**Archived Information**

**U.S. DEPARTMENT OF EDUCATION**

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

**TRANSCRIPT OF PRESENTATION BY DEBORAH BALL**

**TAPE \# 2 OF 4**

**NOLVEMBER 29, 1999**

*Transcript by:*

*Federal News Service*

*Washington, D.C.*

SENATOR GLENN: What does research tell us? Deborah Ball is a professor
of mathematics education and teacher ed at the University of Michigan,
which, Brit, beat Ohio State, I guess, we don\'t admit that this year.
She is currently co-director of a longitudinal study designed to improve
instruction and learning in mathematics in high poverty elementary
schools, and is also directing a study focusing on the practice of
elementary mathematics teaching.

Deborah will build on Nanette\'s presentation with a synopsis of
relevant research results. Deborah, thank you.

DR. BALL: Thank you very much. Is this mike fine now?

**\[See Slide 1\]** The Commission's charge, our charge as
commissioners, is to make recommendations for the improvement of
mathematics and science teaching. One premise of our work, particularly
what we\'re working on at this meeting, is that we can\'t make
recommendations for the improvement of teaching without understanding
better the phenomenon we\'re seeking to improve.

Nanette's presentation offered us the opportunity to consider the
practice of teaching and what the work looks like up close. And her
response near the end saying, what we were doing was different than what
Jim Stigler helped us do last time. She took us inside seven minutes, I
mean seven minutes out of 180 days, seven minutes of class time to look
more closely at some aspects of the work. And I think one thing that she
illuminated was a glimpse of the multiple considerations that teachers
must weigh, the factors with which they must contend. She helped us to
see, we said over and over again, that teaching is much more complex
than some of the cultural images that we hold. A lot of teaching is
invisible to the familiar eye. We think we know teaching and so,
therefore, we think we know what it takes to do it. We take a lot for
granted. And one crucial aspect of our own work is that efforts or
recommendations to improve teaching that ignore some of those more
intricate aspects of the work are unlikely to make a difference. Indeed,
in this country we have a long history of efforts to impact teaching and
learning that have essentially failed to go past the classroom door.

So this meeting is actually a very important one for us. We need to
challenge ourselves to resist the temptation to talk outside of the
work, and to spend some time thinking carefully about what are these
interactions that go on inside of classrooms in mathematics and science
teaching, and what does our understanding of those help us to come to
when we think about what the recommendations need to be.

To do that, I would like to begin with a basic diagram that Nanette had
up in the background of many of her slides that looked like this.
**\[See Slide 2\]** Sometimes people think about teaching or about
instruction as what teachers do to students or with curriculum material.
And the purpose of this diagram, which some of my colleagues and I and
other people as well sometimes use to offer a simple schematic of what
teaching and learning entail, offers a somewhat different way of
thinking about it. It makes plain that what happens in any classroom is
a product of a set of dynamic set of relationships among teachers,
students, and content. And this goes for any kind of teaching. This
model would be a descriptive diagram for lecture teaching. It would be a
descriptive model for what Javier referred to earlier as cooperative
learning. It\'s not implying that students and teachers are talking
equally in this diagram, it simply says that the dynamic among them is
what produces opportunity to learn in the classroom.

**\[See Slide 4\]** Take the problem you saw on the videotape for
example. This one, I think we\'ve seen as much of that as we probably
want to for today. But one thing you can think about in relation to the
diagram is that what happened in that particular class ensued from the
way this particular math problem was interpreted by this teacher and
those students. The way that she used it with her students, the things
her students said, what she made of that, what she gave space for, what
she didn\'t give space for, what her students did with each other\'s
work, how the teacher used and interpreted what students brought up,
what she cut off, and so on.

We could just as easily imagine this same problem taught in another
third grade class, and I think we should do that for a moment just to
remind ourselves that the purpose of looking at a tape is not to suggest
there\'s a single way to do this but, instead, to understand that this
dynamic set of relations that I\'m talking about is what produces
differences across lessons.

**\[See Slide 5\]** So imagine, for example, that another teacher uses
this problem. And what he decides to do is to focus on the how many more
aspect of this question. In our group, which was the content group, we
discussed that in learning subtraction, somebody else mentioned it as
well, "take away" tends to be the dominant model. We use words like
"take away" when we refer to subtraction, and when students encounter
problems that ask how many more, they often find this difficult.

So imagine that a teacher sees this problem, and sees it as an
opportunity to work directly on that phrase, and to think about how to
teach his students that when you see the phrase \"how many more\" it\'s
a key to you that you should subtract. And he might tell the students
this, and explain that, and then he might show them how to set up a
subtraction problem like this, **\[See Slide 6\]** and then he might
carefully review the steps of the borrowing procedure, or the regrouping
procedure. We could see that the children in the other class had also
learned that procedure, but another teacher might use this problem to
directly focus on this procedure. **\[See Slide 7\]** He might explain
how the steps work carefully, going through it, showing, as several
people explained earlier, that you regroup the three tens so that you
now have one ten and two ones, which permits you to subtract the six.
**\[See Slide 8\]** It\'s crucial for students to understand that two
tens and 12 ones is still the same amount as 32.

There\'s a lot to understand here if you want to teach it carefully, and
a teacher might use this problem to do that, and he might then, after
going through this example, assign other problems, giving them a chance
to practice.

