Better Teaching: Only Stuff That Works

Welcome to Better Teaching, Only
Stuff That Works, a podcast for

teachers, instructional coaches,
administrators, and anyone else who

supports teachers in the classroom.

I'm Gene Tavernetti, the
host for this podcast.

And my goal for this episode, like
all episodes, is you laugh at least

once, and that you leave with an
actionable idea for better teaching.

A quick reminder, we will be
sharing only stuff that works.

No cliches, no buzzwords.

So my guest today is Bill Davidson, and
this summer Bill Davidson is entering

his 24th year working in education.

He's taught in Philadelphia, Los Angeles,
Southern Sudan, and Monterey, California.

Yeah.

For the past 12 years, he's worked
independently as as an elementary

math trainer specializing in the
Singapore and math curricula.

Now, you may recognize Bill's
voice, because he's also the

host of the podcast Centering the
Pendulum, a podcast about education,

educators, and teaching practices.

In fact, I was attracted to Bill's
podcast and how I became familiar with

Bill because of the title of the podcast.

As I just said, there are no cliches, and
one of the cliches that I hate most in

education is the swinging of the pendulum.

when I heard the centering of the
pendulum, I said, this is, for me.

And, Bill, before we get started,
let's center one thing on the pendulum.

Let's, agree that we are not going
to talk about the fact that kids

should learn their math facts.

I'll try not to say it

how controversial is that, we
need to even talk about that?

So Bill, I'm excited to talk to you,
and one of the reasons that I wanted to

talk to you is that I met you, again,
the first time I became aware of you

was through your podcast, but then
we had a chance to talk, a little bit

and to share some emails about, Word
problems, story problems, application

of math, and then, you have a new book
coming out, Infusing Dynamism into

Daily Problem Solving, Five Alternative
Methods to Practice Tape Diagramming.

I thought, man, this is a great chance
and a great time to, to catch up with you.

why this book,

there's a couple reasons.

first of all, I, think you mentioned that
I specialize in Singapore math, but also

Eureka Math, which I think has a lot of.

Connections to Singapore math.

The majority of the early writers,
at least of the Eureka math

program, Engage New York were
first Singapore math writers.

So I, I find that the programs are very
similar and they both specialize in

what Singapore math calls bar diagrams,
what Eureka math calls tape diagrams.

I find the same problems arose with
both Curricula and all the schools

and teachers I've worked with.

Why this project?

First, I like projects.

As a teacher, I always
had a project going.

I ran science fairs, I founded a school
newspaper, I did video yearbooks.

I like working towards something
professionally outside of the actual job.

I find it, that does a couple things.

One, it gives me, something to wake
up and do and get excited about.

Two, it helps me articulate philosophies
and ideas and just grow professionally.

It's good self professional
development and I think I become

a better trainer when I do that.

Now, why the book and
why this specific book?

Originally, and I know I said I
wouldn't talk about basic skills,

basic facts, but I was the fluency
writer for Engage New York Eureka

Math, the 3rd through 5th grade.

was the 3rd through 5th grade fluency
writer for Engage New York Eureka Math.

Part of me thought I was getting
typecasted as I, I only work with fluency.

I wanted to do something for problem
solving because I think I have some

things to offer problem solving.

Bar modeling, tape diagramming as well.

But why might it be useful for teachers?

between 2014 and the start of the
pandemic, I was supporting a lot

of schools in the implementation
of Engage New York Eureka Math.

One of the main challenges that
teachers were encountering was

fitting each of the lesson components
into a one hour math block.

The lesson components were, begin
with fluency, an application problem,

so a problem of the day, And then a
concept development with a debrief.

Those were the four components and
most teachers just weren't finding the

time to do the application problem.

And as a result, they
were skipping over it.

I thought this was a mistake, but I
understood why it was taking place.

They were under time constraints.

So I, thought rather than skipping it
entirely, can you find a way to honor

the purpose of an application problem, of
a daily word problem, even if you don't

have the students engage in a redraw
right process for 10 to 12 minutes.

And therefore I started to develop
a few different methods so that the

students could still work with problem
solving, so that they could still

work with tape diagrams without doing
every stage and procedure, in what

you do to solve a word problem, And,
therefore you wouldn't be neglecting

the kind of higher order thinking,
problem solving thinking that you're

doing, and I've I made five alternative
methods in this book, so five different

ways to do problem solving without
just doing a standard word problem.

