October 18, 2023 • 41 mins
Tune in to the latest episode of Math Teacher Lounge where we learn how to intentionally foster fluency with Dr. Art Baroody, Professor Emeritus of Curriculum & Instruction, University of Illinois Urbana-Champaign. Listen as Baroody shares strategies on how to utilize a student’s natural problem-solving skills and desire to learn to build fluency.
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Art Baroody (00:00):
You can get efficiency through rote
memorization, but what youdon't get is efficiency plus
appropriate use and, moreimportantly, adaptive use.
Dan Meyer (00:17):
Welcome to Math Teacher Lounge. I'm your host,
Dan Meyer.
Bethany Lockhart Johnson (00:21):
And I'm Bethany Lockhart Johnson .
Dan Meyer (00:23):
And on today's episode, we are gonna continue
going deeper into the sametopic we've been on this whole
season, the topic of mathfluency. So far we've tried to
establish what it is with oneguest. And then think about how
to measure it and develop itwith other guests. And we're so
excited about our next guesthere today. Bethany, how has
this topic been working on you?
(00:44):
Has it been feeding you andyour ideas about education at
all?
Bethany Lockhart Johnson (00:48):
I mean, Dan, we've already
established that I've been thefluency fan from go.
Dan Meyer (00:53):
Minute one.
Bethany Lockhart Johnson (00:53):
So the better question is, how is
it working on you? I definitelylove our deep dives. I love the
fact that we get to take thistopic and talk about it over
multiple episodes, and hearfrom the audience, and
incorporate their questions.
So, I'm thrilled to be here. Sowhat about you? You're here.
You're ready?
Dan Meyer (01:12):
Yeah. For those who are just popping in for this
episode, Bethany and I have adynamic that pulls in different
directions sometimes. Andthat's always been really
helpful for us. In our banter.
Our repartee. In this season,the tension is that Bethany is
a huge fan of fluency andfluency development. And I have
had experiences, as a secondaryteacher, where kids are coming
(01:35):
to me after eight, nine yearsof having, in some cases, some
pretty negative experienceswith fluency development. Where
they'll come and they've hadthese different moments where
they think like, "Oh, so mathis about doing the same boring
thing over and over again, andgetting some feedback that I'm
smart or dumb. And I generallyfeel dumb with that." So that's
(01:55):
led me to kind of drift awayfrom enthusiasm towards
fluency. Drift away fromfluency. But what's been
working on me a little bit is areally strong strand through
our conversation so far,Bethany, around equity. And how
if I deprive a student ofexperiences of generating
fluency, even if that's infavor of experiences where they
(02:17):
see the beauty of math, let'ssay, or the creativity in math,
I'm making it harder for themto experience, through that
fluency, new ideas asaccessible and approachable. So
that's been, I think, the ideathat's been working on me so
far this season.
Art Baroody (02:33):
I would love to interject here, if you wouldn't
mind.
Dan Meyer (02:35):
Folks, guest Dr. Art Baroody, can't wait for his
intro. He wants inon this! Let's go, Dr .
Baroody! Let's go! Let's talkabout it.
Art Baroody (02:47):
I really appreciate your comment about
how fluency ruins kids. Becauseif it's fostered in the wrong
way, it's exactly what's gonnahappen. Kids are gonna think
that math is all about justmemorizing stuff, and
regurgitating it quickly. Andit does so much damage to
(03:12):
approach fluency in the wrongway, through rote memorization.
But —and this is the wholecentral theme of our talk today
— if you teach fluencyappropriately, if you foster
meaningful memorization, thenyou get the kinds of things
that you wanna see too, whichis an intellectual curiosity. A
(03:35):
looking for patterns andrelations. And an excitement
about math, and an ability toappreciate math. And really
wanna do something about math.
Dan Meyer (03:49):
I'm feeling this so much. And I really want us to
get into this distinction ofmeaningful memorization,
meaningful fluency, shortly.
But, let's tease 'em a littlebit here, Dr. Baroody. Let's
not give away the whole farmright away. Bethany, do
you wanna share who we'reworking with here today
?
Bethany Lockhart Johnson (04:07):
Well, you've already given us quite
an intro, Dr. Baroody. Butlisteners, this is the fabulous
Dr. Art Baroody, ProfessorEmeritus at the University of
Illinois, Urbana-Champaign.
He's someone who's beenstudying math fluency for
(04:30):
decades. And, you're gonna helpmake the case for not only why
fluency is so important, buthow do we do it in a way that's
meaningful. So, thank you forbeing here. And I love that you
just dove right in. You'relike, "Oh, Dan . Oh,
I'm here. I'm ready ."So thank you again for being
(04:53):
here. And we're so excited tohave you on the show.
Art Baroody (04:56):
I just couldn't resist. I've seen so many
teachers frustrated by the waymath is taught. And how kids
come to them in their class andthey're basically not
interested in math. Or they'reeven anxious about math. ...
We've gotta do a better job.
Dan Meyer (05:17):
Fried to a crisp, yup.
Bethany Lockhart Johnson (05:18):
Hey, when you're in the Lounge, let
me tell you, you don't have tohold back . This is a
conversation. We welcome it.
That's why you're here. Wewanna hear from you. I do wanna
ask you, in broadening the waythat we think about fluency,
one of the things that we'reasking all of our guests is,
(05:38):
what's something that you aredeveloping fluency in right
now? Beyond all the amazingwork you've done in
mathematics, what's somethingthat, right now, you either
have recently or are activelytrying to develop fluency in?
Art Baroody (05:53):
Well, at this point in my life, I'm trying to
be a better husband, father,and grandfather . It's
always a work in progress. But,it's especially important to
keep my wife happy.
Bethany Lockhart Johnson (06:11):
You know what? Wiser words have
never... Always alearner. Right? We can always
improve. Let me tell you whatmy husband did yesterday—no,
I'm just kidding. . No,that's fantastic.
Dan Meyer (06:26):
We might jump back in on that too at some point.
Because I love thinking aboutlike, "Is every situation we
have with our significantrelationships, the folks we
want to keep happy, are theyall unique, like snowflakes?
Requiring a deep conceptualwork on the part of the "
learner" — here, you — tofigure out a solution for? Or
(06:48):
are there ways in which yourstudy of fluency can actually
be applied in a meaningful wayto this real-world question of
fluency in relationship?"Please don't answer. This is
going way deeper than weprepped you for. But
it's just a curiosity that Ihave right now. Are there
moments where our work in onearea can actually jump into the
(07:08):
other? I'm gonna let that hangfor a second, Dr. Baroody, and
instead ask you about your workin grad school. What fascinates
me about you is that, when Iwent through grad school, I was
motivated to prove a thing. Igot these ideas! They're never
gonna change! They're correct!I'm gonna find the research and
do the studies to back 'em up!And I feel like you have a bio
that I love. Which is you cameinto grad school with ideas
(07:31):
about fluency, it sounds like.
And then those started tochange a bit, through your
encounter with reality. Withyour empirical work. And I
would love for you to describea bit about your transition, or
what you learned through yourgraduate work on fluency.
What'd you find, as far as theconventional wisdom? And how it
was correct or incorrect?
Art Baroody (07:53):
Sure. As an undergrad, I was trained as a
teacher, and I becameinterested in psychology,
especially the psychology oflearning. So when I went off to
grad school, a natural majorwas educational psychology. And
I had the good fortune ofstudying with Herb Ginsburg,
(08:16):
who was a great mentor. He wasa leading researcher in
mathematical learning. And, inparticular, he was interested
in this new area of children'sinformal mathematical
development, and how itprovided a basis for formal
mathematical learning,including providing an informal
(08:38):
basis for fluency. And sothat's where I got my initial
interest in fluency. HerbGinsburg was an interesting
mentor. What he would do withhis grad students is say, "OK.