So although these were both third grade classes, and although the same
problem was used in both, what students do and have opportunities to
learn differs between these two different lessons. These differences
arise as a function of the teacher\'s interpretation of the problem and
its opportunities for his or her students. The students work on it, the
teacher\'s use of and response to their students work. **\[See Slide
3\]** So think about this diagram that I showed you a moment ago, this
one, think of this diagram as more than a device, in fact, it will be a
device for organizing my presentation this afternoon. But think of it
also as a kind of finding also from research on teaching. That is, that
opportunities to learn and instruction are a product of the interactions
among these elements.

So I\'m going to use this diagram to organize my presentation of a small
set of recurrent findings from research on teaching, from studies of
teaching and learning. I think that the schematic offers us a way to
organize some of what is known from research on teaching. Consider that
each of the points I\'m going to make, which are going to rotate around
this instructional triangle, consider them as contributing to
understanding more about these interactions, and how they play out to
affect the quality of mathematics and science instruction.

I will return to this at the end because understanding the set of
relationships, I would argue, is key to thinking about strategies for
improving instruction.

Let me make a few short remarks about the nature of what I\'m about to
say in this presentation this afternoon. First, I can\'t possibly
present in 45 minutes, no matter how fast I talk, a comprehensive
account of such research. Instead, what I have done is selected a set of
findings that seemed to me to have particular import for our work as a
Commission.

Second, and I think we\'re going to continue to struggle with this, much
research on teaching and learning has, in fact, been generic. Other
research has been specific to mathematics and science. But I drew from
both in an effort to assemble material that bears on the work of this
commission and, in particular, bears on efforts to improve instruction
in mathematics and science teaching.

Third, the body of research from which I drew is highly varied. It
includes international and U.S. studies, it includes qualitative and
quantitative work. It includes recent work, and it includes enduring
findings from studies from the sort of birth of research on teaching,
which was not all that long ago, but was as much as 30 years ago.

Finally, and I would caution you about this, although the findings that
I\'m going to report or talk about may sound, in fact, rather obvious to
you, and not at all surprising, in other words they may seem to you to
be things that anyone who has been paying attention to teaching would
know, consider the fact that for many of the things I\'m going to talk
about few policy initiatives in this country take these particular
aspects of teaching and learning into account in their efforts to
improve instruction. I\'ll return to that throughout my talk because,
although some of these findings may appear to you to be obvious, it\'s
important to realize how rarely they\'re used in planning strategies for
the improvement of teaching.

So I\'m going to begin on one side of this triangle, I\'m a novice at
PowerPoint. **\[See Slide 9\]** I\'m going to start on the side that has
to do with teachers and content. And by content what I mean is both the
subject matter of mathematics and science, as well as the curriculum
materials used to teach it.

# How teachers interact with content

So the first question I\'m going to address in presenting a small set of
findings from research is what\'s known about how teachers interact with
content that bears on instruction.

**\[See Slide 10\]** So the first one that may seem obvious to you, but
is worth mentioning, is that teaching in the U.S. relies substantially
on commercial textbooks. Most science and mathematics teachers use
textbooks as the basis for instruction. They use them as the guide for
what to teach and for the student exercises in math and activities and
informational material in science. Many school districts mandate the use
of a particular text in mathematics and science, and some allow teachers
to choose.

What is meant by \"textbooks\" is more than meets the eye as well.
Curriculum materials in the United States, as many have noted, are
voluminous, including extra materials, teachers' manuals, supplementary
stuff, tools, equipment, counting materials, and so on. And we actually
know little about what teachers actually have access to, what they use
beyond the basic student text.

However, some early research on teachers\' use of teacher's guide
revealed that mathematics teachers often use the teacher\'s guides
principally as an answer key, despite the extensive suggestions for
teaching that were included in the teacher\'s guide. But, in fact,
little work has been done about the ways in which teachers make use of
these voluminous materials provided by curriculum developers. And this
is an important point.

**\[See Slide 11\]** At the same time that teaching in mathematics and
science in this country is heavily based on textbooks, a second finding
is in some paradoxical relationship with that, which is that teachers
make significant choices as they use textbooks. They exercise
considerable autonomy in how they actually use the textbook. They omit
or reorder lessons, they change the suggested materials, they alter
explanations, activities, and examples. They modify problems to make
them more appropriate for their own students, easier or harder. They
change the order of sections, of units, and supplement with other
activities. Thus, textbooks both do and don't provide considerable
structure for instruction in this country.

**\[See Slide 12\]** Third, teachers\' understandings math and science
shape their use of textbooks. So do their beliefs about the nature of
these subjects and how students learn them.

Let me give you a couple of examples. As teachers modify a math problem
to make it appropriate for their students, a quintessential activity of
teaching, how they understand the mathematics shapes what they do with
that problem. They may improve it or they may, in fact, distort it.
Teachers\' sense for mathematical or scientific reasoning affects how
they make use of a problem in class, how they conduct discussions of
textbook activities, how they treat student answers. As they run into
difficulty with a science activity, what they know about the topic
affects how they deal with it. Teachers who don\'t have experience with
scientific inquiry will deal with an experiment in their textbook
differently than will teachers who have considerable sense for such
scientific practice.

