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OCR for page 47

The first full day of the workshop began with a presentation by Deborah Ball designed
to set the stage for the use of teaching practice as a medium for professional develop-
ment. She framed a central question for workshop participants to consider: "How do
teachers in Japan and the United States use practice to work on their teaching?" The
rest of the workshop was spent exploring three different examples of the use of teaching
practice as a medium for professional development: lesson study, video records of a
class, and two cases describing mathematics classes.
The sessions that focused on lesson study included general descriptions of the design
and enactment of a lesson and postlesson rejection by the instructor and observers.
Yoshinori Shimizu provided background on the nature of lesson study, and Makoto
Yoshida extended that background and described one project on the implementation of
lesson study in a U.S. school. Participants viewed two classroom lessons and the follow-
up postlesson discussions. Hiroshi Nakano, the teacher in the fourth-grade lesson, gave
a brief description of how lesson study was carried out at his school and his thoughts
about the value of lesson study for his own professional growth. Video excerpts from the
lesson and a printed version of the postlesson discussion are included in the following
section as part of the proceedings. The day ended with three panelists reflecting on
lesson study, the lessons they had observed, and how the day's experiences related to
their own backgrounds as mathematics educators.
Setting the Stage
Deborah Loewenherg Ball, Professor, University of Michigan
· What can be learned from using practice as a means of (developing teachers'
knowledge of mathematical content and how to teach that mathematics?
· What questions should frame our thinking?

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Lesson Study: What, Why, and How?
Yoshinori Shimiz~, Associate Professor, Tokyo Gak~gei University
· How does lesson study work and what is its role in developing teachers' content
knowledge and understanding of how to teach?
Framing Lesson Study for U.S. Participants
Makoto Yoshida, Professor, Columbia University Teachers College
Lesson Study from the Perspective of a Fourth-Grade Teacher
Hiroshi Nakano, Elementary Teacher, Setagaya Elementary School and
Tokyo Gak~gei University
LESSON STUDY AS PROFESSIONAL DEVELOPMENT

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Deborah Loewenberg Ball, University of Michigan
The question we want to work on
together is, "How do teachers in Japan
and the United States use practice to work
on their teaching?" Why are we so
interested in learning how teachers in
both countries use the practice of teach-
ing itself to work on their teaching? Let
us think about other practices for a
moment, for example, singing opera,
writing poetry, playing soccer, cooking, or
the practice that we are interested in here,
teaching. It is important to notice three
things about learning practices. First, you
do not learn a practice simply by doing it.
For example, poets do not become good at
poetry simply by sitting with paper anti
pencil and writing. You do not learn to
cook simply by taking pans out of the
cupboard and putting a pan on the stove
with some food. Practices also are not
learne(1 simply by acquiring knowle(lge.
No one becomes a good soccer player by
reading books about soccer. Finally,
practices are not learned only by watching
experts do them. If you attend a concert
and listen to an opera singer perform
opera, it is not likely that you wall be able
to perform opera yourself. Each of these
can help. It can help to watch experts
engage in a practice. It can help to acquire
knowledge about the practice. And it can
help to do the practice. But none of these
is enough.
LEARNING PRACTICES
Let me make some points about how
practices are learne(l. First, learning a
practice requires stu(ly. It requires trying
things, anti it requires analyzing how the
things that you tried work. Such analyses
enable improvements. You (levelop new
i(leas of things to try. This is true for
playing soccer. It is true for writing
poetry. It is true for many practices. We
are interested in how the practice of
teaching is learned, and all of these things
are important to learning the practice of
teaching.
Second, there are practices involved in
learning a practice; for example, watching
teaching is not common sense. If you
bring someone into a classroom anti ask
them to watch a lesson, they may not
know what to watch. They may not notice
what the chil(lren are (loin". They may
not know how to listen to the very specific
way a teacher asks a question. There are
things to learn about how to watch teaching
carefully. There are things to learn about
how to (liscuss teaching with colleagues.

