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OCR for page 19

r. rN ~4
1 - 1 ~ ~ ~
The opening session of the workshop included presentations designed to give partici-
pants some sense of mathematics education and teacher preparation and development
programs in the two countries. Deborah Schifter and Zalman Usiskin gave a broad
overview of elementary and secondary teacher preparation and development programs
in the United States. Keiichi Shigematsu and Keiko Hino described a case study of a
middle-grades teacher who had taken a part in a graduate teacher education program in
one Japanese university, while Toru Handa and Mamoru Takezawa spoke of other
examples of professional development efforts in Japan at the secondary level.
Elementary Mathematics Education in the United States
Deborah Schifter, Senior Scientist, Education Development Center
Mathematics Teacher Education in Grades 7-12 in the United States
Zalman ~ Usiskin, Professor of Education and Director of the University of Chicago
School Mathematics Project, University of Chicago
A Study of Teacher Change Through Inservice Mathematics Education Programs in
Graduate School
Keiko Hino, Associate Professor, Nara University of Education
Keiichi Shigematsu, Professor, Nara University of Education
Recurrent Education in Japan: Waseda University Education Research and Development
Center
Toru Handa, Mathematics Teacher, Waseda University Honjo Senior High School
Recurrent Education in Japan: Kanagawa Prefectural Education Center
Mamoru Takezawa, Mathematics Educator, Kanagawa Prefectural Education Center

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:~ ~~ -
Deborah Schifter, Education Development Center
~ have been asked to provide some
background for this workshop on
elementary-level mathematics instruction
in the United States. ~ wall be touching on
four main themes: (~) the tradition of local
control of education policy in the United
States, (2) the fragmentation of teacher
education, (3) current efforts to improve
education, and (4) challenges to this
improvement.
WHO CONTROLS U.S. EDUCATION?
First, and this is not specific to elemen-
tary education, is the question of who
controls U.S. education policy. The
federal government has limited power
over our schools; it does not establish a
national curriculum. Nor is teacher
education or assessment a matter of
national policy. Instead, each of the 50
states has its own policies in such matters
as teacher certification and curriculum,
and in some states, curriculum decisions
are actually left to local school districts or
even individual schools. In spite of the
extreme fragmentation of authority over
U.S. education, there is considerable
uniformity in what is taught largely
attributable to the way textbooks are
produced, marketed, and adopted.
A second mechanism for ensuring a
degree of uniformity is the concern that
students are prepared for college entrance
exams. Although this (Erectly affects only
secondary school instruction, this concern
exerts indirect pressure on instruction in
the lower grades. With such decentral-
ized control over education policy, along
with the tendency of all institutions to
maintain the status quo, it is (difficult to
make fundamental changes in our schools.
However, two arms of the federal govern-
ment, the National Science Foundation
and the U.S. Department of Education, do
provide grants for projects that appear to
promise improvement in e(lucational
practice. Through this mechanism, the
federal government can exert some
influence on policy (recisions.
FRAGMENTATION OF TEACHER
EDUCATION
~ now consi(ler fragmentation of teacher
education and, specifically, implications at
the elementary level. In the Unite(1
States, it is wi(lely believe(1 that a stable
school environment better serves young
children. This is accomplishe(1 by keeping

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one teacher with the same class for the
full school day. In most schools, teachers
are responsible for teaching all subjects to
their students mathematics, reading and
writing, science, and social studies. From
the background reading for this workshop
(Appendix L), ~ understand this situation
is the same as in Japan. In U.S. un(ler-
graduate teacher preparation and
inservice professional development
programs, there is pressure to learn about
instruction in all of these subjects. As a
consequence, most teacher preparation
programs determined by state certifica-
tion requirements generally offer one or
two courses in the teaching of mathematics
(National Center for Education Statistics,
1995; America Federation of Teachers,
1997; Ingersoll, 19981. Some also require
that prospective elementary teachers take
at least one mathematics course, while
others have no such requirement. On the
other hand, there are some states like
Georgia that now require undergraduate
students to take several mathematics
courses to become certified as elementary
teachers. However, it is unusual for the
mathematics courses to be coordinated
with the education programs at the
· ~
university.
In many states, teachers must spen(1 a
certain number of hours in professional
development settings to maintain their
certification. However, once again, there
is no uniform system of professional
development. Some workshops are
organized by the school or the school
system, and these may be led by district
personnel or by consultants who are hired
from outside to work with teachers for a
single (lay. The content might involve a
new teaching technique or assessment
procedure, student behavior or some such
social issue, classroom (liscipline, or time
management. They rarely (leal with
mathematics or how chil(lren learn (Garet
et al., 1999; Shields et al., 1999~. It is
unusual for teachers to come together
regularly to (liscuss substantive or prob-
lematic issues in their practice although
there are exceptions. Some teachers on
their own initiative seek out courses at
universities or summer programs, which
again vary greatly from one another
(McLaughlin, 20001. And there are some
schools or school districts that have
organized and coordinated substantive,
professional development programs.
CURRENT EFFORT TO IMPROVE
ELEMENTARY EDUCATION
Perhaps the most significant point in
our discussion at this workshop is that
until recently, the principal goal of elemen-
tary mathematics instruction in the
United States has been computational
proficiency (National Council of Teachers
of Mathematics, [NCI~M] 19891. The
emphasis has been on remembering facts
and algorithms and being able to produce
correct answers with speed. The curricu-
lum also inclu(le(1 learning the names of
particular shapes, the use of a ruler, the
formulas for the area anti perimeter of a
rectangle, and sometimes the volume of a
rectangular solid. Like the rest of the
mathematics content, word problems have
been treated mechanically.
This emphasis on memorization rather
than sense-making activity is evident in
such errors commonly seen in the
elementary classrooms as shown in
Box I. In these examples, the chil(lren
are applying single-digit math facts but
are not remembering the computational
procedures. Because they are not think-
ing about the size of the numbers they
start with or what the operations (lo, they
form no reasonable estimate of the
outcomes. If neither the chil(lren nor
BACKGROUND CONTEXT FOR TEACHER PREPARATION

