**Archived Information**

**U.S. DEPARTMENT OF EDUCATION**

**THE NATIONAL COMMISSION ON MATHEMATICS AND SCIENCE TEACHING FOR THE
21^ST^ CENTURY**

**TRANSCRIPT OF PRESENTATION BY SUZANNE WILSON**

**TAPE #3 OF 4**

**NOVEMBER 30, 1999**

*Transcript by:*

*Federal News Service*

*Washington, D.C.*

DR. SUZANNE WILSON: Thank you for inviting me to speak with you today.
Today I\'m going to focus on issues having to do with the kind of
knowledge and skill that teachers might need in order to enact the kind
of teaching that you are beginning to envision in your set of
descriptors. In my other life I do other kinds of things, like try to
understand systems of policy that are built to support that kind of
teaching, and the nature of high quality professional development. I\'m
not going to talk about those today, but in the course of your questions
if you have things that you want to ask me, related to what I have to
say about the knowledge that teachers need in relationship to those
policy structures, or the quality of professional development, feel free
to ask them.

I\'m going to begin with a simple claim that might be arguable, but one
that I believe is helpful to begin with, which is that teaching is
falsely familiar. Why is this important to consider? Well, we\'ve all
spent a lot of time in school, and based on that experience we have
opinions, often fierce, about what it takes to teach. Those opinions are
largely based on what worked for us or for our kids, or what appealed to
us as learners and parents. Yet, this presumption is problematic for
several reasons. One, what works for us doesn\'t work for everyone. Two,
experiencing good teaching is not the same thing as understanding good
teaching.

This point was brought home to me recently in a class that I was
teaching of undergraduates. And I taught them all year. The first term I
taught them every day that we met, and I was in control of the class.
The second part of the year I had designed so that they started taking
over the class and practicing teaching the way that I had taught. One
student, Mitra, who really enjoyed the class and participated in all the
discussions, and felt that it was really helping her think about being a
teacher, did miserably when she tried to lead a discussion. She was
under prepared, she didn\'t anticipate what was going to happen, and she
was really upset by this. And she wrote me a little note after class
where she said, teaching is like being a good dancer, you have to make
it look like you\'re not working at all. But trying to lead a class
through the discussion of my book showed me how hard dancing is. My
problem is that I watched you dance all last term, and I never thought
about what it took for you to be able to do that.

Thus, while we might have witnessed good teaching, or while we might
imagine the nature of good teaching, we have little knowledge about what
goes into that practice. And that\'s what I\'m going to focus on today.

Another reason why teaching is falsely familiar, of course, is that we
all teach, we teach as parents, we teach in informal settings, we teach
as politicians, we teach as heads of organizations, and in that practice
we think that we know some things about what it takes to educate people.
But, those practices in many ways suffer the same thing that teaching
does in schools. There isn\'t typically a kind of accountability system,
or a way of looking at your practice and its consequences that help you
understand really what worked and what didn\'t work. So we\'re left,
when we have this sort of false familiar sense of teaching, both because
we teach and because we\'ve all been, as the historian Carl Kastle says,
in fourth grade, with a relatively simple idea about preparing teachers.
If they care about kids, and if they know their subject matter,
everything is going to be okay.

Well, I want to tell you a story, and I\'m going to use this story
throughout my talk. I loved math as a kid. I took every math class I
could. I was always curious about why something happened, and even in
elementary school I remember scribbling on the corners of homework
papers questions to teachers about would this alternative solution that
I was trying out work just as well as the one in the textbook. And
occasionally teachers would sort of pat me on the back and say, that\'s
very nice, good try, but more often than not, people would tell me that
it was inappropriate to do anything that wasn\'t in the pattern that was
established in math class, and they would actually take points off of my
homework for not going through the set of steps that was expected in
math class. So I actually learned in school, I was taught in school,
that the best way to get As was to keep my curiosity under wraps.

As a senior in high school I took a BC calculus class, the most
challenging class I could in the high school that I went to, and for the
first time in my life I needed help with my homework. So I went to my
father, who has a Ph.D. in physics, and an undergraduate degree in
mathematics, and he regularly actually used mathematics at that time in
his work, and I asked him to help me with my calculus homework. And he
struggled with the problems, because they were sort of unfamiliar, he
hadn\'t done them for a long time, but eventually he could show me the
steps of how to solve the problems. But, then when I asked him why,
because I wasn\'t in school and it was my dad, he couldn\'t explain why.
He did exactly the same thing with my physics homework. When I gave him
my problems to help me with in homework, he could help me solve the
problems, but he couldn\'t tell me why.

Fifteen years later, when I was finishing a masters degree in
statistics, the mathematicians and statisticians who taught my classes
seemed more similar to my father than different. No doubt these men, and
every single teacher I ever had in math and statistics was a man, knew
their math. But when I asked why something worked, I was very persistent
about this, they often acted sort of surprised, and sometimes downright
impatient. I remember one statistician telling me, if you read the book
you would understand why this worked. Well, I was a very good student. I
had read the book. More often people would say things like, it\'s
trivial, or it\'s obvious.

I have a theory about mathematicians and scientists as they get more and
more involved in their subject matter. One is that they get farther, and
farther, and farther away from remembering or understanding learning the
subject matter. So the more you know about something the farther away
you are from not knowing about something, and that gap creates a
problem. But, it\'s got to be more complicated than that, because
sometimes I think it has something to do with sort of a compacting of
knowledge. That it gets compressed as you get smarter and smarter about
a subject matter. You experience that sort of thing when you learn how
to ride a bicycle. It\'s awkward initially, you have to think and move
your feet, and try not to fall over. But eventually it becomes second
nature, it gets ingrained, it becomes part of the marrow of your bones.
And in mathematics people talk about developing intuitions. Well, it\'s
a sort of compressing of knowledge. Now, it takes a lot then when we go
to teach children about photosynthesis, or about mathematics, or how to
ride a bike, or how to swim, because we have to back off from our
compressed knowledge and unpack it.

