|
Items in cart [0] |
Questions? Call 888-624-8373 |
| Copyright © 2009. National Academy of Sciences. All rights reserved. Terms of Use and Privacy Statement |
Below are the first 10 and last 10 pages of uncorrected machine-read text (when available) of this chapter, followed by the top 30 algorithmically extracted key phrases from the chapter as a whole.
Intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text on the opening pages of each chapter.
Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.
Do not use for reproduction, copying, pasting, or reading; exclusively for search engines.
OCR for page 111
Participants were divided into five small groups for discussion throughout the work-
shop. These groups were charged with responding to two of four questions using the
experiences of the workshop to ground their thinking and discussion and to draw on the
activities done together as a way to give meaning to their responses. Each group had
assigned coordinators from the United States and Japan as well as a translator. Each
U.S. coordinator was responsible for a summary of the discussion of their group. These
small groups provided the opportunity for the participants from the countries to talk
informally about teaching and professional development. This section contains summa-
ries of the discussions, which varied depending on the focus of the groups, and an
introduction to video clips of highlights of some of the small group work.
Questions acIcIressec! in small group discussion:
I. Teaching mathematics requires that teachers understand mathematics well them-
selves. In particular, what mathematics do teachers need to know beyond the
content of the curriculum that they teach to their students?
2. How do teachers use mathematics together with other kinds of knowledge and skill
in order to connect students with mathematics?
3. How do skilled teachers learn about and make use of their students' knowledge and
capabilities to help them learn mathematics?
What are different approaches to helping teachers develop the mathematical under-
standing they need to teach? What are the advantages and potential problems of
different approaches?
Group ~
Report Coordinator:
Michelle Manes, Project Director, Education Development Center
Group IT
Report Coordinator:
Denisse Thompson, Associate Professor, University of South Florida
OCR for page 112
Group Ill
Report Coordinator:
Susan Beal, Professor, St. Xavier University
Group {V
Report Coordinator:
Ramesh Gangolli, Professor, University of Washington
Group V
Report Coordinator:
Susan Wood, Professor, I. Sargeant Reynolds Community College
Video Small Group Discussion Highlights
SMALL GROUP DISCUSSION
OCR for page 113
Group Members
Jack Burrill, Cindy Connell, Daniel Goroff, Toru Handa *, Keiko Hino ~ *,
Jackie Hurd ~ ~ *, Shunji Kurosawa, Michelle Manes *, Toshio Sawada,
Deborah Schifter, Lee Stiff ~ ~ *, Mamoru Takezawa, Hajime Yamashita
Report Coordinator; ~ ~ Translator; ~ ~ HIS. International Congress on Mathematics Education
(ICME) Travel Group
Teaching mathematics requires that
teachers understand mathematics well
themselves. In particular, what mathemat-
ics do teachers need to know beyond the
content of the curriculum that they teach to
their students?
RESPONSE
The question of what mathematics
teachers need to know was a challenge for
the group. One view is that perhaps that
question is easier to answer for Japanese
teachers than for teachers from the
United States the national curriculum in
Japan defines the content teachers need to
teach and therefore defines at least the
minimal level of content they need to
understand. One example of this from the
workshop discussions: It was striking to
participants from the United States that all
of the Japanese teachers in the room were
familiar with Deborah Ball's "coin prob-
lems," and that there was general agree-
ment that these were "sixth grade prob-
lems." That degree of agreement about
content at each grade level simply does
not exist in the United States. It would be
surprising if a group of teachers were all
familiar with the same problem, much less
agreeing on the curricular purpose for the
problem and at which grade level it
belonged. Perhaps teachers in the United
States get bogged down in particular
pieces of content (both for themselves and
their students) because they lack this
consensus about what happens when in a
child's mathematics education.
However, we agreed that there are
some things teachers need to know, or be
able to do, that transcend any particular
curriculum. For example, it is important
that teachers
· recognize the value of persistence
· understand mathematics as a science,
something you explore
· recognize patterns
· understand mathematics as a system
· have the ability to follow a child's
mathematical ideas and determine if
there are problems with the logic
· have a positive attitude about math-
ematics and the expectation that it will
make sense
There was also some discussion about
the content teachers needed to know.
One point of agreement was that elemen-
OCR for page 114
tar y school teachers probably do not need
to study calculus and it seems, in both
countries, it is unlikely that they do.
There seemed to be consensus that
teachers should know, at a minimum, the
content of the courses that they teach in
addition to the courses before and after
theirs. As we dug into this idea, problems
soon arose, however.
In a district where each school has a
different curriculum, what does it mean to
know the content of the courses before
and after? If a teacher teaches different
grade levels, how does the minimal
requirement change? Some people felt
that simply one year on either side was
not enough, and it should be broadened.
However, a teacher in the group reminded
us that elementary school teachers are
generalists, saying, "You can't make every
elementary teacher into a math professor
and a historian and a literature professor.
Elementary teachers have to teach
everything."
The closest we could come to consen-
sus was to agree on the requirement of
knowing the content of their own course
and one course on either side, along with
some particular pieces of mathematics-
like the base 10 system in deep ways.
An interesting fact was that in both
countries, there are concerns about what
mathematics teachers need to know and
about the weak backgrounds of elemen-
tary school teachers. In Japan, two recent
books have been popular: The University
Student Who Can't Do Fractions (Okabe et
al., 1999) and The University Student Who
Can't Do Decimals (Okabe et al., 2000~. In
Japan, the population (lecrease is making
it easier to get into college, and there is
concern that the elementary teachers are
among the least prepared college students
in terms of mathematical backgrounds. In
the United States, the situation is similar,
with schools of education having propor-
tionally more of the students with the
lowest entrance test scores (IJ.S. Depart-
ment of Education, 2000~.
In both countries, there are claims
about what elementary teachers do not
know. In the United States, there is a
strong research base that shows weak-
ness of mathematical knowle(lge among
prospective and practicing elementary
teachers (Ball, 1990, 1991; Post et al.,
1991; Ma, 1999~. It is important, however,
to talk about this research with respect for
teachers, rather than disparage them.
How do teachers use mathematics
together with other kinds of knowledge and
skill in order to connect students with
mathematics?
RESPONSE
The question of how mathematical
knowledge comes together with pedagogi-
cal knowledge as teachers do their jobs
was in a way easier to a(l(lress.
.
.