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26
+58
714
42
- 29
27
54
x23
162
+103
270
their teachers have learned to approach
such problems with the expectation that
they should make sense, it is difficult to
correct the misconceptions at the base of
these errors.
This brings us to efforts to improve
elementary mathematics in the United
States. For many years, going back at
least as far as the work of John Dewey,
there have been educators devoted to
rethinking K-12 mathematics, designing
materials, working with teachers, and
producing policy statements. The most
dramatic recent change came just a
decade or so ago when some of these
people came together under the auspices
of the NCrM to produce a set of stan-
dards documents. NCrM is a profes-
sional organization of mathematics
educators whose work is mainly conducted
by volunteers and supported by a full-time
staff.
The first three NCrM documents set
out standards for curriculum, teaching,
and assessment respectively: Curriculum
and Evaluation Standards for School
Mathematics,1989; Professional Standards
for Teaching Mathematics, 1991; Assess-
ment Standards for School Mathematics,
1995. The fourth, Principles and Stan-
dards for School Mathematics, 2000, was
E LEME NTARY MATH EMATICS E DUCATION
intended to update, refine, and elaborate
on the issues presented in the first three.
The main principles guiding these docu-
ments are that mathematics is about
reasoning; that children be recognized as
mathematical thinkers; and that eliciting,
assessing, and buil(ling upon their think-
ing should be at the heart of instructional
practice. In particular, the documents
argued that children should be encour-
aged to use a variety of methods for
representing and solving problems and
then present their work to their ciass-
mates for further analysis. These docu-
meets also emphasize that mathematics is
more than arithmetic. Geometry and data
should be made significant components of
the curriculum beginning in kindergarten.
The fe(leral government (foes not set a
national curriculum. These stan(lar(ls are
offered as recommendations, without any
requirement that people should follow
them. However, at the federal level, the
National Science Foundation and the U.S.
Department of Education have supported
these reforms. They have funded research
centers anti professional (1evelopment
programs anti provi(le(1 grants to produce
curriculum materials at the elementary,
middle, and secondary levels. These
materials started to become available
around 1996 anti 1997.
To provide an idea of what the stan-
dards support and the kinds of activities
inclu(le(1 in the new curricula, ~ offer two
examples taken from the newest NCrM
document (2000~. A vignette is presented
of a fifth-gra(le class that ha(1 been given
the homework problem 728 divided by 34.
One chil(l, Henry, presente(1 his solution
metho(1 (Box 2~. Henry explaine(1 to the
class, "twenty 34s plus one more is 21.
~ knew ~ was pretty close. ~ (li(ln't think
coul(1 a(l(1 anymore 34s, so I subtracted
714 from 72S, and got 14. Then I had 21
remainder 14."

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34 X 10 = 340
34 X 20 = 680
680
+34
714
728
-714
14
SOURCE: Aclaptecl from NCTM, 2000,
p. 153.
Another child, Michaela, presented her
solution (Box 3~. Michaela described the
steps of the conventional division aigo-
rithm. "34 goes into 72 two times, and
that's 68. You've got to minus that, bring
down the 8, and then 34 goes into 48 one
time."
Apparently, their teacher had not shown
the conventional division algorithm to her
students, and Michaela's classmates said
they did not understand her method.
Asked to explain, Michaela took the class
through the steps again but with the same
response. Then the teacher asked the
class to identify the similarities in the two
procedures and assisted them by insert-
ing a 0 so that the chil(lren coul(1 more
easily see where Henry's 680 shows up in
Michaela's process. Through the discus-
sion that followed, using Henry's solution
as a point of reference, some of Michaela's
classmates could begin to see the logic of
the steps she hall taken.
Another example is also taken from
Principles and Standards for School
Mathematics. In this fifth-gra(le class, the
students had been given a word problem
that involved adding 1.14, 0.089 and 0.3.
They were asked to work in groups to
come up with an answer. Although they
had done preparatory work on decimals,
a(l(ling them was a new topic. One group
presented this solution (Box 4~.
They explaine(1 that you change all the
(lecimals into regular numbers. Then you
a(l(1 them all up anti get 206. When
challenged by their teacher to consider
whether a number that size makes sense,
they restored the decimal point. And their
final answer was 2.06.
The second group, which had done the
calculation correctly, explaine(l, '~he
reason we (li(ln't line up all the numbers
was because we had to line up tenths with
tenths, and the hundredths with the
21
34 -
-68
48
-34
14
SOURCE: Aclaptecl from NCTM, 2000,
p. 15~.
114
89
+ 3
206
SOURCE: Adapted from NCTM, 2000,
p. 195.
BACKGROUND CONTEXT FOR TEACHER PREPARATION

