It is widely acknowledged that what teachers believe influences their teaching, yet the focus of much professional learning remains on influencing the specific practices and tools that teachers employ in their classrooms. In this article, it is argued that a greater and more explicit focus on teachers' beliefs would be beneficial. To this end, an…

An analysis of the 1990 National Assessment of Educational Progress (NAEP) found that only 46 percent of all high school seniors demonstrated success with a grasp of decimals, percentages, fractions and simple algebra. This article investigates error patterns that emerge as students attempt to answer questions involving the ability to apply…

A lattice is a (rectangular) grid of points, usually pictured as occurring at the intersections of two orthogonal sets of parallel, equally spaced lines. Polygons that have lattice points as vertices are called lattice polygons. It is clear that lattice polygons come in various shapes and sizes. A very small lattice triangle may cover just 3…

In this paper we outline a new theoretical model of expert practice that identifies the importance of attentional skills. We report on a small-scale pilot study of the classroom practice of experienced teachers of mathematics based on this model, through which a method of articulating aspects of classroom practice has been developed. This study…

The purpose of this study was to explore teachers' growth in understanding of algebra using concept maps. The study was set in the context of a five-year National Science Foundation funded teacher retention and renewal professional development project. In this project both beginning and experienced teachers are supported as they increase their…

The aim of this study is to characterize the discourse of two problem-solving courses for prospective teachers. The data, consisting of audio recordings and field notes, were examined from a dialogical approach combined with the theory of contextualization. I show not only the substantial differences between the two classroom discourses but also…

In this study, we consider the potential of multimedia cases as tools for teacher professional development. Specifically, we examined online and face-to-face discussions that occurred within groups composed of pre-service mathematics teachers, in-service mathematics teachers, mathematicians, and mathematics teacher educators. Discussions within…

A video-based program on lesson analysis for pre-service mathematics teachers was implemented for two consecutive years as part of a teacher education program at the University of Lazio, Italy. Two questions were addressed: What can preservice teachers learn from the analysis of videotaped lessons? How can preservice teachers' analysis ability,…

This article arises from a study whose overall purpose was to investigate the relationship between Colombian mathematics teachers' conceptions of "beginning algebra" and their conceptions of their "own" teaching practices. The teachers' understandings of their teaching practices were explored with a view to unravelling their conceptions of change…

Improving mathematics education in the United States has taken many forms. Our work has focused on two aspects: the content knowledge of teachers and a well-articulated coherent curriculum. Our aim was teacher "capacity building" that is enabling teachers to teach to coherent and significant mathematical curricular goals and describe the…

We examine the role of tasks that have the intended effect of teachers re-conceiving the mathematics they teach as comprising a coherent body of meaningful ideas. We ground our discussion in ideas of trigonometry and modular functions and draw from a professional development research project to illustrate our approach. In this project, many…

This paper describes a three-level framework (technical, domain, and generic) which enables some generalisability across mathematics topics for effective teacher-education (TE) task design. It argues that TE tasks which encompass these levels increase preservice and inservice teachers' interest because they transcend the particular mathematical…

In this article we introduce a usage-goal framework within which task design can be guided and analyzed. We tell a tale of one task, the Pentomino Problem, and its evolution through predictive analysis, trial, reflective analysis, and adjustment. In describing several iterations of the task implementation, we focus on mathematical affordances…

Our work is in keeping with the general framework of didactical research in France, where we both studied, and now teach. This paper describes a model training session designed for in-service mathematics teachers, whose purpose is to guide them in their understanding of the relationship between students' tasks and activities in secondary school…

Mathematical tasks, centered on arithmetic word problems, are discussed as the basis of an approach to facilitate preservice elementary teachers' development of mathematical knowledge for teaching arithmetic operations. The approach consists of three groups of tasks that allow students to reflect on their initial knowledge, explore arithmetic word…