[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

in interviews that they hadn’t

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

we asked them. We asked them, can you do

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

see if your answer was correct? And almost everybody we

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

you’ll get — you’ll get — your answer should

match that one to make sure that it’s correct. So yeah. [Teacher] Great. [Student] I’m going to

ask my math teacher today. [Speaker] He’s going to ask

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

of eighth grade teachers. And then instead of giving them

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

things that we fight about and argue about and try

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.