Here's one side of the issue.
Here's the other side of theissue. Here's the research. You
(09:00):
decide which is right." Now,that's highly unusual.
But it helped me a great deal,because what I started to do
was dive into the research onmathematical learning. And one
of the issues that we wereencouraged to explore was, how
do kids learn the basic numbercombinations? When I started
(09:25):
grad school, I believed whateveryone believes: that if kids
don't know the basic facts,then what you have to do is
make sure they have massiveamounts of drill and practice
to memorize those facts. Thatwas certainly true of my
(09:45):
undergraduate training. Infact, I was at a parent-teacher
meeting and one of the fathersstood up and said, "Everyone
knows what two plus two is. Andyou know how we know that?" And
here it is, someone who's hadno training whatsoever, but of
course they know everythingabout how to teach mathematics.
(10:05):
"OK," he said, "Well, rememberback in first and second grade,
what did we do? Your teachersgave you all sorts of practice
and drill on two plus twoequals four. And that's why you
know two plus two equals four."It turns out that a lot of kids
(10:27):
know two plus two equals fourbefore they even get to school.
So, it's not an adequateexplanation. We'll talk more
about that later. One of thethings that comes out of a
traditional or a conventionalview of fluency is that
informal methods, such ascounting, especially finger
(10:53):
counting:"Oh! That's bad!Counting and finger counting,
those are just crutches foravoiding the real work of
memorizing those facts!" Well,that's certainly what I
believed at the time. And I,basically, was looking at the
issue through adult eyes. WhatGinsburg taught me was that, in
(11:17):
order to understand learning,to understand the issues around
learning, you also have to lookat the issues through a child's
eyes. And one of the firstthings I learned was that kids
have their own informal methodsof solving math problems. So,
(11:40):
if you ask a child, "How muchis five plus three?" Kids,
typically, at least initially,what they'll do is they'll
count out five fingers, orthey'll put up five fingers,
count out three more fingers,or put up three fingers, and
then count all the fingers andcome up with the answer. Why is
(12:04):
this important? Because kidsaren't stupid. What they do is
they continuously inventincreasingly efficient counting
procedures to figure out sumsand differences. And, what this
enables kids to do then is tolook for patterns and
(12:28):
relationships among the numberfacts. And this can then be
used to devise reasoningstrategies. For example, I was
observing a kindergarten classwhile I was in grad school one
time. And, we were playing asimple race game, where the
(12:49):
children would throw two diceand figure out what the sum
was, and that's how many spacesthey could move on a racetrack.
And one little girl rolled asix and a one, and she didn't
have six fingers on one hand,so she wasn't quite sure what
(13:10):
to do. And she was really kindof puzzled by the question. And
the little girl next to herleaned over and whispered in
her ear, "That's an easy one,Marcy. It's just the number
after." Now, what thiskindergartner had discovered
(13:34):
was a connection between hercounting knowledge,
specifically her number-afterknowledge, and adding one. When
you add one, the sum is justthe number after the other
number in the count sequence.
So, for six plus one, theanswer is, what number comes
(13:57):
after six when we count? Seven,it's easy.
Bethany Lockhart Johnson (14:00):
I have to say, what I'm hearing
is you seeing the brilliance ofchildren.
Art Baroody (14:07):
Yes.
Bethany Lockhart Johnson (14:08):
And you're seeing the wisdom, and
what they bring in, and theirinnate brilliance, right? And,
so much of the conversation, Ifeel like, when we're talking
to educators, is really wantingto celebrate the way children
think. And I often feel thatthat's separated from the task
(14:31):
of fluency, right? But whatyou're saying here is exactly
what I think excites me aboutit, is that there are patterns.
There are ways to look at it.
There's sensemaking happening.
Even in something that mightseem simple to an adult. Two
plus three — kiddos arebringing so much to that. And,
when we listen, we can respondin ways that respect and value
(14:54):
what they're bringing to thetable. And, it sounds like the
way that you thought aboutfluency and listening and
working with kids reallyevolved.
Art Baroody (15:03):
Yeah, that's interesting. I was once told by
another adult that being apsychologist must really ruin
kids for me. Must take all thefun out of it. No! You
see more. You appreciate moreof what they can do! It's
(15:24):
amazing. And , it'sexhilarating! It's quite the
opposite.
Dan Meyer (15:32):
What I'd love to say about the plus-one example here
is that moment happened in aspan of like five seconds,
right? That exchange betweenstudents. And what folks like
you, and us, and our listenerswho have taught for a while and
thought deeply about teaching —you can see that moment up in
the sports booth playing out inreal time. And say, "Whoa, slow
(15:52):
that down. Back it up. Let's doa replay." And you could do a
whole dissertation —maybe youdid — about that one moment.
And, I just think it's sointeresting how that came from
a peer; it's like a moment thatmight get missed by a
layperson, this person whospoke to you in a meeting about
how kids develop fluencywithout expertise. And I also
(16:15):
think it's so interesting how alot of people might suggest
that what the student neededwas to be told this idea about
one-after, before. But it'samazing to me how much time
students need. Like, I see myown six-year-old counting up to
(16:36):
five. And I'm just like,"Buddy!" I just wanna tell him.
And sometimes I even do. But itdoesn't matter. So much of
these ideas — the informallearning — is so durable and
needs to ... not run its owncourse, but have enough
experiences tossed at it. Sothose moments you described
then become natural. And Iwonder if you'd explain a
little bit about what you'relearning about fluency
(16:58):
development gone wrong. Whatare some common ways ... I feel
like I've mentioned likerushing it—
Art Baroody (17:03):
Can I expand on your other comment, for just a
moment?
Dan Meyer (17:06):
Spin it, yeah. Spin it, please.
Art Baroody (17:08):
What I think is interesting about informal
mathematical learning is thatkids can learn a great deal
from their peers. And, often,their peers are closer to the
situation and understand thesituation better than adults.
We, as adults, have forgottenhow we learn the basic
(17:31):
combinations. And so we — mostadults — aren't even aware that
there's a number-after rulethat can be used to quickly,
efficiently generate the sum ofANY number plus one, for which
you know the count sequence!Most people just are not aware
(17:51):
of that. But this little girlprobably had just made the
discovery, or done so in therecent past. And so, for her,
it was still a fresh issue. Itwas still something in her
mind. And so, she was able toshare that with her peer. And,
w ith any luck, her peerbenefited from that.
Dan Meyer (18:11):
I think it speaks to all these different learning
paths that we've all been on indeveloping our own fluency. And
how easy it's to forget thepath in favor of the
destination, and then try tocreate this paved route for
kids that bypasses what, for somany of us, was necessary kind
of stumbling around a bit. Idon't wanna revel in the
(18:31):
stumbling ... but I guess theinformal-formal dichotomy there
is so important to me. We'renot just, like, discovering or
wandering in the dark. But weare taking stock of our
surroundings, and thinkingabout what resources we have.
And, sometimes, I think, itseems like teachers want to
move to the side of that. Andsay, "Well, actually, there's a
(18:52):
highway over here. It's movingso fast and can get you there
faster." Which I think leavesstudents disoriented to these
connections from informal toformal knowledge. How close is
that to what you studied withGinsburg, about how formal and
informal knowledge is related?
