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.
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.