# 06 Teaching Mathematics for Understanding

[Teacher] I’m a psychologist. I’m a researcher. I’ve spent most of my career
hanging out in classrooms and studying teaching
and learning, mainly in K-12 classrooms. But a few years ago I
started getting interested in what’s going on in
community colleges and working with the Carnegie
Foundation on that work. And as everybody here knows,
there’s a huge problem with developmental
education in this country, in math in particular, which
has been my special interest. Sixty percent of
students nationwide who go to community college,
as you all know, can’t – can’t pass the placement test
for a college level math course. So they get put into
developmental courses where they essentially
retake things that they already took
before in high school. And I was really
interested in trying to understand what was going
on with these students. So I’m just going to start
with a couple of questions that we asked these students. Now we all know they didn’t pass
the placement test, but I wanted to ask them questions
been asked before. So for example, if a is
a positive whole number, which is greater, a
over 5 or a over 8? I don’t want to embarrass anyone
here by asking for the answer, but 53 percent of the developmental math
students we asked this question to, got the correct answer. But remember, if
you just guessed, you would get it correct
50 percent of the time. And what was most interesting
to me is not the answers that they gave, but the
explanations they gave. So for example, a few students
looked at this, and they said, oh damn, you know, I know
I’ve see this before. What do you do in a
situation like this? Well obviously, you
cross multiply because that’s what you do
when you see two fractions. And they got 5a equals
a times 8. And then as soon as you see
that, they go, yes, of course, we can cancel out the
a’s and then they ended up with the answer, 5 equals 8, at which point the
interviewer looked at them and said, does 5 equal 8? And the students checked back through their work
and said, I guess so. I mean, I think I
did everything right. Here is another question
this problem, 462 plus 253. And pretty much everybody
can do that problem. You don’t even have to do
too much to do that problem. And the correct answer,
by the way, which I’ve given you is 715. And then we asked
them a question. We said, how could you check to
interviewed said you take 715 and you subtract 253, and
that should give you 462. At which point we asked, so
why did you subtract 253? Could you also have subtracted
462 and gotten the answer? And I brought an audio clip
of one student answering this. I don’t know if you can hear
it because of your dessert, which I didn’t get,
but I’ll give it a try. So here is one student’s answer. I was — I had a feeling you
were going to ask me that, right, when I filled that in. And I know I’m not going
to give an answer why. I don’t remember why, but
I just know that when — for some odd reason
when checking I think — I don’t know if always true
that you pick the bottom one because I was thinking, why
did I pick the bottom one? How come I didn’t
pick the top one? I went, what happens if
you pick the top one? Let me see that. I never thought about
it until now. Wow, that’s very interesting. I thought about that. Let’s see. So you’re going to
get 3 (inaudible) 6. And 6 from 5 gives you 11. Oh wow, so it doesn’t
even matter. Is that true that it doesn’t
matter which one you pick? Because 253 is in the
original equation. So I guess you’re right. I guess it doesn’t
matter which one you pick. I don’t — I don’t
remember that. I just always remember
picking the last one for some odd reason. This is really interesting
that I learned this right now. But yeah, I guess all you have
to do is get your solution and then subtract it from
any of these numbers. Well, I don’t want to say any because you could be
left with a negative. I want to say you could be. I’m not too sure. But I feel that I’m safe going
by the answer subtracting from the bottom number, and
match that one to make sure that it’s correct. So yeah. [Teacher] Great. [Student] I’m going to
his math teacher about that. [Student] (Inaudible)
[Speaker] I think this is really fascinating. And I think that we put so much
emphasis on trying to help kids or students pass the test,
that we forgot that, you know, I have to say, even people
who pass the test are coming up with answers like this because somehow the way we’ve
educated students all the way through kindergarten
through 12th grade — I mean, I study K through
12 education, and what I see in community college students
is, if you take these students when they’re in K-12, and you
really get them to the point where they can pass the test and
get out of algebra, what happens if you wait two years? What happens is that
the knowledge just comes crumbling down. And the reason is because we’re
not teaching with understanding. It’s because we’re
focusing too much on just the procedures
of mathematics. So what I’m going to do in
the rest of my talk today, is talk about how we can
teach for understanding. And in particular — I’m
going to talk about things that I’ve learned over many
decades of studying mathematics and science teaching
all over the world in different countries,
including our country, but also countries where