Art Baroody (19:10):
Basically, one of the key points that Ginsburg
made was that most mathematicallearning problems in school are
due to a fact that there's agap between the formal
instruction and the child'sinformal knowledge. So, if a
(19:31):
kid is struggling with math —and I have found this with my
own case studies — the problemtypically is not the child. The
problem typically is that theformal instruction is not
connecting with what the childknows already. And, if your
(19:54):
teachers get nothing else outof this talk today, the
Principle of Assimilation,Piaget's Principle of
Assimilation, is that weunderstand things in terms of
what we already understand. So,we understand new stuff in
terms of the stuff that wealready understand. And that
(20:17):
principle is violated way toooften. And when it is, kids
don't have any recourse, otherthan rote memorization. Or
quitting. Not learning it. Andthose aren't good options.
Dan Meyer (20:36):
Yeah, I love it.
Since all new knowledge buildson old knowledge, it leads to
this beautiful rule forlearning, which is that
everybody knows something abouteverything. Pick any topic that
I might one day learn aboutthat's way beyond me.
Aeronautical engineering, let'ssay. That knowledge has gotta
build on stuff that I at onepoint knew, down to knowledge
(20:58):
that I was forming as a smallchild, crawling around. It can
be useful. Puts me on a pathtowards this knowledge. I love
the spirit that Bethany bringsthroughout our work, and what
we've talked about here, thatkids are brilliant. They have
resources that can help all ofus learn. Feels like to me an
important takeaway of ourconversation here. Bethany,
what's your thoughts aboutthat?
Bethany Lockhart Johnson (21:20):
Well, absolutely! I feel like, as a
teacher, we're facilitators tohelp build those connections.
Or to help highlight thoseconnections, and celebrate
those connections, between thenew material and the things
that kiddos already know.
Right? And I feel like whenwe're thinking about fluency,
(21:40):
when we see timed tests orspeed drills, and we see — as
you said — the damage that'scaused by that, not helping
kiddos to build connections. Ifeel like ... I don't know, in
listening to this conversation,I feel really moved. Because I
(22:00):
get so excited thinking aboutdifferent moments when I've
seen kiddos build thoseconnections. And how can we, as
educators, help each other tomake sure that, even in
something that has so beendrilled down to these speed
tests and rote memorization,how can we make those
connections present there, too?
(22:21):
How can we treat that with thatsame level of respect, and the
kids with the same level ofrespect. And I feel like you've
spoken about something calledmeaningful memorization. And I
feel like maybe you can talk abit more about that. Because
there's a lot more awareness ofkids and making it matter and
(22:42):
respecting them than rotedrill-and-kill, right?
Art Baroody (22:46):
Meaningful memorization basically involves
a number of things. Meaningfulmemorization builds on a
child's conceptualunderstanding of an operation,
for example. So a child needsto understand what addition is,
what subtraction is, whatmultiplication is, in order to
(23:11):
achieve fluency with thosefacts. Let me give you an
example. If a child understandsthat one meaning of
multiplication is a groups-ofmeaning. So, five times eight
means I've got five groups ofeight items each. OK? If a
(23:37):
child understands thatconceptually, then five times
zero makes sense. Because whatdo you have? I've got five
groups of no items. So how manyitems do I have all together?
, I have no items. So,contrary to most multiplication
(24:01):
facts, the answer here is notgetting bigger. And
kids can understand it if theyhave this conceptual
understanding of multiplicationas a groups-of meaning. So, it
can help kids then memorize thezero rule for multiplication in
(24:22):
a meaningful way. The otheraspect of meaningful
memorization is discoveringpatterns in relations. Like our
little girl here, whodiscovered the connection
between adding one and hernumber-after knowledge, her
existing number-afterknowledge. So meaningful
memorization involves buildingon both conceptual
(24:45):
understanding and trying tofind new patterns and relations
to enrich that conceptualunderstanding.
Bethany Lockhart Johnson (24:52):
For me, when I hear that, it sounds
so spot-on. And it makes a lotof sense to me. But what would
be a response to folks thatfeel like, "Yes, let's make it
meaningful, let's build theseconcepts, but we still need the
timed test , we still need thespeed drills"? Is there a space
(25:14):
for that?
Art Baroody (25:18):
Speed tests, timed tests, are a tool. An
educational tool. And like anytool, they need to be used
carefully and thoughtfully, andwhere appropriate. The problem
(25:41):
with timed tests as they'reoften used is that they're
overused. Timed tests makesense after a child understands
an operation, after they've hada chance to explore the
operation using counting. Sothat they've had a chance to
(26:04):
find patterns and relationsamong the facts and devised
reasoning strategies like thenumber-after rule. Once a child
has devised a reasoningstrategy, then it would make
sense to have them practice it— even under a timed condition
— to make sure that it becomesmore efficient, that it becomes
(26:30):
fluent. But there's no sensetrying to impose fluency before
a child has constructed thereasoning strategy. That makes
no sense whatsoever. It makesno sense whatsoever to have
timed tests before a child hasconstructed an understanding of
the operation. So, mathematicseducators, for a long time now,
(26:54):
have argued that you need to becareful about premature
practice. That you shouldn'thave kids doing drills before
they have devised means forfiguring out the sum or
difference or product orquotient or whatever. So I'm
(27:16):
not completely opposed to timedtests. But boy, you have to use
them really, really carefully.
When we were developingsoftware for helping kids learn
the basic addition-subtractioncombinations, initially there
was no timing involved. Butonce the child had developed a
(27:39):
strategy, then we wanted tostart introducing the child to
some time limit — a generoustime limit — so that there was
some incentive to use thestrategy as quickly as
possible. Now the thing is,practice doesn't have to be
boring. It doesn't have to beflashcards. It doesn't have to
(28:00):
be boring drills. It caninvolve games. Dice games are
especially important topreschool kids and kids in the
early primary grades. Why?
Because they can see again andagain that two plus one is
three, that two and two isfour. So they can begin to
(28:24):
learn some of the add-onecombinations or facts, and they
can begin to learn some of thedoubles, such as two plus two
is four; three plus three issix. And this then provides a
basis for learning other facts,such as two plus three. Because
if you know the number-afterrule for adding one, and you
(28:48):
know the double two plus two isfour, you can look at two plus
three and say, "Ah, that's justtwo and two and one more." So,
basically, you're building onyour previous knowledge to
figure out new facts.
Dan Meyer (29:07):
So I got a voiceover here for a second. I feel like
you've just helped me have anepiphany here, Dr. Baroody,
which is , well, first of all,timed tests are this topic that
just kinda lurks in thebackground, like a ghost, of
every conversation we've hadabout fluency. And your
perspective is, I think, asomewhat unique one. Not a
(29:27):
hard-line perspective here. Ilike the idea that they're just
overused. They're a blunt-forceinstrument to try to pressure
kids into fluency when there'sso many other interesting
games, where a student'snatural inclination to optimize
or even win can carry thefluency impetus. I dig that.
Number one. And, number two Ithink is this: I feel like
(29:49):
we're just mired in thesedichotomies in math education
discourse around, for instance,conceptual and procedural.
Which one comes first? Whichone comes second? And I think
you've helped make sense of afinding from Bethany
Rittle-Johnson. This articlewas like: They develop
together. And it's given me anew question, which is it's
not—and it's a question I askmyself—it's not like, is this
(30:13):
conceptual or procedural,what's going on here? But
rather , what resources dostudents have? And what
experiences would help themdevelop those? 'Cause what
you've helped me see is thatthe fluency is not just the
formalization of thisburgeoning resource. It
actually creates a newresource. Where when the
student is like, "OK, I get itfive times one, five times two,
five times three, five timesfour." That is not just a
(30:36):
formalization of the kidscounting and grouping. It gives
the kid a resource to then dofive times zero. That is then a
new resource in the bag ofresources to help the kid
create new concepts. Theconcepts create the fluency,
create the concepts. And so, Idig this question of "What are
the resources a kid has? Andwhat do they need to develop
new ones?" Versus, like, "Arewe doing concepts today? Are we
(30:58):
doing fluency today?" No, it'sboth! It can be both! I dunno,
how close is that?