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There are things to learn about how to
make records of one's teaching so that
later one can examine one's own work and
the work of the students, show these
records to other people, and discuss
them. None of these things are automati-
cally known. These are the things ~ mean
by practices important to learning a
practice; and if we talked about opera or
cooking or soccer, we could make a list of
practices important to learning those
practices as well.
In the workshop, we want to learn what
practices teachers in Japan and the United
States use to learn the practice of teach-
ing. What do teachers in these two
countries do that enables them to develop
their teaching practice?
MATHEMATICS TEACHERS'
PRACTICE
Keiko Hino and Keiichi Shigematsu
gave a (definition of teaching practice in
their paper. They said that mathematics
teachers' practice can be construed as a
set of abilities grouped into three parts:
prelesson, midlesson, and postlesson.
Prelesson. The mathematics teachers'
practice is construed as the ability to first, in
the prelesson, organize and transform
mathematical knowledge according to the
goals and purposes of the lesson. Second,
the prelesson requires teachers to have
what they call the "eyes" to evaluate how
that organization of the mathematics works
for the students.
Midlesson. Next, they talked about the
midlesson. They mentioned several things
that teachers must do while enacting the
lesson in class. First, teachers must
execute the plan they have made. They
must move from the piece of paper with the
design and use it with the students in their
class. This means they must create
activities that lead to their goal, and they
must take notice of students' current
situations all the time. In the sixth-grade
classroom we visited at Tokyo Gakugei
University Elementary School the other
day, we noticed that the teacher was
constantly looking at the students, watch-
ing, trying to figure out how the lesson was
being received and experienced. Do the
students understand? Do they know what I
am asking them to do? Do they understand
each other? Sometimes teachers apply
routines they already know, but they must
make judgments; they must decide that this
moment is the moment for that routine.
Sometimes situations arise in teaching for
which teachers have no routine, and at
those times, as Professors Hino and
Shigematsu suggest, teachers must
promptly invent new actions to manage
what they see happening in their class.
This is very complicated work.
Postlesson. In the postlesson, teachers
reflect on what happened in the lesson.
They analyze how the design worked with
their students, and they develop concrete
plans for the next lesson on the basis of
what happened in that lesson.
Summary. This is a cycle of design:
generating designs, using the design with
students, analyzing how it works, revising
the design for the next step. How do
teachers learn to do this well? What
activities and practices do teachers in Japan
and the United States use to develop their
abilities to carry out this practice of teach-
ing? Keiko Hino and Keiichi Shigematsu
indicated that there are several abilities that
are important in a mathematics teacher's
practice. One is the ability to design
lessons, to organize and transform math-
ematical knowledge and have eyes to
evaluate the results of the design. Another
is the ability to enact lessons, to carry them
out in class. This involves creating activi-
ties, taking notice of students, applying
routines, and inventing actions depending
on how students understand the content.
LESSON STUDY AS PROFESSIONAL DEVELOPMENT

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Finally, this indicates the need for yet
another ability: the ability to analyze lessons.
Through such analyses, teachers develop
knowledge about teaching and work out
concrete plans for next lessons. How do
teachers develop knowledge about teaching
so that each day, each year, with each class
of students they become more skillful, more
able to do these things? What practices
enable them to become increasingly more
adept at designing lessons, carrying out
those lessons with their students, and all
the time observing how the lesson is
working with their students?
WHAT DO TEACHERS NEED TO
LEARN?
So ~ offer a short but difficult list. What
do teachers need to learn to engage in
this practice of mathematics teaching?
One thing that repeatedly came through
in their discussion of practice was how to
pay attention to and teach every student
in the class. We were impressed that with
a class of 33 students the teacher was
watching and looking around, trying to
understand what the students were
learning. This is not an easy thing to do.
How do you figure out what the students
un(lerstan(l, whether they are paying
attention, whether they are following, if
the examples make sense, if they are
interested, and if they are learning?
Students are different from one another.
This requires great skill to do what
sounds like a very simple thing.
Second, teachers need to know how to
know mathematics and to use it to help
their students learn. This is not the same
thing as knowing mathematics to do
mathematics by yourself. How do teachers
learn to know mathematics in ways that
enable them to organize the content,
create activities, and adjust the activities
to address the goals of the lesson as well
SETTING THE STAGE
as particular students' interests, needs,
problems, difficulties, and so on? This is
another big area. In the presentations
from the United States, we heard that we
face serious problems of teachers not
knowing mathematics well enough to help
each of their students learn mathematics.
Teachers' mathematical knowledge
equips them to teach all students, so the
first and second points are very related.
Third, anti perhaps a little (lifferent, is
that teachers need to learn how to work
with others on developing knowledge for
teaching. Some of us watched a group of
Japanese teachers discuss a lesson last
week, and today we will learn about
lesson stu(ly. One important practice for
developing teaching is to work with others
on teaching, to learn to do the kinds of
things that ~ have been talking about.
One interesting point in Professors Hino
and Shigematsu's study was how fre-
quently a note was made that Mr. A
engaged in discussions with others about
his teaching. We are interested in what
Mr. A (li(1 in these (liscussions, what he
talke(1 about, what he learne(l. How (lo
teachers learn to work together with
others to develop their teaching?
QUESTIONS FOR CONSIDERATION
~ want to end by suggesting some simple
questions that we should ask ourselves as
we learn about what Japanese anti U.S.
teachers are doing to work on their
practice. As we learn about lesson study,
for example, we want to know what the
teachers actually (lo as they engage in the
practices of lesson stu(ly. What (lo they
work on? What (lo they use to work on
this? Do they look at students' work? Do
they look at mathematics books? Do they
rea(1 articles? Do they bring in other
people with whom to talk? What (lo they

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use to enable this work? Who works with
teachers? Do teachers do this alone? Do
others work with them? What do teachers
seem to learn and how do they learn
these things?
We want to ask these same questions
tomorrow when we examine some of the
practices of teachers in the United States
who also work on practice to develop their
teaching. We hope to leave here with
more knowledge about what it takes to
use practice as the site for working on
something that cannot be learned only
through study, only through watching
experts, or only by working alone and just
(loin" it. What (toes it take to use practice
as a site for (leveloping practice?
LESSON STUDY AS PROFESSIONAL DEVELOPMENT