So one of the, one of the things, Bill,
that, when we spoke that I was so excited,

the hair on my arms was coming up I hate
to tell you this, but you weren't my

first choice to sending this material,
and so I was looking for somebody that

would actually respond because the
people, the math teachers I were sending

to hadn't thought about teaching word
problems in the way that you talked about.

for example, if you could expand on what
you mentioned, one of the methods that

you just referenced was redraw right.

Yes.

Could you, expand on that a little bit,

Yeah, so that's the recommended,
I guess procedure that Eureka

Math, provides for, teachers.

Read a word problem, draw a diagram
or a math picture of what you read,

write a number sentence or equation to
solve, write a number sentence to solve.

And that's how you solve the problem.

Now, and that just provides the
students a loose structure on how

to go about doing a word problem.

it's a loose structure, but it works.

Every time.

I think so.

I saw, there was a book with a let, like
the 11 steps to solving any word problem.

I think that's too much.

I, and I think the kids are
looking at a wall the whole time.

Honestly, and I know a lot
of people disagree with me.

I'm not big on underlining keywords.

I'm, I beginning represent what you read,
and then students may do that in addition.

I, I, think it can be over formulaic.

I like that structure because
it provides a lot of room for

idiosyncrasies within a structure.

what I've found, and, again, I don't
want to give you a blister patting you on

the back of all the things that we agree
upon, but the idea of keywords, underlying

keywords, cross out what's important.

Okay.

What's the key word?

Okay.

What's important?

You don't know that.

To me, what you just talked about, the
read and draw, the draw is what allows

the students to understand the problem.

And now, what's important?

Now, what are the key words?

but the idea that, oh, every time
you see all together, it's that's

not a, that's not a math strategy.

I agree.

And I think it was Leonardo da
Vinci said, to draw is to think.

So we read, how do we
represent what we read?

And, we need students to get in that
practice of I read, I represent.

Now, another, piece of this in my
evolution, this comes from way back,

as a sixth grade teacher, I found,
especially at that age, you'd have,

students would read the problem.

Some would answer it very quickly.

they had a mental model of
everything that was going on.

They'd have an answer within
the first 20 or 30 seconds.

And other students would, I see a
word problem, I put the pencil down.

I quit before I start.

we'll get into complexes maybe later.

And I thought at best I engaged
maybe 40 to 50 percent of the class

in kind of reasoning about this.

But I thought it just lacked dynamism.

It wasn't interesting to be a part of.

I didn't like it.

It was an uncomfortable
drawing to the math flow.

and so I wanted to find ways
to keep everyone engaged.

Ten years ago, I created a five book
collection, what I call problem solving

ladders, where you give students
couple sentences without a question.

They draw what they draw a representation,
and then they answer a series of

problems that escalate in difficulty,
thus the ladder metaphor, right?

And so my goal is every child
will confidently answer a question

correctly, Every child will feel
challenged and have to think.

And then there'll be a couple
other questions in the middle.

And that way you have a below
grade level, above grade level, and

grade level all housed within the
same eight to 10 minute activity.

If I can, I'll get into
another thing I bumped into.

Another thing I bumped into is often
Especially with newer teachers,

you would get 4th, 5th graders to
really begin resisting drawing.

Some would see it as a sign of
weakness, okay, I don't need to

draw, I'm too smart for that.

Some students would, feel, especially
early on in implementation of

Eureka Math, Singapore math.

They wouldn't be that proficient with
the models and comfortable drawing,

so they would just say, I won't do it.

Now, what do teachers do?

you can keep begging them to draw.

You can keep, you can create this
friction, or you can maybe take a

more passively aggressive approach
to getting them to recognize the

relationships that bar diagrams do,
the visualization of word problems.

And, therefore I came up with a
couple methods like two diagrams.

one is a correct depiction of the
word problem, the other is incorrect.

Which one is correct?

Now, as soon as they've got an A and
a B problem, the assignment is to

determine which diagram is correct.

So you have to analyze to
participate in class, right?

Whereas if I just give you a word
problem, don't have to diagram to

participate, to try to solve it, right?

Another method I use in this book is,
here's a diagram, Could you create

a word problem to match the diagram?