Art Baroody (31:02):
Back in 1986, Herb Ginsburg and I wrote a chapter
together to address the issueof procedural and conceptual
knowledge in mathematics. Atthe time, there were two very
popular views. There was theskills-first view, where you
taught kids the skills. Youdidn't bother to take the time
(31:26):
to help them conceptuallylearn. You just taught them the
skills by rote . And then kidswould apply those skills and
eventually understand the mathtoo. The concept-first approach
was you teach for understandingfirst and then the skill
(31:49):
learning will be easier. If youunderstand the conceptual basis
for a procedure, you're muchmore likely to learn the
procedure. Well, Ginsburg and Itook the position that the two
couldn't be separated. Thatbasically it was an iterative
(32:09):
process. That is, you mightlearn a skill and then discover
a relationship, and then usethat relationship to understand
something more, and then devisean even better skill, and so
forth and so on. So I thinkthey go hand-in-hand.
Dan Meyer (32:29):
They're buddies.
They're buddies.
Art Baroody (32:31):
Another view would be the simultaneous view. That
both skills and concepts candevelop together. And that's
another interestingpossibility, I think, of how
they might go togetherhand-in-hand. So, I don't see
this dichotomy betweenprocedural knowledge and
(32:52):
conceptual knowledge. It's notthe best paradigm for trying to
figure out how to teach. Ithink it's really important
that children understand boththe concepts underlying
procedures and the procedures.
And when you do that together,you're much more likely to have
(33:13):
procedural fluency, becausefluency is often defined only
as using a skill or procedureaccurately and quickly, that is
efficiently. The NationalResearch Council published a
book called Adding It Up; andthey argue from the research
(33:36):
that it makes a whole lot moresense to think of fluency not
only as efficiency, that isaccurate and fast use of
procedure, but the appropriateuse of a procedure, and,
perhaps even more importantly,the application of a procedure
(33:57):
to a new situation. Now, youcan get efficiency through rote
memorization, but what youdon't get is efficiency plus
appropriate use and, moreimportantly, adaptive use. So
that's why I think bothprocedures and concepts need to
(34:19):
be taught in an intertwinedway.
Bethany Lockhart Johnson (34:23):
Early childhood ed is my joy, is my
land. And I remember, actually,when I was a student teacher in
a TK classroom. So, thesekiddos are four, right? This is
years and years ago. And wewere talking about numbers. And
this little boy said, "Two plusthree is five." And he said it
(34:45):
in such a way that it was clearthat he's regularly celebrated
for knowing that, right? AndI'll never forget that my
mentor teacher said, "Whoa! Canyou show me with these blocks?"
We had little blocks on thetable. And the little boy burst
into tears. And there was thispalpable panic, right? And, I'm
(35:09):
not exaggerating when I saythat was a transformative
moment for me. Because what Isaw, and I found out later,
this kid had been in — I willnot name it, but a program that
purportedly supports thegaining of math skills. And you
can pay them money to have themwork with your kiddo. So the
(35:30):
little boy clearly knew thisscript, right? But when he was
asked to show it or makemeaning of it, he couldn't. And
there was a fear; and there wasa panic. There was a panic. And
I know that the parents andcaregivers of this child would
never do something ... theythought they were doing good
(35:53):
and right and helping them givea leg up. They weren't trying
to harm their kiddo. They'retaking the time to drive this
kid to a class, probablySaturday morning. And when I
think about that, and I thinkabout the way that the fear
that that little boy had wasalready holding, and the
anxiety that little boy wasalready holding, and I've seen
(36:14):
how fluency can be developed sodifferently, I just ... I feel
like this conversation — and Ihope that for educators
listening, I feel like you arereminding us and giving us
permission to slow down. Toslow down. To attend to
children's thinking. To noticethese little moments when
(36:36):
children are showing theirbrilliance. And I just really
appreciate that. And I reallyappreciate your perspective,
your learned perspective, afterdecades and decades of—not to
like, "Whoa, over decades anddecades!" —and I
celebrate that. So, thank youso, so much for being in the
Lounge with us. Thank you forhelping us make more sense of
(37:00):
this. Really, really, really.
Art Baroody (37:02):
You're absolutely welcome. One of the reasons I
was so passionate about this isfrom my own experience. I was a
first grader at the time. I wassitting there at my desk doing
a worksheet, a math worksheet,and I had my fingers under the
desk, because I was figuringout sums. And my first grade
(37:26):
teacher said, "Arthur, what areyou doing with fingers under
the desk?" I should have toldher anything but the truth.
'Cause I told her I wascounting to figure out my
addition . Well, my normallypatient and kind and
(37:48):
soft-spoken teacher erupted andsaid, "Don't do that! Don't do
that!"
Bethany Lockhart Johnson (37:56):
That makes my heart hurt for baby
Art! I've had parents tell methat they've held their kids'
fingers down. "I've beenholding his fingers down to try
to keep him from using 'em ,but he just keeps using his
fingers." I'm like, "What? Whatare you doing? There's a tool
you carry, a tool around!" Oh,wow. Wow.
Dan Meyer (38:18):
It's a resource.
Art Baroody (38:19):
What was my first grade teacher telling me? What
are the parents who hold theirchild's fingers down telling
the child? Your way of doingthings is wrong. It's inferior.
It's stupid. And that's how achild's confidence in their own
knowledge can be undermined,instead of building on it.
(38:43):
Instead of honoring it. We toooften say, "Oh, well, that's
not right. That's not the rightway of doing it. Here's the
right way of doing it. Theformal way." What it does is
just undermine children'sdisposition to learn
mathematics. It underminestheir confidence to solve
(39:06):
problems. And they stop doingit. They just stop.
Dan Meyer (39:10):
They're learners.
And they eventually learn whatnot to do.
Art Baroody (39:12):
And then they become dependent on the teacher
and they say, "All right , Idon't know how to do this. You
tell me how to do this."
Bethany Lockhart Johnson (39:20):
Yeah . Not to mention all the shame
and the fear .
Art Baroody (39:23):
Yeah. And, you actually hear kids, "Don't
explain to me why! I don'twanna know why. Just tell me
how to do it." It's justabsolute craziness. And that's
what the focus on rotememorization does to kids.
Dan Meyer (39:39):
Yeah, that's the outcome.
Art Baroody (39:40):
It really destroys any disposition to think
mathematically, to enjoymathematics. It can be really
harmful.
Dan Meyer (39:49):
Well, I always appreciate, Dr. Baroody, a
conversation that isfar-reaching and takes very
seriously teaching mathematicsand the gifts that students
bring to us in our classrooms,if only we recognize them as
such. Bethany and I reallyappreciate your time here in
the Lounge. So, thank you. AndI hope we meet again some time
.
Bethany Lockhart Johnson (40:08):
Thank you so much, Dr. Baroody.
Art Baroody (40:10):
Any time . I'd be happy to come aboard any time.
Dan Meyer (40:13):
Thanks so much for listening to our conversation
with Dr. Art Baroody, ProfessorEmeritus at the University of
Illinois, Urbana-Champaign. Letus know what you thought of
this episode. I thought it wenta lot of very interesting
places. Hit us up in ourFacebook discussion group, Math
Teacher Lounge (40:26):
The Community, or on X, formerly known as
Twitter, at MTLShow. Whatquestions do you have about
math fluency? What would youlike to know? Please let us
know and we'll do our best toinvestigate that for you over
the course of this season. Makesure you don't miss an episode
in this new series bysubscribing to Math Teacher
Lounge (40:44):
The Podcast wherever you find fine podcasts
products. You can find moreinfo on all of Amplify shows at
our podcast hub. Go toamplify.com/hub. Thanks again
for listening, folks.