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hundredths to make it come out right,
and," they added, "the thousandths with
the thousandths." However, when the
class came together to discuss their
solution methods, neither group was
convinced by the other's explanation.
Both answers could not be correct, but as
a class, they could not resolve the stale-
mate before the lesson ended that day.
The next day, the teacher suggested
that some of the students work on this
problem with base 10 blocks. Their
representation is shown in Box 5. The
children in the class were convinced by
this demonstration. Not only did it help
them determine that 1.529 is indeed the
correct answer, but it also helped them
interpret the argument the second group
had given you add tenths with tenths,
etc.
These examples illustrate the spirit of
the mathematics education reforms in the
United States. In the past, accepted
practice would have required the children
to memorize the steps of the division
algorithm as presented by Michaela, with
no attempt to explore the reasoning
behind it; or in the past, children would
have had to have memorized two different
rules for adding numbers: For whole
numbers, line them up at the right, but for
decimals, line them up at the decimal
point. In the vignettes, we see children
sorting out the logic of the calculation
procedures. It is noteworthy that in both
examples, the children have been given
responsibility for explaining their reason-
ing to themselves and to their classmates.
Their teacher posed a question or offered
a suggestion that helped them find the
sense in the procedures they were learning.
CHALLENGES TO THE
IMPROVEMENT OF MATHEMATICS
EDUCATION
One of the most difficult issues that the
United States faces in improving elemen-
tary mathematics education is that most
~ ~ ~ ~~ \
1.14
0.089
0.3
Total
SOURCE: Schifter et al., 1999, p. 1 18.
E LEME NTARY MATH EMATICS E DUCATION
1.529

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teachers do not understand the mathe-
matics they teach very well (Ball and
Wilson, 1990; Ma, 1999; National Center
for Education Statistics, 1995; National
Research Council, 2001~. In the past 15
years, there have been many studies that
have documented this problem, which has
become an increasing concern to many
U.S. educators and policy makers. In my
first example about division, the teacher
herself understood the comparable reason-
ing behind the two methods presented by
her students, and so she could draw their
attention to it. In the second, the teacher
understood the principle that underlies
addition of whole numbers and decimals
and so could suggest a representation that
would help her students see it too. How-
ever, such teachers are, at this time, more
the exception than the rule in the United
States. Most, having been educated in
mathematics restricted to memorization of
procedures, have not learned how to
make sense of mathematics. In general,
U.S. teachers have not had opportunities
to learn mathematics from a conceptual
perspective.
The examples ~ have given come from
the most recent Standards document
published in spring 2000. When the first
Standards document appeared in 1989,
there were few instances of curriculum
materials that laid out a coherent math-
ematics program. By the early 199Os, it
became apparent there were serious
problems with the way teachers were
interpreting the Standards documents.
These were largely rooted in a failure to
appreciate that the core idea of the reform
was making sense of mathematics. For
example, one interpretation held that the
standards were about teaching essentially
the same mathematics content, but now
the children were sometimes to work in
small groups, or were to use manipulatives,
as well as paper and pencil (Weiss et al.,
1994~. So there were many professional
development workshops devoted to these
teaching strategies. However, when the
children were put into groups, they were
not necessarily given problems that
required real thinking. And when they
were given blocks to use, they were
merely shown yet another set of steps to
remember.
Another interpretation was that the new
practice was to emphasize problem
solving. So some teachers committed one
day each week to it. Although many of
these problems did require some hard
thinking, they were not necessarily
related to a coherent conception of the
content that the children needed to learn.
In one very disturbing interpretation of
the Standards, some teachers agreed on
the importance of eliciting ideas from
their students but did not understand that
they had a further responsibility to
critically analyze those ideas for math-
ematical soundness. Indeed, having
themselves been taught that mathematics
is memorization, many teachers have
never developed the skills required for
assessing the logic of a mathematical
argument. Nor do they even realize that
this is something that they, much less
their students, should be doing.
There was some research in the early
199Os that alerted us to these problems of
practice in the name of the Standards
(NCrM, 19911. And there were some
programs for college students preparing
to become teachers, and for teachers
already practicing, working to address
them. As a result of these programs, and
with support from new curriculum
materials, there are now a greater number
~ . . . . . ..
ot classrooms in WhlCh the new vision of
mathematics teaching practice has
successfully taken hold (Briars and
Resnick, 20001.
BACKGROUND CONTEXT FOR TEACHER PREPARATION

