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As an initial step to address national,
state, and local issues of teaching and
learning mathematics in the middle
grades, the Mathematical Sciences
Education Board (MSEB) of the Na-
tional Research Council's Center for
Science, Mathematics, and Engineering
Education (CSMEE), the American
Educational Research Association
(AERA), the National Council of Teach-
ers of Mathematics (NCTM), and the
National Middle School Association
(NMSA) co-sponsored a National
Convocation on Middle Grades Math-
ematics. The Convocation was held at
the National Academy of Sciences in
Washington DC, on 25-27 September
1998 with support from the U.S. Depart-
ment of Education and the AERA. The
goals of the Convocation were to
.
challenge the nation's mathematical
sciences community to focus its
energy and resources on the improve-
ment of middle grades mathematics
education and
· begin an ongoing national dialogue
on middle grades mathematics
education, bringing together those
with different perspectives and
responsibilities to jointly consider
the issues.
The Convocation consisted of plenary
sessions attended by all of the partici-
pants and small focused discussion
groups. Over 400 participants inclu(ling
mathematicians, mathematics teacher
educators, state and district mathemat-
ics education policy makers, national
policy makers, mathematics education
researchers, classroom teachers,
curriculum developers, and school
boar(1 members atten(le(1 the Convoca-
tion. Some of the attendees came as
individuals. Manyothersweremem-
bers of the more than 50 (1istrict teams
that addressed the issues in terms of
their own communities anti needs.
Prior to the convocation, attendees
reviewe(1 the following backgroun
materials:
· a paper commissione(1 for the Convo-
cation, ' What is Sth Grade Mathemat-
ics: A Look from NAEP" by John
Dossey,
· the abri(lge(1 version of Turning
Points from the Carnegie Council on
A(lolescent Development,
OCR for page 6
· This We Believe from the National
Middle School Association,
· an article from the Kappa n Articles
online, "Speaking with One Voice, a
Manifesto for Middle-Grades Reform"
by Joan I~ipsitz, Hayes Mizell, An-
thony Jackson, an(1 Leah Meyer
Austin, and
· "Middle Grades Mathematics Educa-
tion: Questions and Answers" a paper
prepared by Andrew Zucker for the
U.S. Department of Education
On the first evening of the Convoca-
tion, Dr. Bruce Alberts, President of the
National Academy of Sciences, wel-
comed the group and spoke of the need
to invest ourselves in improving educa-
tion and to be open to new ways to meet
the challenges of the task. He was
followed by the Honorable C. Kent
McGuire, Assistant Secretary, Office of
Educational Research and Improve-
ment, who brought greetings from the
U.S. Department of Education. The
Convocation Program Steering Commit-
tee Chair, Edward Silver, University of
Pittsburgh, offered brief remarks about
the goals of the Convocation and set the
stage for the rest of the activities. Two
presentations followed, each represent-
ing a different perspective. Glenda
Lappan, Michigan State University,
discussed the importance of laying a
firm foundation for un(lerstan(ling
mathematics during the middle years.
She note(1 the (1ual goals of respecting
the developing capabilities of middle
school students while engaging these
energetic adolescents in learning
mathematics for their own future.
Rather than listing a set of topics, she
proposed a strand approach to the
mathematics students should learn that
would be central to further study of
mathematics or to being a good and
productive citizen. Thomas Dickinson,
In(liana State University, then use(1
"small stories" as a way to characterize
successful middle schools. He dis-
cussed development in the context of
the child as well as development and its
connection to teaching. He gave several
examples of teachers teaching individu-
als, anti teachers teaching mathematics
where development was placed within a
context of individuals and the individu-
als in a context of learning.
CONTENT AND LEARNING ISSUES
After greetings from Luther Williams,
Assistant Director, Directorate for Educa-
tion and Human Resources, National
Science Foundation, the first focus of the
second day of the Convocation was on
content and learning mathematics at the
middle grades. Nancy Doda, from
National-Louis University, spoke of a
crossroads in middle school reform. She
noted the need for a reexamination of the
MATHEMATICS EDUCATION IN THE MIDDLE GRADES
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fundamental philosophy of the middle
school concept to understand what is
needed to ensure academic success for all
children. Doda contended that in many
cases, middle schools have been more
successful in altering aspects of climate
and structure than of the curriculum and
instruction. There is now a pressing
demand to refine the elements of the
middle school concept to build an explor-
atory curriculum that is also intellectually
demanding.
Kathleen Hart, the University of
Nottingham, based her remarks on
projects from the United Kingdom.