You can get efficiency through rote
memorization, but what youdon't get is efficiency plus
appropriate use and, moreimportantly, adaptive use.
Dan Meyer (00:17):
Welcome to Math Teacher Lounge. I'm your host,
Dan Meyer.
Bethany Lockhart Johnson (00:21):
And I'm Bethany Lockhart Johnson .
Dan Meyer (00:23):
And on today's episode, we are gonna continue
going deeper into the sametopic we've been on this whole
season, the topic of mathfluency. So far we've tried to
establish what it is with oneguest. And then think about how
to measure it and develop itwith other guests. And we're so
excited about our next guesthere today. Bethany, how has
this topic been working on you?
(00:44):
Has it been feeding you andyour ideas about education at
all?
Bethany Lockhart Johnson (00:48):
I mean, Dan, we've already
established that I've been thefluency fan from go.
Dan Meyer (00:53):
Minute one.
Bethany Lockhart Johnson (00:53):
So the better question is, how is
it working on you? I definitelylove our deep dives. I love the
fact that we get to take thistopic and talk about it over
multiple episodes, and hearfrom the audience, and
incorporate their questions.
So, I'm thrilled to be here. Sowhat about you? You're here.
You're ready?
Dan Meyer (01:12):
Yeah. For those who are just popping in for this
episode, Bethany and I have adynamic that pulls in different
directions sometimes. Andthat's always been really
helpful for us. In our banter.
Our repartee. In this season,the tension is that Bethany is
a huge fan of fluency andfluency development. And I have
had experiences, as a secondaryteacher, where kids are coming
(01:35):
to me after eight, nine yearsof having, in some cases, some
pretty negative experienceswith fluency development. Where
they'll come and they've hadthese different moments where
they think like, "Oh, so mathis about doing the same boring
thing over and over again, andgetting some feedback that I'm
smart or dumb. And I generallyfeel dumb with that." So that's
(01:55):
led me to kind of drift awayfrom enthusiasm towards
fluency. Drift away fromfluency. But what's been
working on me a little bit is areally strong strand through
our conversation so far,Bethany, around equity. And how
if I deprive a student ofexperiences of generating
fluency, even if that's infavor of experiences where they
(02:17):
see the beauty of math, let'ssay, or the creativity in math,
I'm making it harder for themto experience, through that
fluency, new ideas asaccessible and approachable. So
that's been, I think, the ideathat's been working on me so
far this season.
Art Baroody (02:33):
I would love to interject here, if you wouldn't
mind.
Dan Meyer (02:35):
Folks, guest Dr. Art Baroody, can't wait for his
intro.
Baroody! Let's go! Let's talkabout it.
Art Baroody (02:47):
I really appreciate your comment about
how fluency ruins kids. Becauseif it's fostered in the wrong
way, it's exactly what's gonnahappen. Kids are gonna think
that math is all about justmemorizing stuff, and
regurgitating it quickly. Andit does so much damage to
(03:12):
approach fluency in the wrongway, through rote memorization.
But —and this is the wholecentral theme of our talk today
— if you teach fluencyappropriately, if you foster
meaningful memorization, thenyou get the kinds of things
that you wanna see too, whichis an intellectual curiosity. A
(03:35):
looking for patterns andrelations. And an excitement
about math, and an ability toappreciate math. And really
wanna do something about math.
Dan Meyer (03:49):
I'm feeling this so much. And I really want us to
get into this distinction ofmeaningful memorization,
meaningful fluency, shortly.
But, let's tease 'em a littlebit here, Dr. Baroody. Let's
not give away the whole farmright away
you wanna share who we'reworking with here today
?
Bethany Lockhart Johnson (04:07):
Well, you've already given us quite
an intro, Dr. Baroody. Butlisteners, this is the fabulous
Dr. Art Baroody, ProfessorEmeritus at the University of
Illinois, Urbana-Champaign.
He's someone who's beenstudying math fluency for
(04:30):
decades. And, you're gonna helpmake the case for not only why
fluency is so important, buthow do we do it in a way that's
meaningful. So, thank you forbeing here. And I love that you
just dove right in. You'relike, "Oh, Dan
I'm here. I'm ready
(04:53):
here. And we're so excited tohave you on the show.
Art Baroody (04:56):
I just couldn't resist. I've seen so many
teachers frustrated by the waymath is taught. And how kids
come to them in their class andthey're basically not
interested in math. Or they'reeven anxious about math. ...
We've gotta do a better job.
Dan Meyer (05:17):
Fried to a crisp, yup.
Bethany Lockhart Johnson (05:18):
Hey, when you're in the Lounge, let
me tell you, you don't have tohold back
conversation. We welcome it.
That's why you're here. Wewanna hear from you. I do wanna
ask you, in broadening the waythat we think about fluency,
one of the things that we'reasking all of our guests is,
(05:38):
what's something that you aredeveloping fluency in right
now? Beyond all the amazingwork you've done in
mathematics, what's somethingthat, right now, you either
have recently or are activelytrying to develop fluency in?
Art Baroody (05:53):
Well, at this point in my life, I'm trying to
be a better husband, father,and grandfather
always a work in progress. But,it's especially important to
keep my wife happy
Bethany Lockhart Johnson (06:11):
You know what? Wiser words have
never
improve. Let me tell you whatmy husband did yesterday—no,
I'm just kidding.
Dan Meyer (06:26):
We might jump back in on that too at some point.
Because I love thinking aboutlike, "Is every situation we
have with our significantrelationships, the folks we
want to keep happy, are theyall unique, like snowflakes?
Requiring a deep conceptualwork on the part of the "
learner" — here, you — tofigure out a solution for? Or
(06:48):
are there ways in which yourstudy of fluency can actually
be applied in a meaningful wayto this real-world question of
fluency in relationship?"Please don't answer. This is
going way deeper than weprepped you for.
it's just a curiosity that Ihave right now. Are there
moments where our work in onearea can actually jump into the
(07:08):
other? I'm gonna let that hangfor a second, Dr. Baroody, and
instead ask you about your workin grad school. What fascinates
me about you is that, when Iwent through grad school, I was
motivated to prove a thing. Igot these ideas! They're never
gonna change! They're correct!I'm gonna find the research and
do the studies to back 'em up!And I feel like you have a bio
that I love. Which is you cameinto grad school with ideas
(07:31):
about fluency, it sounds like.
And then those started tochange a bit, through your
encounter with reality. Withyour empirical work. And I
would love for you to describea bit about your transition, or
what you learned through yourgraduate work on fluency.
What'd you find, as far as theconventional wisdom? And how it
was correct or incorrect?
Art Baroody (07:53):
Sure. As an undergrad, I was trained as a
teacher, and I becameinterested in psychology,
especially the psychology oflearning. So when I went off to
grad school, a natural majorwas educational psychology. And
I had the good fortune ofstudying with Herb Ginsburg,
(08:16):
who was a great mentor. He wasa leading researcher in
mathematical learning. And, inparticular, he was interested
in this new area of children'sinformal mathematical
development, and how itprovided a basis for formal
mathematical learning,including providing an informal
(08:38):
basis for fluency. And sothat's where I got my initial
interest in fluency. HerbGinsburg was an interesting
mentor. What he would do withhis grad students is say, "OK.
Here's one side of the issue.