That's harder, but it's also to
participate in the activity, you need

to critically analyze a math picture.

And I think those things may Make the math
experience for students, the math problem

solving experience for students, much
more dynamic than here's another three

sentences, read it, draw it, write it.

It gets very mundane if we do that.

Mm

as you talk about that, and as
I was, looking at your book,

I think two things happen that
are very powerful for teachers.

The first thing is that, what don't the
kids know how to do, and what don't the

kids want to do, because nobody wants
to do what they don't know how to do.

They only want to do
what they know how to do.

You just articulated the fact that
students have an issue, it's, they

haven't practiced, they have a, a
phobia, or it's too easy, whatever the

reason, they don't do the, diagram.

The visual representation, but the
way that you've presented it and the

way that you've just described it,
where, okay, here's a, representation

and here's one that's wrong.

Now they can get repetitions
of just that part.

And the way that you've presented it with
problem after problem, If that's this part

that's stopping the students from being
successful, this is what I tell teachers.

If you know the part that's being,
that's stopping them, practice that.

Forget the rest of it for now.

You could just take this problem, practice
it, and then the other thing before I lose

my train of thought, which is interesting
to me about, teachers not asking kids to

do the diagrams because they think that
they are uncomfortable or, whatever.

I don't know a lot about engineering,
but I know my brother's an engineer

and they sketch out everything.

So I, don't know how you could be in a
STEM school or a STEAM school and that.

And that we don't demand.

part of the way we do things in
this school is, we're going to draw

some sort of visual representation.

And it also creates common
diagram for discussion.

teachers, do the students need
to become expert diagrammers?

I think it's ideal that they do, but
Sarah Cottingham, one of my guests

at my podcast, she would have the
right word, and I'm just, I don't

have the word with me right now.

She would say, you, students need to
take in information the same way so

that you have a shared understanding.

But you can hold a group of students
together in a discussion about

a word problem, if you have a
common model to work from, right?

I, I don't, I'm, I regret this, that
I'm working on a podcast right now, I

recently introduced, in, interviewed
Scott Baldridge, a research mathematician

at LSU, and he said that, they're
creating a problem out of a math

graphic is something mathematicians do.

Now I don't have the details,
but that is another method,

that I'm using in this book.

And so he said it has a lot of
mathematical relevance longitudinally

as students go to greater, to
work with higher levels of math.

I think one of the things that I found,
and I don't know if you found this, but

I've worked, for the past 20 years in
schools throughout, California, a few

out of state, and I can go to a school
and work with the math department.

I go to any middle school in California,
regardless of economic status, overall

scores, state scores, and say, oh,
what do your kids have trouble with?

fractions, integers, and
they can't do story problems.

and so that tells me, if it's happening
everywhere, that tells me it's not the

kids, that there's an issue with your
methodology of teaching word problems.

I, remember, As working with a high
school staff and there was a time

during the training that had a chance
to talk one on one with this very

experienced teacher and he was very nice.

I don't know how you are, working
with some very experienced teachers

there, they've been to a million
trainings and they're very nice to you.

and then they'll tell you,
here's what I really think.

And so we're chatting and he says,
what my kids can't do the simplest.

word problems.

And I said, so how do you, teach him?

He goes, let me show you a problem.

And he showed me a problem and
he says, that's pretty simple.

That should be easy for an algebra
student, but they can't get it.

So I asked him, so how do you teach it?

He goes, then I give them this problem.

and the point is there was
no strategy for teaching.

And I think that starts in elementary.

I think, we talk about, there are
people talking about, should we have

math specialists teaching elementary?

I don't know about the level
of math that people teach.

I, think, minimally, if you're teaching,
you should have at least an understanding

of what's going on in algebra.

If you're teaching second grade to
be able to know that, but you should

also be able to teach word problems.

not, here's a strategy to work for this
concept we're working on, and then, oh,

and here's another one next time, and
then here's, no, and again, that's what I

thought was so powerful about your book,
that you have problems starting from

you have it advanced through the grade
levels and then even within the grade

levels and the content areas, and also,
from less difficult to more difficult.

And so for me, another big benefit
of this book for teachers would be

the teachers can see the repetition.

The teachers, it's a
meta learning for them.