they have especially high achievement. Countries like Japan
and China and so on, certain European countries. And in Japan I was
seeing that differently. And to me this is really the big
— the big thing that I learned from observing classrooms
in different countries. So let me just talk a little
bit about the largest studies that I’ve done looking at math and science teaching
around the world. And these are the
TIMSS Video Studies. It stands for Trends
in International Math and Science, I believe. So this was two large studies. And what we did is, we went
out and did a video survey. It’s a combination
of two methodologies. A survey where you take a
national probability sample. So we took national samples
a questionnaire or something, we actually sent a videographer out to videotape them teaching
a lesson in their classroom. And what did we learn
from this work? First of all, we
began to get a sense of what average teaching
looks like. So we all think we know
what’s going on in classrooms, but actually we don’t. We only know what’s going
on in the classrooms where we have experience. But this was exciting
because we could look at lessons being
videotaped and coming in from national probability
samples of teachers. The other thing, of course,
is to compare teaching across countries and discover
what — what’s going on. How do people teach
math differently. And then finally, this was
a U.S. government study, and they were especially
interested in trying to understand what it is
that they do in the classroom in these very high
achieving countries. So for many years we’ve known that in certain countries
they just totally kill us in mathematics and
science understanding. And it’s not just
the procedures. It’s also the understanding. So I just want to talk a little
bit about what I’ve learned from this work over time. The most important thing I
think that we’ve learned is that teaching is a
cultural activity. And this was actually
surprising, especially if you’re
an American, you think everybody
teaches their own way. You go in that classroom,
and you get to decide what — what you’re going to
do in the classroom. Somebody else may
determine what the goals are or what textbook
you have to use, but you get to decide
how to teach. And what we found is that
within countries and even within our country,
which is so diverse, everybody pretty much
teaches math the same way, even though they don’t really
know that they’re doing that. But then you look at other
countries, and they teach math in completely different ways. And in fact, every country
we’ve ever looked at has sort of a unique cultural script
that plays out in the classroom. So let me just talk a little
bit about the cultural script that plays out in our
classrooms, and I’ll compare it to Japan because it’s such an
interesting comparison to me. First of all, if you ask
teachers before you videotape the lesson or right afterwards, what was the main thing
you wanted students to learn from today’s lesson? And this graph shows responses
from three different countries. In our first study Japan
or Germany, the U.S., Japan and the U.S.
in that order, red and whatever those
other two colors are, you can see a big difference. There were two main answers that the teachers would
give to this question. One was they would say skills. I want them to learn how
to calculate the area of a triangle, or how to convert
a fraction into a decimal. Whereas in Japan, the answers
would all focus on thinking. So they wouldn’t say I
want students to know how to convert a fraction
to a decimal. They’d say things like, I
want students to understand that the division of any two
integers could be expressed as a fraction. It’s a very different thing. In fact, most of the things that the Japanese teachers
told us were things they wanted students to understand. So based on these goals, we
have very different scripts that determine what a
math lesson looks like. So in the U.S. a math lesson
pretty much always starts by going over the
homework from yesterday. Then the teacher will go
through an example problem and tell you step by step how do
you solve this kind of problem. And then the teacher will say,
okay, why don’t you try one? And the teacher walks around
to see how you’re doing. And then you have a little time
left, and the teacher says, why don’t you get started with
your homework for tomorrow. So a lot of the sense you get in American classrooms is
teachers managing the flow of practice and homework. If you go to Japan,
it’s almost backwards. They start the lesson
generally with a problem that you’ve never been
taught how to solve. And then you have to
try to figure it out. And that’s often really hard because no one taught
you how to do it. It’s very interesting
that if you were to ask American students, which
I’ve done plenty of times, to work on a problem that
they have not been taught how to solve, the first thing
they’ll do is raise their hand and say, we haven’t had that. Why are you giving me
this problem to solve? You didn’t tell us
how to do that. That’s in the next chapter. In Japan they’ll start with a problem they haven’t
been taught how to solve. After working on it for 15, 20
minutes, something like that, all the students will
come up to the board and share the different
methods they came up with because they were
all using, you know, had not been taught a particular