It is clear from the Japanese lessons
that great care is given to planning
things like blackboar(1 use, order of
problems presented, anti thinking about
what might happen (luring a lesson. All
of the plans for "if students do this, I'll
react this way..." anti even knowing
what students are likely to say and do
with particular problems bring
together both mathematical knowle(lge
anti the un(lerstan(ling of what students
know anti how they think about math-
ematics.
Helping to connect students with
content requires that teachers un(ler-
stan(1 the content in (jeep ways anti that
they can think flexibly about it.
Examples given include making topics
like simultaneous linear equations more
SMALL GROUP DISCUSSION
OCR for page 115
and what they can skip. No course
covers this.
CONCLUSIONS
.
.
interesting by setting the problems in a
social science or physics context and
connecting solving equations to a
broader notion of "doing and undoing"
which students understand. This goes
beyond contextualizing problems to
answer the question of "What is this
good for? When wall ~ use it?" to
helping students better understand the
content by showing the bigger picture
and connections to things they already
understand.
In planning a lesson, teachers need to
move beyond what they know as teachers
and think about what the students know
and understand. There is a difference
between the way a teacher understands
a piece of content and the appropriate
level for students. Understanding a
piece of content as a teacher means
being able to talk about it appropriately
for different levels of students.
When an experienced teacher listens to
a student's explanation, they are able to
repeat back what the student said (an
example of careful listening) and pick
out the essential mathematics (bringing
the careful listening together with the
leacher's consent knowledge). Inexpe-
rienced teachers are unable even to
repeat what the students have said.
Even if they know the mathematics,
they are unable to really listen to their
students. If they can not understand
what their students are saying, they can
not present the mathematics appropri-
ately or address misunderstandings.
· In reading students' journal responses
and problem solutions, experienced
teachers are able to pick out the good
ideas and provide guidance to students,
whereas less experienced teachers may
have difficulty making sense of what
students write.
Teachers need to know enough to make
decisions about what is useful to pursue
.
G R O U P
One major concern, discussed during
two of the three meetings, was the fact
that lesson study is used primarily at the
elementary level in Japan. According to
the Japanese participants in the group,
lesson study exists but is less common in
middle school and not used at all in high
school and college. The Japanese teachers
explain this by saying that the high school
teachers graduate from the mathematics
department; they like math and have an
easier time teaching it. Yet they also said
that exploration of mathematics content
by the teachers may be part of lesson
study but is not always. Another explana-
tion is the tracked high school system in
Japan. In high school, the focus is prepa-
ration for university and entrance exams;
this is much less of a worry at the elemen-
tary level. It seems that in Japan, elemen-
tary teachers take to lesson study because
of an opportunity to learn content. It is
not the only purpose, but it is a big
attraction.
To the U.S. participants, it was clear
that a lot of pedagogical learning went on
in lesson study: how to plan a lesson, how
to anticipate student responses, how to
use the blackboard, and so on. The
purpose of lesson study as described by
one Japanese participant was to "look at
the delta the difference between what
was planned and what actually happened
in a class." That seems like a valuable
experience for a teacher at any level, and
in fact Daniel Goroff described a lesson
study like program at Harvard University.
The program is for mathematics professors,
and the goal is to improve teaching, not
content knowledge.
OCR for page 116
Another issue was around the evalua-
tive nature of classroom observation in
the United States. Teachers wall still be
observed as part of their evaluation, so
how is this formal observation different
from a lesson study experience, and how
can it be made clearly different for the
teachers? This brought up issues around
evaluation of mathematics teachers: Some
members of the group felt strongly that in
evaluating teaching of mathematics, you
have to have mathematics experts (and
not just administrators) involved. The
Education Development Center has a
project called Lenses on Learning focused
on designing materials and videos to help
administrators look at teaching practice
and understand what they should look for
in a mathematics classroom.
In discussing the two main types of
teacher professional development pre-
sented in the workshop, participants
identified clear positives and negatives in
both approaches.
One Japanese participant explaine(1 that
one negative aspect of lesson study is that
teachers try to create lessons that no one
would criticize. That kind of pressure can
keep a teacher from taking risks and from
exercising creativity and originality. In
addition, the point was made that when
watching a real lesson with real kids, the
issues that come up may not be about the
lesson. Whatever happens happens, anti
you can not always predict or arrange to
discuss particular ideas based on study
lessons. Another point about lesson
study, or at least its potential implementa-
tion in the United States, is that teachers
see being observe(1 as a "showcase."
They (lo something they (lo not usually (lo
with their students. The feedback received
on a study lesson may not be useful if it
does not apply to daily practice.
In thinking about records of practice,
one participant compare(1 that form of
professional development as training
people to be opera critics rather than
singers; he claimed that watching a video
of teaching does not help teachers with
their teaching practice. Defenders of the
mo(lel, however, claim that it's not about
being a critic, but rather to learn skills
that are important in teaching. It is true
that records of practice are removed from
actual practice, but there are things you
can learn: You can learn to examine
students' talk, for example. What are they
really saying? Do their mathematical
icleas make sense? Also, records of
practice are less threatening than having
· ~
someone come in ant examine your own
practice. Using vicleoancicases might be
a way to cultivate the kind of atmosphere
that wouic! allow lesson stucly to happen.
Deborah Schifter clescribec! a professional
clevelopment exercise she uses: Teachers
write a narrative of a classroom conversa-
tion. They react each others' narratives,
en c! professional clevelopment staff react
en c! comment on them. Some teachers
say that the exercise makes them hear
their students' mathematics clifferently.
They need to attend clifferently to recorc!
it. This kind of activity can serve as a kind
of bridge between examining videos en c!
cases to examining one's own practice.
In comparing the two methocis, there
are some interesting points of contrast. In
using records of practice for professional
clevelopment, you can specifically select
records for particular learning goals. In
essence, you can build a curriculum
around these records. However, it takes
an outside expert to collect, examine, en c!
catalog the records, and possibly to
cleliver the professional clevelopment.
With lesson stucly, you cannot select for
particular learning goals. Whatever
happens is what there is to discuss. But
lesson stucly can be clone in a school
without the participation of outside
SMALL GROUP DISCUSSION
OCR for page 117
professional developers. Lesson study
groups require leadership and organiza-
tion but that can come from among the
teachers in a school.
RESEARCH ISSUES
In looking at the issues raised in our
discussion, some interesting areas for
research emerged:
.
What is the potential for implementing
a lesson study or similar professional
development structure at the high
G R O U P
school level? Is there potential benefit
for teachers at this level?