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`'A
Yoshinori Shimiz~, Tokyo Gak~gei University
Lesson study is a common element in
Japanese educational practices. At the
outset, however, there are differences in
cultural background that should be
considered as we discuss lesson study at
the elementarylevel. Professor Usiskin
indicated, for example, that student
teachers in the United States usually have
eight to ten weeks for training in the
classroom, but the Japanese student
teachers spend three or four weeks or in
some cases just two weeks in classroom
work. Also, in Japan in elementary
schools and junior high schools there is a
very large teachers' room where every
teacher has his or her own desk. Those
teachers who are not giving lessons spend
their time in this room. Most U.S. teachers
have their own room where they spend
their time when they are not teaching.
Consider one other example. About ten
years ago, ~ visited a middle school in San
Francisco. When the lesson started, a boy
began to eat an apple during the lesson.
"Why was this boy eating?" ~ asked the
teacher after the lesson. She said, "He
must have been hungry." What ~ wanted
to ask was why he had to eat an apple in
his class during a lesson, because it would
never happen in a Japanese classroom. So
things that we take for granted in our own
culture may be some things that are not
natural at all on the other si(le of the ocean.
We have to keep that in mind when we
consider any cultural activity like teaching.
What follows is a brief outline of lesson
study with a special focus on the role of
lesson plans. Sometimes this is called the
agenda or schedule, but whatever its name,
for Japanese teachers it is something that
is taken for granted, although they do not
always prepare the lesson plan.
Lesson studies are held at (1ifferent
levels, and there are different types as
well. Lesson stu(lies are con(lucte(1 as
part of the preservice teacher training
programs for student teachers. There is
another type, calle(1 intraschool lesson
studies, where maybe three times a year
lesson studies are held within a particular
school. Lesson studies are also held on a
prefectural level, city level, or a school
(listrict level, anti consequently organiza-
tions anti programs vary, which is an
important consideration to remember.
Finally, lesson studies are held at the
national level, open to outsiders. ~ just
listed four (1ifferent types, but this, of
course, is not an exhaustive list. There
may be some other types as well.
Generally a lesson study consists of the
following: the actual classes taught to

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Before
· Deciding a "theme" (and organizing a team)
· Selecting a particular topic for the study
· Writing a lesson plan (analyzing the topic to be taught, assessing students' learning,
examining the task to be posed, thinking teacher's roles, etc.)
· Discussing and revising the lesson plants)
· Tried by other teachers, or in another class
· Reflecting on the lesson and re-revising the plan
pupils, observation by others, immediately
followed by intensive discussion called the
study discussion. Designing, enacting,
and analyzing are the three stages that
evolve before, during, and after the
lesson, in other words preparation,
implementation, and analysis (Box 11.
There is extensive preparation made
before the class, and there wall be exten-
sive work done after the lesson study as
well, which wall be used as a follow-up and
as a preparation for the next lesson
studied. These events form a cycle.
Lesson studies also have different
objectives and aims. One is to educate
student teachers, and a second is to
monitor and instruct novice teachers. In
the late 1980s, a new system of teacher
education programs was introduced in
Japan for new or novice teachers. Newly
hired teachers are closely supervised for a
one-year period by the (1eputy school
a=,'
.c.~.~.l,.~er.tr=~_~er~el`~.~.l,~.~ml~e
During
Teaching/observing the lesson
· Recording what the teacher and students said, how students worked on the task
during their seat work, and what was written on the chalkboard
Extensive discussion on the lesson
· A se~f-ref~ection by the teacher
· Discussion on the task, students' responses, teacher's roles, and so on
Comments and suggestions by a mathematics educator or an experienced teacher
After
· Ideas are used in the following lessons
· Next theme may be identified
· A report of the lesson is sometimes shared with outside people
LESSON STUDY AS PROFESSIONAL DEVELOPMENT

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principal or by well-experienced teachers.
As part of the program, novice teachers
have to take various classes, and the
contents of these courses are well defined,
and often lesson study is a part of these
courses. Another objective for all teachers
is to improve teaching skills, which is
what counts most of all. All teachers also
need to learn the roles of maintaining and
managing the school, and therefore
collaborations among teachers are
needed. Lesson studies are conducted to
maintain such collaborations. Lesson
study can be used to improve teacher
content knowledge. In addition, the
national-level study programs that are
organized by a group of teachers or by a
school are sometimes used to share new
ideas, new methods, test new materials, or
new approaches, as well as to demonstrate
those new approaches.
Usually lesson studies begin by choos-
ing a specific theme. For example, a focus
in the current movement of educational
reform is on helping children develop
their own thinking ability. This focus may
guide the selection of the theme. More
general themes may also be chosen, such
as "teaching pupils how to live." A team of
teachers is organized as part of the
process. When the size of the school is
small, the entire school wall often be
involved in the team. A particular topic is
selected for the study, and the lesson plan
is written by the team. One thing to be
emphasized here is that by writing and
revising a lesson plan, we work on the
lesson plan, refining it in an iterative
manner. The content knowledge, the
pupiT's learning level, the specific tasks to
be presented to that pupil are part of the
lesson plan. "Are you going to use the
number 10 or 12 for this particular task of
multiplication?" It can make a big differ-
ence, and such minute details are well
planned before the class is given. Usually
LESSON STUDY: WHAT HOW AND WHY?
the class duration is 45 minutes long, but
hours of preparations are made before-
hand. Sometimes the same topic is taught
by other teachers in other classrooms for
trial purposes. Then you revisit and reject
on the lesson to rerevise the lesson plan.
The activities above occur before the
class. During the study lesson in class,
the observers will take very detailed
notes. What are the responses of the
pupils to the given task and what did the
teacher say? What were the questions
raised by the pupils? What the teacher
wrote on the blackboard is recorded as
well. In other words, many things happen
during the observation phase.
During the postlesson (liscussion that
follows the study lesson, the teacher who
taught the class would share his or her
own impression or reflection about the
class with the observers. This is followed
by intensive discussions on the tasks,
students' response, teacher's role, anti on
anti on. An invite(1 principal, mathematics
educator, or experienced teacher may give
comments and suggestions about the
class as well.
After the stu(ly lesson, the feedback
from this class woul(1 be use(1 for the next
class, and the theme for the next class will
be identified as well. Sometimes a report
is put together, the ideas from a lesson
stu(ly are presente(1 in journals, or the
materials are (listribute(1 within the school
or within the school (listrict to be share
by fellow teachers.
That is the basic outline of lesson study,
but ~ woul(1 like to say a few wor(ls about
the lesson plan. Throughout the lesson
studies, the lesson plan serves as a
medium for communication among
teachers (Box 2~. Lesson plans have
various purposes or objectives as well.
Box 3 shows the common framework for
lesson plans. The matrix shows the steps
that shoul(1 be followe(1 (luring a 45-