When you put this finding, this third one, together with the recurrent
and unfortunate finding that many American teachers lack deep and
flexible understandings of core content in mathematics and science, it
becomes clearer that curriculum materials alone do not and cannot
determine what students have opportunities to learn. In fact, a good
question is how much one might want textbooks to shape the enacted
curriculum in math and science. On one hand, common textbooks could
provide a common ground for all students. On the other hand, good
teaching depends on teachers being responsive to their students and
adapting instruction accordingly.

The question of what to recommend about curriculum materials and their
use is an important question and one with many angles, not a simple
question.

## How students interact with content

**\[See Slide 14\]** I'll shift now to a different side of the triangle,
the one between students and content. And I\'ll, again, go through a
short set of recurrent findings.

**\[See Slide 15\]** First, students develop and bring to school
scientific ideas from everyday experience. They bring a host of
well-established ideas to instruction. Their experience in the world has
led them to develop theories about how things work. Some of these ideas,
in fact, provide a useful foundation on which instruction can build.
While others are in tension with scientific explanations, and may, in
fact, be quite resilient and persistent.

For example, one of my favorite lines of research trace the ways in
which young students understood how plants make food. They derived their
ideas about this from watering plants at home, and from feeding plants
with fertilizer. And they quite reasonably come to think of plant
fertilizer as food for plants, and don\'t have a sense about
photosynthesis as the process by which plants make their own food so to
speak.

Since this notion of food is not intuitive from their everyday human
experience with food, they may not easily exchange their ideas for
scientific ones. And, in fact, considerable research in this area
suggest that these ideas about food for plants and photosynthesis are
quite resistant to even what might appear to be quite good instruction.

Other topics in science reveal similar theories that can be
counterintuitive. For example, how we see how light works or how gravity
works.

**\[See Slide 16\]** There\'s a parallel finding in mathematics, which
is that some mathematical topics are surprisingly and persistently
difficult for students to learn, and these don\'t necessarily meet the
eye as obvious. In fact, I think Nanette made a point with us, trying to
get us to go back in time to understand how it is that this procedure
for subtraction that we\'re all perfectly skillful at using is, in fact,
a very difficult procedure for young students to learn. I remember when
I was first teaching and explaining very carefully the meaning of the
procedure, and going through it and giving my students practice with it,
and having them exhibit the fact that they could do several of these on
Friday correctly, and on Monday again have forgotten how to do it, and
again be subtracting up. It\'s just not an easy procedure, and there are
other topics like this. Integers, for example, and operations with
integers. What are negative numbers, how do we operate sensibly with
them? Rational numbers, fractions, decimal numbers, percents, ratios,
and connections among those and operations with them. The notion of
variable, concepts of function, definitions and proof, all turn out to
present some persistent challenges for students' learning.

These two sets of findings in science and in math make plain how much
learners' knowledge shapes what they bring to instruction and what they
make of it. Although these may seem common place to you, lots of
mathematics and science instruction all levels proceeds with little
design for addressing in special ways the sorts of ideas that students
develop that are, in fact, at odds with mathematical and scientific
understandings. We persist in believing that if we say it again, or say
it louder, that somehow students will give up ideas that they\'ve
developed. And, in fact, one of the basic challenges for teaching is how
to take account of some of these more resilient ideas that students
develop and find ways to work with them to change their minds.

**\[See Slide 17\]** A more generic, and probably the most banal thing
I\'m going to say this afternoon, but still worth saying I decided, is
that the time engaged in worthwhile academic tasks makes a differences
for learning. You can laugh about this but, in fact, this is one of the
earliest findings from research on teaching. And what it suggests is
that the amount of time that students spend engaged in worthwhile tasks
has a significant impact on what they learn.

This is a complicated finding because it\'s dependent on the quality of
the task. Time spent obviously on poor academic tasks or on work that\'s
too easy or too hard does not have these sorts of effects. But there\'s
not agreement on what too easy or too hard is. It depends a lot on your
conception of what there is to be learned. Some research on teaching
suggested that appropriate work given to students was work that was easy
enough that they could proceed without getting stuck at all. However,
there are views of mathematics that would suggest that becoming stuck,
working your way through that stuckness, is a critical part of learning
to be mathematically proficient. So, generic research on teaching that
suggested that activities should be within reach of students without any
further assistance could, in fact, be at odds with good mathematics
teaching.

As I said, this may seem the most banal of all possible research
findings. Who needs research for this. But the reason I think it\'s
important to bring this up is because, in fact, when you put this
together with the finding that mathematics and science are not taught
every day, at least in our elementary schools, and that time allocated
to these two subjects in this country is often much lower than it is in
other countries, and that, in fact, both subjects suffer from a lack of
academic time.

One simple improvement we might consider is that time is one of the
highest predictors of student learning, and that we should think
carefully about the way in which time is spent in elementary classrooms,
and middle school classrooms.

**\[See Slide 18\]** Finally, a finding that\'s really somewhat
different than any of the other three up here, is that mathematics and
science learning depends on integrating abstract and concrete models.
Sometimes learning starts with concrete and moves to the abstract, which
is, I think, a commonplace assumption that people make, things we often
hear people say. But sometimes it\'s the other way around. Sometimes,
for example, first graders come to school already reading two digit
numbers competently and having no idea what they mean.