Here's the other side of theissue. Here's the research. You
(09:00):
decide which is right." Now,that's highly unusual
But it helped me a great deal,because what I started to do
was dive into the research onmathematical learning. And one
of the issues that we wereencouraged to explore was, how
do kids learn the basic numbercombinations? When I started
(09:25):
grad school, I believed whateveryone believes: that if kids
don't know the basic facts,then what you have to do is
make sure they have massiveamounts of drill and practice
to memorize those facts. Thatwas certainly true of my
(09:45):
undergraduate training. Infact, I was at a parent-teacher
meeting and one of the fathersstood up and said, "Everyone
knows what two plus two is. Andyou know how we know that?" And
here it is, someone who's hadno training whatsoever, but of
course they know everythingabout how to teach mathematics.
(10:05):
"OK," he said, "Well, rememberback in first and second grade,
what did we do? Your teachersgave you all sorts of practice
and drill on two plus twoequals four. And that's why you
know two plus two equals four."It turns out that a lot of kids
(10:27):
know two plus two equals fourbefore they even get to school.
So, it's not an adequateexplanation. We'll talk more
about that later. One of thethings that comes out of a
traditional or a conventionalview of fluency is that
informal methods, such ascounting, especially finger
(10:53):
counting:"Oh! That's bad!Counting and finger counting,
those are just crutches foravoiding the real work of
memorizing those facts!" Well,that's certainly what I
believed at the time. And I,basically, was looking at the
issue through adult eyes. WhatGinsburg taught me was that, in
(11:17):
order to understand learning,to understand the issues around
learning, you also have to lookat the issues through a child's
eyes. And one of the firstthings I learned was that kids
have their own informal methodsof solving math problems. So,
(11:40):
if you ask a child, "How muchis five plus three?" Kids,
typically, at least initially,what they'll do is they'll
count out five fingers, orthey'll put up five fingers,
count out three more fingers,or put up three fingers, and
then count all the fingers andcome up with the answer. Why is
(12:04):
this important? Because kidsaren't stupid. What they do is
they continuously inventincreasingly efficient counting
procedures to figure out sumsand differences. And, what this
enables kids to do then is tolook for patterns and
(12:28):
relationships among the numberfacts. And this can then be
used to devise reasoningstrategies. For example, I was
observing a kindergarten classwhile I was in grad school one
time. And, we were playing asimple race game, where the
(12:49):
children would throw two diceand figure out what the sum
was, and that's how many spacesthey could move on a racetrack.
And one little girl rolled asix and a one, and she didn't
have six fingers on one hand,so she wasn't quite sure what
(13:10):
to do. And she was really kindof puzzled by the question. And
the little girl next to herleaned over and whispered in
her ear, "That's an easy one,Marcy. It's just the number
after." Now, what thiskindergartner had discovered
(13:34):
was a connection between hercounting knowledge,
specifically her number-afterknowledge, and adding one. When
you add one, the sum is justthe number after the other
number in the count sequence.
So, for six plus one, theanswer is, what number comes
(13:57):
after six when we count? Seven,it's easy.
Bethany Lockhart Johnson (14:00):
I have to say, what I'm hearing
is you seeing the brilliance ofchildren.
Art Baroody (14:07):
Yes.
Bethany Lockhart Johnson (14:08):
And you're seeing the wisdom, and
what they bring in, and theirinnate brilliance, right? And,
so much of the conversation, Ifeel like, when we're talking
to educators, is really wantingto celebrate the way children
think. And I often feel thatthat's separated from the task
(14:31):
of fluency, right? But whatyou're saying here is exactly
what I think excites me aboutit, is that there are patterns.
There are ways to look at it.
There's sensemaking happening.
Even in something that mightseem simple to an adult. Two
plus three — kiddos arebringing so much to that. And,
when we listen, we can respondin ways that respect and value
(14:54):
what they're bringing to thetable. And, it sounds like the
way that you thought aboutfluency and listening and
working with kids reallyevolved.
Art Baroody (15:03):
Yeah, that's interesting. I was once told by
another adult that being apsychologist must really ruin
kids for me. Must take all thefun out of it. No
see more. You appreciate moreof what they can do! It's
(15:24):
amazing. And
opposite.
Dan Meyer (15:32):
What I'd love to say about the plus-one example here
is that moment happened in aspan of like five seconds,
right? That exchange betweenstudents. And what folks like
you, and us, and our listenerswho have taught for a while and
thought deeply about teaching —you can see that moment up in
the sports booth playing out inreal time. And say, "Whoa, slow
(15:52):
that down. Back it up. Let's doa replay." And you could do a
whole dissertation —maybe youdid — about that one moment.
And, I just think it's sointeresting how that came from
a peer; it's like a moment thatmight get missed by a
layperson, this person whospoke to you in a meeting about
how kids develop fluencywithout expertise. And I also
(16:15):
think it's so interesting how alot of people might suggest
that what the student neededwas to be told this idea about
one-after, before. But it'samazing to me how much time
students need. Like, I see myown six-year-old counting up to
(16:36):
five. And I'm just like,"Buddy!" I just wanna tell him.
And sometimes I even do. But itdoesn't matter. So much of
these ideas — the informallearning — is so durable and
needs to ... not run its owncourse, but have enough
experiences tossed at it. Sothose moments you described
then become natural. And Iwonder if you'd explain a
little bit about what you'relearning about fluency
(16:58):
development gone wrong. Whatare some common ways ... I feel
like I've mentioned likerushing it—
Art Baroody (17:03):
Can I expand on your other comment, for just a
moment?
Dan Meyer (17:06):
Spin it, yeah. Spin it, please.
Art Baroody (17:08):
What I think is interesting about informal
mathematical learning is thatkids can learn a great deal
from their peers. And, often,their peers are closer to the
situation and understand thesituation better than adults.
We, as adults, have forgottenhow we learn the basic
(17:31):
combinations. And so we — mostadults — aren't even aware that
there's a number-after rulethat can be used to quickly,
efficiently generate the sum ofANY number plus one, for which
you know the count sequence!Most people just are not aware
(17:51):
of that. But this little girlprobably had just made the
discovery, or done so in therecent past. And so, for her,
it was still a fresh issue. Itwas still something in her
mind. And so, she was able toshare that with her peer. And,
w ith any luck, her peerbenefited from that.
Dan Meyer (18:11):
I think it speaks to all these different learning
paths that we've all been on indeveloping our own fluency. And
how easy it's to forget thepath in favor of the
destination, and then try tocreate this paved route for
kids that bypasses what, for somany of us, was necessary kind
of stumbling around a bit. Idon't wanna revel in the
(18:31):
stumbling ... but I guess theinformal-formal dichotomy there
is so important to me. We'renot just, like, discovering or
wandering in the dark. But weare taking stock of our
surroundings, and thinkingabout what resources we have.
And, sometimes, I think, itseems like teachers want to
move to the side of that. Andsay, "Well, actually, there's a
(18:52):
highway over here. It's movingso fast and can get you there
faster." Which I think leavesstudents disoriented to these
connections from informal toformal knowledge. How close is
that to what you studied withGinsburg, about how formal and
informal knowledge is related?
Art Baroody (19:10):
Basically, one of the key points that Ginsburg
made was that most mathematicallearning problems in school are
due to a fact that there's agap between the formal
instruction and the child'sinformal knowledge. So, if a
(19:31):
kid is struggling with math —and I have found this with my
own case studies — the problemtypically is not the child. The
problem typically is that theformal instruction is not
connecting with what the childknows already. And, if your
(19:54):
teachers get nothing else outof this talk today, the
Principle of Assimilation,Piaget's Principle of
Assimilation, is that weunderstand things in terms of
what we already understand. So,we understand new stuff in
terms of the stuff that wealready understand. And that
(20:17):
principle is violated way toooften. And when it is, kids
don't have any recourse, otherthan rote memorization. Or
quitting. Not learning it. Andthose aren't good options.
