,like, thinking about one of the
first baguettes that you made,right? Right. And compared to
(07:22):
like what you are putting onthe table now, how has that
process changed or how hasthat, like your thinking about
the process changed?
Speaker 1 (07:31):
Right . I definitely have improved. For example,
I've got, you know, the , theproofing down. So yeah, there's
definitely, I , I can, youknow, I wish I did the before
picture right . Graduate ,because I'm definitely getting
better. So now, like Imentioned, maybe it's just the
water is that one part that'snot quite perfect. But
otherwise, yeah, I mean, youknow, made leaps and bounds. So
(07:53):
we , we get better at somethingwe spent , um, a lot of time
on.
Speaker 3 (07:57):
Um, we are so excited to talk to you because
one, you're fantastic, and two,you're an expert on problem
solving. Uh , before we delveinto the nitty gritty, I'm just
curious at a high level, howdoes an enthusiast on problem
solving think about fluency?
Speaker 1 (08:13):
Well, first what , anytime I hear the word fluency
, um, it always takes me to thecontext of what it means to be
fluent in a , a language,because I'm an English learner,
right? So to have flow, notstumbling, not being stuck. And
I finally can make a claim liketwo days ago that I can speak
English fluently. I mean , it'sjust right . So I'm, I'll
(08:35):
always be an English learner. Ifeel like that. And , um, now
sadly, however, I have to sayI'm no longer, I was just in
Vietnam last month and I am nolonger fluent in Vietnamese.
Right. 'cause I spent so muchtime I left when I was so
early. I spent so much time ,um, trying to learn English.
Oh, wow . And now that I haveit, yeah. So I , I can't say
I'm, I'm fluent in Vietnameseanymore. You know what's weird
(08:58):
though , uh, Bethany and Den isI woke up one morning and I
wish I'd written down the date.
I woke up one morning andrealized my thoughts were in
English, my thoughts inEnglish, right? It's a language
I'm trying to learn. I'm tryingto learn. I'm, I wanna get this
right. And then all of a suddenit's like, wow, I'm my thoughts
in English. Oh, wow. So that'swhen I knew I'd switched over.
(09:20):
It's weird. Right now, you knowwhat remains completely intact,
however, completely intact ismy Times table is in
Vietnamese, because I hadlearned that in Vietnamese,
right? So , oh . So even thoughI've, you know, yeah, I , um,
well maybe this had everythingto do with it, which was, it
(09:41):
was literally beaten into me.
And I've shared this storyhundreds of times, and you can
ask any Vietnamese , uh, whowent to Catholic school. So we
would stand and , uh, so we'dall stand up with our palm face
up like this one palm face upas , as we all stand. And the
teacher would walk around witha stick to hit us and just
point at us and say like, youknow, seven times eight. And if
(10:04):
the answer does not comepromptly, you know,
automatically, then we get hit.
The problem is, well, it , ithurts a lot. But the problem
is, what normally happens isone of us in the room, and it's
like, who is gonna be is, youknow, we're humans, right? We,
we , we naturally just kind ofpull back, so we flinch and
pull back. If you pull yourhand back and the teacher
(10:26):
misses your hand, then you'regonna get twice. Then you get
two . Yeah. So that happened afew times. So what we do is a
lot of us do, is we take theother hand and we grab our
wrist just to hold it in placeand then look away. But then if
we look away, we don't know ifthe teacher's pointing at us.
So it's like, oh, so that's,that's traumatic, right? That,
that is, that's how I learnedthe times table by just, it was
(10:50):
literally beat into to us. Uh ,yeah , I don't joke about that.
That's just, yeah. So it isterrifying. I remember missing
it a few times or, you know,I'm just taking my time just
trying to figure out what itis. And it didn't come out fast
enough for the teachers andthen Yep . Got hit.
Speaker 2 (11:06):
What an interesting, like, connection to fluency.
Like the , the , you know, whatan interesting connection to,
to thinking about how you learnsomething and to thinking about
how your relationship with mathdeveloped, right? Like , right.
Speaker 1 (11:21):
Yeah. And so I , I don't know if that's why I'm,
so, I'm literally scared. Imean, I freeze up anything.