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· Sharing ideas
· Discussing various aspects of the lesson
· Used as a frame of reference for the lessons
· Shaping the lesson flow (script)
Steps
Posing a problem
Students' problem solving on their own
Whole-class discussion
Summing up
~ Exe rc ice/Extension
Main
Learning
Activities
Anticipated
Students'
Responses
Remarks
on Teaching
minute class. These give the teacher a
certain image of how the class is going to
move forward. Sometimes an exercise or
an extension will be provided as the final
step.
Keeping the common framework for
lesson study in mind, we can interpret
some of the fin(lings of the Thir(1 Inter-
national Mathematics anti Science Stu(ly
videotape classroom study conducted by
Stigler anti his colleagues that compare
eighth-gra(le mathematics lessons in
Germany, the Unite(1 States, an(l Japan.
One of the biggest (lifferences among the
three countries was, for example, the
alternative solutions presente(1 by the
teachers anti by the students (luring one
class (Figure I).
As Figure ~ indicates, more alternative
solutions are presente(1 by students in
Japan than in Germany anti the Unite
States. This is naturally interprete(1 as
closely relate(1 to the lesson plan because
it is reflected in considering the antici-
LESSON STUDY AS PROFESSIONAL DEVELOPMENT

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FIGURE 1 Lessons that included alternative solution methods.
Teacher Presented
45 ~
40 ~
35 -
~n
o
In
o
a)
<~ 20-
a)
c'
a)
15 -
Student Presented
42
30 -
25 -
~9
O-
12 =
14
Germany Japan United States
SOURCE: Data adapted from Stigler, 1999.
pated response of the children and in the
course of discussing and revisiting the
lesson plan. In fact, the anticipated
students' responses make up a large part
of the lesson plan Appendix D).
Finally, throughout the discussion on
lesson study, the teacher's content knowI-
edge and understanding of teaching
practice wail improve, and through the
entire lesson stu(ly this will be further
refined (Box 4~.
As was mentioned above, lesson study
is a common element in Japanese educa-
tional practices. Also it is a necessary
element for improving teachers' content
knowledge and understanding of how to
teach.
_=R ~er.~.~.~i.~.~=r~c~.~~l~r~i~~.~.~ MAR:_
· Interwoven in a certain way
· Reflected on his/her anticipation of students' response to the task to be posted
· Developed through examining and discussing lesson plans and by observing and
reflecting on the lesson
· Elaboratec~ in the process of lesson stuc~ies
LESSON STUDY: WHAT HOW AND WHY?

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on the school. However, in interschool
study, each school is required to conduct
in-schoollesson study. During the
prelesson study, the teachers teaching
first and second grade form one group,
third- and fourth-grade teachers form
another group, and the fifth- and sixth-
grade teachers form a third group for the
discussion. The research promotion
committee chairperson sometimes joins
those groups, and sometimes we would
call in an external adviser to have the
discussions. Then we conduct the class
and have a discussion afterwards, in
which the teachers within the school
participate.
~ was primarily engaged in war(l-
initiated lesson studies and school-
initiated lesson studies. For me, what is
lesson study? It is an indispensable field
of training. When ~ was in the university,
did practice teaching just like lesson
study. The teacher responsible for my
teaching training told me that it is very
important to have lesson study, and that
idea has not changed since then.
~ would like to make my class be
enjoyable for children's thinking. ~ want
the class to operate so that the children's
thinking can be recognized by others and
also by the teachers. ~ also like to make
the class feel that they can find out about
the similarities and differences of their
ideas in relation to others. To realize
these wishes means training. That is how
~ see lesson study. Lesson study is where
you can express your pleas and also you
can improve your position and status.
Through these lesson studies you can
make presentations about your teaching
within the mathematical education com-
munity and establish your own ideas.
What we learn from lesson stu(ly
changes as we accumulate experiences.
When you have little experience, you
learn methodology and how to run the
class. You are taught by many people
through the prelesson studies. As you
gain experience, rather than learning how
to conduct the class, in prelesson stu(ly
you can get to know the value of math-
ematics anti the value of the materials.
Accumulating such experiences was a
great asset for me. Through the
prelesson study ~ am taught by others.
Having others point out my weaknesses,
un(lerstan(1 what they are. This leads to
motivation to improve for the next occa-
sion. However, lesson stu(ly is rather
(difficult even if you plan ahead very well.
Even though you think you did well,
others might point out what went wrong.
So lesson stu(ly is both (difficult anti
rewarding.
LESSON STUDY AS PROFESSIONAL DEVELOPMENT