way to solve the problem. They’re all using
different kinds of methods. And then the teacher
would lead a discussion where they would compare
the different methods. Some of them were correct. Some of them were incorrect. And in the end there would
be a discussion or a summary that sort of connects the
different methods student are using to the — to the concepts
that are important in class. So in the U.S. it’s very much
a focus on following steps and doing it the way the
teacher taught you to do it. In the Japanese classrooms
it’s see if you can solve the problem. And then see if you can
understand why some solutions work and others don’t. Some Japanese researchers
wrote an article, which I really liked, where
they summarize the difference between U.S. and
Japanese classrooms. They said U.S. classrooms
are quick and snappy. Japanese classrooms
are slow and sticky. So, some characteristics
of cultural activities. First of all, they’re
learned implicitly. So even though we go to teacher
training programs to learn how to teach, and actually in higher
ed, we don’t even do that. Most people teach the
way they’re taught. Not the way they were taught
in some training program or professional development
and so on. Cultural activities
are learned implicitly. They’re multiply determined, which means teaching
is a system. You know, it’s not — the reason
we teach the way we do is not because we decided
to teach that way. It’s because over time we
evolve to teach that way. And everything is going to try
to keep us teaching that way. So the textbooks, the
policies, the administration, the students — like you
go to the students and say, we’re going to do
something different. Like when my students say, you didn’t post the
PowerPoints for your lecture. And I say, yeah, I’m not going to post the PowerPoints
for my lecture. I want you to come and
listen to the lecture. They believe that’s the most
unfair thing they’ve ever heard. It takes courage to
change the way you teach because everybody
is trying to get you to teach the way
you’ve always taught. Cultural activities
are hard to see. That was sort of the
story I started out with. You know, you see cultural
activities by comparison, but most of us don’t even
realize what cultural routines we’re — we’re participating in. And then cultural activities
are very hard to change, which is the story of school
reform in this country. Almost every effort
to make widespread and sustainable changes
in the way we teach or learn has failed
in this country. And the reason is
because there’s just so much force pushing
back the other way. Okay. So that sounds
a little depressing. We have a limited
way of teaching, and we can’t change it. So now what do we do? Well I have a few minutes left. The next thing I want to look at though is how do
high-achieving countries teach? So in the subsequent studies
that we did, we went back — And we looked at not just Japan,
but we looked at other countries in Asia and also in Europe where
they have high achievement. So for example, we looked
at lessons in Hong Kong. We looked at lessons
in the Netherlands and the question this
time was, does everybody where they have high
achievement, teach the same way
they do in Japan? Or do they all have
different ways of teaching? And the answer to this question
I think is extremely important and that is, they don’t all
teach the same way they teach in Japan. As a matter of fact, all these
high-achieving countries teach in different ways. If you look at the
strategies they use, it differs from country
to country. Yet they all are able to
achieve higher performance than Americans do. And the question is why? The things they differ in are
to change all the time. Like for example, is
teaching lecturing, or shall we have the
students break into groups and work on their own. It turns out high achieving
countries, you know, in Hong Kong it’s all lecture. In the Netherlands they break
the students into groups for almost everything. But they both are
very high achieving. Should you teach math with
real world problem situations or just symbolic notation? Well, we’re pretty sure
in this country that oh, you’ve got to get
real world situations. The more you have those the more
you connect to the students. It turns out some countries
do that, like Switzerland and other countries don’t do
that, like the Czech Republic. So all of the things
that we looked at varied across these countries, which
makes it very difficult to know, oh, if you just do this, you’re
going to get high achievement. What we found on
digging deeper is that it’s not the things you do that define high quality
teaching for understanding — but it’s the opportunities
you’re able to create for students. Well, what are these three
learning opportunities? There are three of them,
and they not only come out of the international
work, but they also come out of a whole body of
work that’s been done in cognitive psychology
and the learning sciences. So the three are — I’m
going to talk a little bit about each one —
productive struggle. Students have to struggle and
work if they want to learn. Explicit connections — so we have to find a way to
connect students’ thinking to the concepts, the core
concepts of the domain that we’re trying
to teach them about. And deliberate practice,
which is not the same as repetitive practice.