And I think the light bulbs would
just absolutely go on and say,

wow, now I know how to teach this.

so let me start here.

I don't think many students will become
proficient at solving word problems unless

their teachers are regularly presenting
them with robust, rich word problems.

Now, assuming that's in place, I still
see several reasons why what you've

described might occur with students
still not getting, becoming good at them.

I, think solving word problems
can be very tiring for students.

They require a lot of
thinking, a lot of persevering.

But there's also not an algorithm
or formula to master when it

comes to solving word problems.

I can teach students 3 digit plus 3
digit addition, and through a procedure

they can get, a lot of repetition
with it, and they can master it.

Same with long division, same with adding
fractions with unlike denominators.

You do this, now you do
this, you have an answer.

But that isn't the same
with word problems.

I find that, every one of them
is a little bit different.

There's not a set formula.

There may be a set approach
to solving word problems.

Read, draw, write.

So if I get the problem in a practice
set wrong and I get corrective feedback,

I can't necessarily use that corrective
feedback to solve the next problem.

Does that make

Yes, it does.

Yes.

Yes.

now, pedagogically, I think too often
I see teachers, they present a word

problem, and then they start talking
to students through the word problem.

I'm going to draw this, now I'm going to
draw this, I'm going to put the question

mark here, I see that it's 354 fewer, so
now I know to subtract, and now I solve

this and I add that, and they talk them
through a two, Step word problem, and I

think the kids by and large are hearing
blah blah blah blah blah blah blah So I

think you know if solving word problems
requires critical thinking, right?

They need to present the approach to
solving them through critical thinking.

I believe in asking the students
questions, to guide the modeling of

it, and not telling the students,
this is how I solve the word problem.

I think teachers, and I know I've been
very guilty of this, Anytime I present a

word problem, I first ask the students,
depict, draw, diagram, what you've read.

And even if that's 30 to 45 seconds,
I need students to write something

down so that they're beginning to work
on it actively rather than passively.

Too often I see teachers We read a
problem and then the teacher is the

first one to talk and the students watch.

I think that's, flawed, flawed teaching.

And then I think at some point the
teacher, they do need to model, they, the

students need to see an expert diagram.

They need to see an expert model it.

but they need to begin the process
of drawing themselves, and, on

audio, we, I can't demonstrate what
I'm describing, but, even in some,

let's take a simple problem, a farmer
collects a thousand chicken eggs.

He collects 300 fewer duck
eggs than chicken eggs.

which bar or which tape would be longer,
the chicken eggs or the duck eggs?

I'm asking a thinking reasoning
question for the students

to craft a diagram, right?

Instead of, doing it for them.

Lastly, I think the place where a lot of
schools bump into problems or challenges

with students not doing word problems
well is, and I don't have data to back

it up, but there's usually an inverse
relationship between the age level of

students and their comfort with word
problems, meaning that A higher percentage

of students shut down when they see word
problems as they advance in the grades.

And the reason for this, I think,
is usually in kindergarten, first

grade, they try some problem solving.

They're very willing to try it.

But as the grades pass on, the students
start to associate word problems with

a time that they can't be successful.

And it increases as
the grade levels go up.

And, by the time they hit fourth or
fifth grade, they see a paragraph,

and they put their pencil down.

I've been here before.

I know I can't do it.

And to me, that's a complex and a
very real complex, but I'm not a

researcher, but I believe if a complex
can be formed, it can also be reversed.

But the longer the complex holds, and as
adults, we have them in our lives, right?

yes,

You can reverse it, but it takes
a lot of work to reverse it.

My belief is elementary school teachers
need to start reversing them as quickly

as possible, and hopefully prolong
them taking hold in the first place.

And I think that begins in kindergarten.

I absolutely agree with,
a lot of what you said.

And, I'm going to layer on something
when you talked about when you,

the teacher presents a problem.

Before the teacher does it, there should
be some, should be the student attempting

to draw the diagram to do whatever.

And before that, what the teacher
has done is just what you said.

When I see this problem, this is
the first thing I think about.

Is this, bar greater or anything?

which, bar is going to be longer?

So as the students now start
to take over it, That language

should echo in the student's head.

So it needs a lot of repetition.

The other thing I want to talk
about one of my, buzzwords that

I don't like at all, and how it
relates to this, and this is grit.

you got to persevere.