Middle grades are recognized as the
time students should move from the
concrete to the abstract, but the transi-
tion must be carefully developed so that
understanding emerges. She gave
examples where the children she
studied did not connect concept devel-
opment with formal work with aIgo-
rithms, often due to the fact that teach-
ers did not make strong connections
between the two. She noted how
important it is for teachers to under-
stand where mathematical ideas are
leading and to be ready to build on the
different pieces of knowledge individual
children take away from a lesson.
Following the plenary session, partici-
pants engaged in a mathematical task in
small (liscussion groups (Marcy's (lots,
see page 581. They were asked to
renect on the mathematics involve(1 anti
EXEC UTIVE SUMMARY
to analyze student responses as a
backdrop for their discussion about
content and learning mathematics in the
mi(l(lle gra(les.
TEACHING ISSUES
The afternoon sessions were focused
on teaching in the middle grades. In the
plenary session, participants viewe(1 two
video clips. Nanette Seago, California
Mathematics Renaissance Project,
showed a videotape of an eighth grade
class during a lesson on patterns in
algebra. The viewers consi(lere(1 how
listening to conversations among
students enables teachers to learn about
student understanding. Groups dis-
cussed the decisions the teacher made
as she pursue(1 the lesson anti renecte
on the impact of these (recisions on the
outcome of the lesson.
Line Foreman, Portian(1 State
University, showed a videotape of
middle grades students making a
presentation at the 1998 National Coun-
ci} of Teachers of Mathematics Annual
Meeting in Washington, DC. The
students spoke about what they hall
learned in mathematics and what their
teacher had done to enable that learning
to happen. In particular, the students
recognized and supported the notion
that learning (li(1 not take place without
some "(lisequilibrium" anti that strug
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gling with learning was a natural part of
the process. These students had been
taught by Foreman for four years, and
she suggested that structuring learning
so students and teachers stay together
over time is one way to create a success-
fu} community of learners.
A pane} consisting of Hyman Bass, a
mathematician from Columbia University,
Deborah Ball, mathematics educator from
the University of Michigan, and Sam
Chathn, a middle grades teacher from
William H. English Middle School in
Scottsberg, Indiana, reacted to each
videotape. Bass observed the blend of
algebra and geometry in both videos and
noted that use of video might be an
appropriate too} to help bridge the gap
between mathematics as content and
mathematics in practice. ChatUn's com-
ments related to the environment estab-
lished by the teacher in each case as
evidenced by the kinds of questions and
answers and by the confidence students
displayed about their work. Ball framed
her remarks around the interplay between
the mathematics to be learned and the
role of discussion where the teacher's
decision about how to frame a question
and how to respond drives what students
learn. In the small group discussion
session that followed, the participants
were asked to resect on teaching issues
raised by the videotapes as wed as on the
use of videotape as a means for stimulat-
ing resection and discussion.
ORGANIZATIONAL ISSUES
The last theme of the Convocation was
the organization of schools for middle
grades students and its relation to
teaching and learning mathematics. A
pane} presentation framed the issues for
(liscussion. Craig Spilman,a principal
from Canton Middle School in Baltimore,
MD, emphasized the need to make
intelligent use of data to promote articu-
lation among elementary and middle
grades teachers to understand where
students are in their learning. The
design and implementation of programs
should be flexible enough to accommo-
date students as they grow. He spoke of
the nee(1 for principals to communicate
with anti counsel teachers to ensure that
their mathematics instruction is centered
around student learning.
Mary Kay Stein, a research scientist
from the University of Pittsburgh,
hypothesized that the developmental
approach teaching mathematics with a
focus on the whole child an(1 the
subject matter approach teaching
mathematics with a focus on the con-
tent each have news. She propose(1 a
middle school organization and structure
that is jointly informed by subject matter
anti (developmental concerns. Stein
argued that the mathematics for middle
school students should take into account
the (levelopmental needs of adolescents.
For this to happen, professional develop
MATHEMATICS EDUCATION IN THE MIDDLE GRADES
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ment should be part of the day-to-day
administrative functioning of the school.
Robert FeIner, National Center on
Public Education, University of Rhode
Island, spoke about project work that
focused on high performing learning
communities. A key finding was that if
educational improvement efforts do not
attend to the full ecology of the setting,
they will ultimately fail. Although some
schools have success in raising student
scores for a time, the success is not
sustainable, and, in fact, according to this
research, raising student achievement
over time is related to the degree of
implementation of key structural
changes in the school. He also men-
tioned that parent involvement that
correlates with gains in student achieve-
ment is "sending home information
about how to work with and talk to your
children." In response to questions,
panelists pointed out that raising expecta-
tions is a critical part of raising student
achievement, that structural changes
should be accompanied by thoughtful
support for teachers, and that even
though we continue to improve, the task
keeps changing, masking the gains.