Now, one thing you might notice about my father, you might not know
this, but my father cared deeply about my learning. He wanted me to
understand math and physics, he wanted me to be a mathematician. He also
knew his subject matter. But, he could not teach me. So what I want to
talk about today is what more do we need to know besides simply caring
for kids and subject matter. I\'m not going to skip the subject matter
part, but what more is there about knowing something in order to be able
to teach it, that it goes beyond simply caring, and simply having a
command of the subject matter. There are a lot of different things that
I could talk about, the various organizations that delineate the
knowledge base of teaching have lists and lists and lists. But I\'m
going to focus on three today.

# Subject matter knowledge

The first one I\'m going to focus on is what\'s the nature of subject
matter knowledge that might inform good teaching, the kind of teaching
that you were describing. The second point that I\'m going to consider
is, is there a way of thinking about subject matter in a pedagogical
way? People in teacher ed, and education sometimes call this pedagogical
content knowledge. I don\'t know, sometimes these labels obscure the
point. But is there a way of thinking about subject matter that is
unique to teaching? And then the third part of a teacher\'s knowledge
that I\'m going to consider is what kind of knowledge of instructional
strategies does one need to have in order to teach well, or at least in
the ways that you were beginning to list.

So let\'s start with subject matter, because that might seem like the
most obvious. Teachers should know their content, however this domain
isn\'t really obvious for several reasons. One is, there is not a clear
match between the K-12 curriculum that teachers are going to be
responsible to teach and what they learn if they major in a subject
matter in the university. So if you\'re a math major you don\'t learn
Algebra 1, Algebra 2, trigonometry, plane geometry, those are things
that you encountered in your or arithmetic, those are things that you
encountered as a child in school, but the math curriculum is something
else entirely. Often you start with calculus and you move forward from
there.

So there isn\'t a lot of occasion in one\'s undergraduate career where
one encounters the subject matter that then you\'re presumed to be able
to teach. It\'s sort of mysterious. If you know that subject matter,
then somehow or another you\'ll be able to know the stuff in the
textbook, and you\'ll have an understanding of it that allows you to do
that. But there\'s not a clear match. And some recent research that\'s
being done collectively between a teacher educator and a mathematician
suggests that there are actual topics that are central to teaching, like
equivalents, that pops up repeatedly over the curriculum, the K-12
curriculum, that\'s never addressed explicitly in a math major. But if a
teacher doesn\'t understand equivalents, in all of its robust and
complicated nature, then she\'s going to be hard pressed to teach kids
equivalents across the K-12 curriculum.

In science it\'s a slightly ‑‑ they have the same problem, but it\'s
even more complicated as I\'m sure you all know that. The kind of
science degree that we highly value is a degree that focuses in a
science, biology, astronomy, physics, chemistry, but, science teachers
like social studies teachers get assigned to teach everything under the
umbrella of science, not just what they majored in. So there\'s a ‑‑ the
first problem with knowledge of subject matter, when we think about what
kind of knowledge of subject matter do teachers need to know, is that
what they encounter in the university is not clearly mapped onto what
they encounter K-12. But then it gets more complicated.

Much as we would like to believe that our undergraduates in mathematics
and science experience the kind of teaching that you imagined, in fact
that they are going to create opportunities for their kids to learn,
many undergraduates in science and mathematics in this country don\'t
ever have the experiences that you\'re going to charge them to have to
create for their kids. They don\'t inquire, they don\'t have access to
experiments, they don\'t have access to the problems of practice of
mathematicians and scientists, for a lot of different reasons.

My stepson was a math major at the University of Michigan. And the
University of Michigan has a rather prestigious math department. He got
straight As, and he loved his major. But, he never encountered a
problem, he never encountered something that was not readily, quickly
solved, not mechanically solved through sort of going through the
textbook and understanding how somebody else had created knowledge,
until one summer he wanted to stay in Ann Arbor, so he got a job working
with a mathematician in the math department. And the mathematician was
working on an NSF project that had to do with trying to figure out how
to make sure that satellites don\'t collide with one another. And all of
a sudden all of these things that he was learning in school took on a
different kind of life, and had a different kind of meaning.

He was quite content as a math student. It wasn\'t as if he wanted to be
experiencing those problems. He did very well. He did like I did in
math. He was very, very happy, but he never experienced that kind of
problem. Another thing that happens in math and science departments in
some schools, not all schools is that as soon as you declare your
interest in teaching, in those departments you become a second class
citizen. So there are formal and in some cases informal policies about
what courses people who are math majors who want to be teachers can\'t
take. There are courses that science majors who want to be teachers are
counseled out of. And so the teaching population in the department
becomes the sort of cash cow for the real work of the department, which
is to produce a small number of elite people who then want to get
masters degrees in mathematics, or chemistry, or physics. But it\'s the
same sort of ‑‑ the same thing happens at the university more generally.
As soon as you say it\'s a college of education, it doesn\'t have the
prestige of a department of arts and letters, or a department of
medicine. And within departments the same thing happens.

And so there are these policies within the system of higher education
that almost purposefully try to figure out how not to give
undergraduates in math and science the kind of education in the subject
matter that they would need to have in order to then even begin to think
about how to replicate that for students. The problem gets a little bit
more complicated.