I remember when my daughter was learning to use a microwave, and she
could type in quite easily 45 seconds on the microwave. She would go to
the microwave and type 45 and say 45, and then one day I saw her in the
kitchen, and she typed 405. I thought, what is this, this hotdog is
definitely going to get really scorched. But what I recognized in a very
interesting way is that what looked like regression or some very
peculiar sort of error was, in fact, progress. Because it was at that
moment where Sarah had suddenly realized that 45 is 40 and 5, not 4-5.
So, in fact, years before she understood that, she could write 45 and
read 45.

So examples like this permeate student learning, cases where they
develop abstract formula or symbolic knowledge before they develop
concrete understandings. So the notion that it goes in one direction or
another is actually not what research has shown. But what is crucial is
the capacity to link among them. And this is a place where instruction
has often been the weakest. How does one actually map across
representations. How can concrete materials and mathematics be used
effectively together with symbolic and other forms. This is essentially
mathematical as well as cognitive. And mathematics, as Nanette said
several times, mapping multiple representations, inspecting the ways in
which they\'re similar or different, is an essential mathematical
activity. It turns out also to be core for learning.

Again, this may seem altogether so obvious as not to be worth saying,
but much instruction continues to proceed without engaging students in
any concrete experience, on one hand, on the other hand, we\'ve also
come through a period, probably still are in such a period, where
excessive faith in concrete materials has led us to ignore the
challenges entailed in helping students make good use of concrete
experience to move in the direction of both abstract and concrete
understanding. **\[See Slide 19\]** So that\'s this side of the
triangle.

## How teachers and students interact

We\'ll move to the last side, the teacher student side. **\[See Slide
20\]** Here the findings that I'll report really emphasize the fact that
the arrows in this diagram are bidirectional. I haven\'t said much about
that so far, but if you think back many things I\'ve been mentioning go
both ways. I\'ve talked about students understanding of the content and
I\'ve also talked, for example, how the content does and does not yield
to student learning. But on the teacher-student side, it\'s really
crucial to understand that some of the most important findings about
teaching and learning have to do with the bidirectional nature of this
relationship.

The first one, therefore, that I report is that students shape
instruction. **\[See Slide 21\]** They shape instruction in some in
fundamental ways. And this is a finding that may surprise people, or not
exactly surprise them, but one which we consider only rarely. Certainly
few efforts at improving math or science instruction reflect a serious
awareness of what this means for teachers or for classrooms.

One way in which students shape instruction from some bodies of research
has been referred to as bargaining. Students when confronted with
difficult work, ask teachers for help, complain about the level of
demand, and as teachers respond in various ways to these pleas for help
and these complaints, researchers have been able to trace real changes
in the nature of the academic tasks which students are doing. The
intellectual challenge of the work is degraded or reduced, sometimes to
the point that the work is substantially different from that which was
first assigned. Students\' resistance, complaints, pleas, all work to
affect whether and how they engage with what sort of academic task. A
task isn\'t the task that\'s simply given at the beginning of a period,
it\'s what students make of it, and some of this work on bargaining
illustrates quite profoundly how what appears to be intellectually
challenging work can be reduced into something very different.

Another way in which students shape instruction lies in teachers\'
fundamental dependence on students. Teachers cannot make students learn.
If teachers hold a discussion, for example, and no one answers questions
or volunteers ideas, no discussion can occur. If students don\'t do the
work that teachers create for them or assign, or don\'t engage in it as
the teacher intends, then the intended lesson cannot happen. This can be
attributable to motivation, or tailoring instruction, or to other
factors, but this finding is robust whatever the explanation that
students affect instruction.

And a third way in which that students affect instruction that didn\'t
come up very much in our discussion of the video is that they learn from
one another. They\'re shaped by their peers\' attitudes and ideas for
better and for worse. One very interesting thing that Jim Stigler has
been reporting is something that fits with this notion of culturally
different views of teaching and learning, and that is that in Japanese
classrooms the notion that a few children up at an overhead performing
an explanation or talking is, in fact, an opportunity for all students
to learn. That students learn to create these presentations to their
peers, that other students in the class see listening as a critical part
of learning, as a critical part of the work to do in classrooms, is a
very different idea than permeates many American classrooms where, if
the teacher is not talking, students assume it\'s not for them, nothing
they have to pay attention to. And the notion that we could take the
fact that students attend to one another and turn it into a productive
source of learning is a provocative and important derivation of this
particular finding.

**\[See Slide 22\]** A second finding in this category is a very old one
and a robust one, which is that teachers\' expectations shape what
students learn. Teachers who hold and convey high expectations of their
students learning get much better results than those who don\'t. This is
an important finding also cross-culturally for there is some research
that suggests that Americans think of mathematical ability as inborn or
innate in some way, whereas in Japan mathematics learning is thought to
be mainly a product of diligent and concerted effort.

**\[See Slide 23\]** Third, teaching depends on teachers\' capacity to
understand their students' understanding. Some of you exercised that a
little bit in watching the video, trying to figure out what did
Bernadette know, what did Ronya mean, what did those different students
think or what were they learning. This capacity to understand what
students understand is affected by teachers\' own content knowledge as
well as their knowledge of students.