I think that's one of the, one of
our, math practices, perseverance.

here's the thing.

I can ask a student to persevere
if they have a strategy.

Okay, all right, you give a
problem, you're walking around,

kid hasn't done anything.

When I tell teachers the first
question to the student is, so

what's the first thing you do?

Oh, I should draw a tape diagram.

I'll be back.

And then you come back and then
you say, how did you know to do it?

How did you know that was the longer one?

And again, you should hear the words echo.

telling a student, Oh,
come on, try harder.

Keep, thinking.

No, we have to give them a strategy.

If they've tried the strategy, that's all
I can, that's all I can do, because I do

agree, just because they work through this
structured approach doesn't mean they're

going to have everything, be able to
draw on everything to solve the problem.

But you know what?

you've done your best.

It's to me, it's like writing, you
learn the principles of writing when

you're young, you don't become a
great creative writer till much later.

And I think the same thing
in math, but you have some,

structural skills to fall back on.

Yes, I think I have high expectations,
so I give my students hard

problems, and I don't help them.

I think that's false expectation.

and so I think one thing everyone needs
to understand in learning anything

in school or outside of school, You
have to feel some level of success or

you're probably unwilling to persevere.

I know that I was in danger
of flunking out of college.

I, I was higher than a fourth grade
reading level in 10th grade, but I scored

that because of the time constraints.

And I, I, got to the point, I
didn't think I could read a book.

And so I went to the school, Landmark
College in Putney, Vermont, where

they was specialized in helping
students with learning disabilities.

And a teacher helped me realize I could
read an adult book and get through it.

but I, but my complex had overcome me.

And so I was able to
do more than I thought.

And I don't know how conscious of it I
was, but problem solving letters, which

was the original alternative strategy.

which is a part of this book.

is the first step to solve a word problem.

I always think, students say, I
don't, I can't solve a word problem.

You might not be able to solve what
is being asked, or that might be a big

challenge, but more than you think you do.

And if you can piece together what you
know, you're building a ladder towards

being able to solve complex things.

I want to make sure that I mentioned,
I guess I likened it to solving a

word, a standard word problem as an
athletic contest or a performance.

You do that once or twice a week, but you
have to practice the little pieces, right?

And the, isolated skills in between.

So I always recommend to teachers
do a standard word problem once

or twice a week and spend the
other days tinkering with it.

with the pieces, isolating skills
that students need to be able to

solve word problems independently.

But we almost need a coach's or a,
director's mentality when looking at this.

and permission not to do it.

I don't know about Eureka Math or
New York Math, but most of the texts

that I see the teachers are using,
there's a, skill that's introduced,

they do a little practice, and then
here's your application problem.

Same day.

Yeah.

and, so how do they do
those application problems?

they'll give them a trick that works
for, again, instead of the structure.

I liked, Who cares if I liked it?

But I think it was real important
what you said about, maybe you can't

solve it, but what do you know?

Again, getting back to, a STEAM or
a STEM program with science, what

are we teaching kids in science?

Scientific method.

what's one of the steps?

What do we know already?

and I think that's the same
thing because again, cue that a

teacher could ask a student who's
struggling, what do you know already?

I'll be back,

Yeah.

And if we forget test scores, standards,
grades, everything, ultimately what do

we really want is a child to realize
their potential on a regular basis.

And so if we give the students
opportunities to succeed on a regular

basis, I don't know whether you get
to grade level or not, but I'll sleep

well at night as a teacher knowing
I gave you a chance to succeed.

And there are problems that are
greater than an individual teacher.

But for me, I always thought,
I'm teaching sixth grade.

I've got a second grade, a
child on a second grade level.

It's unrealistic for me to say
they're going to enter seventh grade

prepared for seventh grade math.

What can I do?

Make sure they're on a third grade
level by next year or, beyond.

it may seem very simple, but I think we
can often lose that as teachers because

we're overwhelmed by other pressures.

and, and often not having
technique or, the content knowledge

to know how to, get there.

a lot of times teachers will ask me
questions during trainings or, what about

these kids that are, below grade level?

And I'll say, you know what, if you
do what we're talking about, they'll

do better, but nothing's magic.

if somebody is expecting you to
do that, it's, not, but what can

you do to help them get better?