# Pedagogical knowledge of subject matter

So before I go onto ‑‑ actually, this will be my segue to what might we
want to think about in terms of a pedagogical understanding of the
subject matter. So what happens when you have that sort of mastery of
the algorithms and fundamental understanding of the basics in math, and
you\'re expected to teach? Now, this is a contemporary problem in the
U.S., because we are emergency credentialing all sorts of people in math
and science because we need them in classrooms. So I\'ll tell you more
about my story.

So I really loved math, and I really wanted to be a math teacher. I went
to Brown, because it had an applied math department, and all of the
teaching was done by graduate students from other countries, who didn\'t
speak English very well. None of the teaching was ‑‑ all of the teaching
was done in large classes with small sections with those same
undergraduate ‑‑ those graduate assistants, who didn\'t speak English
very well, and had no education in teaching. I was 18 years old. I had
had a high school math education, which meant for the most part my
understanding of mathematics was relatively fragile. I didn\'t have any
kind of depth of understanding. So I go into a department where people
tell me that my questions are trivial, or obvious, the teaching isn\'t
very good. I can\'t understand what perpendicular is, and I am one of
the statistics.

There was a report done in, I think 1995, by a group that was thinking
about mathematics for the year 2000, and they found that women drop out
at extraordinary rates, out of mathematics. About 50 percent of the
women every year after a certain point are gone the next year, that one
of the problems in undergraduate mathematics is that the departments, in
order to stay within their budgets, have to use teaching assistants,
that a lot of the teaching assistants are graduate students from other
countries, that there isn\'t very much preparation that\'s done of these
teaching assistants. So I was a statistic.

Then I become another statistic. I became a history major, and loved
history. Nobody told me that my questions were obvious or trivial. It
turns out that, of course, in mathematics that is a technical term, it
simply means that there is nothing interesting left in the proof at that
point. It\'s not that I was being trivial, it\'s not that I was being
stupid, it\'s that there was nothing left in the proof. But, I\'m not a
member of the discourse. So a mathematician or a statistician says to
me, that\'s obvious, I think, God, I\'m so stupid; I\'m such an idiot.
So I went on. I majored in history, but I couldn\'t get a job teaching
history. You can never get a job teaching history. So I got an emergency
credential in the State of New Jersey, because my high school math
teacher and my father were so upset about me not going into mathematics.
I took my GREs after college, I got an 800 in mathematics. I\'ve always
done really well. I knew all the answers to all the problems in all the
books, so they thought I could teach math. And I taught everything from
remedial mathematics to ninth graders, to pre-calculus for seniors.

The whole time, even though I was only 21 years old, I knew I was a
charlatan. I knew all the answers in the back of the book, but I was a
charlatan. And this is why I knew I was a charlatan.

Can you get to the second overhead, the one with the word problem?

You were saying in your list of things that you want teachers to be able
to do in math and science is to make it real, to make math and science
real. So in the math curriculum, the problem of math and science, I\'m
not so sure about science teaching, I don\'t know the history of science
teaching as well as the history of science teaching as well as the
history of math teaching. The problems of math teaching are problems
that have a history of over 100 years. People have been saying it has to
be real and relevant and applicable for over 100 years. So word problems
were invented in order to make it applicable. But what happens to
problems well intended in school is that their realness gets turned into
nonsense, and mechanics, and procedures. So this is actually from a
lesson that I was teaching when I was an emergency credentialed teacher.

The problem read, you\'ll all be familiar with this kind of problem,
"Susie and Beth both have cookies for their classmates, Susie has six
more cookies than Beth does, between them they have 100 cookies, how
many cookies does Susie have?" Well, you know, kids they bring cookies
to school, there are 100 cookies, everybody cares about whether or not
they get one or two, or how many cookies they get, so this is like real.
What happens in math? Well, this is what I did, because I didn\'t know
any better, I didn\'t know what to do. The kids blanch as soon as you do
word problems. They want to skip that section of the book; they hate it;
it doesn\'t make any sense; it\'s very, very scary. So I\'m thinking,
I\'m kind and I care; I want them to engage. So I say, Beth has C
cookies, C cookies a variable of cookies. Okay. Silence. So Susie has
six more cookies. Silence. More means plus, so Susie must
have ‑‑ silence ‑‑ C plus six cookies. This is what math teaching
becomes.

All together we\'re reading the problem, that means plus, so they must
have? Silence. C plus, C plus six equals 100. And then it goes through
sort of a familiar script, as soon as you get the formula on the board,
then you\'re in math class, it\'s still a little scary, but we know what
to do. Okay. You move the numbers on one side, the variables on another,
you do your subtraction, you go from 2C equals 100 minus 6, to 2C equals
94, you divide each, both sides by two, because that\'s a rule. So C is
47. What\'s the answer to the problem? Silence. Silence. For most
students math is like this, it\'s like, it doesn\'t make any sense, if
you just give me the rules, this is not about meaning, I\'ll be able to
survive if I can just memorize stuff. There\'s a lovely poem called
"Arithmetic" by Carl Sandburg. He writes, if you have two animal
crackers, one good and one bad, you eat one and a striped zebra with
streaks all over him eats the other one, how many animal crackers will
you have if somebody offers you five, six, seven, and you say, no, no,
no, you say nay, nay, nay, you say nix, nix, nix. That\'s what math
class sounds like and feels like.