Understanding how someone other than yourself thinks is a challenge, and
requires a flexibility of knowledge that\'s not automatically part of
any adult\'s content understanding. It\'s not part of any
mathematician\'s understanding automatically, it\'s not automatically
part of a physicist's or other scientist's understanding. The capacity
to be curious about and engaged with the way someone else thinks,
particularly someone who is understandings are less developed, less
compressed, less mature than your own, it requires a kind of flexibility
and capacity to unpack that we only understand in small ways so far.

Being able to figure out what\'s going on in a student\'s understanding
depends on this capacity to unpack from your own finished understandings
of a topic back into the depths of what it looks like as it\'s
developing. This also depends on teachers\' knowledge of what are likely
student difficulties in a content area. Something that one can actually
learn and have in one\'s mind to anticipate. The fact that I know that
borrowing is a difficult topic for second graders would affect the way I
could teach that in way that someone who had never encountered second
graders before would simply miss. **\[See Slide 24\]** So that\'s a few
things about that side of the triangle.

## How the environment affects instruction

Up until now, my comments about this instructional triangle has made it
seem as though all instruction occurs in a vacuum, isolated from its
surrounding contexts. **\[See Slide 25\]** But, just as one thing that
research has consistently shown, that the dynamic among teachers,
students, and content is central to classroom teaching, another crucial
understanding is that the environments in which instruction takes place
permeate the classroom. And although this may make sense, few
discussions of policy, standards, frameworks, or instructional
environments seem to take account of the way in which the external is
really internal to instruction.

For now, I\'m just going to make a couple of points about this, because
I want to end my talk by saying a few things that understanding this
dynamic implies for the improvement of teaching.

**\[See Slide 26\]** First, environments affect classroom instruction
through teachers and students. What I mean by this is that as teachers
and students perceive, interpret and respond to features in their
environments, these things affect the interactions among them in
classrooms. Take, for example, a child who brings home math homework and
finds a parent who says, what is this, what are you bringing home, this
doesn\'t look like anything that I brought home when I was in third
grade, is going to affect something about the way that child views the
work of school, what that child does with the homework, what happens in
class the next day. And here I\'m not judging one way or the other, I\'m
illustrating that things that are going on in the community actually do
come in through students into the classroom.

Another way in which these "external", so-called \"external\" factors
permeate classroom instruction could be seen in the way that multiple
signals and messages about goals and outcomes enter into teachers\'
minds and work through the multiple sources of curricular guidance with
which they must contend in this country. Textbooks, local district
guidelines, state frameworks, standardized tests, multiple signals mean
that teachers have mixed messages about what it is that they\'re to
teach, and this shows up when they\'re sitting making lesson plans, when
they\'re figuring what to do in the next month.

What\'s crucial to understand here is not just that these external
forces find their way inside of the classroom, but that, in fact,
they\'re also interpreted differently by different players. Two teachers
teaching in the same school with the same principal in the same school
district can have entirely different views about the extent to which
they\'re constrained, not constrained, urged to do something, not urged
to do something, because people differ in the way that they interpret,
perceive, respond to such messages.

So to think of it as purely linear, that is to think of the environment
arrow as going only one way would be inaccurate. Research shows
persistently that actors: teachers, students, and others in the
environment, their interpretations of the environment and or of policies
affect how they use them.

I want to stop now by returning to where I began, to the need to
understand teaching and learning in order to make sensible
recommendations about how to improve it. If we look back at the
triangle, I don\'t know if it\'s worth going back to it now, I think
you\'re familiar with it, so I won\'t go back. If you think back about
the triangle ‑‑ sorry, I told you I wasn\'t experienced at this. Well,
there it is, okay **\[See Slide 2\]** and what we know about some of the
factors that shape the dynamic in this triangle, it becomes clearer that
efforts to improve instruction must target the interactions within the
triangle, they must improve the effectiveness of the interactions among
teachers, students and content.

What would that mean? My colleague, David Cohen, and I have been working
on this question, and what we\'ve come to ask ourselves is, what are
problems implied by this diagram with which teachers have to contend and
that are, therefore, also central to anybody\'s work who wants to
improve instruction. There\'re three major factors on which we focus,
now I will have to leaf through. You don\'t want to do the Joshua
problem again, do you?

## Ways to improve teaching

I\'ll start talking while I get there again. Three problems on which we
focus we call use of knowledge, incentives for performance, and
instructional coordination. And each of these derives from a central
challenge of teaching and learning that bears on the probability that
students will learn what we actually want them to learn. And I'll say a
little bit about each of them. Let\'s see if I can this time more
patiently get to the right spot.

**\[See Slide 27\]** Okay, first, I want to talk about knowledge use.
Instruction is shaped not just by what teachers know, but how they use
the knowledge. Consider the example I gave you at the beginning about
two different teachers teaching the same subtraction problem because
they thought about mathematics differently, they thought about what
there was to learn differently. They used knowledge of students and of
mathematics differently to produce a rather different lesson.