There's time when something istimed. Even games like Kahoot,
which is very popular amongteachers, right? Kahoot games
when I'm playing a , a game ,um, I don't know all those
apps. And if there's a timefactor, I I just lose it as I
(11:42):
give up. I don't even tryanymore. So that's how , um,
yeah, I don't know if that tiesback to being hit, but
something, right. It's , it'spretty good evidence.
Speaker 2 (11:51):
I would , I would say there's probably a
Speaker 1 (11:52):
Link. Yeah . Right ?
Yeah. Uh , alright , well, so Ithink math , um, back to the
math fluency, it's , it's quitesimilar to language fluency,
right? It's doing mathematicswith a flow, not fast, I didn't
say fast, but flow, it's likefluid. So it's having
confidence in knowing efficientand accurate ways to compute.
(12:12):
It's being flexible in one'sthinking and applying
strategies. And school math, Iwanna say seems to tie fluency
more to basic facts andcomputation. And this is
evidenced , um, by the explicitlanguage that I see in the
grade level standards, right? Imean, I don't recall, I don't
know if you , correct me if I'mwrong, but I don't recall
seeing fluency standards inhigh school. So, which further
(12:37):
sends a message that, hey,you're in high school now, you
should have these proceduresdown, right? So it , it puts a
stop to fluency to me that,that this seems, when it's not,
it's not mentioned as often,you know, you should be fluent
in high school and in uppergrades. And I think that's just
something, I don't want it tobe that way. I don't want that
to be the case. I think you canalways learn new strategies and
(12:58):
learn more efficient ones. Sothe , the intersection for
problem solving that youbrought up, I think, you know,
the three words that we alwaysuse to describe fluency, right?
It's , um, efficient. We wantto be efficient with problem
solving so we don't get boggeddown in too many steps and lose
track of our reasoning. And ,uh, we certainly wanna be
(13:21):
flexible in flexible in problemsolving. Just like in fluency
is knowing more than onestrategy, knowing more than one
approach. That way you canchoose in order to pick the
right one, we have to have morethan one to choose from. And
also we might use one strategyto problem solve and a
different strategy to check theresult. And then the third word
(13:42):
we use to describe fluency thatyou can also describe problem
solving is to be accurate,right? So of course we wanna be
accurate during problemsolving. Uh , we wanna do
careful. That means carefulrecording, mindfully systematic
in our recording and alwaysdouble checking our work. I
mean, all of those habits ofmind or just , um, those
(14:03):
quantitative qualitativemeasures apply if I take all
the problem solving strategiesor the common problem solving
strategies and put it side byside with fluency strategies,
right? If side by side theyhave quite similar, for
example, or very powerfulproblem solving strategy is to
do a simpler problem andfluency. That's what we're , we
(14:26):
are doing when we aredecomposing and breaking up
numbers, right? To do a simplerproblem. Another one is working
backwards in problem solving .
It's helpful to sometimes thinkabout applicable to use working
backwards. And that's inverseoperations. You know, I, my ad
in a subtraction problem, I'madding, I'm moving forward, you
know, all these inverseoperations , um, make a table
(14:49):
in problem solving. That'sliterally the organization of
the Times table , right? Andthen this one I love, this is
from , uh, Jocelyn Zucker. I ,I love this strategy for
problem solving. It's calledwishful thinking, right? The
problem's difficult, we're notquite sure. So we, we wish it
were this part or we, we wishthis constraint wasn't there.