Students, the confident ones like I was in school march through the
steps. They remember you subtract six from both sides, divide both sides
by two. But, it\'s always clear to the teacher how much no one is
thinking, because the answer is C plus six. Now, that seems trivial,
right. Okay. So it\'s not 47, it\'s 53. But it\'s not trivial. There is
no drive to understand. For some reason we just sort of get complacent.
Either we\'re badgered into it or something. We just accept the fact
that math class isn\'t going to have meaning. And as it turns out, not
only do the students accept that and sort of want it, but teachers
accept it, too.

Let me ask you, what\'s seven divided by zero? Seven divided by zero?
Undefined, infinity, okay. How would you figure out what the answer is
of seven divided by zero, how would you go about doing it? You remember
you were told, okay. So now if I could just remember the truth of what I
was told, undefined or infinity. The problem is, if I can just remember.

MEMBER: Isn\'t part of mastery of teaching as well as mastery of
learning starting with concepts, in other words rules like seven divided
by zero, or any number divided by zero is undefined, then you go onto
other concepts. And then at some point, you come back to the theoretical
or the why. But, in order to learn a student has to have some building
blocks as a framework to start to communicate, or to learn?

DR. WILSON: Well, in the best of all possible worlds you\'re absolutely
right in many ways. But it\'s not what happens in school. What happens
in school is that math gets reduced to a bunch of things to memorize. So
that when you\'re asked something like what\'s seven divided by zero, or
one of my favorites, what\'s a half divided by five, how would you solve
that? Oh, come on.

MEMBER: Lots of ways.

DR. WILSON: How do you solve it? You were all good students, I\'m sure.
You wouldn\'t be here otherwise.

MEMBER: Five times what equals one-half. Just reformulate, and it\'s,
you know, trivial.

MEMBER: The rule says invert and multiply, that\'s what the rule says.

MEMBER: The rule says, the algorithm.

DR. WILSON: And it is trivial. The problem in American schooling is
that, we did a survey ‑‑ we did a large scale survey and interviews of
student undergraduates with mathematics majors across the country, and
undergraduates with mathematics majors could not explain why and could
not build representations, accurate representations of either of those
problems. So both in interview data, where we pushed them to see whether
or not maybe there was something wrong with the survey, and in the
survey, you find that undergraduates with the majors can\'t represent
those problems. Yes, they are trivial. Why can\'t they do it? Probably
because the math teaching doesn\'t have the logic that you offer, have
it be conceptually based, so that you understand and can work with these
things.

So we have a problem with the content knowledge part. Teachers need to
know the subject matter. It\'s not entirely clear what the match is
between the undergraduate major and what they need to be able to know in
order to be able to teach in school. As many of you might know, there\'s
considerable controversy over who owns the undergraduate preparation of
teachers in the subject matters. And many people believe that the
problem of teacher education, or the education of teachers, is the
problem of teacher education in general, the existence of teacher
education. If we got rid of teacher education, just put those people in
the departments, we\'d be okay. I\'m sorry. It\'s not going to work,
because these people who are in teacher education got their degrees in
these disciplines, and they still can\'t answer trivial problems. So
even within the knowledge of subject matter, there are problems.

But now let\'s think about pedagogical sort of aspects of subject
matter. I\'m participating in an NSF study where we\'re studying teacher
induction in Shanghai and France. And in Shanghai undergraduates major
in mathematics. It\'s a middle school mathematics and science project,
but we\'re in charge of the math part, and Senta Raizen is in charge of
the science part. And as it turns out in Shanghai, where we\'re studying
mathematics, undergraduates get a degree in math, so they understand the
math, they major in math. But when they\'re inducted into teaching they
learn two new things about mathematics. They are taught two new things
by teachers about mathematics.

One thing is called the important points. And there\'s a word for it,
zhongdian or something like that. And the other thing is the difficult
points, nandian. These are two different aspects of the subject matter.
One allows them to prioritize the content, so what\'s important to
teach, so that you might learn that concepts are important to teach, and
you might not look at this landscape of facts and concepts and skills,
and just try to teach everything equally, and not do anything well. The
second kind, nian ghin, which is difficult points, is about what kids
have trouble with.

And as Jim Stigler might have told you about Japan, the Shanghai
teachers have a similar kind of cultural conception of teaching. They
only teach about 10 periods, 10 50-minute periods a week. They are in
charge of a home room, where they have pastoral care for a group of
students, and they teach math about 10 times a week. And they complain
about being busy and overworked all of the time. And the reason why
they\'re busy and overworked all the time is because they are doing a
scholarship on tjung zhim and nian ghin. They are doing research and
investigating, developing materials, trying to understand the important
points and the difficult points of their subject matter.

We have something like this in the United States, but it\'s very uneven
and it\'s not talked about in that way. There\'s been considerable
research, actually some of the most interesting research started in
science education with conceptual change literature. And I\'ll just tell
you some of the findings from that literature. Researchers have found
that children have various ideas about matter, in many cases gases and
liquids aren\'t seen as being matter, or having weight. Very tiny solid
particles are also not seen as having weight. Anything that the
children ‑‑ that students can\'t feel the weight of, they don\'t
understand as matter or weight. Research also shows that telling
students very clear, careful explanations doesn\'t break these beliefs.
And that some of the more powerful pedagogy around trying to help
students undo these sort of misconceptions about matter involves having
kids weigh things, having them do it.

Similarly, researchers have found that students' ideas about the shape
of the Earth are closely related to their ideas about gravity and the
direction of down. Students can\'t accept, even though we tell them,
that gravity is center directed when they don\'t understand that the
Earth is spherical, nor can they believe that the spherical Earth,
they believe in a spherical Earth without some knowledge of gravity to
account for why people on the bottom don\'t fall off. So actually it
turns out that kids, even like in fifth grade, sometimes think that the
Earth is up there, it\'s another planet, and we\'re on a flat space,
because people down there aren\'t falling off, so they can\'t actually
be on this round thing. And students believe this after having been
taught otherwise. These beliefs persist because they make sense to
children.