Second, knowledge use is also more complex the more ambitious are the
aims for instruction. A form of instruction which doesn\'t require
teachers to contend with unexpected student responses or to deal with
complex content demands less in terms of knowledge use than a form of
instruction that does. But, in any case, there are high demands for
using knowledge, using knowledge of students in context, using knowledge
of content in context. It\'s not just what teachers know, although
that\'s critical, it\'s whether and how they can make use of it in the
particular contexts of their work.

**\[See Slide 28\]** A second issue has to do with what we call
incentives for performance. And here, I don\'t want to talk about
incentives in the way that they\'re typically talked about. I want to
think about that triangle particularly. If you think about the dynamic
among teachers, students, and content, in particular between teachers
and students, and recall the finding that I was reporting about the way
in which students shape instruction, teachers expectations shape what
students learn, and so on, then one has to consider that the principal
way in which teachers can feel successful about their work is if their
students learn.

Now, consider what that means. That means that when faced with the
possibility of teaching the subtraction problem to your students,
what\'s the incentive to teach in a way that opens up subtraction such
that multiple representations are exposed, that different children\'s
understandings are exposed, that the time is taken to actually unpack
the meaning of it. There\'s very little incentive to do that in many
ways.

If what you want to do is to be assured that your students have learned
something, there\'s a powerful incentive to help your students to
constrain the task in a way such that your students get the right
answer. There\'s nothing more pleasing to those of us who teach than to
hear our students do well. Consider what it takes to have the patience
and the courage, in a sense, to be willing to consider student errors,
to open up discourse in such a way that allows the content to be dealt
with in a more intellectually serious way. And here, again, I\'m not
referring to an approach to instruction, I\'m talking about what the
incentives are to bear down carefully into the content itself.

So the finding here has to do with the fact that teachers face competing
incentives for performance. They, on one hand, want kids to get the
right answers and to do well on tests. And they also, perhaps, may want
them to engage in complex work that will prepare them better for the
21st Century. But if we\'re serious about expecting teachers to do that,
we need to consider what the incentives would be to take the risk of
asking students to do intellectually more challenging work. Students
will resist, teachers will feel worried about their capacity to be
successful. We\'ll have to think carefully about what the incentives are
to raise the standard of instruction in ways that move beyond very
conventional and closed forms of content. It\'s worth underscoring here
that this problem of incentives is only exacerbated when standards are
raised, and when accountability is heightened. And simple minded ideas
about incentives are not the answer here.

In fact, what often happens is, when students fail, is that teachers
find other people, students to blame, or other people to blame about why
it\'s not working. We have to figure out how to get the incentives right
for teachers to care about high level instruction.

**\[See Slide 29\]** The third finding, and the last one, has to do with
what we call instructional coordination. If you think about the American
system and the fragmentation that we know about in the policy
environment, the same thing is true inside the classroom. You have many
students, Javier said many classrooms have 30 to 40 students, this is
true, so imagine the problem of coordination between having an
instructional goal, having 30 to 40 students who each have somewhat
different understandings of the content that you\'re trying to help them
learn, different interpretations of the work. And somehow, as teacher,
you\'re trying to coordinate among your students to move the lesson
along. That\'s a tremendously difficult problem of coordination, and
that\'s just at the level of inside the classroom.

If you move beyond that to consider teachers\' efforts to link up with
what the next grade teacher expects, or what\'s happening in a parallel
class, to think about coordinating between what they\'re trying to do
and what the test will expect, between what they\'re trying to do and
what the parents and the community are wishing for, the challenges of
coordination are immense. And efforts to improve instruction that don\'t
pay attention to the incredible lack of coordination and the threats to
coordination that exist in a dynamic as complicated as this one also
will fail.

I\'m going to close by repeating something I said but didn\'t support at
the beginning of my talk, which is, we have a long history in this
country of efforts to improve instruction. And many different people
have written about our recurrent failure to impact instruction. We\'ve
impacted lots of other things, many kinds of things that we\'ve managed
to impact in improving instruction aren\'t inside the classroom. We\'ve
changed from ungraded schools to graded schools, for example. Structural
things we\'ve been able to change. We\'ve not been successful at
changing the dynamic that\'s represented in that triangle. And if this
Commission is serious about improving mathematics and science teaching,
we\'ll have to find a way to consider carefully that this dynamic set of
relationships is much more complicated than putting different people in
the classroom or changing the textbook. If you take seriously the set of
findings, banal as they may be, you\'ll understand that this dynamic set
of relations is core to the quality of instruction, and that efforts to
improve it and recommendations about it have to take that seriously.

SENATOR GLENN: Any questions or comments?

MEMBER: You made reference before that a lot of curriculum is driven by
textbooks and sadly that's true. Sometimes, no matter how good a teacher
is, they don't have a choice. Very often, it's dictated by what Senator
Glenn talked about, the 2000 Boards of Education that are responsive to
parents. And parents of kids coming home without textbooks are convinced
that kids are not learning anything. And the fact that this is a
political issue flies in the face of really good science teaching. All
too often they have to use the books.

MEMBER: Is it possible to expand on the piece of research that you
looked at where the students bargained? In other words, the teachers
that confront the unwilling learner, is there something that you have
that might lead us to look at how teachers can address the diverse
classroom?