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l l ~ l a-l
~ ~ 1 -
LESSON STUDY
Workshop participants viewed two
elementary classes and the follow-up or
postlesson discussion for each lesson. As
part of the Ninth International Congress
on Mathematics Education CCME-9) that
preceded the workshop, workshop
participants had the opportunity to visit a
s~xth-grade class during an introductory
lesson on functions. Makoto Yoshida
translated the lesson and the postlesson
discussion for those in attendance. The
lesson plan and transcripts of the selected
portions of the lesson and postlesson
discussion are available in Appendix E.
During the workshop itself, participants
viewed a video of a fourth-grade lesson on
place value along with a video of the
postlesson discussion by the observers.
The lesson plan, a description of the
content, and a summary of the postlesson
discussion held by the Japanese, both
'1~-1
translated by Makoto Yoshida, and a
transcript of the selected portions from
the actual lesson and discussion can be
foun(1 in Appen(lix F.
VIDEO SELECTIONS
A Demonstration Lesson: Function
Thinking at Sixth Gra(le taught by
Shunji Kurosawa,
Tokyo Gakugei University
Setagaya Elementary School
Lesson anti Postlesson Discussion among
S~xth-Grade Lesson Observers
A Study Lesson: Large Numbers at
Fourth Gra(le taught by
Hiroshi Nakano,
Tokyo Gakugei University
Setagaya Elementary School

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A panel addressed the following questions related to the fourth- and s~xth-grade
lesson:
· How do the two lessons compare?
· What are the differences and the similarities?
· What was the mathematical content and how did the lessons develop student under-
stan(ling?
Panel Moderator: Keiichi Shigematsu, Nara University of Education
Jacqueline Goodioe, Resource Teacher, Burrville Elementary School, Washington, DC
Jerry Becker, Professor, Southern Illinois University
Ichiei Hirabayashi, Professor Emeritus, Hiroshima University

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COMPARISON OF THE TWO
LESSONS
"lacqueline GoocIloe, Burrville
Elementary School
First ~ would like to say that this has
been a very rewarding experience, both
the Ninth International Congress on
Mathematics Education (ICME-9) and
this workshop. ~ am an elementary
mathematics resource teacher. A math-
ematics resource teacher in the District of
Columbia public schools does not have
responsibility for one classroom but
services all the teachers in that building in
mathematics. Our school has about 360
students, a relatively small enrollment
with classes from all day prekindergarten
through sixth grade.
~ want to point out some of the similari-
ties that ~ saw in the two lessons. Both
lessons addressed important mathematical
ideas, getting a sense of large numbers,
posing questions, finding functional
relationships. Most elementary ciass-
room teachers do not often discuss these
mathematical ideas. The teachers in the
videos often related the task to concepts
with which the students were familiar.
Time was allowed for students to talk and
explain their thinking. In both examples,
class size was large, between 30 and 40.
That is large for some U.S. classrooms.
The planning was evident. Another
similarity that is familiar to some of us was
the teachers' genuine delight in student
learning and student understanding.
The strength ~ see in lesson study is the
growth that develops through the collabo-
ration and discussions with other teachers.
This is not always evident in U.S. schools-
thinking about "why we do what we do."
Reflecting on practice is an area in which
would like to see some improvement by
teachers in the United States.
REFLECTIONS ON VIDEOS: PANEL
In both videos we saw the teachers
moving around in the classes. However,
we did not discuss a lot today about what
specifically the teachers were looking for
that was going to help them assess the
student learning. They were moving
throughout getting an un(lerstan(ling of
what the students were thinking, but what
were the teachers thinking as they
watched the students?
Much of the lesson study in the
preplanning deals with anticipated student
responses, the student results. ~ would
like to know more about how much time
is spent developing that part of the lesson
anti the source of that information. Is it all
from the teacher's experiences? Is it all
from the study group's experiences?
Where do we get the notions of how
students will respond and how can that
benefit teachers? As we look at the
videotapes anti sit in (1iscussion groups,
something else about lesson study
emerges. There are implications about
what "teacher talk" is about. The last
(liscussion about new teachers, preservice
teachers, and those that are new to lesson
study and how they must know the
curriculum, the textbook, and the math-
ematics was important. When does this
happen anti over how long a perio(1 of
time are we talking about? If lesson stu(ly
is such a powerful tool, ~ am wondering if
videotaping of the sessions shoul(1 be
done so that other teachers can get the
benefit of the experience. There are other
questions as well. For example, what
does lesson study look like at the high
school level? Do skille(1 teachers, inexpe-
rience(1 teachers, anti preservice teachers
all have (lifferent perspectives on the
lesson study?
This past week ~ have heard more and
more about the Japanese intent to engage
students in interesting anti motivating
activities and, in fact, one of the speakers