Researchers have also found that after some years of physics instruction
students don\'t distinguish well between heat and temperature when they
explain thermal phenomena. They\'re belief that temperature is a measure
of heat is particularly resistant to change. Again, being repeatedly
told that they\'re wrong doesn\'t seem to be effective. As those of you
who are parents know, children are actually very highly skilled at
nodding their heads and looking like they understood what we said and
then believing what they will. And it is no different in school.

Photosynthesis is one of my favorite examples. As it turns out, you
might not know this or you might know it, that plants get their food
from their roots, they eat through their roots, and they drink through
their leaves. And when children are taught photosynthesis they continue
to believe that food comes through roots, and plants drink through their
leaves. And people have tried, they\'ve actually had sort of
quasi-experimental designs where they\'ve looked at what happens when
teachers try to directly teach photosynthesis, and what other kinds of
strategies seem to help. And one of the things that they\'ve found is
that if you create opportunities for kids to find out, kids also then
believe that there is all sorts of food for plants. I guess plants eat
bugs, too, or something. But there\'s all sorts of food for plants. They
don\'t understand that plants have one food, and that it\'s created
through photosynthesis.

So science, actually the science literature, it\'s very uneven. So it\'s
spotty, and it\'s not a massive literature, but it does turn out that in
the United States we have a growing knowledge base about things that you
might call the difficult points for students. And so when you think
about subject matter knowledge, and it turns out in mathematics it\'s
the same sort of thing, they found out that kids, when they read time
distance graphs, they think this is the actual trip that you took. They
don\'t understand the abstraction. They think that this graph of slope
is actually a map of the maximum point and the minimum point. They don't
understand. They think that axis, like an X axis on a graph, it doesn\'t
really matter if you have one increment over here and another increment
over there. It\'s okay. And these beliefs are really resistant to
change. So they persist and they\'re found in kids who are older.

I don\'t know how many of you know this videotape that was mentioned in
one of your readings of the Harvard undergraduates after graduation
where, like Harvard undergraduates, I have to say that because I went to
Brown, they very confidently and happily gave answers to the question,
why is it colder in the summer. And they said things like, because the
Sun is closer to the Earth. Now, some of these people, embarrassingly
enough, had majored in science, not all of them. But many of them were
well educated graduates of Harvard and of the kinds of prestigious high
schools that send kids to Harvard. And all of these things are sort of
trivial, these questions are trivial. It\'s not the point whether or not
they can answer that question, it\'s whether they can reason. But if you
push, through interviews in educational research, what you find
underneath those answers is no reasoning.

So they can\'t even come up with, given the time, the right answer to
the question. My hunch is that this is because our teachers don\'t know
very much about the difficult points. It also seems to me that, I\'m a
fan of trying to understand the disputes in mathematics, as I\'ve been
studying math reform in California for about 15 years. One of the things
that math educators will sometimes do is say, you know, mathematicians,
why do they think it\'s their right to determine the curriculum of the
school. I\'m thinking this is a little curious, mathematicians know
math, maybe they should have some say. But, I\'m beginning to think that
maybe part of the problem is that we don\'t have this distinction in
understanding math. That, of course, mathematicians might be in a very
good position to understand or make judgments about the important
points, what\'s critical. But, who\'s going to find out what\'s
difficult? Teachers. Teachers are going to find out what\'s difficult.

So we have, sort of, if you\'re going to teach the way that you have
this list, emergent list, you\'ve got to know the subject matter. But,
that\'s not entirely clear what you mean by that. You have to have some
kind of pedagogically sensitive knowledge of the subject matter, in
terms of what students understand, have problems with. The gentleman
this morning said, if you change behavior you can change beliefs.
Actually, there\'s a lot of dispute in psychotherapy about the
relationship between changing behaviors that lead to belief, or vice
versa. I\'m of the mind that you have to do both, and that there isn\'t
one best way to do that, and this sort of leads me to the third kind of
knowledge that I want to talk about, and that\'s a sort of knowledge of
instruction. I\'ve alluded to a lot of this in the discussion of the
children\'s misconception literature.

# Knowledge of instruction

Faced with the persistence of kid\'s ideas, even when they are told
clearly and carefully otherwise, teachers need to have a sense of a
range of instructional strategies that can help them create
opportunities to help kids work through their beliefs and come out the
other end with more valid beliefs. With knowledge that will stick. And
it\'s not really that there\'s a best practice. And it\'s not really
that there are these sort of generic practices that you can lecture, you
can use small groups, you can use simulations, you can use role plays,
you can use laboratories.

It\'s more that teachers need some knowledge about the interaction of
those things. They need some way of understanding: what is my purpose in
this particular case, what do I know about kids in this particular case,
and what different kinds of strategies might I use in order to help kids
learn this thing. So sometimes telling kids will work, because sometimes
it\'s the appropriate strategy. But sometimes telling kids won\'t work,
because they will leave the room believing what they will, just like you
will. Kids are just like you. No matter how much I smile when I try to
tell you what some people believe, you will leave believing what you
will, some of you, and some of you might have changed your minds about
something. Kids are the same way. Instruction then has to be tailored to
that.