(15:10):
We wish this, we wish that, sothat , um, we can control some
stuff, make it easier. Andthat's just when we do in
fluency, right? I for example,don't multiply by seven ever. I
don't, I I , there actually,there are many, honestly, if
you ask me to multiply a threedigit or more , um, I stop at
(15:32):
two. I stop at two. Uh , if I,I take a three digit number,
multiply by. If I had tomultiply by three, I will
multiply by two and then addthat whole , you want me
multiply by four? I do double,double , right? I tell my
students this , I don'tmultiply by five, I multiply by
10 and I divide by two. 'causethere's , there's just easier
operations. So that's what Imean by wishful thinking is ,
(15:56):
um, we do a lot of rounding. IfI had to multiply by eight and
nine-tenths, eight andnine-tenths, okay, that's a
little taxing on my brain. I'mgonna try multiplying by nine
instead or by 10 and take awaythat kind of thing. So just a
ton of strategies. And for meit's more, it's fun because I ,
you know, it , it , it's fun toplay. Why not? Why not play
(16:18):
with the numbers in that way,
Speaker 2 (16:19):
The language that you're using to describe
problem solving ? You talkabout those elements of fun. I
love the, I haven't heard yousaid Josh Zucker's wishful
thinking. I hadn't heard thatidea of, well, imagine it in
another way, right? And I'mimagining the strategies that
students are using to solveproblems and using derived
(16:41):
facts, using known facts tofigure out facts that they
don't know. And when I'mthinking about students in my
class or when I'm thinkingabout games that they're
playing, students are actively, like, I want them actively
playing around with thenumbers, right? And trying out,
well, let's see, let's see ifthat works. Or how did you
(17:01):
think about that in sharingtheir strategy and those
conversations that arehappening in fluency practice.
I so see that connection toproblem solving. Because in
problem solving, I want mystudents to think about, well,
what did you use to solve that?
Or what do you understand aboutthe problem? And how did you
get that part? Or where didyour thinking come from? And
(17:21):
those conversations and thatmeaning making it, it, the
playfulness with which you talkabout problem solving, I think
a hundred percent, it's sooften missing from fluency
practice. And so I love thoselinks that you're building.
Speaker 3 (17:38):
I think it often feels like that fluency is
there to support problemsolving is how it's often
positioned where kids need tobe drilled on their basic facts
and had to have to , uh, haveeverything down to automaticity
and then they can engage insome problem solving
experiences. But like Bethanysaid, it's really nice to see
the bidirectionality here whereyou can inject elements of
(18:02):
problem solving into fluency.
How they can support oneanother. If you had, you know,
any kind of way other ideas toinject that energy of problem
solving into fluency work, I'msure we'd love to hear it.
Speaker 1 (18:14):
Well, first, I, I wanna mention what you , you
said earlier about , um, youknow, that you have to be
fluent in order to problemsolve. And I unfortunately, I ,
I hear that a lot. And actuallypeople would just abruptly say
to me , um, how does my studentbe able to engage in this
problem if they don't knowtheir facts? Right ? And I, I
think out of all the thingsthat I hear, that one breaks my
(18:36):
heart. It really breaks myheart because somehow we're
equating computational andmemorization with be able to
think, be able to enjoymathematics, right? And then
the intersection betweenproblem solving fluency is
symbiotic. There's, you know,if I can think of it that way,
right? One helps the other, itreally, you don't need, it's
(18:57):
not a prerequisite. And ifanything , um, when you have a
kid engaged in the problem,problem based , that, that
engagement, right? We want kidsto be interested in the
problem, then that would justbring out, now they're
interested, now they wannaspend time working on
something. And then so it , it, you know, one should be
(19:17):
playing off the other . And Iremember , um, in Dr. Staley's
podcast, I really appreciatehearing you say, Dan, about the
binary, right? That when we saythat the kids can do this or
not do able to do something,and I think, you know, where's
the progression when we saythey can't do something that ,
that's just, that just makes,you know , me just wanna die
(19:41):
because Yeah. Well , it , it'ssuch a heavy label that you
can't do something. I mean, Iknow we say that as adults
often, but when we're, whenwe're labeling children that
they can't do something and,and something such as fluency,
which is, is supposedly take ,um, this lifespan, right? This
K through 12 at least. Andagain, I keep mentioning Dr.
(20:01):
Staley just because I, that'sthe last episode I had listened
to, and I so much appreciateeverything he said. He
mentioned the story and math asa story, and how is this
vertical, how mathematics ismore vertical. And then we
have, we owe it to thechildren, we owe it to the kids
that they get the time toprocess. And it just seems,
(20:22):
yeah. It's like what? Theycan't be exposed to this
interesting problem justbecause they don't have their
number facts.
Speaker 3 (20:30):
Yeah. My feeling is that when, when people say that
students can't do a particularproblem solving experience,
it's often because of poordesign of that experience, not
because of any kind oflimitation on the kids
themselves, like students ,right ? Right . Kids, people
are all sense making mammals,like they look at things and
make sense of them without evenhaving to, in lots of ways. And
(20:50):
when, you know, I've worked onsome of the problems that, that
you've produced for people.