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talked about developing a sense of wonder
and a new approach for mathematical
thinking. ~ would like to know, if lesson
study is so powerful a resource for
professional (levelopment, why aren't
more lesson studies done in all the public
schools as much as is done in schools
attached to a university? It would seem
that the public school teachers could
benefit from this powerful resource as
well. The workshop has opened my eyes
to some wonderful and powerful ways of
looking at mathematics and of looking at
mathematics teaching. Teaching is hard,
and thinking about teaching is even
harder.
FOURTH GRADE AND SIXTH
GRADE: OBSERVATIONS OF THE
TWO LESSONS
"ferry Becker, Southern Illinois
University
~ feel very fortunate to have had the
opportunity to see Mr. Kurosawa's lesson,
the film of Mr. Nakano's lesson, and to
have had the benefit of the discussion of
these lessons. If ~ put both of these
lessons together for a moment, that is, if
just speak in aggregate of some of the
observations ~ made, the first thing ~
would say is this, Teaching mathematics
is really a big job! Overall ~ thought the
tasks in both of the lessons were very
good. To begin with ~ was not so sure
about the task around which Mr. Nakano
(levelope(1 his lesson plan, but eventually
through the discussion ~ came to see that
it was also a very good task. One of the
things ~ noticed is that the productions
and the observations of the students
during the lesson are written on the board
for everyone to see. They are left there so
they can be referred to and so students
have enough time to write them down in
their notebooks and to refer to them. We
saw in each of the lessons that the teachers
wrote down a number of important
observations made by the students.
In the s~xth-grade lesson ~ think the
number of observations on the part of the
students was nine or ten, and none were
suggested by the teacher. The teacher
rather skillfully set the learning situation
and then provided time for the students to
use their own natural thinking abilities. So
the lesson proceeded on the basis of the
productions of the students, which ~ think
is significant. Another thing ~ noticed is
that in both of the lessons not only (li(1 the
students seem to be enjoying them anti
sometimes getting excited about what was
going on, but they felt perfectly comfort-
able in sharing their observations. The
significance of this to me is that the class-
room situation has all the makings of a
community of scholars. These young
people feel perfectly comfortable once the
learning task is set, and they are given
time to make their observations, to share
them, and to get reactions from the other
students. It was also very clear to me that
the teachers were experts on the problem
task anti in han(lling anti managing the
lesson so that when various responses
were given by the students the teacher
knew how to respon(1 to those anti how to
(leal with them. Clearly they were very
knowle(lgeable about what was going on.
~ think the students hall time to work
on the tasks, to use their own natural
thinking abilities. The teacher took time
to view the work the students were doing
so that the responses that fit the objective
of the teacher coul(1 be written up on the
boar(1 for everyone to see, where it coul
form the basis for a discussion. There
was also encouragement to the students
to write (town their observations, writing
out their observations in wor(ls as well as
LESSON STUDY AS PROFESSIONAL DEVELOPMENT

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writing down what they were thinking in
mathematical symbols.
~ saw a very nice contrast with what we
commonly see in the United States.
There is a difference between verbalizing
observations and verbalizations. Verbaliza-
tions come as a consequence of learning
mechanical procedures, but here in the
lessons we saw the students commenting
on and explaining the reasons for the
observations that they made. The teacher
asked for clarifications. Students were
asked to verbalize their thinking. To me
this forms the basis for developing the
language of mathematics in the class-
room, and ~ think this is very important.
At the end of the lessons, the teacher
asked students, for example, to think of a
general expression for a given number of
minutes later. The common approach is
to go right to that expression, but here the
focus was on all of the processes that lead
up to that. Also, we found the teachers
making a conscious attempt to put the
problem situation into the reality of the
students, and ~ thought that was very
goo(l.
In discussing the lessons, it occurred to
me that what we are talking about here is
the assessment of lessons. During {CME-9
~ heard several talks in which the points
were made that good problems are crucial
in assessment, the assessment of lessons
as well as the assessment of individual
learning. It is the case that the teacher
can get whatever the teacher asks for, and
that means the richer and the more open
the problems, then the more potential
they have for revealing students' under-
standings and abilities. For example, at
the beginning of Mr. Nakano's lesson, ~
thought he was going to engage in direct
teaching, and ~ wondered why he was
going to do that. It took me a while to
realize that was not the case. In fact, he
was seeking to find out what the students'
REFLECTIONS ON VIDEOS: PANEL
understandings and abilities were, and he
had a systematic way of approaching the
situation so he could get some insight.
We saw that insight also in the discussion
of the lesson. So there was time given to
the students to show what they could do
with these learning tasks, and in so doing
we could get some insight into the difficul-
ties they had, the ability that they seemed
to be able to demonstrate, and so on.
Since my first visit to Japan many years
ago when ~ talked about the things ~
thought ~ had learned from observing
numerous class lessons, my colleagues
back in the United States commented,
"But, Jerry, you have to remember that is
Japan. The culture is very (lifferent from
ours. You cannot import what goes on in
another culture." ~ (li(ln't believe that
then, anti ~ (lon't believe it now. ~ think we
can look at the tasks, for example, what
was on the blackboard (Appendix F). We
can have very good insight, regar(lless of
culture, about the problem the students
are given and the different ways the
students approach it. While we watch the
students and listen to them develop their
own ways of (leafing with the problem
situation, we can see that they exhibit
responses that are qualitatively different.
The students generate a situation where
they can (liscuss, in a mathematical way,
many responses anti perhaps even come
to a consensus about which are the better
ones from a mathematical point of view.
don't think that is specific to any one
culture. What that is specific to in my
judgment is mathematics, anti teaching
anti learning mathematics.
Finally, in our small group (liscussion
we were considering the question: How
(lo skille(1 teachers learn about anti make
use of their students' knowle(lge anti
capabilities to help them to learn math-
ematics? We have seen (luring the
discussions here that the skilled teacher