Is there an impact of participation in
lesson study on daily practice? One
point made in the workshop is that
Japanese teachers do not spend the
same amount of time on a regular
lesson as on a study lesson. So is it
true that study lessons are "showcases"
with little relation to a regular class, or
does going through the process of
carefully planning some lessons affect
how a teacher presents all lessons? If
there is an impact, what is it?
OCR for page 118
Group Members
Hyman Bass, Phyllis Caruth ~ ~ *, Frances Curcio ~ ~ *, Alice Gill, Ichiei Hirabayashi,
Tadayuki Ishida, Carole Lacampagne, Carol Malloy, Nob~hiko Nohda, Semi Soga *,
Akihiko Takahashi, Denisse Thompson *, Hiroko Uchino, Tad Watanabe
Report Coordinator; ~ ~ Translator; ~ ~ HIS. ICME Travel Group
The group did not make any specific
assumptions in order to answer the
assigned questions. However, some
misunderstandings among the American
educators that were not clarified until
Tuesday may have impacted the discus-
sions. In particular, there was confusion
about what was meant by lesson study.
Not until Tuesday did it become clear that
Japanese teachers might participate in
several lesson study groups throughout
the year on a regular basis to discuss a
variety of instructional materials and
content; participation in a long term lesson
study occurred much less frequently.
From the discussions and from addi-
tional information provided by Tad
Watanabe, it appears that lesson study is a
practice that occurs typically at K-6
schools. Some of these may focus on
developing a lesson over a period of time,
and others may focus on providing mutual
support for (laity problems or for more
broad-based issues, such as how to
implement a new national course of study.
At the lesson study sessions focusing on
the development of particular lessons,
different individuals may bring plans and
have others critique these plans. Lesson
stu(ly affords teachers the opportunity to
talk about different aspects of lessons or
to explain what happened when they tried
a lesson in the classroom.
Formal lesson stu(ly with open-house
stu(ly lessons occurs only a limited
number of times during the school year
and is for the explicit purpose of research-
ing the lesson. These lessons often take
place at elementary schools attache(1 to
universities; education researchers at the
universities might serve as outside
observers anti reactors to the lesson
study. The availability of lesson study at
the elementary schools affiliate(1 with
universities seems to fulfill one of the
missions of these schools, namely, to
provi(le national lea(lership in improving
teaching in all subjects inclu(ling math-
ematics. It appears that not all Japanese
teachers regularly participate in a formal
lesson study as a teacher being observed
teaching a lesson to children.
A(l(litionally, lesson studies appear to be
practices that occur primarily at the K-6
levels. Very little (liscussion occurre(1
about professional (levelopment practices
for teachers at gra(les 7-12.
In responding to the charge to (liscuss
the two questions liste(1 below, the group
focused on both questions simultaneously.
OCR for page 119
Although responses do not fit neatly with
only one of the two questions, this sum-
mary attempts to sort responses to
provide some coherent comments to each
question.
How do teachers use mathematics
together with other kinds of knowledge and
skill in order to connect students with
mathematics?
RESPONSE
In the course of discussing this ques-
tion, it became clear that the mathematics
knowledge of the teacher is crucial at all
points in the instructional process.
Teachers design lessons, enact lessons,
respond to students' ideas and make use
of those ideas in furthering the lesson,
analyze lessons, and study student work.
All of these points require considerable
mathematical knowledge. For instance, in
the sixth-grade lesson the teacher explic-
itly wanted the children to construct the
task of the lesson the teacher's knowI-
edge guided the lesson by guiding the
children until the task for the day emerged.
That is, students raised ideas related to
area models; the teacher guided the
children to consider other ideas until the
linear growth model emerged. Clearly the
teacher had to know what mathematical
structure was inherent in his goal.
Other issues that arose in relation to
the question are discussed below.
Teachers use their knowledge of
mathematics as they develop appropriate
tasks to use with their children. The
teacher's mathematics knowledge is used
in conjunction with his or her knowledge
of children to develop tasks that will be of
interest to the children and motivate them
to learn. This intersection of the knowI-
edge of children and mathematics was
GROUP 11
evident in the fourth-grade lesson in
which the teacher used a quiz show
format in an attempt to motivate children
to deal with issues of large numbers.
Teachers use their knowledge of
mathematics to consider real-life prob-
lems and examples that people need for
everyday life. Japanese K-6 mathematics
education seems to focus on "mathematical
literacy," so that Japanese teachers
attempt to begin a lesson with a context
that children might encounter; this was
evident in the fourth-grade lesson in
which the teacher dealt with large num-
bers related to size and costs. It was also
evident in the Japanese teachers' concerns
about the U.S. third-grade lesson dealing
with the number of pennies, nickels, and
dimes to equal a given amount of money.
Japanese teachers thought the problem
woul(1 be more compelling if the teacher
had the coins in her pocket to show to
students or to have students consi(ler
what the coins might purchase.
Teachers use their knowledge of
mathematics to un(lerstan(1 the (1ifferent
aspects of the mathematics in a problem
that may impede the work of the students
on the given task. Teachers need to be
aware of the cultural dimensions that
provide gates that may have an effect on
learning, either positively or negatively.
For instance, in the fourth-gra(le lesson
on large numbers, the numeration system
involves grouping by ten thousands in
Japanese, rather than by thousands as in
English. The different mental structure
require(1 to comprehen(1 the numbers in
this system was a source of difficulty for
some of the American educators watching
the lesson; focusing so har(1 on compre-
hen(ling the numbers represented in an
unfamiliar way ma(le it more (difficult to
focus on the mathematics embe(l(le(1 in
the lesson. Although this (lifference in
grouping was not a problem for the
OCR for page 120
Japanese children, the cultural differences
caused a problem for the American
educators. Given the cultural diversity of
children in U.S. classrooms, it serves as a
potent reminder of the need to be sensi-
tive to children's perspectives. When
cultural backgrounds cause children to
think differently, it is often easy to dismiss
their thinking as incorrect; however,
student explanations and good question-
ing on the part of the teacher can help
both students and teachers to bridge the
cultural divide and enhance learning.
Teachers use their knowledge of
mathematics when they take a simple
problem from their textbooks and con-
sider how the problem might be adapted
or extended to explore the mathematics
more deeply. This was evident in the
sixth-grade lesson as the teacher took a
basic problem dealing with growth by four
per minute and had children look for
patterns and attempt to express those
patterns by means of functions and
variables. As part of this work with
functions, the teacher had children
considering the different interpretations
of 4 x 48 + ~ and 48 x 4 + ~ in the context
of the situation. This functional work
foresha(lowe(1 anti lai(1 the foundation for
more advanced work in later grades.
