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Contexts for Learning Mathematics Series
Contexts for Learning Mathematics and Young Mathematicians at Work Resource Packages
Each unit is related to previously released CD-ROM-based professional development materials funded by the National Science Foundation, also available from Heinemann. These are interactive, digital learning environments with rich video of children and teachers at work, using investigations and minilessons from the units. The CD-ROMs can be used in study groups, in in-service (or pre-service) courses, and in online venues.
Digital production projects have traditionally used the technology to accompany texts by illustrating examples such as lessons, coaching models, and interviews (Carpenter, Franke, and Levi, 2003; West and Staub, 2003) or in other cases for modified lesson study (www.lessonlab.com). Although these uses can be helpful in providing illustrations, they are not powerful enough for systemic, transformative professional development.
Our digital lab environments go well beyond these uses; they are interactive, multilevel learning environments for the professional continuum with built-in questions, notepads, and suggested inquiries. They bring the context of the classroom to the fingertips of the learner. Immersed in the study of children over time in exemplary classrooms, users can examine the teacher's didactical employment of context; inquire about and analyze the pedagogy; clip and paste moments from footage and build learning trajectories (or "landscapes of learning," as we prefer to call them) that show children constructing big ideas, developing strategies, and using mathematical models as tools. Asked to solve mathematical problems in several ways, users can anticipate and subsequently examine student strategies; design investigations and minilessons for the next day; examine how the teacher in the environment continues; and subsequently analyze children's work to assess the effectiveness. Users can even add clipped footage as hypertext evidence to support arguments and provide examples in term papers and literature reviews or as sample evidence of the NCTM standards. The CD-ROMs are cross-platform, thus enabling users to work at home on assignments as well as in on-site workshops.
We designed these video mat erials this way for several reasons. A large divide still exists between professional development programs and classroom-based practice—a gap between theory and practice. Even when programs include much in-classroom work, it is difficult to dramatically transform teachers' practice. This is true because teachers' beliefs about learning and teaching are based on their prior experiences of being students themselves, and they act and interpret what they see based on these beliefs (Hutcheson and Ammon 1986; Dolk 1997; Schifter and Fosnot 1993). At the same time, successful professional development is needed that improves teaching practice in large numbers of classrooms (Elmore 2000), and this need is particularly systemic in large urban areas where teacher shortages are at their highest (Darling-Hammond 2000). Digital lab environments for professional development have the capacity to reach large numbers of teachers and provide a realistic context for teacher learning, merging professional development workshops with school-based classroom practice. Digital technology allows users to re-look and discuss, examine and analyze, in contrast to realtime observations and on-site lesson study where moments cannot be replayed and reexamined. As pre-service and in-service teachers reexamine digital video clips and share their observations and interpretations, they experience disequilibrium and deepen their understanding of mathematics teaching and learning. They learn to seek evidence for their statements, to understand the mathematical ideas children are constructing, and to assess learning. As teachers work within collaborative communities of discourse, analyzing teaching and learning, doing mathematics, and examining mathematical development of children, the norms of practice, forms of justification and evidence, what counts as the doing of mathematics, and even their own professional identities all potentially can shift. These shifts we captured in our pilot and field-test data as we developed the Contexts for Learning Mathematics materials and led workshops with them.
The materials have two different developmental journeys. As we fieldtested and piloted, we realized that a critical component of using the digital environments to their full impact is knowledge of a developmental trajectory of teacher development, specifically situated in the context of the use of digital learning environments. For example, we initially assumed that teachers should use the Landscape of Learning CD-ROM (Dolk and Fosnot 2004) first. This CD-ROM has short film clips to enable participants to examine and discuss big ideas and strategies, in contrast to the other CD-ROMs, which depict long-term investigations over several days. We conjectured that knowledge of the development of mathematical ideas and strategies was a critical first step in becoming a good mathematics teacher. But we learned instead that this step was too difficult for teachers unaccustomed to teaching with a workshop model or thinking about mathematics knowledge as developmental landmarks. Often they focused on definitions and spent too much time with language (e.g., unitizing, making tens, constant difference, distributive property), trying to apply what they were reading, instead of really thinking about what the children were doing. The teachers' ability to do "kid watching" needed to be developed, as did their ability to analyze and seek evidence for their opinions. Many teachers needed to deepen their own mathematics in order to understand the work of children. They interpreted and generalized too quickly, often focusing on affect only, seemingly unaware of the potential mathematical development. Many teachers, who had never seen classrooms where children were actively exploring and talking about their thinking, wanted to explore full investigations over time in these classrooms, rather than short clips. Their own questions regarding these issues distracted them from analyzing the big ideas as they wondered what investigations looked like over time, how the teacher enabled the children to work in such a community, what the teacher would do next, and how it might impact learning. For all these reasons, we built two developmental journeys into each of the CD-ROMs in order to provide opportunities for differentiation.
Five CD-ROMs are available for each level of Contexts for Learning Mathematics. Facilitator guides for workshop leaders accompany each one. To view the Young Mathematicians at Work grade-specific resource packages click on the following links:
Workshops and Seminars
Two-day institutes are offered once a month at Mathematics in the City, City College of New York. Information is available on our website, www.mitcccny.org (click on DELTA Institutes-DELTA stands for Digital Environments for Learning, Teaching, and Assessment).
On-site offerings and online courses taught by the authors are also available. For information on these, go to www.newperspectivesonlearning.com.
For related workshops and seminars, go to pd.heinemann.com