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However, since 1995, a movement has
emerged to put a stop to these reform
efforts. This movement has arisen out of
several concerns. Some people became
justifiably alarmed when they saw some of
the problematic practices ~ have just
described. And even where the mathe-
matics was being taught wed, some parents
were made uneasy by the unfamiliar look
of the mathematics their children were
bringing home. Others felt that the new
practices would interfere with children
learning basic arithmetic and algebraic
skills, without which they believe math-
ematical understanding cannot develop.
Still others were distrustful of teachers
who now aspire to a more complex,
intellectually, and demanding practice. In
any case, some reformers now confront
groups of parents supported by some
professional mathematicians and backed
by wealthy funding sources who have
succeeded in blocking local and state
reform efforts (Battista, 1999; Becker and
Jacobs, 2000; Grossman and Stoduisky,
1995~. Although this conflict has made its
way into the media, society at large-
including the journalists covering the
conflict is unaware that the questions
concerning how learning takes place or
how children put mathematics concepts
together are complex and profound.
The one mathematical specific that
seems to come up regularly in the debate
is the role of conventional computation
algorithms. In the public arena it is posed
baldly: Should teachers encourage children
to devise procedures that make sense to
them for example, to add 45 + 36,
children will often find the sum of 40 and
30 first, then add 5 and 6, and then total
the partial sums: 70 + Il = 81 or should
teachers demonstrate conventional
algorithms? As my earlier example
suggested, the dichotomy is a false one.
E LEME NTARY MATH EMATICS E DUCATION
Nonstandard procedures can render the
meanings of the operations transparent
and the reasoning behind the standard
algorithms accessible. However, as the
debates get played out in public, sensible
and deep discussion of such issues
becomes increasingly (lifficult.
To summarize the context in which we
are working to educate teachers and
support their professional development,
many teachers have yet to be introduced
to the idea that mathematics is about
reasoning, that children, anti they them-
selves, have mathematical ideas and the
capacity to explore and critically assess
them. Once they come to appreciate that
doing mathematics is a sense-making
activity, there is much mathematics
content for them to learn. Then they must
create a teaching practice that is (lifferent
from the practice they know, that helps
their students understand the mathematics
too. Finally, they must do all this while
keeping up with comparable (levelop-
ments in the teaching of literacy, science,
and social studies. And yet there exists
no coherent teacher education and
professional development infrastructure
to support these efforts.
Furthermore, we are trying to do this
work in a period in which basic directions
in education policy are the subject of
contentious political (rebate. Wanting to
(lo the best for their students, teachers are
caught in the middle, attacked for their
students' failures, hel(1 accountable for an
impossibly difficult task. However, as ~
anti my colleagues have (liscovere(l, once
teachers begin to learn mathematics in
settings that support the (levelopment of
their own powers of reasoning, they
become eager to learn more mathematics
for themselves and to provide their
students with opportunities they missed
out on as children.

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~ ~ ~ ~ ~ 1 ~
Irk
row
51e'
1~.1
r ram I
~T~ ~
Zalman I? Usiskin, University of Chicago
Good afternoon. It is a pleasure and an
honor to present this overview of second-
ary mathematics teacher education in the
United States.
At least since the 1960s, when the
results of the International Study of
Mathematics were announced and Japan
scored at the top and the United States at
the bottom, Japan has been seen as a
place to which we might turn to improve
mathematics education in the United
States. In the early l980s, at the University
of Chicago, we ha(1 the opportunity to
translate Soviet and Japanese high school
texts, and we were able to see the algebra
and geometry taught in junior high school
and how virtually an entire population can
be brought to higher mathematical
performance. The recent third Inter-
national Mathematics and Sciences Study
suggests that we may be able to learn not
only how to raise student performance but
also instructional techniques from Japan
(National Center for Education Statistics
[NCES], 1996).
In turn, the size of the United States
and the length of time we have been
working to improve mathematics education
has led to numerous studies and projects
in the United States. We hope that some
of the ideas we have been working on wail
be informative and stimulating, and we
look forward to sharing them.
In the workshop materials, we presented
data regarding mathematics teacher
education in the Unite(1 States. These are
pages 31-34 of a longer (1ocument that
was distributed at the Ninth International
Congress on Mathematics Education
(Dosseyan(lUsiskin, 20001. Rather than
repeating what is there, ~ will elaborate on
the picture the (lata present with regar(1 to
mathematics teacher education for those
who teach mathematics in grades 7-12.
With respect to the first two questions
consi(lere(1 in Deborah Schifter's remarks,
the situation is the same at gra(les 7-12 as
at grades K-6. Decisions are left to states
anti schools, anti education is fragmented.
To un(lerstan(1 the picture of mathe-
matics teacher education, it helps to have
knowle(lge of the mathematics curriculum
at the secondary school level. There has
never been a national curriculum in the
United States. But at a given time, most
schools follow much the same curriculum.
This is because, among the most-use(1
textbooks, there is a strong tendency to
have the same content anti approach.
~ begin with a typical curriculum of 50
years ago for high school anti the first
year of college for well-prepare(1 students

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(Table I). At that time, algebra and
geometry were avoided almost entirely
until ninth and tenth grades. Only a
minority of students took mathematics
beyond tenth grade (National Advisory
Committee on Mathematical Education
[NACOME], 1975~.
The most-used and most influential of
the new math curricula, the School
Mathematics Study Group (SMSG)
curriculum, and Me demands for increased
performance in mathematics and science
modified the 1950s curriculum in a
number of ways. Some content was moved
into lower grades, and other content was
compressed and integrated. Also, many
schools maintained a second curriculum
for slower students. As a result, by 1975 a
typical curriculum was bifurcated as
shown in Table 2 (NACOME, 1975~.
In this 1970s curriculum, there was
some algebra and geometry in eighth-
TABLE 1 Typical Curriculum for High School anc! First Year of College, 1950
Grade
Content
7th
8th
9th
1 0th
11th
12th
College
Arithmetic
Arithmetic
First-year algebra
Geometry
Seconcl-year algebra
Solicl geometry; trigonometry with logarithms
Analytic geometry; first-semester calculus
SOURCE: NACOME, 1 975.
TABLE 2 Typical Curriculum, 1975
Grade Most Students Slower Students
7th
8th
9th
1 0th
11th
12th
College
7th-gracle mathematics
8th-gracle mathematics
First-year algebra
Geometry
Seconcl-year algebra with functions
Functions and trigonometry
Calculus
7th-gracle mathematics
8th-gracle mathematics
General mathematics (arithmetic)
Consumer mathematics (financial arithmetic)
SOURCE: NACOME, 1 975.
M AT H E M AT I C S T E A C H E R E D U C AT I O N I N G R A D E S 7- 1 2