Teachers use their knowledge of
mathematics to anticipate various student
solutions that may arise during classroom
discussions about a task. Anticipating
student solutions helps teachers deter-
mine where the discussion may go and
enables them to make appropriate deci-
sions on how to procee(1 with the lesson.
For instance, in the sixth-grade lesson,
children's ideas about area were leading
to a quadratic relationship. By anticipat-
ing students' solutions, the teacher was
able to steer the discussions until the
linear relationship that he wanted the
chil(lren to investigate was generated.
Teachers use their knowledge of
mathematics as they make connections
between mathematics and other disci-
plines.
What are different approaches to helping
teachers develop the mathematical a~nder-
standing they need to teach? What are the
advantages and potential prohiems of
different approaches?
RESPONSE
A number of issues arose regarding the
various professional development ap-
proaches presente(1 at the seminar-
research study lesson, video case record,
written cases, and professional develop-
ment in general.
The discussions before and after the
stu(ly lesson help teachers know anti
un(lerstan(1 the mathematics embe(l(le(1 in
the lesson. The (liscussions are important
because they help teachers reflect on how
to fix structural errors in the lesson. For
instance, in the fourth-gra(le lesson, the
fact that each successive question in the
task increase(1 by a factor of ten cause
some students to focus on the pattern
being generate(1 rather than the un(lerly-
ing mathematical relationships, thereby
obtaining an incorrect result for the comic
book problem. Through (liscussions
about the lesson after its presentation,
teachers are able to think about this
structural problem anti how it might be
resolved prior to teaching the lesson
again.
One approach that is particularly
helpful for teachers is to use vertical
articulation of topics. That is, teachers
shoul(1 stu(ly topics anti consi(ler how
they change anti grow across the various
gra(les anti levels. If elementary anti
lower secondary teachers understand
SMALL GROUP DISCUSSION
OCR for page 121
where topics eventually lead in high
school, then they might teach those topics
differently. For instance, the s~xth-grade
teacher taught his lesson to lay founda-
tions for more formal work with functions
in a later grade. It appears that in Japan,
teachers are encouraged to participate in
lesson study at various levels so that they
have a good sense of the mathematics that
is taught prior to and following the level at
which they teach. The same is not
necessarily true in the United States.
Although the recently released Principles
and Standards for School Mathematics
(National Council of Teachers of Math-
ematics, 2000) encourages this vertical
articulation through the content stan-
dards, it is not clear that such articulation
typically occurs in practice. U.S. teachers
have little opportunity to interact with
teachers at levels other than their own to
gain a better understanding of the math-
ematics taught at the different levels.
The national course of study in Japan
helps teachers make assumptions related
to the issue. This course of study helps
teachers make assumptions about the
mathematics that children should know,
both in terms of instruction at their grade
level and in terms of prerequisite knowI-
edge. U.S. teachers are not able to make
such assumptions quite so readily.
Materials need to be available in
teachers' editions that show how ideas are
related across levels. The materials in the
Japanese teachers' editions are not the
same as in the pupils' editions. The
teachers' editions contain supplementary
problems and discuss how a topic relates
to previously taught ideas and to ideas
that wall appear in later grade levels. The
Japanese teacher materials provide
support to teachers, pointing out the
important ideas of the lesson and where
students might have problems; in addi-
tion, the teachers' edition may provi(le
GROUP 11
different solutions or ways that children
think about a problem. Some of these
same instructional approaches are present
in the teachers' editions of newer U.S.
textbooks that provide additional prob-
lems, ideas for review, reteaching ideas if
students have difficulties, or extensions
for more advanced work. A big difference
may be that the course of study in Japanese
education helps focus on connections to
earlier and later courses that are not
possible in U.S. textbooks; hence, Japanese
materials often seem more focused than
corresponding U.S. teacher materials.
Research is needed to evaluate the
relation between school mathematics that
teachers are expected to teach and college
mathematics that students are expected to
learn. University mathematics does not
necessarily help in teaching school
mathematics. The mathematics teachers
stu(ly shoul(1 begin with the mathematics
in the textbooks of the curriculum. This
perspective closely relates to the work-
shop paper delivered by Zalman Usiskin
in which he raised the issue that school
mathematics is a rich source of study in
its own right. Japanese anti U.S. e(luca-
tors agree(1 that there was a need to stu(ly
mathematics for teaching and not just
mathematics for mathematicians.
The three different professional devel-
opment approaches seen at this workshop
facilitate the integration of content with
methods. This integration is important
because teachers' pedagogical content
knowle(lge comes into play in un(lerstan(l-
ing how students (levelop their math-
ematical knowle(lge.
The purpose of the formal stu(ly lesson
is to provide research on the lesson and
its effectiveness. However, written cases
anti videos could be use(1 in lesson stu(ly
groups anti coul(1 help in the (development
of a research lesson. Published study
lessons coul(1 also be use(1 in stu(ly
OCR for page 126
daily lessons is important as is the need to
plan for every lesson.
Video records as an approach to
professional development can focus on the
details associated with the lesson.
Materials available to be studied consist of
detailed documentation of teaching and
learning, although all of these are not
necessarily analyzed. The materials are
used to encourage teachers to learn how
to observe, discuss, and make records-
all important in learning a practice such as
teaching as described by Deborah BaD in
her opening remarks. Her study took place
over an entire academic year in two classes.
The richness of the detail allows explora-
tions of what took place in the teaching-
learning process over a period of time.
Lesson Summary
All three approaches for professional
development have strengths and weak-
nesses. Many study lessons are tran-
scribed, which is very time-consuming.
One participant noted that he has one of
his students copy what is on the black-
board and write about his or her thoughts
on the lesson, rotating the task so that all
students have the opportunity. In lesson
study, the blackboard can be used as a
record for research. Participants observed
that it takes a long time in constrained
circumstances to watch a video, while
opportunities to read a case study were
more flexible. All agreed that it is nice to
have a summary of the lesson. However,
whether it is video or written, a point of
view is always present, depending on the
person who videotaped the lesson or the
one who summarized it. Since lesson
stu(ly has many observers, there is not
one point of view. Because each approach
for summarizing or recording a lesson has
strengths and weaknesses, the group
thought that it would work well to com-
bine the methods.
