In this article, we report two studies investigating what mathematicians value in a pedagogical proof. Study 1 is a qualitative study of how eight mathematicians revised two proofs that would be presented in a course for mathematics majors. These mathematicians thought that introductory and concluding sentences should be included in the proofs,…

This study extends prior research on both individual self-regulation and socially shared regulation during group learning to examine the range and quality of the cognitive and behavioral social regulatory sub-processes employed by six small collaborative groups of upper-elementary students (n = 24). Qualitative analyses were conducted based on…

We discuss an emerging program of research on a particular aspect of mathematics learning, students' learning through their own mathematical activity as they engage in particular mathematical tasks. Prior research in mathematics education has characterized learning trajectories of students by specifying a series of conceptual steps through which…

This study examines presentations of the distributive property (DP) in two widely used U.S. elementary text series and one main Chinese text series along three dimensions: problem contexts, typical problem types within each problem context, and variability in using the DP. In general, the two U.S. texts were found to resemble each other but to…

This study reports findings from comparative samples of sixth-grade Chinese and U.S. students' interpretations of the equal sign. Ninety-eight percent of the Chinese sample correctly answered 4 items indicating conceptions of equality and provided conceptually accurate explanations. In contrast, only 28% of the U.S. sample performed at this level.…

The present study contrasts mathematical knowledge that two sixth-grade teachers apparently used when teaching fraction multiplication with the Connected Mathematics Project materials. The analysis concentrated on those tasks from the materials that use drawings to represent fractions as length or area quantities. Examining the two teachers'…

Recent mathematics education reform efforts call for the instantiation of mathematics classroom environments where students have opportunities to reason and construct their understandings as part of a community of learners. Despite some successes, traditional models of instruction still dominate the educational landscape. This limited success can…

The theoretical perspective outlined here offers an alternative to explanations of knowledge transfer that posit its source in the construction and application of abstract, context-independent knowledge structures. A case study analysis of an undergraduate student's attempt to solve a series of problems related to an elementary statistical…

Previous research (e.g., De Bock, 2002) has shown that--due to the extensive attention paid to proportional reasoning in elementary and secondary mathematics education--many students tend to overrely on proportional methods in diverse domains of mathematics (e.g., geometry, probability). We investigated the development of misapplication of…

In this article, I present a study in which 12 pairs of 8th-grade students solved problems about a physical device with algebra. The device, called a winch, instantiates motions that can be modeled by pairs of simultaneous linear functions. The following question motivated the study: How can students generate algebraic models without direct…

The premise of this article is that the study of representation is valuable and important for mathematics and science students. Learning about representation should go beyond learning specific, sanctioned representations emphasized in standard curricula (graphs, tables, etc.) to include principles and design strategies that apply to any scientific…

Analogical reasoning has long been believed to play a central role in mathematics learning and problem solving (see Genter, Holyoak, & Kokinov, 2001); however, little is known about how analogy is used in everyday instructional contexts. This article examines analogies produced in naturally occurring U.S. mathematics lessons to explore patterns…

Investigated the ways teachers' content knowledge affected implementation of mathematics education reform by observing teachers during a reform-based algebra lesson. Found that teachers learned during this process by modifying existing content knowledge and developing new content knowledge and appropriate strategies, and that teachers negotiated…

Compared two types of case-based hypermedia for teachers' professional development: one in which cases are episodes of classroom teaching and learning, the other in which cases are narratives that structure the episodes. Comparison showed that while both types helped teachers learn the material, access to narrative cases afforded greater…

Explores school districts' responses to recent mathematics reforms. Notes that district leaders' understanding focused on piecemeal changes that often miss the disciplinary particular of the reform. Advocates inclusion of implementers' interpretation of the reform message with more conventional variables, such as local resistance to reform and…