All fields of research need to reconsider, on an ongoing basis, their standards for research quality. At the present time in mathematics education, such discussion is particularly critical to the ongoing growth and strength of the field. Mathematics education research has evolved significantly in the last 20 years. The most sweeping change has…

In this article we investigate how preservice elementary school (K-7) teachers understand the concept of prime numbers. We describe participants' understanding of primes and attempt to detect factors that influence their understanding. Representation of number properties serves as a lens for the analysis of participants' responses. We suggest that…

Many children read mathematics word problems and directly translate them to arithmetic operations. More sophisticated problem solvers transform word problems into object-based or mental models. Subsequent solutions are often qualitatively different because these models differentially support cognitive processing. Based on a conception of problem…

Some years ago, Gila Hanna offered the very insightful and useful distinction between mathematical proofs that prove and those that also explain. Proofs that explain not only state the deductive logic that justifies their mathematical claims, they lay bare the mechanisms and structures that underlie that logic, making content and logic more…

We articulate and explicate a mechanism for mathematics conceptual learning that can serve as a basis for the design of mathematics lessons. The mechanism, reflection on activity-effect relationships, addresses the learning paradox (Pascual-Leone, 1976), a paradox that derives from careful attention to the construct of assimilation (Piaget, 1970).…

This study was prompted by the current availability of newly designed mathematics curriculum materials for elementary teachers. Seeking to understand the role that reform-oriented curricula might play in supporting teacher learning, we studied the ways in which 8 teachers in the same school used one such curriculum, Investigations in Number, Data,…

Information processing researchers have assumed that algebra symbol skills depend on mastery of the abstract rules presented in the curriculum (Matz, 1980; Sleeman, 1986). Thus, students' ubiquitous algebra errors have been taken as indicating the need to embed algebra in rich contextual settings (Kaput, 1995; National Council of Teachers of…

This article considers the question of what specific actions a teacher might take to create a culture of inquiry in a secondary school mathematics classroom. Sociocultural theories of learning provide the framework for examining teaching and learning practices in a single classroom over a two-year period. The notion of the zone of proximal…

Widespread agreement exists that U.S. teachers need improved mathematics knowledge for teaching. Over the past decade, policymakers have funded a range of professional development efforts designed to address this need. However, there has been little success in determining whether and when teachers develop mathematical knowledge from professional…

Discusses the nature of tasks used to elicit the development of such systems by middle school students. Analyzes mathematical reasoning development of students across tasks and the diversity of thinking patterns identified on problem tasks. Discusses student reasoning about the relationships between and among quantities and their application in…

Explores some of the possible contributions of complexity science to discussions of the teaching of mathematics. Focuses on the use of the vocabulary of complexity in the re-description of mathematical communities and the application of principles of complexity to the teaching of mathematics. Highlights compatible and complementary discussions…

Presents a case study in which two 8th grade students developed knowledge for modeling a physical device called a winch. Demonstrates that students have and can use criteria for evaluating algebraic representations. Explains how students can develop modeling knowledge by coordinating criteria with knowledge for generating and using algebraic…

Reports results of a study on instructional practices that relate to student achievement in high school classrooms. Uses a standards-based curriculum called Core-Plus and identifies teachers' background characteristics, behavior, and concerns associated with growth in student achievement. Indicates the importance of professional development…

Explores the way that mathematics and secondary education mathematics majors read and reflected upon student-generated arguments purported to be proofs of a single theorem. Suggests that such undergraduates tend to focus on surface features of arguments and that their ability to determine whether arguments are proofs is very limited. Discusses…

Reports on a 2-year study about teaching and learning mathematics for social justice in an urban, Latino classroom and the role of a National Council of Teachers of Mathematics (NCTM) standards-based curriculum. Uses qualitative, practitioner-research methodology. Indicates that the standards-based curriculum was also important in promoting…