One characteristic of lesson study is
that the physical records of the lesson are
on the blackboar(l, which can help as
teachers analyze the lesson. In fact, when
(liscussing the sixth-gra(le lesson, we ha(1
the physical "blackboar(l" to help us in our
analysis and recollection. The blackboard
is organize(1 to help students keep notes.
Furthermore, we can categorize what the
teacher said by using the notes from the
blackboard.
CONCLUSIONS
Cultural Differences and Their
Impact on Professional
Development
How Teachers Are Known to Be
Good Teachers
In Japan some teachers are known for
their teaching and become known because
they have been observed in the classroom,
perhaps through lesson stu(ly. Many also
write about their ideas in one of the many
magazines for teachers. If the idea appears
in a mathematics-specific issue, comments
are ma(le by mathematics educators. Also,
Japanese bookstores have many books
written by teachers to share pleas.
The U.S. participants felt that U.S.
teachers become known by their profes-
sional affiliations anti accomplishments
anti not necessarily by their teaching.
The professional journals, for example
those publishe(1 by the National Council
of Teachers of Mathematics, include
articles by teachers that (lescribe their
i(leas or classroom activities, but it is very
(difficult to know from this how a teacher
actually teaches. An(1 although there are
magazines such as Teacher, which might
mirror those in Japan, these magazines (lo
not necessarily support the mathematics
education community.
SMALL GROUP DISCUSSION
OCR for page 127
Direct Comparison of Teachers
Direct comparison of one teacher with
another does not happen in Japan, as in
the case study example with the compari-
son of Catherine's and David's classes. In
Japan, the lesson study consists of a set of
hypotheses on how to conduct the lesson,
which are then tested. It is these hypoth-
eses that are critiqued during the discus-
sion following the observation of the
lesson. Furthermore, to prevent over-
whelming the teacher with too many
suggestions during the discussion of the
lesson study, focal points are established
and the participant observers categorize
what the teacher said, using the black-
board as a point of reference. Usually, the
lesson has been planned with others, so
the emphasis is on the work done by the
group. The notion of "we will help you
improve" in Japan starts as early as when
you are a student and so naturally comes
to the teaching process, but this is not so
in the United States.
Working Harc! in Mathematics
Versus "Math People"
The attitude that all students can learn
mathematics seems to be more prevalent
GROUP 111
in practice in Japan than in the United
States, even though NCrM (and the
Mathematical Association of America)
specifically include this in the literature.
Being a Teacher
In the United States, teachers go back
to college to get advanced degrees but not
in Japan. Becoming a teacher is very
difficult in Japan. Many teacher appli-
cants fail the screening to become a
teacher. The test to become a teacher is
to teach in front of a set of examiners. As
in the U.S., supply and demand has a lot to
(lo with how rigorous the requirements
are to become a teacher.
In Japan there is also a first-year
teacher training program in lieu of student
teaching. The year includes 30 days in
school with a mentor to help. Every five
years teachers go back to the education
center for continued professional develop-
ment. In the United States, 33 states
require a small amount (minimal) of
professional development (Dossey and
Usiskin, 2000~. In most states teachers
have to do some form of professional
(levelopment to retain certification.
OCR for page 128
|1~.
Le]ni.Ill ~
Group Members
Jerry Becker, Shelley Ferguson ~ ~ *, Ramesh Gangolli *, Beth Lazerick ~ ~ *,
Shinichiro Matsumoto, Judith Mumme, Hiroshi Nakano, Yasukiro Sekig~chi *,
Kiyoaki ShodFa, Margaret Smith, Shinichi Suzuki, Daniel Teague ~ ~ *, Miho Ueno
Report Coordinator; ~ ~ Translator; ~ ~ HIS. ICME Travel Group
Although an attempt had been made to
focus the group's discussion on the ques-
tions suggested at the workshop, given
the central position occupied by lesson
study in the plenary sessions, it was
perhaps inevitable that much of the small
group discussion was related to assessing
the role and effectiveness of lesson study
as a tool of professional development.
What follows is a summary of responses
to the questions as they relate to Japanese
teachers taken from the interaction at the
workshop. It is based partly on what
transpired in the small group sessions but
also on information obtained through other
opportunities (formal as well as informal)
as they arose during the workshop. The
remarks are directed toward mathematics
(and not toward any other subject), and
those that might seem to apply more
generally to other subjects should be
viewed as limited by this restriction.
BACKGROUND
Elsewhere in these proceedings one
can find a full description of lesson study:
what it is, how it is implemented, the
contexts in which teachers encounter it,
and the role it plays in their professional
development. In this summary, it is not
necessary to repeat that information. It
will nevertheless be useful to mention
here a few aspects of the process of lesson
study that might run contrary to the
conception of lesson study that some may
have formed. These aspects will serve as
a backdrop against which some of the
remarks that follow may make more sense.
.
Lesson study sessions are not as
pervasive a part of the daily life of
teachers in Japan as one might sup-
pose. The extent to which individual
teachers engage in formal lesson study
sessions is quite limited. Thus, many
elementary teachers might do so only a
few times a year at their school. They
might also observe such sessions as a
part of professional development
activities at professional meetings, etc.
Secondary school teachers engage in it
to an even lesser extent. Schools
attached to university departments that
do research in mathematics education
are exceptions. There, lesson study is a
far more recurrent feature of life.
Moreover, the lesson study sessions in
which teachers participate might deal
OCR for page 129
only qualifying proviso
.
.
with any subject (not necessarily
mathematics).
There is a tremendous range in the
type of experience afforded by such
lesson study sessions, depending on
grade level, context, and purpose.
Sessions might range Tom very informal
ones in which a couple of colleagues
might casually observe a small change
of presentation and comment on it, to
very formal heavily attended "study
lessons" in which new investigations or
major changes in existing lessons
might be launched or dissected. The
latter type of session is typically held in
a university context. The end product
of such sessions (after many iterations)
might well end up as a published lesson
made available to teachers.
Great care is clearly taken in the
preparation and implementation phases
of a lesson study session. Thus, there
is continual stress on the importance of
(a) clearly identifying the conceptual
and cognitive goals of a lesson,
fib) designing interesting questions that
will promote those goals, (c) weighing
and selecting from different pedagogi-
cal devices that might be useful in
pursuing the problem, anti ((1) anticipat-
ing a variety of student approaches
and/or misconceptions. Notw~thstand-
ing the scrupulous attention paid to
these elements, the postdiscussion that
follows a study lesson sometimes
concentrates on conceptual content and
at other times on individual pedagogical
style. Discussions may range from
polite noncommittal ones to long
substantial debates.
