These professional development materials support elementary teachers in extending their level of understanding about how students reason when they perform arithmetic operations. The two sets of materials are identical in design and differ only in the emphasis on addition and subtraction in the one case and multiplication and division in the other. Each set includes a 55-page Teacher's Study Guide and two videotapes.

These materials are intended to assist elementary teachers who have been "working hard to rethink their classroom practice in order to support the development of their students mathematical thinking" (p. 5), through examination of classroom episodes where students use a variety of whole number computation procedures and exercises designed to help the user revisit the arithmetic operations.

The author's aim is to impact teachers' planning and implementation of addition and subtraction/multiplication and division lessons. An equally important purpose of the materials is to help teachers think through the mathematics content of the operations in order to consider what underlies the development of student ideas.

The Teacher's Study Guide contains three types of information: (1) detailed lesson plans for professional development providers (six sessions in the Addition/Subtraction package and eight in the Multiplication/ Division package), with each lasting approximately one and one-half hours; (2) summaries of students' solutions from videotape 1; and (3) six readings for further consideration between sessions. A list for further reading offers selections focusing on these operations.

The first videotape in each set of materials shows students in diverse classrooms engaged in reasoning about mathematics problems, and the ensuing discussion about their thinking and strategies. Videotape 2, "Teaching and Learning about Addition and Subtraction/ Multiplication and Division," includes narration and interviews with teachers that offer perspectives on the types of learning environments that support students in building an understanding of arithmetic operations.

The study guides are clearly organized to provide the user with a preview of the session, specific information for the facilitator to structure the sessions, proposed timelines, and discussion questions related to the videotapes. The plans are informative but not overly scripted. The videotapes illustrate students' mathematical thinking by providing vignettes of students solving problems and interviews with teachers about the students' thinking. The interviews with the teachers also demonstrate some of the issues that emerge when teachers embark on changing practice.

The videotapes are of good technical quality; the guide is easy to read and generously formatted with headings to provide for ease in implementation. The selected readings are relatively short, practical, and relatively free of jargon.

The materials are specifically designed for professional development use, and could be used effectively in pre-service settings as part of a methods course. In-service professional development could take on various structures, according to the needs of the target audience. While the materials could be used for self-study, they are "intended for use by a group of teachers working together with the videotapes" (p. 7).

The sessions highlighted in the materials give users an opportunity to explore and understand students' solutions to a few well-chosen problems. Numerous examples of questions, exercises, and tasks that can be presented to the participants during the professional development sessions provide a framework for challenging teachers to rethink their current approaches to teaching whole number computation. Included readings raise questions about the importance of notation in mathematical communication, computational fluency, student-driven procedures, and expanding mathematics beyond traditional algorithms.

One reviewer noted that the time estimates suggested in the plans may be low, and, as suggested in the guide, sessions can easily be extended to help teachers fully understand the students' solutions and their own thinking about instruction. Another reviewer recommended that professional development providers using these materials pay particular attention to how discussion and related activities impact the pedagogical understanding of the teachers participating in the sessions.

One reviewer noted that the range of instructional strategies depicted was narrow. Since the videotapes most often portray students reporting on their solution strategies and don't show the small group and pair work that precedes these whole class sessions, viewers might get an incorrect impression that most work is done with the teacher directing the whole class. Facilitators may want to emphasize that, as the guide points out, students have spent a long time working on these problems with their peers before reporting to the whole class.