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FIGURE 1 Framework of inservice mathematics education program at the graduate school of the Nara
University of Education.
Enrichment
in Specialty Area ~ ~ Improvement
of Practice
Information
Exchange
| of Practice |
Enrichment in | Improvement | Self-Expre';sion 1 Improvement T Informal
Specialty Area of Practice of Practice Exchange
Lectures and practices in Reflection Presentations at Reflection Attending study meetings
graduate school seminars
Alternative view Alternative view Visiting schools and
The acquirement of requisite Presentations at study institutions
knowledge for research Lesson analysis groups Lesson analysis
Attending conferences
u, Specifying the research Lesson Presentations at Lesson
.~ theme, reviewing the organization conferences organ~zabon
preceding research, planning,
preliminary research Contribution to the
university bulletin
Plan and practice of research Reflection Research towards Reflection Attending study meetings
master's thesis
Getting results in shape Alternative view Alternative view Visiting schools and
Writing thesis institutions
Examining conclusions, Lesson analysis Lesson analysis
o suggestions, and tasks for the Submitting thesis Attending conferences
~ future Lesson Lesson
u, organization Presentation and organization
contribution to the
university bull Tin
schools, not only in our prefecture but the
neighboring prefectures and teachers in
those prefectures. Through such ex-
change and participation in academic
meetings, teachers can further cultivate
their professional skills.
Figure ~ does not show lesson study,
but teachers aggressively participate in a
lesson stu(ly program. In the two-year
program, the first year takes place at the
graduate school level. However, in the
second year teachers go back to their
original school for further training or
further education.
With respect to the second purpose of
our research characterizing the school
teacher's change, we found that a teacher's
growth in mathematics takes place in four
phases. Phase one is consideration of
their own teaching practice; teachers have
to be aware of the framework for their
own teaching practice. In other words,
they have to be aware of the issues in
their own view of their teaching. Unless
they are aware of their problems or
issues, it is very (difficult to motivate them
to improve their teaching ability. Second,
a clear un(lerstan(ling of their issues,
BACKGROUND CONTEXT FOR TEACHER PREPARATION

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however, wall not be sufficient to move to
the next step. They need to think about
what they should do. So they must search
for another implicit framework (phase
two). In phase three, that framework is
internalized, and finally through that
process an improved framework should
be developed (phase four). These four
phases show that improvement in profes-
sional ability, for example, having more
students come to enjoy mathematics,
happens only when teachers construct
their own framework for mathematics
teaching. An explicit diagram is advanta-
geous (Box I). In our research we found
that if a teacher can notice improvement,
the teacher wail continue to further
improve his or her professional ability.
Now we go into more detail about our
research concerning the teacher's
r: =-1-1-1~i-~~
Based on the result, Mr. A's change was characterized by interviews conducted three times during and
after the participation in the program.
Activities
Reviewing recorded materials on his lessons
Taking lectures in graduate schools
Having discussions with various people
Phase 1 Confrontation with Mr. A's implicit view of mathematics teaching
1
Taking lectures in graduate school
Having discussions with various people
Teaching practice in the club activity in his school
| Phase 2 Seeking alterrnative views of mathematics teaching l
Lesson practices along with successive observation of
experienced mathematics teacher's class
Having discussions with various people
Construction of the framework in his master's thesis study
Presentation of the framework in meetings and conferences
Phase 3 Concretizing the alternative view by querying the ideal of mathematics teaching and
key words for approaching this
.
Master's thesis study
(classroom teaching practice)
Teaching practice in an "optional mathematics class" in this
school
Phase 4 Construction of Mr. A's own framework for mathematics teaching
A STUDY OF TEACHER CHANGE