· In spite of these variations in form and
~ i!
s that in order
to be useful, this process must be
undertaken on a continuing basis over a
sufficiently long period.
Bearing in mind these aspects, a
summary of responses to four questions
follows.
What do teachers do?
RESPONSE
The simple answer is that they continu-
ally engage in the process of trying to
become more effective teachers, thinking
about what they teach and how they teach
it. But this is a glib summary that does
scant justice to a number of institutional
anti cultural forces that promote anti
indeed ensure this behavior on a continu-
ing basis. Notable among them are the
following.
Preservice Acculturation
Although classroom-based preservice
training (e.g., as a student teacher) is
short typically just three or four
weeks we were told by our Japanese
colleagues that throughout the prepara-
tory years certain expectations of what is
regarded as professional behavior are
built in as an integral part of preservice
education. Among these are (a) the notion
that as a teacher, one is expected to strive
for "continuous improvement" throughout
one's career and fib) that lesson study or
other activities like it (involving reflection
and analysis of one's teaching) will form
the most important tool for achieving
continuous improvement.
substance, there is a universal accep-
tance of the proposition that the pro-
cess of reflection exemplified by the
lesson study experiences is a powerful It seems to be a widespread practice
tool for professional improvement. The that some structured mentoring is pro-
GROUP TV
Initial Mentoring
OCR for page 130
vided for the beginning teacher. This is
especially the case for elementary teachers,
who are usually not specialists in math-
ematics.) It was not clear whether this
mentoring is subject oriented (e.g., toward
mathematics). The precise manner in
which this is done was not clarified in our
discussions, but it seems that it is usual to
assign a fledgling teacher to a senior
colleague who becomes responsible for
mentoring that teacher for one or two
years. A policy adopted in 1998 requires
that such mentoring shall be provided as a
rule to all beginning teachers. It is not
clear whether this policy is fully imple-
mented as of today, but it surely sends a
strong message to the beginning teacher
that her or his teaching effectiveness is of
immediate concern to the school system.
The system of mentoring seems to be an
effective vehicle for transmission of the
professional culture of the school and its
cadre of teachers.
Inservice Integration
The mentor who is a part of the
school's professional culture reiterates the
expectations clearly conveye(1 (luring the
preservice phase during the initiation
phase. Simultaneously, the beginning
teacher is also integrated as a part of a
team of teachers in the school. Most of
our Japanese colleagues seemed to
indicate that this is the prevalent model.
The integration is facilitated by the
existence of a number of support struc-
tures that promote the implementation of
the model. Among these are
a. Availability of time for teachers to
prepare anti (liscuss lessons among
themselves.
b. Opportunities for getting expert
advice; this may come via an ongoing
or occasional liaison with a faculty at a
university or by means of other specific
opportunities (e.g., at lesson study
sessions at the ward, prefectural, or
national level).
Stability of employment; after an initial
probationary period, a teacher acquires
tenure de facto. Implicitly, the other
part of this social contract is the
assumption that the teacher will fulfill
the expectations of professional
behavior described above. Fluctua-
tions in the total number of teachers
needed are handled by attenuating
and adjusting replacements needed
due to retirements or resignations,
rather than by lay-offs. This encour-
ages in(livi(lual teachers to view
themselves as a part of a permanent
value(1 structure, contributing signifi-
cantly to a national need, rather than
as functionaries who are hired and
fire(1 base(1 on random (demographic
· ~
exigencies.
d. Standardized, unambiguous curricular
materials, supplemente(1 by explicit
implementation schedules and sup-
porte(1 by (letaile(1 teachers' gui(les;
the implementation schedules are
expecte(1 to be observe(1 scrupulously,
so much so that we were tol(1 that in
any given week, essentially all stu-
(lents at a particular gra(le level will
be studying the same mathematical
topic throughout the country. On the
other han(l, the extent to which an
in(livi(lual teacher woul(1 follow
exactly the (letaile(1 lesson plan for
teaching that topic as lai(1 out in the
teachers' gui(le varies. The variation
seems to depend on the experience
anti initiative of the teacher, as well as
the gra(le level. The adherence to a
stan(lar(1 curriculum, both as to the
sequencing of topics anti the time
spent on them applies without excep-
tion, so far we can tell, to elementary
anti mi(l(lle school gra(les. There
SMALL GROUP DISCUSSION
OCR for page 131
seems to be much more variation at
the high school level, both in regards
to the lessons used as well as the
pacing. However, the order of devel-
opment of the topics and the inter-
connections that are to be stressed
seem to be governed by the collective
wisdom of the national curriculum
and is fairly uniformly observed.
What do teachers work on?
RESPONSE
The honing of specific lessons in order
to make them more effective is the motif
that seemed to run through the efforts of
the individual teachers. Observations of
lesson study sessions indicate that teachers
devote considerable attention to (a) clearly
articulating the specific intellectual goal of
the lesson (it is noteworthy that the goal
is often phrased as an abstract goal of
achieving understanding of a concept
rather than the mastery of a skill or
technique); (b) selecting a problem or an
investigation that will help students to
arrive at an understanding of that goal;
(c) seeking the most effective way of
posing the problem to the class;
(~) anticipating student responses,
including alternative approaches as well
as misconceptions; and (e) encouraging
or eliciting generalizations.
Although these features are at the
forefront in the lesson study sessions, one
question that arose was to what extent
does this type of thinking and reflection
pervade the day-to-day activity of teachers?
Necessarily, the role most often playe(1
woul(1 be that of an observer rather than a
presenter, especially when the teacher is a
novice, gaining experience. Presenters
may be more experience(1 or repute(1
teachers. Thus, an important function
GROUP TV
served by lesson study sessions seems to
be to provide each teacher with an oppor-
tunity to experience and absorb the
ingredients of a successful model as an
observer but with the privilege of being
an active participant as a friendly critic.
Of course, soon a time comes for every
teacher when that teacher needs to play
the role of a presenter. By this time,
however, the tension of being on the spot
is dissipated by the previous experiences,
in which they have observed that there is
an impersonal protocol of criticism
directed more at the content of the lesson
(what works anti what (toes not) or
suggestions about how one might more
effectively introduce a specific detail of
the lesson, rather than at the inadequacies
of the teacher.