CLOSING REMARKS
In his closing remarks, Edward Silver
noted the dual commitment of the
Convocation participants: to enhancing
the quality and quantity of mathematics
EXEC UTIVE SUMMARY
learning in the middle grades and to
a(l(lressing other nee(ls of young a(loles-
cents. He suggested a major concern is
how to make mathematics interesting
and important to young adolescents.
Silver pointed out that the examples
presented during the Convocation
indicated students can be interested in
the mathematical tasks we give them, in
the mathematics itself, or in the process
of struggling with the tasks, and that the
role of the teacher is to cultivate this
interest. He challenged the audience to
contrast their own view of algebra with
the algebraic ideas that were presented
in the Convocation sessions and to renect
on what it means to say that students are
learning algebra and what it would mean
for ah students to learn algebra. He
advocated a systematic examination of
different instructional and curricular
arrangements designed to have ah
students learn algebra. His closing
comments addressed the issue of using
the generalist/specialist notion as a way
to set up a false dichotomy and caped for
thinking about ways to form a commu-
nity with a joint identity that moves the
Convocation agenda forward.
ACTION CONFERENCE ON THE
NATURE AND IMPACT OF
ALGEBRA AT THE MIDDLE GRADES
The agenda for the Action Conference
was (lesigne(1 to bring attention to
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different possibilities for algebra in the
middle grades and to the issues in-
volve(1 in implementing any of these
possibilities. Discussion was framed by
six questions presented by the Confer-
ence organizer, Hyman Bass, a math-
ematician from Columbia University.
The questions covered the following
topics: attention to subject matter vs.
attention to students; algebra as the
language of mathematics; real world
contexts vs. generalization and abstrac-
tion; covering mathematics vs. uncover-
ing mathematics; situating algebra in
the mathematics curriculum; materials,
design, selection criteria for mathemat-
ics curriculum. km Fey from the
University of Maryland suggested that
the important aspects of algebra are the
concepts and techniques for reasoning
about quantitative conditions and
relationships. With this as a theme, Fey
claimed that moving high school aIge-
bra into the middle grades wait not be
sufficient. Al Cuoco, Director of the
Center for Mathematics Education at
EDC, presented a view of algebra that
relied more on symbols and problems
from the world of mathematics, with an
emphasis on the ways of thinking that
can emerge from reasoning about
calculations and about operations. Bass
suggested that Fey's and Cuoco's
approaches were two (lifferent aspects
of the same thin". OritZaslavsky,a
mathematics education researcher from
Israel, postulated that learning is about
constructing meaning that can change
over time, across learners, anti across
contexts. Learning algebra in the
mi(l(lle gra(les is just a beginning, where
examples play a critical role and the
issue of representation is inherent.
The second day of the conference
featured approaches to a middle grades
algebra curriculum from the Connected
Mathematics Project, University of
Chicago School Mathematics Project,
Mathematics in Context, and Saxon
Mathematics in which the presenters
described the nature of their algebra
stran(1 anti what works in practice. As
part of a pane} on general implementa-
tion issues, Anne Bartel, from the
Minnesota project SciMath, pointed out
that many of the issues are tied to
people's belief systems about whether
algebra is focused on skills or thinking
and about what "algebra for all" really
means. She closed with a discussion of
the characteristics of effective profes-
signal development, including the need
to make the algebra content an(1 corre-
spon(ling instructional strategies ex-
plicit. Vern Williams, Gifted anti Tal-
ente(1 Coordinator from a Virginia
mi(l(lle school, emphasize(1 that some
children nee(1 more than the norm with
an emphasis on theory, structure, an
problem solving. These students nee(1
to be challenge(1 every (lay, anti a gifted
and talented course in algebra opens the
MATHEMATICS EDUCATION IN THE MIDDLE GRADES
OCR for page 11
universe of mathematics for them.
Nancy Doda, National-Louis University,
disagreed with Williams and advocated
that all students needed to be chal-
lenged and raised issues of equity. In
closing, participants discussed algebra
in relation to mathematics content,
curricular design, and use of research in
the context of a search for guidance on
how to scale up promising programs to
realize improved mathematics learning
for more students.