DR. BALL: It\'s a very big area. I mean, you\'re pointing at something
that\'s very crucial, because what\'s called bargaining has good reason
behind it, too. I mean, it has a funny name to it, but when students get
work that's difficult for them, it\'s reasonable that they might say
it\'s too hard, ask for help, and what the research shows that may have
been an unexpected sort of thing to notice is that, what may have
started out as a very good piece of academic work turns out to be
something much more mundane, or not really what the teacher had in mind.
So it\'s not even just the question of unwilling learners, it\'s also
finding ways to help students get engaged with work that don\'t
necessarily distort or degrade the task, unless that\'s what you intend.
If you intend to make the work easier, fine. But part of this research
suggested that academic work declined in intellectual quality without
necessarily that having been the intent.

So there are two different things going on here. One has to do with
what\'s sometimes called motivation, and there\'s quite a lot of work on
things about motivating learners, to care about learning and so on, and
the other is, how does one manage the subtle work of helping students to
become engaged in their academic tasks, keeping an eye on whether what
you\'re helping them do is what you had in mind in the first place or
whether you\'re changing it. Those are somewhat different issues, and I
think there is work on both of those that we might try to refer to. But
I think taking seriously how much capacity the students have to shape
what the lesson or the unit becomes is a critical part of considering
all of this.

MEMBER: It\'s somewhat telling that, I think through TIMSS, and what Jim
had talked about, even in his book, is that American children tend to
love mathematics and think it\'s easy, and the Asian children did not
like mathematics and found it difficult. And so I\'m wondering, in the
context of your research with students constantly bargaining, whether
there\'s an answer to embedding in our culture that the standards in a
regular classroom need to be higher and that students would look upon it
in a kinder manner?

DR. BALL: Well, Paul, if you put what you\'re saying together, which
perhaps other people would agree with you, that is, how can we raise the
standards and the willingness of students and teachers to engage in
difficult work, if you put that together with what I said at the end
about incentives, it\'s important to understand that there are
disincentives to get kids stuck, struggling, parents complain that the
work is too hard, kids complain, kids don\'t do well because the work is
hard. The patience required to let kids struggle, all of those things
are bound up together. So, in some sense, what you\'re pointing to, if
we were serious about that, we\'d have to think more carefully about how
you support teachers\' efforts to do that, how, in fact, you could
expect that you\'d find classrooms where kids were struggling with what
any adult could say in two sentences, but for kids might take some work.

So all of those thing are wrapped up together, and I think you\'ve got
your finger on something that would be important if we understood how to
think about it in a larger picture.

MEMBER: I have two observations and some questions. The first is, has
this body of research actually been embodied in any kind of
longitudinal, one might call, it if not experiments, at least controlled
conditions, so that one can see really the long-term effects on student
learning? Have we created an environment to bring it all together?

And the second, I think they\'re related, is, if the Commission were to
make recommendations to try to make sure that this research is used in
policymaking on implementation, what would be the most effective point
for those recommendations to be made, in schools of education, school
boards, federal government, where?

DR. BALL: Well, on your second question, all of those, right.

MEMBER: But the most effective.

DR. BALL: Well, we\'ll have to ‑‑ I mean, I think that\'s a question
that my work being what I talked about today doesn\'t really address yet
and is a crucial one for us to talk about as a Commission. Your first
question ‑‑ I mean, I\'m not ducking it, I\'m happy to talk more, I just
don\'t think I really talked about that. I do think all of those groups
will have to be involved, but simply calling for their involvement
won\'t do anything. I think what your question suggests is, what sorts
of roles could different constituencies and stakeholders and
institutions play, and that is an important question.

But your first question, when you asked has there been longitudinal
work, which work were you referring to now?

MEMBER: The bringing together this whole body of observations to which
you have alluded. Have we seen them put in practice in a way that one
can see the effects of bringing them altogether, or do they come from
really isolated observations in different environments.

DR. BALL: That\'s a very interesting question. If I understand your
question correctly, and I might not so I\'ll answer it quickly in case
I\'m misunderstanding you. These findings that I assembled actually come
from rather disparate areas, disparate fields, and one might argue that
teaching itself, as I was trying to assemble it for us, hasn\'t been
looked at with all of those angles. Research on student learning hasn\'t
always been looked at instructionally. Research on curriculum hasn\'t
always been looked at from the point of view of teaching. And part of
what I was trying to do by putting it in the triangle was to say to us,
if we\'re going to talk about the improvement of teaching, we probably
need to think about these things together. I wasn\'t yet saying what we
should recommend, but that we\'d better think about them together or
I\'m not sanguine that our recommendations will do much.

So perhaps the things I\'m saying is something pretty simple, and again
rather banal, if you want to improve instruction, you\'ve got to make
recommendations that are about instruction, and instruction, what I\'m
arguing, is this set of relationships. So the recommendations that are
only about one element aren\'t very likely to make much impact.

MEMBER: That answered it.

DR. BALL: Does that answer it?

MEMBER: Yes, thanks.

DR. BALL: Okay.

MEMBER: May I just quickly follow on that. I mean, are you saying that
we would be wasting our time to look around the country for best
practice schools that are built up from these studies that we should
replicate around the country?