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change. Improvements in the teacher's
teaching practice has been mentioned
several times, but what do we mean by
teaching practice? There are three
perspectives from which we looked at the
teaching practice (Box 2~. One is the
class program itself. In other words,
preparing for the class organizing the
class is one practice. Another is during
the class, the kind of practice the teacher
carries out. And the third aspect is after
the class, making improvements in the
practice the teacher has just finished. In
other words, prelesson, midlesson, and
postlesson are the three different aspects
pertaining to the teaching practice.
In our study we surveyed teachers,
interviewed teachers, and collected class
observation cards from the participants of
the class. Using these different methods,
we tried to analyze the improvement in
these three different levels of teaching
practice of the teacher. Mr. A is a junior
high school teacher, who admitted that he
had very little interest in mathematics
education (Box 3~. He was more inter-
ested in guiding his students in the
extracurricular activities of rugby games.
After 15 years of experience, Mr. A
wanted to improve his skills and his
performance as a school teacher. How
slid his teaching practice actually change?
The improvement of Mr. A's teaching
practice and his professional development
was initially assessed by observing his
class before he took this graduate course.
Then we went back to his class, after he
took this two-year course at the graduate
school. The contents of the two classes
that were observed were almost identical.
We tried to compare what we noticed had
changed over the two-year period (Box 31.
We analyzed these observations from
three perspectives. First, we analyzed
~ _
1. To characterize a school teacher's change through participation in mathematics
education program at the graduate level.
2. To assess the change of the teacher from the perspective of teaching practice in their
mathematics class.
Mathematics teachers' teaching practices are construed as the ability that enables
teachers to conduct activities such as:
· Prelesson: To organize and transform content knowledge along with the purpose. To
have the "eyes" to evaluate the result of such a process.
· Midlesson: To execute the plan. To create activities that lead to the goal, taking notice
of a student's situation all the time, sometimes by applying various routines and other
times by inventing them promptly.
· Postlesson: To reflect on the lesson using as a basis information about the attainment
of a teacher's goal. To work out instructions concretely for the next lesson.
BACKGROUND CONTEXT FOR TEACHER PREPARATION

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The teaching practice of a teacher was assessed twice, before and after the participa-
tion in the graduate school program offered by Nara University of Education.
Mr. A;'s Profile
· A junior high school teacher working at a public secondary school.
· In addition to teaching mathematics, he arranges student club activities every Jay
aher school and also on the weekend.
· He does not participate in any mathematical study groups.
· "I have been teaching for 15 years. To keep up the times, I want to study how to
make use computers toward mathematics education in new ages. I also need to
develop my teaching practice in class through this program. When I return to my
school I need to show my improved teaching practice to my colleagues./'
Lesson Observation Dates ant! Participants
Before: February 17, 1998, at a junior high school first-graders' class (seventh-grade)
and a seconJ-graJers' (eighth-grade) class.
· Observers: 5 persons (2 university instructors, 1 high school mathematics teacher, 2
graduate students).
· Target students: 6 first graders and 6 second graders.
Alder: February 15, 2000, at a junior high school first-graders' class.
· Observers: 5 persons (2 university instructors, 1 high school mathematics teacher, 2
graduate students).
· Target students: 5 first graders.
Data Collection
Pre/esson
· To teacher: interview questions on mathematics and ~earning.
· To target students: interview on their belief and interest of mathematics.
Lesson Observation
· Recording of the lesson by videotape recorder and microphones.
· Observation by a script form from two aspects: the flow of lesson and students'
activities.
Post/esson
· To teacher: interview on se~f-assessment of the lesson.
· To target students: interview on their interest and motivation toward the content they
learned in the lesson.
A STUDY OF TEACHER CHANGE

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an
ad 1.0-
~5
O 0.8-
Q
In
an
o
o
an
Tar
0.6 -
0.4-
0.2 -
o-
from the observer's viewpoint. Observers
were fellow graduate students, the teach-
ers who had already finished the graduate
school program, and ourselves, the
researchers.
We made three findings. We did not
see any dramatic change; however, Mr. A's
attitude with respect to the students had
clearly changed. There are some specific
data to indicate this change in attitude,
which is shown in Figure 2. This figure
shows the rate of correspondence among
different observers before, during, and
after this graduate course.
Before Mr. A took the graduate course,
most observers agreed that Mr. A was just
teaching with a one-way method, teacher
to students. This is shown by the overall
high rate of correspondence among the
observers. After the two-year course, this
approach changed dramatically to become
more interactive (Box 41. Notice the
decrease in the rate of correspondence in
most items. Mr. A's teacher-(lirecte(1 way
of teaching ease(1 a little. That was a
unanimous observation.
The second perspective is the observa-
tion made by the students themselves
(Table I). Here again, (lepen(lence on
textbooks has change(l. (In a (lifferent
research study, we found that the Japanese
teachers who do not like mathematics
would inevitably depend on the textbooks
in teaching the mathematics classes.~.
But as you can see, after this two-year
course, Mr. A (lepen(le(1 less on textbooks,
which meant that he was trying to cater to
the actual needs of the students in his
class, and students noticed the difference.
An(1 the thir(1 perspective is the self-
assessment by Mr. A (Table 2~. Through-
out the graduate course, Mr. A learned
that what is most important is for the
students to learn mathematics, not for the
teacher to learn mathematics. What is
most important in organizing the math-
FIGURE 2 In the eyes of the observers.
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1—1 1—1 1 1
1 4 7 10 13 16 19 22 25 28
Item
Before
After
NOTES:
Mr. A's teacher-directed ways of teaching eased a little. Communication with students was brought into the
class. The time for students' activities was put in the lesson. Rate of correspondence among the observers on
each item, or what was the percent of observers who agreed that the item was oberved in the lesson. (1)
items No. 1 to 14 concern of the teacher, (2) items No. 16 and 17 concern of the classroom atmosphere and
(3) items No. 19 to 28 concern students in the class.
BACKGROUND CONTEXT FOR TEACHER PREPARATION