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poses a problem situation and lets the
students show what they can do. That is a
good way to get insight. We also dis-
cussed or had described to us the impor-
tance of and the meaning of a detailed
lesson plan. So before the lesson is ever
taught, the teachers sit down and discuss
the problem situation and try to list all of
the anticipated responses of the students.
If the three of us are the team that is
getting ready to develop a lesson, then
am going to learn quite a lot about that
problem situation from my colleagues,
and perhaps they will learn something
from my insights on the problem too.
While the lesson is in progress, the
teacher has the option of purposefully
scanning the work of the students. By
looking at their work, the teacher can
decide which responses everyone should
see. The teacher knows which responses
should be shared on the board and why
they should be discussed. The reason, of
course, is because the discussion wall tie
in with the objective of the lesson that the
teacher has in mind.
LESSON AS A DRAMA AND
LESSON AS ANOTHER FORM OF
THESIS PRESENTATION
Ichioi Hirabayashi, Hiroshima
University
Prelucle
~ remember it was around 1975 that ~
happened to be visited by two American
professors: one of them was in science
education whose name ~ have forgotten,
and the other was in mathematics educa-
tion named M. Vere Devault. ~ hall seen
his name in the 32nd Yearbook of the
National Council of Teachers of Mathe-
matics anti other publications, but it was
my first time to meet him. At that time
was a young professor at Hiroshima
University, and it was very impressive to
hear from him that there were two
features in the research on curriculum in
mathematics education: technology and
humanity. Ithink the same two features
would be in lesson stu(ly.
Since then, in Japan, and perhaps in
America, the research on technological
features has developed remarkably, but
not much attention has been paid to the
humanistic feature. This feature is far
more important than the technological
one because of its external effect on
children's future life. There also are the
same two features in the lesson, and
would guess in a workshop on lesson
study as a method of research about
mathematics teaching, the technological
feature would also be emphasized far
more than the other. In this atmosphere
of research, it is important to stress the
humanistic feature of the lesson, anti this
is the very reason that ~ wrote this short
paper.
A Reminiscence: Complexity of a Lesson
~ wish to start with a reminiscence of
my own when ~ was mathematics teacher
in the lower secondary school soon after
the graduation from university. In this
school ~ ha(1 the same two courses for two
classes in the same gra(le. ~ taught them
the same topics almost every time in the
same way in each class. But in the final
examination, ~ was surprise(1 to fin(1 that
the achievement of the class taught in the
second lesson was far better than the first,
although each class was equally divided
according to their ability at the start of the
term. The reason seemed evident: The
teaching ability of the teacher (me) ha
progressed from the first lesson to the
second. But at that time it was difficult,
anti still is, for me to analyze this reason
clearly anti persuasively. The lesson is a
LESSON STUDY AS PROFESSIONAL DEVELOPMENT

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very complicated phenomenon with many
subtle factors interrelating with each other:
the teacher's ways of speaking, asking,
responding, and time, place, gestures, and
so on. Among these factors, there may be
some vital importance in deciding the
success or failure of the teaching. Lesson
is a complex phenomenon in a "classroom
culture," as pointed out by Heinrich
Bauersfeld (Bauersfeld, 19961.
The advantage of lesson study as a
method of research in mathematics
education is that it is the way to grasp the
true state of affairs of the problem in its
whole and to bring a synthetic, totally
recognized interpretation to it, being
aware of many factors that are subtly
interacting with each other as if it were in
one organism. The lesson, as ~ mentioned
above, is a very complicated process, and
its effect on students' learning is too
subtle to express in a simple literal thesis.
It can be evaluated only through close
observation about what is actually going
on; a mere written report is not only
unable to describe the actual state of
affairs but often overlooks the kernel and
essential points of the lesson.
Lesson Demonstration in Conference
Before War World IT, we had two or
three normal schools for the education of
primary school teachers in each 47
prefectures of the country and several
higher normal schools for the education
of the secondary school teachers in some
districts of the country. One or two
primary or secondary schools were
attached to these teacher training schools
for the training of student teachers and
the practical research of education. In
each attached school it was a custom to
have a conference usually once or twice a
year to present educational ideas to the
teachers of the (listrict. It also was a
custom to have a session of lesson observ-
REFLECTIONS ON VIDEOS: PANEL
ing, where an expert teacher from this
school showed a lesson, which was a
model for the participating teachers.
During a renowned teacher's lesson, the
class might be surrounded by many
attending teachers occasionally number-
ing more than the pupils in the class.
There was one thing to be noticed here:
The observing teachers seemed to believe
that a unique best teaching method was
embodied in this expert teacher's demon-
stration of teaching and could be learned
only through observing this model lesson
(Erectly.
At that time we had a national curriculum
(as is still the case), and in primary
school, textbooks were exclusively edited
by the Ministry of Education. Un(ler
these e(lucational-political circumstances,
it was natural that teachers at that time
believe(1 in the existence of a best metho
of teaching and seriously wished to
acquire it. Certainly it would be an
obvious falsehood that there is a unique
best teaching method. However, there is
a profound implication in believing that a
teaching method, whether goo(1 or ba(l, is
embo(lie(1 in a teacher's performances or
even embe(l(le(1 (1eeply in the teacher's
character. The only way to know is to see
the lesson.
This tra(lition of lesson observing
(luring a conference has been maintaine(1
over a long perio(1 of time in every school
attache(1 to a university or its (1epartment
of education. An(1 even more than this,
custom has spread to almost every
regional educational conference of
teachers. Most teachers in Japan, in
transmitting their educational ideas to
others, seem to prefer (demonstrating an
actual lesson to a formal oral or written
presentation. To show their beliefs
through an actual lesson is far easier than
to express them on paper. ~ think this
may be the reason we see the "(lemonstra-