Mm-Hmm .
question without even reallyneeding to state it out loud,
right ? For instance, theyinvite me to experiment and to,
to push things around and seewhat happens next , um, to talk
to people. These are all justlike basic fundamental, you
(21:12):
know, attributes of being aliveand as a human. These are very,
like, natural competencies.
We're not talking aboutknowing, you know, the long
division algorithm here, whichis different of course. Um , so
I'm just wondering, I guesslike when we think about the ,
the kinds of drill sheets thatpeople often produce and other
kinds of fluency experiencesthat students could be having,
(21:32):
that involved everything I justmentioned, that's in your
problem solving work, you know,experimentation, visuals, et
cetera, communication. Likewhat are ways that those
attributes can be transferredonto fluency exercises? Have
you seen any, have you producedany that you think capture
elements of problem solvingbetter than the fluency?
(21:53):
Exercise Worksheets do.
Speaker 1 (21:55):
I mean, right. There are really smart worksheets,
really smart worksheets, and I,I don't get to see them often,
but they're smart because they,they focus, they have the kids
focused on, it's calledvariation I might be familiar
with, right ? It's a term,right? It's variation, whereas
it's not just a bunch ofvariety of problems, but it's
(22:15):
just, it's smart practicebecause it's having kids honing
onto one thing that'sdifferent, right? So those are
wonderful. I mean, worksheetsare not bad. Tests are not bad.
It, it's what's on there,
know, right. It depends on thecontent and, and the context of
it. But , um, yeah, I mean, youcan turn a , a , um, a
(22:36):
numerical test into problemsolving. So I'm, I'm less of ,
um, you know, thosenon-routine. I used to be
really love those non-routinetasks. But I think you can take
anything from the textbook,from, from text and turn that
into problem solving simply bywhen and how you introduce it,
when, if you just do it earlierprior to teaching it, there you
(22:58):
go. It becomes, right. So if Igave you a multiplication
problem, solve it , it's a ,it's an exercise for us. But if
we gave a multiplicationproblem to a child who has not
learned it, that's problemsolving. So there's that,
there's that basic, just, youknow, the definite problem
solving simply means, I don'tquite know how to do this, but
I understand multiplication, orI understand repeat addition .
(23:21):
I understand something aboutit. Right? And, and , um, yeah,
and, and you can put it incontext, but it , it's just all
around us. You don't have tosearch for, I think the degree
to which , um, somethingbecomes problem solving is
simply how familiar we are withit. Whether we have a
prescribed set of steps.
Speaker 3 (23:40):
I'd love to push a little bit more on this right
here, where, like, if I gave a, a calculus question to a
third grader, like they, thatwould be earlier than they'd
been taught it , obviously. And, um, so , but like, it , it
wouldn't feel like a problemsolving exper experience, I
would imagine so much as anexperience of , of frustration
and futility, right ? So Ithink that there's like maybe
something extra, I'd love tohear from you about what makes
(24:01):
the experience a problemsolving experience.
Speaker 1 (24:04):
I'm gonna steal what Terrence Tao , uh, I heard him
say once, and I love that hesaid just out of reach, just
like , so it's right there,just right there, right? So for
example, I've, I've been inelementary classroom where the
teacher says they haven'tlearned , um, multiplication
yet. So I thought, okay, that'sperfect. That's what I'll do,
right? That's , or they'll,they'll learn , um, about area
(24:27):
in the spring, like perfect.
That's then that's , I'll do,that's in the fall. Just , just
, they're gonna get to it soonenough. But I just know they
have, you know, have enough,have enough so that it becomes
problem solving.
Speaker 2 (24:40):
I, I think about in, you know, particularly like
thinking about work with Megan, Frankie and cognitively
guided instruction. Yeah . I II've seen so many examples of
students who, like, forinstance, in conversation, not
in like a specific roteassessment, but in conversation
saying to a student, like Ieven my kindergarten student
(25:02):
saying, you know , uh, you havethree packs of gum and in each
pack is four pieces. Like, howmany pieces of gum do I have?
Right? Mm-Hmm .