Politics, there\'s a sort of pedagogy in politics that\'s sort of
similar, I think, that you don\'t always act the same way as a
politician. You don\'t always debate, sometimes you listen and try to
hear. You don\'t always shake hands, sometimes you lecture, so that you
show that you know something. And when you make those decisions in
politics, sometimes what people define their work to be as educating
their constituents in the issues. So in some ways it\'s a sort of
teaching, and it\'s a teaching that requires knowledge and practice over
time of, in a particular circumstance, with a particular purpose, what
do you do.

So with photosynthesis, what it turns out really works with more kids is
to create an experiment where they have to find out what food is in the
plant, and where it is. And you can create a circumstance where they
find out that there\'s starch everywhere in the plant. So this flies in
the face of their belief that there are multiple kinds of food, and that
plants have lots of different food. And it creates, the theory goes, a
sort of need to understand this point about learning is the ‑‑ I forget
what it is on the list, but learning is the result of sort of pursuing
your curiosity. That often it\'s something that simple that what one has
to do is try to figure out, what does this person believe or care in?
How can I create some kind of dissonance in an educational environment
that leads them to puzzle and leads them to search for understanding.

I\'m sort of curious about why it turns out that seven divided zero is
undefined. But, school didn\'t enable that kind of curiosity and the
pursuit of those kinds of questions. And that\'s the problem that if we
don\'t create a pedagogy that is sensitive to what kids believe and what
they have trouble with, then we\'re not going to dislodge their
misconceptions, and we\'re not going to give them knowledge that sticks.
And there is a growing body of research that tries to be subject
specific in a very detailed way, so that you don\'t presume that all
physics teaching about all topics will be the same, that there isn\'t
one way to do it, but that there might be a best practice for a
particular topic.

I\'m sort of suspicious of best practice. I think best practice is a
really important idea, but I don\'t know many people who, if they just
acquire practices, survive in their careers. That a practice allows you
to do something, but the world, especially the world of school, is an
unpredictable, undetermined space, where there are new technologies, and
there are different kinds of kids, and there are new kinds of mandates
about what you\'re supposed to do. And if all I know is a set of best
practices, that doesn\'t put me in a position to adapt to those best
practices to a rapidly changing and very unpredictable world, human
world of schools. So I think best practices are a good idea, but I
don\'t think that it\'s a very powerful idea unless underneath those
best practices are some sense about what knowledge would you have to
have about why those practices work so that when they don\'t work any
more you can invent a new practice.

So there are lots of other kinds of knowledge, but those are three sort
of critical kinds of knowledge. I was going to talk about some of the
other ones, but those are sort of three critical kinds of knowledge,
given the list that you are beginning to develop. And if you look across
the INTASC standards for K-12 beginning teachers, where they\'re trying
to list kinds of knowledge that teachers need, and you look across the
professional organizations and they try to list different kinds of
knowledge that teachers need, the lists look different but there are
some patterns. Teachers need to know the subject matter. They need to
have some sort of pedagogical awareness of the subject matter. They need
to know things about community. And you can have people do that for you
and look at those things. But there\'s one thing that I haven\'t talked
about that I think I really need to sort of conclude with.

# Professional Knowledge

In the United States, as I said, our knowledge of this sort of
pedagogical ‑‑ this pedagogically sensitive subject matter knowledge,
whether it\'s a knowledge of instruction or it\'s a knowledge of kids
and their ideas is really uneven. And I don\'t think that we\'re going
to invent it in universities, and I don\'t think that we\'re going to
invent it by imagining what was difficult for us when we were first
learning something. I think the place we acquire that knowledge is
school. So teachers are right when they say, for the most part, teacher
education didn\'t teach me what I needed to know, what I need to know I
learn in school. Well, I think somebody said this yesterday, I wasn\'t
here, but that\'s a little problematic, because I\'m alone in my
classroom with a bunch of kids who often are telling me what I want to
hear, and who I don\'t have a relationship with that offers me sort of a
critical view of my practice.

So as a teacher I might acquire knowledge, but I might simply acquire
belief. I might just learn to believe some things about kids. Girls
can\'t think abstractly, otherwise known as, I haven\'t been able to
figure out how to teach girls to think abstractly. But in a profession
where there is no critical discourse, and where you can close that door
and learn anything you want to learn, there is a danger in presuming
that a professional knowledge grows simply out of schools as they exist.
Those Shanghai teachers go to each other\'s classrooms. They are
observed regularly, and they observe regularly. They write papers and
books that are peer reviewed by their colleagues. They live in a culture
of disagreement and debate, and inquiry, a culture not unlike most
disciplinary cultures or politics, public cultures. Schools don\'t tend
to be, teachers\' practice doesn\'t tend to be made public.

So, yes, this knowledge that is uneven, we need more of, and I think
that probably we need to find a way to support its development in
schools, but not in schools as they currently exist, but in some other
kind of environment where those ideas get tested and documented, and
then disseminated.

So, I was sort of lying when I said teaching was falsely familiar. It\'s
not really that teaching is falsely familiar, because we are familiar
with it. But we often don\'t ask the right question, or somebody
doesn\'t ask us the right question. When you decide to teach somebody
something, or you see good teaching, it\'s not sufficient to stop there.
It\'s not okay to just say \"I know it when I see it\" if what you want
to do is improve it for generations to come.

I understand that Jim Stigler made a distinction, I was reading your
notes, between investing in teachers and investing in professional
knowledge. I think you need to do both, because the teachers create the
professional knowledge. But, if we had more knowledge, if we had more of
this sort of pedagogically sensitive knowledge of subject matter, then
future generations of teachers could have access to that knowledge. So
we have to start asking ourselves another question. We have to start
asking a question, and not be content by saying, "I know it when I see
it, or I feel it when I do it." The next question is to say, "what did
you do," "how do you know," "what did you know," or "what were you able
to do that enabled that to happen?" So that in another circumstance,
with another group of students at another time you have more power over
being able to do it again, just knowing it when you see it isn\'t
sufficient.