Thus, lesson study sessions seem to
provi(le a model for emulation rather than
an active (lay-to-(lay operating procedure.
Nevertheless, because the model is highly
valued and the lessons produced and
perfected by the model can be used on a
national basis, it exerts a powerful exem-
plary force. The idea of continuous
improvement through reflection and
analysis of specific pedagogic decisions is
implicit in the model, and the general
agreement seems to be that it is a tried-
and-tested operational model worthy of
adoption on a routine basis.
What do teachers apse?
RESPONSE
At the elementary level, the standard
curricular materials, containing specific
lessons, together with the detailed teachers'
guides seem to provide the foundation on
which classroom lessons rest. For the
novice teacher they provide a safe, secure,
and acceptably effective map that can be
OCR for page 132
followed while that teacher is acquiring
and honing professional skills. For the
more experienced or enterprising teacher,
they provide a template from which that
teacher may (lepart by providing innova-
tive variations of approach and treatment,
while maintaining the conceptual goals of
the lessons.
A striking feature of the lesson study
sessions that we observed was the use of
the blackboard as a vehicle for recording
the ideas generated by students, thereby
acting as an archive of the ideas under
discussion in class, rather than as a
written repository of the ideas sought to
be conveyed by the teacher.
It was not clear what role manipulatives
played in lessons at the elementary level.
There was a short, oblique discussion of
this at one point. The impression one gets
is that they play a peripheral role, subject
to the mathematical goal of the lesson.
to works with teachers?
RESPONSE
Our Japanese colleagues said that
during the preservice years interactions
between faculty and students are formal
for the most part. However, there seems
to be some mechanism by which the
student becomes aware quite early in the
pre-service years that continuous improve-
ment is the professional ideal, and that
lesson study (or other similar reflective
activity) is a powerful anti effective tool
for continuous improvement. The impor-
tance of the central role played by this
cultural agreement about professional
norms cannot tee overestimated. Japanese
colleagues made statements such as, '~he
importance of continuous improvement
though lesson study is taken for granted
in Japan," "One can always improve by
studying one's own lessons with the help
of colleagues", "How can you be a func-
tioning teacher if you do not care to
improve?," anti "Even excellent teachers
can always improve. One must continu-
ally seek to improve. Teaching is 80
percent confi(lence anti 20 percent (loubt."
During the initiation/mentoring phase,
we were told that experienced teachers
work with new entrants to the profession.
This seems to have been a common
practice of the profession, now adopted as
official policy by the Ministry of E(luca-
tion since 1998.
During this novitiate phase as well as
later in the teacher's career, teachers
seem to have some opportunities to get
advice directly or indirectly from disciplin-
ary experts such as mathematicians or
mathematics educators in higher e(luca-
tional institutions. In schools attached to
university departments charged with the
preparation of teachers, this happens
quite regularly as a part of the ongoing
implementation of lesson study in the
research program of the university. In
other schools it seems to happen less
systematically. When it does happen, it
would probably be in the context of lesson
study events at the ward or prefectural
level. On the whole, the development of
deep conceptual understanding of the
concepts that teachers teach seems to
come from this continuing process of
careful lesson planning and implementa-
tion, followed by reflection about its
effectiveness, informed by constant
analysis of student ideas. Few research
mathematicians seem to be involved in
school mathematics education.
CONCLUSIONS
The process of lesson study seems to
be a key ingredient in the professional
SMALL GROUP DISCUSSION
OCR for page 133
development of Japanese teachers,
especially at the elementary and middle
school levels. However, the precise
manner in which this process facilitates
the acquisition of disciplinary as well as
pedagogical content knowledge by
teachers is subtler than one might assume
at first sight. Lesson study sessions are
not numerous enough to act as vehicles
through which teachers can acquire deep
content knowledge of the great many
mathematical topics they need to teach-
teachers must necessarily seek other
ways in which this knowledge may be
acquired. Likewise, lesson study sessions
are not frequent enough to ensure that
teachers will absorb their methodology
through force of habit. However, lesson
study sessions serve as exemplary
models for several aspects of practice.
The thoroughness of preparation and
presentation, the stress laid on analysis
and reflection, and above all the single-
minded focus on the effectiveness of the
lesson (rather than the effectiveness of
the teacher) are ideals that the teacher is
encouraged to emulate in daily classroom
practice. By the avowed acceptance of
lesson study as a long-term professional
development tool, the individual teacher
affirms a belief in the value of these
practices and a commitment to put them
into practice. The opportunity to engage
in lesson study sessions on a regular basis
has a practical as well as symbolic value.
A lesson study session forces the partici-
pating teacher to hone her or his skills to
an edge sharp enough to withstand critical
evaluation by colleagues and to continue
to employ those skills in daily practice.
On the other hand, due to the high status
collectively accorded by the profession to
the process of lesson study, the individual
teacher can justly regard his or her
successful participation in it as evidence
of professional growth and competence.
This periodic affirmation of exemplary
GROUP TV
professional values and practices greatly
reinforces the teacher's image of himself
or herself as a professional engaged in a
process of continual improvement. Thus,
in a roundabout way, the process of lesson
study seems to have the effect of enabling
the majority of teachers to arrive at and
sustain a view of themselves as members
of a professional community engaged in
continualimprovement. On-thejob
acquisition of deep understanding of
content and pedagogy depends on this
attitude more than any other single factor.
In the final analysis, this effect may be as
valuable as any other effect of the process
of lesson study.
RESEARCH ISSUES
In the short time span of the workshop,
it was (difficult to get an idea of the extent
to which individual teachers implement on
a day-to-day basis the practices exempli-
fied in the lesson study sessions. A study
of how lesson study effects day to day
practice would be useful.
Also in (liscussions the Japanese
indicated that teachers were generally
expecte(1 to have a "good un(lerstan(ling"
of the mathematics they are expected to
teach. An examination of various certifica-
tion requirements coul(1 give us an idea of
the formal (leman(ls that are ma(le on
teachers in the way of technical content
knowle(lge. How (lo teachers acquire the
(leeper conceptual un(lerstan(ling (involv-
ing major ideas anti their interconnections)
as well as the pe(lagogical content knowI-
e(lge (such as effective strategies for
communicating those ideas, awareness of
common misconceptions anti strategies of
(leafing with them, etc.) ? Is it on the job,
through a commitment to the i(leal of
continuous improvement through reflec-
tion anti analysis, as (lescribe(1 above?