Dan Meyer (20:36):
Yeah, I love it.
Since all new knowledge buildson old knowledge, it leads to
this beautiful rule forlearning, which is that
everybody knows something abouteverything. Pick any topic that
I might one day learn aboutthat's way beyond me.
Aeronautical engineering, let'ssay. That knowledge has gotta
build on stuff that I at onepoint knew, down to knowledge
(20:58):
that I was forming as a smallchild, crawling around. It can
be useful. Puts me on a pathtowards this knowledge. I love
the spirit that Bethany bringsthroughout our work, and what
we've talked about here, thatkids are brilliant. They have
resources that can help all ofus learn. Feels like to me an
important takeaway of ourconversation here. Bethany,
what's your thoughts aboutthat?
Bethany Lockhart Johnson (21:20):
Well, absolutely! I feel like, as a
teacher, we're facilitators tohelp build those connections.
Or to help highlight thoseconnections, and celebrate
those connections, between thenew material and the things
that kiddos already know.
Right? And I feel like whenwe're thinking about fluency,
(21:40):
when we see timed tests orspeed drills, and we see — as
you said — the damage that'scaused by that, not helping
kiddos to build connections. Ifeel like ... I don't know, in
listening to this conversation,I feel really moved. Because I
(22:00):
get so excited thinking aboutdifferent moments when I've
seen kiddos build thoseconnections. And how can we, as
educators, help each other tomake sure that, even in
something that has so beendrilled down to these speed
tests and rote memorization,how can we make those
connections present there, too?
(22:21):
How can we treat that with thatsame level of respect, and the
kids with the same level ofrespect. And I feel like you've
spoken about something calledmeaningful memorization. And I
feel like maybe you can talk abit more about that. Because
there's a lot more awareness ofkids and making it matter and
(22:42):
respecting them than rotedrill-and-kill, right
Art Baroody (22:46):
Meaningful memorization basically involves
a number of things. Meaningfulmemorization builds on a
child's conceptualunderstanding of an operation,
for example. So a child needsto understand what addition is,
what subtraction is, whatmultiplication is, in order to
(23:11):
achieve fluency with thosefacts. Let me give you an
example. If a child understandsthat one meaning of
multiplication is a groups-ofmeaning. So, five times eight
means I've got five groups ofeight items each. OK? If a
(23:37):
child understands thatconceptually, then five times
zero makes sense. Because whatdo you have? I've got five
groups of no items. So how manyitems do I have all together?
(24:01):
facts, the answer here is notgetting bigger
kids can understand it if theyhave this conceptual
understanding of multiplicationas a groups-of meaning. So, it
can help kids then memorize thezero rule for multiplication in
(24:22):
a meaningful way. The otheraspect of meaningful
memorization is discoveringpatterns in relations. Like our
little girl here, whodiscovered the connection
between adding one and hernumber-after knowledge, her
existing number-afterknowledge. So meaningful
memorization involves buildingon both conceptual
(24:45):
understanding and trying tofind new patterns and relations
to enrich that conceptualunderstanding.
Bethany Lockhart Johnson (24:52):
For me, when I hear that, it sounds
so spot-on. And it makes a lotof sense to me. But what would
be a response to folks thatfeel like, "Yes, let's make it
meaningful, let's build theseconcepts, but we still need the
timed test , we still need thespeed drills"? Is there a space
(25:14):
for that?
Art Baroody (25:18):
Speed tests, timed tests, are a tool. An
educational tool. And like anytool, they need to be used
carefully and thoughtfully, andwhere appropriate. The problem
(25:41):
with timed tests as they'reoften used is that they're
overused. Timed tests makesense after a child understands
an operation, after they've hada chance to explore the
operation using counting. Sothat they've had a chance to
(26:04):
find patterns and relationsamong the facts and devised
reasoning strategies like thenumber-after rule. Once a child
has devised a reasoningstrategy, then it would make
sense to have them practice it— even under a timed condition
— to make sure that it becomesmore efficient, that it becomes
(26:30):
fluent. But there's no sensetrying to impose fluency before
a child has constructed thereasoning strategy. That makes
no sense whatsoever. It makesno sense whatsoever to have
timed tests before a child hasconstructed an understanding of
the operation. So, mathematicseducators, for a long time now,
(26:54):
have argued that you need to becareful about premature
practice. That you shouldn'thave kids doing drills before
they have devised means forfiguring out the sum or
difference or product orquotient or whatever. So I'm
(27:16):
not completely opposed to timedtests. But boy, you have to use
them really, really carefully.
When we were developingsoftware for helping kids learn
the basic addition-subtractioncombinations, initially there
was no timing involved. Butonce the child had developed a
(27:39):
strategy, then we wanted tostart introducing the child to
some time limit — a generoustime limit — so that there was
some incentive to use thestrategy as quickly as
possible. Now the thing is,practice doesn't have to be
boring. It doesn't have to beflashcards. It doesn't have to
(28:00):
be boring drills. It caninvolve games. Dice games are
especially important topreschool kids and kids in the
early primary grades. Why?
Because they can see again andagain that two plus one is
three, that two and two isfour. So they can begin to
(28:24):
learn some of the add-onecombinations or facts, and they
can begin to learn some of thedoubles, such as two plus two
is four; three plus three issix. And this then provides a
basis for learning other facts,such as two plus three. Because
if you know the number-afterrule for adding one, and you
(28:48):
know the double two plus two isfour, you can look at two plus
three and say, "Ah, that's justtwo and two and one more." So,
basically, you're building onyour previous knowledge to
figure out new facts.
Dan Meyer (29:07):
So I got a voiceover here for a second. I feel like
you've just helped me have anepiphany here, Dr. Baroody,
which is , well, first of all,timed tests are this topic that
just kinda lurks in thebackground, like a ghost, of
every conversation we've hadabout fluency. And your
perspective is, I think, asomewhat unique one. Not a
(29:27):
hard-line perspective here. Ilike the idea that they're just
overused. They're a blunt-forceinstrument to try to pressure
kids into fluency when there'sso many other interesting
games, where a student'snatural inclination to optimize
or even win can carry thefluency impetus. I dig that.
Number one. And, number two Ithink is this: I feel like
(29:49):
we're just mired in thesedichotomies in math education
discourse around, for instance,conceptual and procedural.
Which one comes first? Whichone comes second? And I think
you've helped make sense of afinding from Bethany
Rittle-Johnson. This articlewas like: They develop
together. And it's given me anew question, which is it's
not—and it's a question I askmyself—it's not like, is this
(30:13):
conceptual or procedural,what's going on here? But
rather , what resources dostudents have? And what
experiences would help themdevelop those? 'Cause what
you've helped me see is thatthe fluency is not just the
formalization of thisburgeoning resource. It
actually creates a newresource. Where when the
student is like, "OK, I get itfive times one, five times two,
five times three, five timesfour." That is not just a
(30:36):
formalization of the kidscounting and grouping. It gives
the kid a resource to then dofive times zero. That is then a
new resource in the bag ofresources to help the kid
create new concepts. Theconcepts create the fluency,
create the concepts. And so, Idig this question of "What are
the resources a kid has? Andwhat do they need to develop
new ones?" Versus, like, "Arewe doing concepts today? Are we
(30:58):
doing fluency today?" No, it'sboth! It can be both! I dunno,
how close is that?