ACTION CONFERENCE ON
RESEARCH IN THE TEACHING
AND LEARNING OF
MATHEMATICS IN THE MIDDLE
GRADES
The focus of the conference was to
help define research required to better
understand and articulate the assump-
tions that underpin activities aimed at
improving the mathematics curriculum,
teaching, and learning in the middle
gra(les. The group was charged with
making suggestions to those in the field
about what further research is needed
and advising the Department of Educa-
tion and National Science Foundation on
where strategic investments in research
might be made. Invited talks were
organized to move from broad theoreti-
cal and practical issues on how to
address research to specific research
EXEC UTIVE SUMMARY
efforts to the applications of research
knowledge to recent curriculum devel
opment projects.
From the first perspective, lames
Hiebert, University of Delaware,
described a tension between solving
practical problems and doing good
research and offered a framework to
better understand and resolve these
tensions. Alan Schoenfeld, University
of California-Berkeley, responded to
Hiebert by arguing for a stronger
theoretical base for and through
research. Mary Kay Stein illustrated
how research can grow from attempts
to solve problems of practice. Richar
Mesh, In(liana University-Pur(lue,
commented on how thinking can be
change(1 over time anti the nee(1 to be
explicit about the big ideas in the
middle grades curriculum. lames
Fey, University 0 f Mary}an(1, and
Koeno Gravemeijer, Freudenthal
Institute, the Netherlands, described
ways that the extent research on
rational number anti proportional
reasoning shape(1 (1esign (1ecisions in
their respective curriculum projects.
Ju(ly Sow(ler, University of California-
San Diego, (liscusse(1 ways to (1eepen
teachers' knowledge of mathematics.
The participants ma(le recommen(la-
tions in three areas: teaching anti
teacher learning; student learning;
anti communicating with a variety of
interested constituencies.
OCR for page 12
ACTION CONFERENCE ON THE
PROFESSIONAL DEVELOPMENT
OF TEACHERS OF MATHEMATICS
IN THE MIDDLE GRADES
The Action Conference was designed
to afford an opportunity for participants
to examine promising approaches to
professional development by creating an
analytic and practical conversation about
the sorts of opportunities in professional
development most likely to lead to
teachers' learning and improvements in
their practice. Deborah Ball, University
of Michigan, conference organizer,
framed the discussion from the vantage
point of teacher educators considering
sites through which teachers might
most profitably learn mathematics
content needed in teaching, based on
tasks which teachers regularly do as
pert of their leaching. Participants
discussed what is known about profes-
sional (levelopment, teacher learning,
anti the improvement of practice. As an
initial example of how teachers use
knowledge of content to shape their
teaching, Ball together with Joan
Ferrini-Mun(ly, Center for Science,
Mathematics, and Engineering Educa-
tion engage(1 participants in reformulat-
ing a mathematical task and considering
the mathematical knowledge used to
create anti evaluate these new tasks.
Margaret Smith, Pennsylvania State
University, involved the participants in
a case stu(ly of student work anti
(liscusse(1 how analyzing student
responses led to a discussion of the
importance of the mathematical knowI-
edge of the teacher. Karen
Economopoulos, TERC, posed two
questions for reflection and discussion:
How might curriculum materials offer
professional (levelopment opportuni-
ties for teachers and how can these
materials influence or support teach-
ers' daily decisions? Nanette Seago,
Mathematics Renaissance Project,
using a video of an eighth grade math-
ematics lesson, facilitated a discussion
of the use of videotape as an instruc-
tional medium in professional (levelop-
ment. The closing session feature(1 a
pane} that presented their reflections
on the improvement of professional
(levelopment. Iris Weiss, from Horizon
Research, argue(1 for the nee(1 to help
teachers (levelop some way to filter and
make (recisions, anti raise(1 a concern
about how to scale up professional
development models. John Moyer
from Marquette University, reflected
on professional (levelopment with
urban, large city middle grades teach-
ers using teacher responses to an
observer's comments to promote
teachers' reflection on their practice.
Stephanie Williamson, Louisiana
Systemic Initiative, described the work
MATHEMATICS EDUCATION IN THE MIDDLE GRADES
OCR for page 13
done to build collaboration among
school systems and universities in
Louisiana that led to the development
of a document used to guide decisions
about professional development pro-
grams. Participant's comments at
EXEC UTIVE SUMMARY
the end indicated that the Conference
did take seriously professional devel-
opment as a field and attempted to
create a frame for thinking about
theoretical, research, and practice-
based learning.
Representative terms from entire chapter:
mathematics education