DR. BALL: Oh, no, I wasn\'t saying that. That would be more a question,
have there been practice sites where people try to assemble these
things. I think that there are, and I think that that would be useful to
look. Again what the sorts of data are on those, we\'d have to learn
something. But I think there are a lot of efforts to put together some
assemblage of these things. But even then some of the efforts to create
model schools or hold school reforms haven\'t always assembled all these
parts of the system. They\'ve targeted one element or another. For
example, some whole school reforms target professional community, and
get teachers working together professionally. That doesn\'t necessarily
affect curriculum. It might.

So the notion of the difficulty of putting these things together into
one lens, I think there are examples. But I think what Walter, what Dr.
Massey was asking me had to do with was there research that had
assembled all these and studied them longitudinally. I think typically
not. So, I think, yes, to you and no to Dr. Massey.

MEMBER: Yes. I may be asking almost the same question, but take for
example one of your findings, time engaged in worthwhile tasks makes a
difference for learning. Do we have a body of knowledge on what are
worthwhile tasks? You know, if we don\'t know what worthwhile tasks are,
then the statement ‑‑

DR. BALL: No, it\'s interesting because the early research on teaching
was a time on task finding without much regard either for content or for
the quality of the content. And more recent work has focused more on
quality of task. And, you\'re absolutely right that what counts as a
quality task is something on which there\'s been less work, partly
because it\'s related to what one counts as worthwhile learning, right?
So depending on what your conception is of mathematics and science, and
what your goals are for student learning, your conception of what counts
as a worthwhile task would differ.

So some worthwhile tasks would be quite constrained and close and
focused on a single element of mathematical skill. Another mathematical
task might embody both mathematical computation and reasoning about sort
of a solutions space. And those are rather different goals. So, the
problem with trying to stipulate what counts as a worthwhile task is the
fact that it\'s dependent on what the outcome is that you want for
students. If the Commission fixes on that, then you may have an easier
time locating work that speaks to that. But as long as that\'s open,
then the nature of tasks will differ.

MEMBER: Question, in your research, when we talk about quality of math
or science instruction, we also talk about standards. So, within this
research, have you find much written about when separate school
districts decide content, maybe at a state level we establish standards,
and then from an outside source we purchase an assessment instrument,
and whether the quality of our instruction is impacted, or how much by
lack of alignment between those three.

DR. BALL: Yes, that\'s part of what the coordination point at the end
was, is that there are these sort of multiple messages that teachers end
up contending with. And, as a result, the fact that teachers actually
have a fair amount of discretion means that these things vary, and the
research ‑‑ I was reporting about much more than anything I\'ve done.
But in our own research on standards and teachers\' interpretations of
those, what we see is immense variation; in fact, greater variation in
some ways because teachers are contending with these standards, new
assessments, new texts, new curricula, and their own knowledge and the
fact that they have some discretion means that they make use of those in
somewhat different ways, just the opposite, in fact, of what you might
expect by instituting standards.

SENATOR GLENN: Time for just a couple more questions, and then Linda is
going to go into wrap up.

MEMBER: Is there any research on the teaching for teachers of
mathematics and science, especially in grade schools, in the context of
the very rapid development of Internet technology particularly many
youngsters, they are very skilled on the Internet, and you gave the
example about your daughter, Sarah, about the use of the microwave, the
timer and so on. How can we introduce this kind of thing more into our
instruction?

DR. BALL: I don\'t know a lot about this, and perhaps there are some
members of the Commission or others who can help us with this, but, the
advance of the access to the Internet, especially in science instruction
has been an interesting new phenomenon, perhaps following some of the
same path that manipulatives felt in math, which is, it\'s there, it\'s
available, people put a high faith in its capacity to be educative. But,
like any curriculum material, it depends a great deal on how it\'s used.
And there is some early research now on teachers teaching with the
Internet and, again, suggests that, like these other sorts of materials
before them, in some cases it\'s highly educative, and in some cases not
at all. It, in itself, is not educative. I mean, that\'s perhaps one of
the big findings: materials by themselves don\'t teach.

MEMBER: Is there any research that has been done on science curriculum
which is not based on textbooks? Is there anything that that tells us if
there is such a curriculum?

DR. BALL: I don\'t think I can answer that as fully as you\'d probably
like. The main thing I\'m aware of there is studies about how teachers
make use of those type of curricula, and I think I reported something
briefly about teachers\' use, for example, of experiments that
activity-based science curricula that engaged students more in
scientific inquiry will be used differently as a function of how
teachers themselves understand the scientific content and the activity.
So, again, you have this mediation of the teacher with the curriculum
material, so that in studies that would directly trace curriculum
material to learning are difficult to do because the interaction is
among the students, teachers and curriculum.

EXECUTIVE DIRECTOR: Thank you very much, Deborah.

Luckily we still have many more opportunities to think about the ideas
today, and more ideas that will come forward. If I could remind you,
Senator Glenn opened the meeting saying that we really had two goals,
one was to come out with a set of core premises that were descriptors of
high quality teaching in math and science. If the commission is to come
out with a set of recommendations that would help ensure that high
quality teaching is going on in classrooms across the country, we\'d
better be sure that we know exactly what that is. The planning group in
many, many, many conference calls talked about this notion of core
premises, and we began to ask ourselves if we really understood what we
meant. So let me, for the moment, we\'re getting Deborah off and the
overhead on, we think, and we thought this might help. And that is
looking at some core premises of parenting.

\[END OF TAPE\]