Art Baroody (31:02):
Back in 1986, Herb Ginsburg and I wrote a chapter
together to address the issueof procedural and conceptual
knowledge in mathematics. Atthe time, there were two very
popular views. There was theskills-first view, where you
taught kids the skills. Youdidn't bother to take the time
(31:26):
to help them conceptuallylearn. You just taught them the
skills by rote . And then kidswould apply those skills and
eventually understand the mathtoo. The concept-first approach
was you teach for understandingfirst and then the skill
(31:49):
learning will be easier. If youunderstand the conceptual basis
for a procedure, you're muchmore likely to learn the
procedure. Well, Ginsburg and Itook the position that the two
couldn't be separated. Thatbasically it was an iterative
(32:09):
process. That is, you mightlearn a skill and then discover
a relationship, and then usethat relationship to understand
something more, and then devisean even better skill, and so
forth and so on. So I thinkthey go hand-in-hand.
Dan Meyer (32:29):
They're buddies.
They're buddies.
Art Baroody (32:31):
Another view would be the simultaneous view. That
both skills and concepts candevelop together. And that's
another interestingpossibility, I think, of how
they might go togetherhand-in-hand. So, I don't see
this dichotomy betweenprocedural knowledge and
(32:52):
conceptual knowledge. It's notthe best paradigm for trying to
figure out how to teach. Ithink it's really important
that children understand boththe concepts underlying
procedures and the procedures.
And when you do that together,you're much more likely to have
(33:13):
procedural fluency, becausefluency is often defined only
as using a skill or procedureaccurately and quickly, that is
efficiently. The NationalResearch Council published a
book called Adding It Up; andthey argue from the research
(33:36):
that it makes a whole lot moresense to think of fluency not
only as efficiency, that isaccurate and fast use of
procedure, but the appropriateuse of a procedure, and,
perhaps even more importantly,the application of a procedure
(33:57):
to a new situation. Now, youcan get efficiency through rote
memorization, but what youdon't get is efficiency plus
appropriate use and, moreimportantly, adaptive use. So
that's why I think bothprocedures and concepts need to
(34:19):
be taught in an intertwinedway.
Bethany Lockhart Johnson (34:23):
Early childhood ed is my joy, is my
land. And I remember, actually,when I was a student teacher in
a TK classroom. So, thesekiddos are four, right? This is
years and years ago. And wewere talking about numbers. And
this little boy said, "Two plusthree is five." And he said it
(34:45):
in such a way that it was clearthat he's regularly celebrated
for knowing that, right? AndI'll never forget that my
mentor teacher said, "Whoa! Canyou show me with these blocks?"
We had little blocks on thetable. And the little boy burst
into tears. And there was thispalpable panic, right? And, I'm
(35:09):
not exaggerating when I saythat was a transformative
moment for me. Because what Isaw, and I found out later,
this kid had been in — I willnot name it, but a program that
purportedly supports thegaining of math skills. And you
can pay them money to have themwork with your kiddo. So the
(35:30):
little boy clearly knew thisscript, right? But when he was
asked to show it or makemeaning of it, he couldn't. And
there was a fear; and there wasa panic. There was a panic. And
I know that the parents andcaregivers of this child would
never do something ... theythought they were doing good
(35:53):
and right and helping them givea leg up. They weren't trying
to harm their kiddo. They'retaking the time to drive this
kid to a class, probablySaturday morning. And when I
think about that, and I thinkabout the way that the fear
that that little boy had wasalready holding, and the
anxiety that little boy wasalready holding, and I've seen
(36:14):
how fluency can be developed sodifferently, I just ... I feel
like this conversation — and Ihope that for educators
listening, I feel like you arereminding us and giving us
permission to slow down. Toslow down. To attend to
children's thinking. To noticethese little moments when
(36:36):
children are showing theirbrilliance. And I just really
appreciate that. And I reallyappreciate your perspective,
your learned perspective, afterdecades and decades of—not to
like, "Whoa, over decades anddecades
celebrate that. So, thank youso, so much for being in the
Lounge with us. Thank you forhelping us make more sense of
(37:00):
this. Really, really, really.
Art Baroody (37:02):
You're absolutely welcome. One of the reasons I
was so passionate about this isfrom my own experience. I was a
first grader at the time. I wassitting there at my desk doing
a worksheet, a math worksheet,and I had my fingers under the
desk, because I was figuringout sums. And my first grade
(37:26):
teacher said, "Arthur, what areyou doing with fingers under
the desk?" I should have toldher anything but the truth.
'Cause I told her I wascounting to figure out my
addition . Well, my normallypatient and kind and
(37:48):
soft-spoken teacher erupted andsaid, "Don't do that! Don't do
that!"
Bethany Lockhart Johnson (37:56):
That makes my heart hurt for baby
Art! I've had parents tell methat they've held their kids'
fingers down. "I've beenholding his fingers down to try
to keep him from using 'em ,but he just keeps using his
fingers." I'm like, "What? Whatare you doing? There's a tool
you carry, a tool around!" Oh,wow. Wow.
Dan Meyer (38:18):
It's a resource.
Art Baroody (38:19):
What was my first grade teacher telling me? What
are the parents who hold theirchild's fingers down telling
the child? Your way of doingthings is wrong. It's inferior.
It's stupid. And that's how achild's confidence in their own
knowledge can be undermined,instead of building on it.
(38:43):
Instead of honoring it. We toooften say, "Oh, well, that's
not right. That's not the rightway of doing it. Here's the
right way of doing it. Theformal way." What it does is
just undermine children'sdisposition to learn
mathematics. It underminestheir confidence to solve
(39:06):
problems. And they stop doingit. They just stop.
Dan Meyer (39:10):
They're learners.
And they eventually learn whatnot to do.
Art Baroody (39:12):
And then they become dependent on the teacher
and they say, "All right , Idon't know how to do this. You
tell me how to do this."
Bethany Lockhart Johnson (39:20):
Yeah . Not to mention all the shame
and the fear
Art Baroody (39:23):
Yeah. And, you actually hear kids, "Don't
explain to me why! I don'twanna know why. Just tell me
how to do it." It's justabsolute craziness. And that's
what the focus on rotememorization does to kids.
Dan Meyer (39:39):
Yeah, that's the outcome.
Art Baroody (39:40):
It really destroys any disposition to think
mathematically, to enjoymathematics. It can be really
harmful.
Dan Meyer (39:49):
Well, I always appreciate, Dr. Baroody, a
conversation that isfar-reaching and takes very
seriously teaching mathematicsand the gifts that students
bring to us in our classrooms,if only we recognize them as
such. Bethany and I reallyappreciate your time here in
the Lounge. So, thank you. AndI hope we meet again some time
.
Bethany Lockhart Johnson (40:08):
Thank you so much, Dr. Baroody.
Art Baroody (40:10):
Any time . I'd be happy to come aboard any time.
Dan Meyer (40:13):
Thanks so much for listening to our conversation
with Dr. Art Baroody, ProfessorEmeritus at the University of
Illinois, Urbana-Champaign. Letus know what you thought of
this episode. I thought it wenta lot of very interesting
places. Hit us up in ourFacebook discussion group, Math
Teacher Lounge (40:26):
The Community, or on X, formerly known as
Twitter, at MTLShow. Whatquestions do you have about
math fluency? What would youlike to know? Please let us
know and we'll do our best toinvestigate that for you over
the course of this season. Makesure you don't miss an episode
in this new series bysubscribing to Math Teacher
Lounge (40:44):
The Podcast wherever you find fine podcasts
products. You can find moreinfo on all of Amplify shows at
our podcast hub. Go toamplify.com/hub. Thanks again
for listening, folks.
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