Okay, so you have this, this structure
that again, I'll be, I have to be honest

with you, the first time we spoke and you
were talking about, this, I didn't get it.

Okay.

I needed the repetitions.

Once I saw it in the book and I
saw the problems, and I saw the

explanations why one graph worked.

And the other one was it?

You go, okay, and guess what?

And then they're similar,
and then I see a pattern.

And I think that's one of the things
as in your book, they're patterning,

there are patterns that emerge beyond
the patterns that we usually hear

teachers talk about with respect to,
oh, look at, I add three and look at

what the digits add up to, or whatever
it is, looking for patterns that way,

versus What you're doing is developing
patterns in this problem solving.

Am I off base or is that something

No, it is correct.

and I think, maybe if it's possible.

You could share a link or a PDF
in the podcast notes, it'd be good

if teachers could preview visually
what we're doing, or listeners could

preview what we're doing visually,
as I talk, if it's possible.

but basically, so the book consists
of 152 problems from 2nd grade.

through maybe late second
grade through early sixth.

Okay, originally I wanted to do it
K to six but logistically I didn't.

It just wasn't going to work out.

Basically that's 152 problems but
there's six different ways that

you can use each of those problems.

So it's actually, in a way, you could
say there's 912 problems in the book.

so yeah, there is a logical progression.

I used a popular math curriculum
as a structure for me to write

problems, to work the students
through second through fifth grade.

And we're touching on every major
topic, arises in tape diagrams and

bar models, and basically the problems
gradually become more sophisticated

as we work our way through the grades.

Initially, they're standard
single bar diagrams.

Or comparison diagrams, and then gradually
we get into fraction, of a set, fractions

times fractions, divided by fractions,
and then pre algebra problems as well.

I'm not sure if that got to what
you were talking about or not,

Yes, it did.

Because wherever I am as a teacher, I
think it helps to see that progression

and that modeling I think again, the
way the book is presented, and we'll

get something we'll talk about, But
what to put in the notes, but again, it

is a good for teachers who don't feel
confident about doing the instruction.

And I've done.

That's what I found.

you know that phobia that
you talked about, I think

you catch that from teachers.

Sometimes, and so it's important
that they're competent.

I've worked, with teachers before, new
teachers and we do, a one day training

and there's, she goes, Oh, thank you.

I did not know how I
was going to teach math.

I had no idea how I was going to teach
math now is my way, the only way, the

best way, but it's a way that works.

And so now the teacher no longer has that.

That beer, and it becomes a meta for them.

I'm excited about this because, again,
I was obsessed with, with word problems.

In fact, I was on a campaign.

I think you know that to change them from
word problems to story problems, because

so that the kids could help understand
that, there's something going on here.

There's a flow.

We have some, some things going on.

And in fact, I had done a lot of reading.

I had done a lot of research
how people teaching then I

had a chance to talk to you.

I said, no, I'm not writing anything.

I think Bill has got a lot of
this covered in the same way.

I think those folks who,
struggle teaching word problems.

I think this is going to be a
great episode for them, because

that's what it was about.

It, this was about story problems,
not talking about them or research.

No, this is how you can do it.

Yeah.

I guess it may seem out of place, and
I don't know whether it'll be edited

out or it could be patched earlier, but
I should have began with this, I, for

probably close to 15 years, I've, often
delivered workshops on bar modeling

or tape diagramming, and I found it
very normal for teachers to do this.

to leave getting excited
about teaching bar diagrams.

But then they would get in
front of students and it would

be scruffy in their delivery.

The students would not share the same joy.

And I it took maybe a while, a
couple years of reflecting on it.

But the teachers already had the
prerequisite skills to solve everything.

So this seemed new and
interesting and dynamic, but

students are like, they're eight.

They, I don't care.

This is just, This is
now my normal, right?

And so they weren't finding that
their students found the same joy

in, in, in these representations.

And in order for students to
get what we want out of them,

we need to change the way they
experience, problem solving sessions.

Bill, thank you.

I think

Yeah,

like I said, teachers are going to get a
lot out of this and, we'll put in the show

notes, how to, access your book and we'll
put a little bit of a sample in there.

We'll talk about should be.

And, thank you so much for, being on
Better Teaching, Only Stuff That Works.

My pleasure.

I really appreciate the opportunity.