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tion lessons" in many conferences of
teachers.
But recently ~ began to feel some
anxieties: Has "lesson observing" become
something like a mere ritual of the confer-
ence without any reflections on its func-
tion as a method of studying the problem
of mathematics education?
Functions of Lesson Stuck
~ think we can notice two functions for
lessons open to other teachers in an
educational meeting:
as a method of research and
as a place for presenting new ideas or
findings.
In the first, the lesson is open to
colleague-teachers in a school or in a
regional conference, and the analysis is
divided according to its aim:
to ask fellow teachers to see a typical
lesson to become aware of the teaching
problems through the many eyes of the
participants and
· to find causes of failure or success of
the lesson or to focus attention on some
particular problematic factors.
In both cases it would be better to prepare
a written lesson plan for the participants'
reference.
The second function shoul(1 be a
carefully plannedlesson. Observers
should be informed about its aim, topic,
teaching process, suppose(1 pupils'
activities, and intended results before-
han(l.
Traditionally in research in mathematics
education, we present the results of our
research in a paper, but ~ think there is
another way of presentation. The demon-
stration lesson as in the second function,
would like to say, is a form of research
report that can be compared to the thesis
in a paper. Our findings during lesson
studies could be ones that are (difficult to
inform others about using the usual form
of thesis in a paper. They may be trans-
mitted best through the demonstration
lesson. Such a lesson, ~ think, should
qualify as a written paper. These lessons
are different only in form and have the
same values as the written thesis. They
have the same quality as the performance
of a musician or the masterwork of an
artist, which should be considered as a
whole, not broken into pieces.
Conclusion and Aciclitional Comments
Lesson study is a synthetic method of
research in teaching anti learning. We
should regard this method as legitimate
as the usual analytic method of science.
As mentioned above, lesson stu(ly is a
very complicated phenomenon and not
easily studied by a strict analytic method
of investigation. The natural complexity
of this originates in our thinking or
learning activities. An analytic method is
difficult to use in treating such a complex
phenomenon and often overlooks some
subtle anti essential factors.
The (demonstration lesson is another
form of thesis presentation, and we should
consider how such a lesson should be
undertaken. Among many things to be
considered, the most important is how to
make a lesson plan. Here ~ give some of
my ideas about how to write lesson plans
as ~ am not satisfie(1 with the current ways
of (loin" so:
.
The anticipated process of teaching
may be a mere outline of the lesson, but
the teacher's intention should be clearly
(lescribe(l. In Japan we often see a
written lesson plan with a very (letaile
(lescription of the process, including
the teacher's behavior anti pupils'
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.
.
activities. But actually the lesson wall
not go so well as in the planning. ~ was
often bored reading lesson plans and
thought ~ would enjoy them if they were
written more like a work of literature.
A lesson plan may be like the playbook
or the scenario of a drama which
permits the actor's timely digressions
or ad fibs in a large scale. In fact, a
teacher may be an actor or actress, as
Polya (1963) said: The classroom may
be the stage of a theatre, and a good
lesson may be a good performance of a
drama.
But in many schools in Japan, a lesson
may be seen as the time to cram
knowledge into the brains of pupils to
prepare for the entrance examination to
the upper school. The lesson plan may
necessarily turn to a recipe for cram-
ming mathematics effectively.
PostIucle
In Japan, we have traditional arts called
raka~go, which is something like an
entertainment held in a small theatre
telling a short comic story. Some years
before, we invited to the annual confer-
ence of the Japan Society of Mathematics
Education the famous rakugo teller
named Katsura Beicho; he had a high
reputation for his refined way of talking.
The subject of the story was tsa~ho-zan
(bottle math), with a very simple and very
stupid plot.
REFLECTIONS ON VIDEOS: PANEL
A man went to the bottle shop and
bought a bottle for 100 yen. But on the
way home he changed his mind and
wanted a larger one. He returned to the
shop and changed to the larger one for
200 yen. When he was going out the shop,
of course the master of the shop asked
him to pay 100 more yen. But the man
said to him, "I had already paid 100 yen
when ~ bought a smaller one and just now
~ paid with another bottle worth 100 yen.
In total, ~ paid to you 200 yen, and ~ can
have a large bottle for 200 yen." The master
was very confused and could say nothing.
That's the full story, and to tell the
story takes only a few minutes, but this
rakugo teller attracted the entire audience
throughout one solid hour. It is the art
not the technology that attracts the
attention of the audiences.
Mathematics teachers should have
such art in the classroom. For instance,
the first-gra(le teacher has to teach
2 + 3 = 5 to pupils taking at least one hour
without boring them. Or, to tell the
answer to the problem "how many remain
in a box of 10 candies if 2 are eaten" needs
only a few seconds. But it would be a
marvelous thing for chil(lren to know
eight candies remain in the box without
opening the box and counting them. To
make them understand this marvelous
thing as such may take more time, and it
is not only the technology of teaching but
also the art of the teacher to do so.

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