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A description is given of a 3-year research and development project designed to address the fundamental problems of preparing elementary teachers to teach mathematics for understanding. The use of a new technology is proposed. Hypermedia combines elements of multimedia environments for learning and teaching with a recent development in computer software called "hypertext," a representation of multiple and flexible links between discrete pieces of data that allows users to navigate along multiple paths through a network of chunks of information and to build and store their own links. A hypermedia tool can make it possible to access and link video representations of a year's worth of teaching and learning in two mathematics classrooms from teachers' and students' perspectives. It will include video and textual representations of teachers' thinking about both the mathematical content and the pedagogical decisions involved in their work, graphic and textual representations produced by students of their mathematical reasoning processes, and video and textual annotations which analyze lessons from the perspective of relevant academic disciplines. Hypermedia can give teacher education students an experience that is close to observing good mathematics lessons in real classrooms and then give them the opportunity to analyze those lessons with the teacher who taught them. (JD)

Assuming guided inquiry as a pedagogical ideal in mathematics education implies that teaching must connect students' thinking about a subject with curricular agendas and instructional goals. Because students are not typically inclined to consider their active inquiry as a route to acquiring the knowledge that is valued in school, such teaching must simultaneously elicit students' engagement in inquiry and legitimate inquiry as a route to learning. Within this conceptual framework, the empirical research reported in this paper describes several strategies used by secondary school geometry teachers as they attempted to practice a pedagogy of guided inquiry using the "Geometric Supposers." The teachers' strategies are discussed in terms of sociolinguistic theories about the teacher's role in defining the meaning of mathematical knowledge in the classroom. (Author)

This paper is an attempt to draw on the thinking of a group of secondary school geometry teachers who are participants in the Laboratory Sites Study of the Educational Technology Center (ETC) at Harvard University. The purpose of the Lab Sites Study is to understand the process of implementing technology-enhanced guided exploration in school classrooms. Teachers and researchers are working together to understand what it takes to use computer-based technology to support student exploration of mathematical and scientific ideas in ordinary classrooms. The data analyzed in this paper was collected as a substudy of the Lab Sites project, which looked at comprehensive questions of implementation in relation to materials produced at ETC for teaching science, mathematics, and programming. The substudy reported herein was concerned with teachers' points of view about using one piece of educational technology--The Geometric Supposer--to substantially change the way they teach geometry. The Supposer is designed to fundamentally change the way instruction is delivered in classrooms by enabling students to engage directly in the exploration of subject matter. What is reported here, therefore, is the teacher-users' thinking about that broader change in the way they do their work, as well as their thoughts about the technology. (TW)

The concept of multiplication is described and illustrated using several different representational systems. A conceptual approach to teaching mathematics is compared with the procedural approach commonly found in the school curriculum. Four different methods of representing the multiplication process with numbers larger than ten are presented: combinations of coins, stories in which quantities are counted by grouping them, recording numbers in a way that is congruent with operations on quantities, and using multiplication as a procedure to solve problems involving estimations of weights. Each of these methods is related to the conventional algorithm for multiplying large numbers. Children's work is used to illustrate how they respond to each approach. (Author)

This paper describes a research and development project in teaching designed to examine whether and how it might be possible to bring the meaning of knowing mathematics in school closer to what it means to know mathematics within the descipline by deliberately altering the roles and responsibilities of teacher and students in classroom discourse. The project was carried out as a regular feature of lessons in fifth-grade mathematics in a public school. The teacher modeled a form of social interaction appropriate to making mathematical arguments in response to students' conjectures. Students were taught to examine the hypotheses about mathematical structures that underlie their solutions to problems. The outcomes of this teaching are described in terms of the abilities and propensities students develop to create mathematical arguments in a particular episode of classroom interaction. (Author)