So the trick is not to sort of say, suspend your assumptions that you
know teaching, it\'s to say, sort of take advantage of the fact that you
know teaching and look deeper into that practice and try to understand
what\'s underneath it. And my hunch, and the hunch of a lot of these
professional organizations, including the national board, is that\'s
what is under good practice is a kind of subject matter knowledge,
that\'s sort of about the content, a kind of subject matter knowledge
that\'s about kids, a kind of knowledge about instruction and the
chemistry that one can create between a set of pedagogical practices,
and kids, and the teacher and the subject matter.

So that\'s it. That\'s all I know, and I don\'t really know it.

(Applause.)

SENATOR GLENN: Questions, questions, comments?

You were so good nobody has any comments.

MEMBER: We talk almost generically math science. So my question
basically is, is there a difference in terms of your understanding of
the issue, a difference in the way preparation of math teachers and
science teachers needs to be considered?

DR. WILSON: I am an expert in neither field. But, that never stops
anybody in universities from having an opinion. So I\'ll say a little
bit about that. My inclination is to believe that we know very little
about subject specific differences. But, there are deep subject specific
differences. And I think people at this table could teach each other
more about those differences than I can. But, for instance, the nature
of knowledge in mathematics is a different kind of knowledge than in
science. Although in schools that knowledge gets sort of presented as
all the same. My understanding of science is that, while it\'s not like
history, which was my field of study, it bears some resemblance to
history in the sense that there is theory and that the things that we
think we know, I mean, the things I learned in science in high school,
as it turns out, most of them are wrong; that a lot of things don\'t
look the same now, because science has changed, and understandings have
changed.

My understanding in mathematics is that it\'s not the same thing. That
when you solve something, and it\'s tested by the community, at some
point or another it\'s a kind of permanent knowledge. Is that right,
Deborah? In some forms of knowledge. That\'s not to say that there
isn\'t sort of a form of inquiry that goes with both, but I think the
inquiry is different. So I would imagine that if the inquiry is
different, and you\'ve got ‑‑ and the way that nature is created is
different, and you\'ve got teaching as being inquiry driven, or having
something about the discipline at its heart, then the preparation of
teachers in those fields probably needs to have something different at
its heart, too. But I\'m not in a position to say what it is.

Yes?

MEMBER: I\'m having a little bit of difficulty differentiating, or maybe
you can tell me whether I even need to differentiate between I guess
creating that chemistry of learning, the pedagogy, and understanding the
content area. In my past I\'ve done substitute teaching. And where I
thought I was going to be at the end of a class period was always an
adventure. I could always tell when I got the point across, because the
students were asking questions beyond which I\'d prepared for the day.
So I always tried to prepare beyond, in other words, immerse myself
enough to where I could anticipate that creative moment, the discovery
time, which leads me to a question overall of teacher preparation and it
comes back to the role of a professional teaching standards board,
whether national or state or whatever. The balance between subject
matter and knowledge in that subject matter, and the other side of the
balance being the methodology or the process, or what\'s known as
pedagogy. And here would be my question. For teacher preparation
institutions, would you encourage the student to major in the subject
matter and minor in education, or should they major in education and
minor in the subject matter?

DR. WILSON: This is a matter of great debate, but I have a strong
personal view. And my strong personal view is that it is not a
sufficient condition, but it is a necessary one that teachers love the
subject matter, and that they themselves have experienced a kind of
learning of the subject matter that is robust, and flexible, and
growing, and alive, not static. But I come from ‑‑ you know, I am also
quite aware that I come from an upper middle class family that sent
seven kids to Ivy League schools for liberal educations. And so, of
course, my view of the first step of a teacher probably ought to be
mine. I mean, I have to be honest and say that I understand that there
is a ‑‑ that my own education really influences how I think about that.

That said, everyone thought I was a naturally born teacher. Everyone I
ever met thought that I was born to teach, and I thought that it was the
only thing I could do. And I taught for six years, and I was caring and
considerate, and I worked very hard to make sure my kids knew the
subject matter. I think I was a horrible teacher. I was caring, but I
don\'t think I knew a tenth of what I needed to know in order to enable
learning until I became a researcher of teaching, because as a
researcher of teaching I learned to ask questions about practice and
about kids and what they were thinking that I was never taught to ask. I
learned how to listen, I learned how to see and hear things in
classrooms that my undergraduate education in history never did. So my
answer to your question is, I personally believe that if we could give
everybody a high quality education, that that would go a long way toward
improving practice. But that subject matter can blind people in ways
that we don\'t understand a lot, but I really believe are true.

I think that ‑‑ I have a graduate student right now who is finishing a
dissertation on two social studies teachers. This one just loves
history; he knows history; he\'s so excited about history, and he\'s so
excited about history that whenever he teaches something that he knows a
lot about the kids have no space to talk, because he feels the need to
tell them everything he knows. And he is so caring, and so excited, and
so wants to teach and make a difference, that subject matter is a
necessary but not sufficient condition, and there are ways in which just
loving the subject matter, while it can enable some things in classes,
can also limit some things in classes about teaching.