Yeah . And looking at the, I'mnot expecting the student to
use multiplication, right? Butthinking about how does the
students solve it and what arethey using to solve that
problem, right? Right. And sosometimes students are saying,
(25:25):
well, I know four and four iseight, or they're , they're ,
you know, we have tools on thetable. And so students are
sometimes pulling out thetools, or, or another example
that I do in my fluency work aproblem. We have Dr. Garino on
the , uh, podcast, and we hadworked with my students on this
problem where Dr. Garino hasfive bunnies in the hutch.
(25:47):
Sometimes the bunnies areinside the hutch and sometimes
the bunnies are outside of thehutch. What are all the
different ways the bunnies canbe, you know, she could come
out right and see the bunnies,right? And there are ways that
students could solve it that, Imean, there's, they could make
up, you know, equations, all ,all sorts of things. I'm sure
there are very advancedmathematical, right ? You know,
if that you could extend fromfive bunnies to a million
(26:08):
bunnies, right? Like , andBethany,
Speaker 1 (26:10):
That's a kind of lesson that I lose sleep. I
mean, I am so excited. I, Iloosely yes . Can't wait
because what , what you'resaying, I can't wait to see
their strategies. I mean, yes,I think that's, you know, my
blog will will , you know, if,if anybody reads my blog, it's
just those types of lessonsthat it's just, I can't wait to
share this because it blows, itgoes in a direction. I'm super
(26:31):
proud. It's, you know, I don'twant class to end kind of
thing. You know, I've done, youknow, Dan Dan's, the taco card
brings to mind just pennypyramid. Just those things
where, you know, I, I , youknow, there's some calculus
problems, but no, we can dowhat we can do, right? We can
do, and they can take it as, asfar as they can, but just so,
so right there, it's, it'sproblem solving when it's just
(26:54):
a little outreach.
Speaker 2 (26:54):
But I give those examples too, because what I
am, as you're talking about thelink between problem solving
and fluency, I'm feeling like,let's say I'm doing that
problem in my class. What Ithink I hear you saying is then
you're attending to thestrategies that students are
using or the meaning makingthat students are using. And
are you saying that then thatcan be transferred to the way
(27:18):
that they're building fluency?
Or are you saying fluency ishappening as they're
Speaker 1 (27:23):
Both, I, I want to say both.
Speaker 2 (27:25):
Yeah . Yeah. I'm , I'm imagining that the teacher
that's listening to this andthinking about their fluency
work is solved, right? They're,they're hoping to develop
fluency in a way that willsupport problem solving. And
I'm wondering, you know, I lovehearing the way you think about
those links.
Speaker 1 (27:42):
Yeah. So for example, if by engaging that
task, and with classmates,right? They're always doing
these in groups. When they see,they're motivated by when the ,
the group's strategies is , oh,we're building this table. And
they can hear, I mean, when ,when we, especially when we
randomly group kids, they willknow who has what facts, right?
(28:04):
And then when it , it's helpingeach other out, and they say,
okay, you know, I'm, I'm, I'mdoing this my way, I'm doing
this my way. But somebody elsenext to them is like, oh, I'm
doing this way. So hopefully,you know, they're seeing the
efficiency, right? They'reseeing , oh, that's more
efficient. And that's where thenumber talks come in also is so
(28:25):
powerful. The number talks. Ithink the most powerful thing
about the number talks, whichis mental math for me, when I
say number talks, is the sharedstrategies, the share
strategies is my favorite part.
Not just , um, not justbecause, okay, I'm hearing
different things, but the bestpart is they're saying, I just
learned a new way. Bethany didit this way. Oh my gosh, I just
(28:46):
learned it. And then the nexttime they name it after you,
Bethany Mm-Hmm ,
or I really like how Danstarted the problem, and I
started the same way. So it ,it's, we have mathematicians ,
um, right, have names, andthey're sitting right in front
of us, all of them. And, andit's empowering. We talk about
wanting to , um, empower thekids. How, how do they know
(29:07):
that they have a place inmathematics? It's, it's only
because they have a voice. Andso the number talks not just
in, in, in the fluency skills,but building those skills,
build confidence, but also justthe fact knowing that they have
a voice is, is , um, one feedsoff the other, right? When ,
when you're, you're this good,and then when you're told, yay
(29:29):
, you know, you applauded, you,you , um, you , you respected
for your thing, and that just,it just feeds on, its its own.