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~ or..
s~.~.~.~.I~.1881.'.r.~el~_
Before 1998 ~
The teacher does not suggest students try various approaches to solve a problem.
The teacher does not relate mathematics with other subjects or daily life.
The teacher Joes not instruct to develop individually.
The teacher teaches the content exactly as it is in the textbook.
The teacher gives a teacher-centered telling of the lesson.
Many students do not volunteer to state their opinions.
Many students do not think in various ways.
None of the students in the class chat among themselves.
All of the students in the class put their textbooks and notebooks on their desks.
After 2000
· The teacher doesn't relate mathematics with other subjects or daily life.
· Communication with students is seen.
· Communication among students is not seen.
· Some students in the class have private charter.
· Many students tackle to compute, J raw figures and graphs, or use compasses and
rulers.
TABLE 1 In the Eyes of Students
Items Recognized By Students
Before (°/0) After (°/0)
Many of his lessons go with textbooks
Materials are ohen used in his lessons
I ask him questions frequently
Pace of his lessons seems fast
Frequently students attempt to first solve the problem by themselves
We ohen have a discussion
Two teachers have taught me a lesson
I write the lesson summary with my own words
100
o
67
83
100
17
o
25
80
20
~0
20
100
60
o
60
NOTES:
Fewer students feel that the pace of his lessons is fast. More students think they have discussions in the class.
A STUDY OF TEACHER CHANGE

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TABLE 2 In the Eyes of the Teacher
Before 1998
After 2000
First Year Second Year First Year
of Junior of Junior of Junior
Items for the Interview High School High School High School
Aims of the lesson To understand To understand To promote
how to construct the Miclpoint stuclents'
figures. Connection exploratory
To learn how to theorem and activities.
use the compass. to use it.
Special attention To make all To make Tovaluestuclents'
students learn students voluntary statement.
the proper use understand Not to force teachers'
of a compass. the theorem. views.
To give comments on
contents that require
direct explanations.
Diclyou value stuclents' cliscussion? No No Yes
Dicl you arrange time for stuclents No No Yes
to solve problems by themselves?
Diclyou giveteacher-centerecl lesson? Yes Yes No
Dicl you communicate with the Sometimes Sometimes Yes
stuclents?
Dicl you consider connecting the No No No
subject with claily life examples?
Dicl you con lecture possible Yes Yes Yes
questions the stuclents may have
prior to the lesson?
Dicl you make use of wrong answers No No Yes
proviclecl by stuclents?
NOTES:
He gave a better evaluation to his "aher/' lesson than "before/' lesson. The aims of the lesson focused on
stuclents' thinking and attitucle. Special attention to teaching instructions can be seen in the "aher/' lesson,
while they were restricted solely to the teaching contents in the "before/' lesson.
BACKGROUND CONTEXT FOR TEACHER PREPARATION

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emetics class is to make sure that the
students learn, that the class is more
student focused, and student oriented.
And through his self-assessment and
through interviews, we can see that he
has actually learned that difference. The
biggest change within Mr. A was that he
learned that he needs to communicate
with the students (Box 51. He learned
that what is often termed the mathemati-
cal activities of the students need to be
supported. And to do this, you have to
provide enough time for the students to
engage in mathematical activities. Mr. A
learned that as well. As we saw at the
beginning of this discussion, these
changes did not take place at once.
Rather, the changes were supported by
Mr. A's long-term investigation into his
own framework of mathematics classroom
practice characterized by the four phases
of growth.
While this is a case study involving just
one teacher, we are continuing our
research involving elementary school
teachers and senior high school teachers.
Our preliminary findings indicate that we
can see similar improvements in teaching
practices and professional development
based on the fact that the teachers need to
have an open min(l, identify and recognize
their shortcomings and areas for improve-
ment and be walling to try to cope with the
changes. The responsibility of the gradu-
ate school is to help teachers tackle those
issues and challenges.
Conclusions
· Change in the teacher's teaching practice in mathematics class was mainly observed
in two aspects.
The teacher came to communicate with his students with respect to mathematics.
The teacher came to have more time for the students to work on his or her own.
· The emergence of this "pedagogical reasoning" was supported by his long-term
investigation into his own framework of mathematics classroom practice. The process
of investigation was characterized by four phases.
· The graduate program influenced his change by providing opportunities in observing,
practicing, and discussing mathematics classes in junior high school and in planning
and conducting a research study for the purpose of writing a master's thesis.
Tasks for the Future
· By accumulating the information from more cases, the process of teacher change
needs to be clarified further.
· Construction and modification of the classes and the program that enhance the
smooth shift along the four phases need to be investigated from different angles.
A STUDY OF TEACHER CHANGE