OCR for page 134
~;TTT"
Group Members
Angela Andrews ~ ~ *, Deborah Ball, Toshiakira Fujii, Henry Kepner, Jr., Jean Krusi,
Marilyn Mays ~ ~ *, Keiichi Nishimura, Ottawa Sakai, Kayo Satou *, Mark Saul ~ ~ *,
Keiichi Shigematsu, Yoshinori Shimiz~, Lucy West, Susan Wood
Report Coordinator; ~ ~ Translator; ~ ~ HIS. ICME Travel Group
The group asked and answered the
questions listed below. Much of the
discussion centered around the details of
Japanese lesson study.
The group discussed the three models
of records of practice presented in the
workshop: video (Ball), written cases
(Smith), anti Japanese lesson stu(ly as
seen through the video of a fourth-grade
lesson and study group discussion and
observation of a s~xth-grade lesson and
study group discussion.
How do teachers use mathematics
together with other kinds of knowledge and
skill in order to connect students with
mathematics?
How do skilled teachers learn about and
make use of their students' knowledge and
capabilities to help them learn mathematics?
RESPONSES
Both questions were addressed
through the discussion of the record of
practice feature(1 (luring the workshop.
Both video anti written cases provi(le
records of practice for long-term use.
Video records of practice can recor(1 the
learning of the same student over time. In
one viewing, a spectator can see lessons
from different parts of the year. The
author of a written case study may affect
how useful it is. Japanese colleagues
found video an easier medium to work
with than the case stu(ly. To them video is
a familiar medium and easily allows
forming of images. If a teacher poses a
question and does not get any of the
expected answers, live observation might
provide insight into why the expected
answers did not occur. In Japan, the
lesson plan, student responses, and
teacher's reflections form a sort of "case
study," although not in written form.
Overview of "lapanese Lessons
In the lesson write-up, like a script for a
play, the left column gives student activi-
ties, the right side gives "cautious points"
and "evaluation points." Usually, a middle
column in the plan contains expected
student responses.
The evaluation of the lesson must be
consistent with the lesson's aim. Stan-
OCR for page 135
cards for evaluation are in terms of
students'
· willingness, intent, and attitude toward
learning;
mathematical thinking displayed;
representing and processing origi-
nally called skills of content; and
· understanding.
Each lesson is like a mountain, with a
slow climb to the peak. Each lesson has a
rhythm. Teacher personal satisfaction is
not enough. To help students gain
mathematical satisfaction, the teacher
must sum up the mathematical objective
of the lesson in which the teacher moves
from naive student solutions to those that
are more sophisticated. Japanese teachers
like whole group discussion, not the one-
to-one discussion between student and
teacher. Students often explain other
students' ideas.
The parts of the lesson are introduction,
development, and summary. Lessons
must be set in a clear context. The use of
the board is carefully planned and makes
a summary of the whole lesson. Students
can see what was learned, even if they do
not completely understand.
Students' Notes
Students' notes and errors are valued
by teachers. Japanese students are taught
to take notes in first grade. They copy the
task and conclusion and record their own
work in the middle. They are encouraged
to write their own ideas. The levels of
student note taking are
· animpression "it was fun";
why ~ am interested in the lesson the
math content "it was about x";
classmates' ideas; compare my ideas
and friends' ideas see myself objec-
tively; the student is the owner of an
Plea; and
GROUP V
· self-reflection generalize the problem
beyond the content of the lesson.
Lesson Stucly arc! Teachers'
Mathematical KnowlecIge
In Japan, teachers' mathematical
knowledge is important in enabling
teachers to anticipate student responses
which in turn strengthen that knowledge.
Teachers also need mathematical knowI-
edge to build on unanticipated responses
and be ready to adapt to the students'
responses and misunderstandings.
Teachers select student examples that are
close to their goal, then build on them to
help accomplish the purpose of the
lesson. Teachers must be alert to stu-
dents' ideas that extend the lesson or for
opportunities to probe for deeper under-
standing of the content. Working on and
revising the teaching plan is one way to
buil(1 mathematical knowle(lge. The
teacher must think mathematically while
creating the plan. Much attention is given
in lesson design to the specifics of the
lesson, even to details such as using the
number 12 instead of Il. Emphasis is
placed on finding a suitable task, with the
structure of the task being very impor-
tant. Teachers must stay focused on the
goal of the lesson.
What is the nature of the postlesson
discussion?
RESPONSE
The postlesson (liscussion focuses on
· the gap between the plan anti the
imple me nte (1 le s so n ; anti
· the gap between lesson stu(ly anti the
general sense of mathematics education.
There can be confusion between these
two aspects.
OCR for page 136
Teachers of the same grade cooperate-
they are given the opportunity to look at
existing lesson study plans and read
related books. Japanese teachers have
very different backgrounds, much the
same as U.S. teachers.
Lesson study focuses on the "whys" of
lesson design. In lesson study, the
conversation immediately following the
lesson is crucial to understanding and
improving the lesson. The Japanese
believe that observing the class (seeing
with your own eyes) is best for staff
development.
What is the role of the advisor in lesson
study?
RESPONSE
The advisor's role in lesson study is to
identify key points to improve the lesson
and teaching, to identify the most valuable
things mathematically and pedagogically
even in a disastrous lesson. In lesson
study, the quick feedback from and to the
teacher is very powerful. Advisors can
ask "Why (li(1 you write in that place on
the blackboard?" A brief record of the
entire lesson should be on the black-
boar(l. The teacher evaluates the advisor
and does not invite an advisor back if the
comments are not deep.
How are teachers taught to he good
observers during lesson study?
RESPONSE
Observation skills for live observation
are very important. Teachers need to
know how to observe well. When the
teachers are students and first visit
classes, they cannot even take notes-
they do not know what to look at. They
are taught to look at the relationship
between the class purpose for that day
and what happens in the class. They
study one student, sometimes standing
beside them to observe all they do, to
(letermine whether the stu(lent's actions
reflect the purpose of the lesson. Obser-
vation is from the teacher's side of the
classroom, not the back of the classroom,
in order to view the students' faces.
Observers consi(ler what the teacher
asks, discriminating one question from
another, anti (1iscerning the teacher's
moves. Itis(lifficultto teach teachers to
observe well. A goo(1 advisor is nee(le(l.
SMALL GROUP DISCUSSION
Representative terms from entire chapter:
professional development