And then now you wanna workmore on that. So, so there's
that balance. Whereas aworksheet doesn't do the same
thing for you. It doesn't giveyou that feedback that other
classmates can give feedback to. In addition to number talks,
(29:50):
which I do as warmups , um,every other day, by the way, we
do number talks every otherday. And , um, you know, I , I
want teachers, I alwaysencourage teachers to always
ask for estimates all the time.
'cause that builds number senseand play games. And I know you
had , um, Jennifer on , um, thelive session that was, that's
fun, right? And , um, so playgames , uh, strategic games
(30:13):
like she had. Um, and then Iwant to also mention that
numeracy routines, becausenumeracy is part of fluency
that's important. I thinkfluency is, that's why fluency
is such a bigger category forme. And , um, one of my
favorite activities withfluency also is , um, different
bases working other than Base10 and James Tanton exploding
(30:36):
dots come to mind when kidslearn, you know, to work in
base five, base three, all of asudden they really pay
attention, especially in thedecals unit, right? And now,
now we get to the decals andit's like, yeah, it's, it's on
our Base 10 system. What if wewere, you know, an alien
dropped in and they onlyfunction in base seven, that
kind of thing. So , um, andthen , um, school-wide, nothing
(30:58):
should happen in isolation,right ? I don't want math to
just happen within the, themath classroom. We wanted that
to , um, be far reaching asmuch as we can. So I was
thinking, you know, I wouldlove to have, you know, schools
adopt where over the paannouncement that happens every
morning that they maybe justend with a mental math problem.
Just, just toss it out thereand, and whatever class wants
(31:19):
to pick it up, pick it up. Butit has that, that school kind
of theme that , um, communitysense and , um, newsletters,
home, newsletters home,encourage parents to play with
mathematics with their kids.
And , um, yeah, justschool-wide things. And, you
know, I, I would love to starta staff meeting with a number
talk. Really just, just drop iteverywhere, everywhere you can
(31:43):
because , uh, once you promotesomething so that it , it's
natural and , um, you know, itbecomes embedded. And it's not,
oh, I'm going to math. I onlydo this in a math class. It
really should be borderless
Speaker 3 (31:56):
Fa . That was , I really appreciate you adding
those bo those , uh, agendaitems for, you know, the fa
Winn School of the Future atthe
Speaker 1 (32:04):
Just a little bit more.
Speaker 2 (32:05):
Well, I love that the way you're talking about
those ideas, fawn, are , you'renot talking about something
that requires massive overhaulof like school systems or
massive budget, reallocation,
you could start incorporating.
Um, and you know, I feel likethe work that, and the , what
I've heard you talk about, I'veheard you talk about work like
(32:26):
not only in the school, but atdistrict levels. And, you know,
we, here we're three matheducators all based in
California, and you know,California has a brand new math
framework, which it's goingbeyond just fluency, but it
places a greater emphasis onreal world application, right?
(32:46):
And so I'm curious what standsout to you in, in the new
framework as you start to divein
Speaker 1 (32:53):
There? There are several things. One of the key
things, and I really applaudthe framework, is it's the most
equity-based focusedrecommendations, right? And we
are in a state, we have 6million students in a state .
So I'm glad equity is at theforefront. Also, the advocates
for collaborative inquiry-basedlearning. And I think , um,
(33:16):
back to the binary, I I, I'mhearing it around Twitter in
terms of, you know, it's, it'sdirect instruction versus where
it's, it's a blend. And so I'mglad to hear that. No, we're
not saying it's one or theother, but it's, it's problem
based . But that doesn't meanit's all just, you know, give a
kid a problem and let 'em go atit. But it has directed
(33:36):
instruction also. And , um, thepush, because of the push for
collaborative inquiry-basedlearning, and along with the
equity-based, it just, it makesthis very wonderful promise
that we will support allstudents from diverse
backgrounds, include Englishlearners, like I am and
(33:56):
neurodivergent learners to knowthat we all belong in a math
class. Another thing that , um,the framework does is the big
ideas, right? The , the bigideas is a big thing. And we
have data science along withthe fluency, but the data
science and math content itselfsupports the equity based in a
way that when the students canexamine the important issues
(34:19):
and address the inequities thatthey may see in their own
communities. So that part is,is wonderful. I think what we
all need now is , uh, whatwe've always needed is the
teachers providing that supportbecause , um, we, it took a
bunch of years, right? Three,four years for that to get
finalized, get approved, and ,um, we need to focus on
(34:41):
providing teachers with thesupport, not just in the
pedagogy, but the, the mathcontent to bring that framework
to life.