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COG
EM
~ ~ ~ · _a—_r ~
C.l,{.~.
my
Toru Handa, Waseda University, Honjo Senior High School
Waseda University has an annual
summer seminar for inservice teachers at
what we call recurrent education centers.
In 1996, Waseda University Education
Research and Development Center
launched this program. Since then, four
or five days in August are dedicated to
this seminar, with about ten courses. The
basic course is for beginning users of
personal computers. There is a class
management and an attainment manage-
ment course; a course of English with
personal computers; a course of Japanese
language with personal computers;
courses in mathematics, social studies,
anti sciences; a course to make computers;
and a course on networks in schools. This
seminar is for inservice teachers. We now
have information technology everywhere,
but teachers need to use the technology
and information media to make their
classes more interesting and easy to
understand by students. Therefore, for
each subject, teachers must first improve
their computer literacy; however, Japanese
teachers are busy in daily routines. From
early in the morning to late in the evening,
they have to manage extracurricular
courses. Before classes anti after classes,
there are teachers' meetings. So they (lo
not have enough time for studying infor-
mation technology.
So how can teachers change their
classroom teaching to use technology anti
the information me(lia? First, how can
they get information that will be useful in
their teaching? How can they introduce
the technologies and information me(lia
into the classrooms? How can they create
self-ma(le successful classes? To achieve
these things, they need the basic knowI-
edge and expertise for the new technolo-
gies and information processing media.
Specifically, some of my remarks focus
on how to improve and change the
mathematics classes. The two mathematics
courses I selected are an introduction to
MATHEMATICA, a software package that
processes numerical formulas and does
mathematical integration, and how to
make a web page. A web page is a future
textbook, and that means the teacher
must be able to use HTML.
On the first day of the seminar, basic
MATH EMATICA and the basics of HTML
are presented exclusively for the real
beginners. So these are really the basics.
And on the second and third days they
use MATH EMATICA to process formulas
and to enjoy the program. At the end of

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the third day they learn about software
other than MATHEMATICA. On the
fourth day they create a web page using
MATH EMATICA and then develop the
teaching materials for their own classes.
Finally they make presentations and have
a discussion about their work.
Word processing software in Japan for
information technology literacy is Excel
and Ajitalo. Our participants vary in skill
level. We had one person who had never
touched the keyboard of a computer, 72
percent of the participants didn't know
about MATH EMATICA at all, and about
83 percent had no web page experiences.
Despite this, the participants finish the
seminar with some achievements. The
computer beginners in last year's seminar
created a spiral structure of shells and
used formulas to make simulations. By
changing the value of a constant, one
participant created different types of
shells. Some showed a tangent of the
function as an example of simulation. Last
year we had a three-day seminar, and after
completing the three days, participants
could create such simulations.
Finally, ~ would like to address some of
the problems or the challenges that we
have now. The budget for the education
center is limited. We do not receive many
grants, so the participants have to pay
approximately $100 for four (lays. Maybe
this is too expensive. A second problem is
that the number of trainee teachers does
not increase in the way we would like to
see. There could be several reasons.
First we do not have a preparation initia-
tive tied to promotion. In Japan this kind
of recurrent training is not so popular, and
it is not generally accepted. Also in Japan,
once teachers get a teaching certificate,
they can stay there without studying or
learning anything new until retirement
age. In addition, even during the summer
holidays, Japanese teachers are very busy
because they have to take care of extra-
curricular activities. Teachers also have
to ask for permission of the principal or
head teachers to participate in this kind of
a seminar, which introduces all sorts of
procedures. Another possible reason is
that there are some people who take this
course every year, many of whom are
very experienced teachers. We probably
need to prepare several different levels of
the courses anti a follow-up service. We
have to develop the mailing list of the
participants anti make a (1atabase system
so that we can continue to provide infor-
mation about teaching and education to
the teachers. For these reasons, it is
(difficult to motivate the teachers to
participate in this kind of seminar. The
important thing is considering how we
can create new Pleas anti how we can
create a good image about participation in
the seminar. Demonstration lessons may
be an effective method in this respect.
RECURRENT EDUCATION IN JAPAN: WASEDA UNIVERSITY

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Mamora' Takezawa, Kanagawa Prefectural Education Center
The Kanagawa Prefecture, the local
government, carries out programs for the
inservice programs for teachers. The
education center is located in the middle
of the prefecture so teachers in Kanagawa
Prefecture can come to it easily. This is
where teachers receive training, both
during summer vacation and during the
school terms. They also come for training
as a long-term researcher or student for
one or two years. We provide training for
between 40,000 to 50,000 teachers from
1,644 schools. The center has a staff of 50.
belong to the information technology
and education department.
The education center is organized into
four divisions: Japanese art, social science
or social studies, and foreign languages
for English and Spanish (Kanagawa has a
relationship with Peru and Brazil in South
America, and we have many children from
Peru and Brazil); mathematics; informa-
tion technology and education depart-
ment; and principal and vice principal
school management. The basic programs
in information technology anti education
start with general seminars, followe(1 by
specialized seminars like networks,
multime(lia,robot programming. For
example, Logo or Mindstorms can be
used. We also have a leader seminar to
create information technology coordinators,
anti we nurture lea(lers throughout the
year. There is a license seminar for the
new evaluation of the national curriculum.
We also act as a help desk for supporting
other (lepartments such as science, social
studies, an(l Japanese language in how
they can utilize information technology.
What is the license seminar? In 2003,
Japan will have a new national curriculum.
The subject of information or informatics
will be a(l(le(1 to the high schools' pro-
grams in the new curriculum. However,
we have no teachers to teach that subject.
So for the next three years, we have to
train or develop teachers in informatics.
The program is targeted at math and
natural science teachers and is a three-
week summer seminar to give them the
license to get really for the new curricu-
lum in 2003. One of the issues is that
many of the better mathematics teachers
are using computers. They will receive a
license for information technology, anti
when they are ready to teach it, the
number of goo(1 mathematics teachers
will be (leplete(l.