workshop series provides an interactive context for teachers,
administrators, and other interested adults to explore issues
about learning and teaching mathematics. Central to each session
is a 60-minute videotape that offers a sequence of episodes
showing children and/or teachers engaged in authentic mathematical
activity and discussion, consistent with state and national
standards for teaching and evaluating mathematics. These episodes
come from a variety of sources in diverse school communities
and across grade levels from pre-kindergarten to 12th grade.
The episodes and accompanying narratives in each videotape
- students and
teachers actively engaged in doing mathematics;
that encourage meaningful mathematical activity; and
for learning, teaching, and assessment.
The materials and activities presented in the sessions have
been developed in long-term research programs about mathematical
thinking that share certain presuppositions about learning
and teaching. Key to this perspective is that knowledge and
competence develop most effectively in situations where students,
frequently working with others, work on challenging problems,
discuss various strategies, argue about conflicting ideas,
and regularly present justifications for their solutions to
each other and to the entire class. The role of the teacher
includes selecting and posing the problems, then questioning,
listening, and facilitating classroom discourse, usually without
direct procedural instruction.
Each videotape contains episodes from a 12-year research
study carried out in the Kenilworth, New Jersey public schools
within a partnership with the Robert B. Davis Institute for
Learning at Rutgers University that began in 1984. This partnership
included an intensive, classroom-based staff development program
in mathematics for teachers and administrators at the K-8
Harding Elementary School. Classroom sessions in which students
frequently worked together in small groups on meaningful problem
activities. One classroom of children was followed from first
through third grades by regularly videotaping small-group
problem-solving sessions, whole class discussions, and individual
task-based interviews. From 1991 to the present, with periodic
support from two National Science Foundation grants, the research
team has continued documenting and studying the thinking of
a focus group of these children through grade 12. The videotapes
from these later years document the students as they participated
in problem-based activities developed by the university researchers
in classroom lessons, after-school sessions, a two-week summer
institute, and individual and small-group task-based interviews.
In the workshop videotapes, participants will see some of
the same children solving mathematical problems at various
grade levels over the years. In addition, the activities they
engage in have been used in other communities and at other
grade levels. In June 2000, the focus group of students in
this study graduated from Kenilworth's David Brearley Middle/High
participants will explore particular questions about learning
and teaching mathematics based on the shared experience of
watching the videotapes. Key questions are:
Each videotape includes
episodes of children engaged in mathematical problem activities.
The goal is for participants to become able to recognize what
is mathematical in students' activity by attending very closely
to what they do and say. As they observe, study and discuss
what they see on the tape from the perspective of the questions
listed above, participants will gain insights about learning
and teaching. As preparation, participants need to build their
own solutions to the central problems in each tape with assignments
given between sessions and during the first hour of each workshop.
They are encouraged to select and use appropriate problems with
their own students and to read further about the learning and
teaching of these ideas.
- How do children
(and adults) learn mathematics?
- How do children
(and adults) learn to communicate about mathematics and
to explain and justify solutions to problems?
- What conditions
(environments, activities, interactions) are most helpful
in facilitating this development?
- What does it
mean to be a teacher of mathematics?
- What is the
connection between learning and teaching?
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