MEMBER: One quick follow up I guess I would observe is that past
practice seems to have overemphasized the methodology or the process,
and under-emphasized content knowledge, and immersion in the subject, in
other words, love of knowledge of the subject, because what I see
enabled through technology today is that the student can be
self-initiated and even self-directed in learning, which means they
don\'t have to understand why they learn, they just need to know what
they learned. It\'s a lot like learning to ride the bike, as you
described it. You don\'t need to know why gravity happens, you know the
effect of falling down, you know the kinesthetics in your body, but you
don\'t have to explain it biologically, because your goal is to ride the
bike, to gain the knowledge of riding the bike, and then you can come
back to it.

So immersion in the subject matter seems to be a critical item, along
with methodology, but is there a trend? I guess I\'ll phrase it this
way, at the University of Wyoming teacher prep program the shift is
toward enrolling in arts and sciences and math and science, and then
getting the courses in education pedagogy, rather than enrolling in
education, and picking up basically what I consider to be more of a
minor in the subject matter. Is that a trend that you see from your
position on the national board?

DR. WILSON: Well, I worked with the national board. I\'m not on the
national board, I was a ghost author of the standards, the early draft
of the standards. Historically there is a trend, there are different
kinds of teacher preparation, philosophies of teacher preparation. At
large universities, state universities across the country I think the
trend has generally been to overemphasize, especially for elementary
school teachers, process and the department of education. There are lots
of reasons for that, not the least of which I think it\'s important for
committees like this to understand that not a lot of people value
subject matter. Parents of perspective teachers don\'t value subject
matter. They come to the university and they say, my kid doesn\'t need
to major in the subject matter, my kid wants to be a teacher. The
students don\'t value subject matter.

I mean, you can love ‑‑ people around this table can love subject
matter, but you live in a culture that doesn\'t necessarily love subject
matter. And the NCTM sort of "let's have a campaign to sort of have fun
with math" is an example of you know working on the, that if schools are
part of a society and the society is sort of a petri dish for what
happens in schools, that petri dish of society doesn\'t have within it a
sort of valuing of subject matter. So in large, especially land-grant,
but in large state universities the trend has been process over content
for a lot of political, cultural, social factors. Hence, the need to
sort of, like the England example, come up with a multiple method
strategy for change. Hence, this sort of fondness for thinking about
systemic reforms, or for thinking about big picture reforms where you
come up with policies that come in at various parts of the system. So I
think, my answer to your question is, yes, the trend has been there
historically in large schools, but for reasons that are deep and not
easily solved by recommending that we love subject matter, or have
people major in subject matter.

MEMBER: Well, I have one personal observation, and then one general
comment. On the personal side, I\'m one who learned my elementary
mathematics and science in the best Shanghai elementary school many,
many years ago. I really appreciate your point on two elements that you
say ‑‑ zhongdian and nandian ‑‑ that\'s the most important concepts, you
ask a teacher only focus on two elements, what are the most important
concepts; the others, what are the most troublesome spots for the
children in terms of their learning. I think that helped tremendously.
I\'ve been spending my whole academic life on the topic of temperature
and heat, I still got confused a lot. But I always keep thinking, what
are the important concepts, what are the troubling spots, that kept me
over to even now in many, many cases. So that was my personal
observation. I think somehow we should keep it simple, not require
teachers to do a lot of things. Those two elements would help a lot in
terms of teaching and student learning.

My other more general observation, which is also from yesterday, last
night\'s discussion, I feel we have not even talked about another
element. I did mention to Deborah last night during the dinner time, we
have the triangle: student, teacher, and content. I think we should not
forget there is another dimension, that is family, and parent. I think,
Suzanne, your point, because of your family background and so on,
especially for science and math learning and teaching, that is so
important, that element. So I would say, for any teachers, if they can
find a way to interact with the parents or families to bring up this, so
that parents and family can also show appreciation about the math and
science in their everyday home life, talking to the students and so on,
I think that will really carry through a lot. I hope we can talk a
little bit more, how can we generate more family and parent
participation in the whole teaching and learning process.

MEMBER: Any other burning questions, or we\'ll move along here.

A burning question, there we go.

MEMBER: It\'s not a burning question, it\'s a personal comment. To our
presenter, I guess some people would consider me an old warhorse, in
that I\'ve been around for a long time, and I\'ve been in the classroom
for almost 30 years, but for me in your presentation you did the
ultimate, I guess what a teacher is supposed to do for the student. You
made me excited about going back into that classroom, and developing
that ‑‑ I mean, you opened a whole new world of curiosity for me, as you
talked about the difference between subject matter that helps in
teaching, and knowing what you need to know in order to impart that. And
just your little examples, it hit so home.

I had a coach back many years ago and he coached me in basketball, and I
later found out that he had been a tremendous baseball player. And I
asked him, so why are you coaching basketball instead of baseball. And
he said, well, you know, in baseball I didn\'t know how to teach someone
to have a level swing, I always had one, and I could hit well. He said,
but in basketball everything I ever achieved I had to work very hard,
and I always thought I could be a better teacher in basketball than in
baseball. And that\'s what I heard you talking about. For many of us who
are in math and science, the subject matter came very easy for us. And
we enjoyed doing it, and our success was highly reinforced for that. And
so then when you get into how do you teach first grade math, the work is
really easy, the problems are easy in first grade. And so we have these
people who think teaching first grade math ought to be a lot easier than
teaching pre-calculus, because the subject matter is so easy. But, in
reality it\'s far more difficult to understand how that belief system ‑‑
so I\'ve gone on a long time. But I wanted to say to you as a presenter
that you provided me something very special. And that doesn\'t often
happen, and I thank you.

SENATOR GLENN: On that note, thank you, Suzanne, very much.

DR. WILSON: Thank you. Thanks for inviting me.

\[END OF PRESENTATION.\]