Speaker 3 (34:48):
Yeah. Right? The framework. I mean, it is just
always so interesting whenthese documents come out, the
common Core math framework andothers like, like teaching is
the single largest professionin the us . There's just so
many teachers of so manydifferent capacities, and from
novice to veteran, from, youknow , people who have math
degrees and who don't, who aregeneralists and specialists and
so on. And it really, it's avery lovely marker to put out
(35:10):
there a flag on the ground ismy feeling about it. And now
I'm left to wonder, yeah . Sohow do we, how do we back those
words up with deeds with, likeyou said, fawn , um, with ,
with, you know, the pedagogicalsupport for teachers and what
good is pedagogical supportthat teachers don't have useful
materials to work with thatbring out, right . From
students, that invite fromstudents. So it's exactly, the
(35:32):
work is just beginning here isis what it seems. But I love
how you highlighted what's so,so good, good about this
document.
Speaker 1 (35:38):
Thank you. And , um, this is something I read and I
just want , um, and it stickswith me that I love. And , um,
I don't know who said it, but ,um, I love it. It says, it's
not that students don't needmath, it's that math needs
students, right? All of themmath needs students. 'cause we
always say, you know, you needmath to do this and that. I
(35:59):
said , no, mathematics needskids, needs humans. All of them
Speaker 3 (36:05):
Shout out to, I think , uh, Rochelle Gutierrez
has made that a part of her,her messaging about
mathematics. I love that.
Speaker 1 (36:11):
I could've guessed that
Speaker 3 (36:13):
Matches, right? Uh , well th thank you so much for
coming on the show and , andtalk to us again about problem
solving fan favorite, but alsohow to link that up with
fluency and help the two ofthose nourish one another
rather than being kinda thesecompeting priorities in a
class. It's been wonderful tochat, so thank you so much. Fun
. Thank
Speaker 2 (36:28):
You, fa
Speaker 1 (36:30):
Thank you so much.
Appreciate you guys always.
Thank you. Thank you. Pleasesee you in real life soon.
Speaker 3 (36:35):
Soon enough.
Speaker 2 (36:38):
Thanks so much for listening to our conversation
with Von Nguyen specialist onthe math teaching and learning
team at Amplify.
Speaker 3 (36:45):
If you folks are interested in delving deeper
into the future of math inCalifornia, we're excited to
announce that Bethany and Iwill both be a part of a brand
new webinar series exploringjust that beginning of
February. You can catch the twoof us hosting conversations as
part of a free series all aboutmath in California.
Speaker 2 (37:03):
That's right. We'll be breaking down the new
California Math framework ingreater detail. We're talking
about building equity andengagement in California's math
classrooms. And we'll answerall the questions you've ever
had
learning. Actually, Dan, wecan't promise all , all of the
questions will be answered, butwe're gonna do our best. I
think together. We can at leasttackle a few of 'em , right,
(37:25):
Dan? Yeah , let's
Speaker 3 (37:26):
Do it.
Speaker 2 (37:27):
For more information and to register for the
webinars, go to amplify.com/ca.
Math webinars, that'samplify.com/ca. Math webinars,
all one word. And we're alsogonna put a link into the show
notes. And speaking of shownotes, that's where you can
find links to follow Fawn andwatch her first appearance in
(37:48):
the Math Teacher Lounge wayback in season two. Next time
on the show, we are wrapping upour fluency focus season. We're
gonna be talking about whatwe've learned, what questions
do we still have, and try tosynthesize some of all these
amazing thought leaders andwhat we've learned together.
(38:10):
Find that episode andeverything else we've done this
season in the Math Teacherlounge, feed . Wherever you get
podcasts, you could find moreinformation on all of Amplify
shows at our podcast hub. Go toamplify.com/hub.
Speaker 3 (38:24):
Thanks for listening.
Speaker 2 (38:25):
Bye.
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