Functions are all around us, disguised as actions on concrete objects. Composition of functions, too, is all around us, because these actions can be performed in succession, the output of one serving as the input for the next. In terms of Gray and Tall's (2001) "embodied objects" or Lakoff and Nunez's (2000) "mathematical idea analysis," this…

Our primary purpose in this article is to propose an interpretive scheme for analyzing the identities that students develop in mathematics classrooms that can inform instructional design and teaching. We first introduce the key constructs of normative identity and personal identity, and then illustrate how they can be used to conduct empirical…

Sets of mathematics problems are generally arranged in 1 of 2 ways. With "blocked practice," all problems are drawn from the preceding lesson. With "mixed review," students encounter a mixture of problems drawn from different lessons. Mixed review has 2 features that distinguish it from blocked practice: Practice problems on the same topic are…

What factors affect transfer of knowledge is a complex question. In recent research, the authors demonstrated that concreteness of the learning domain is one such factor (Kaminski, Sloutsky, & Heckler, 2008). Even when prompted and given no time delay, participants who learned a concrete instantiation of a mathematical concept failed to transfer…

Mathematics teachers' selection and implementation of instructional tasks were analyzed before, during, and after their participation in a professional development initiative that focused on selecting and enacting cognitively challenging mathematical tasks. Data collected from 18 secondary mathematics teacher participants included tasks and…

Solving arithmetic and algebraic word problems is a key component of the Singapore elementary mathematics curriculum. One heuristic taught, the model method, involves drawing a diagram to represent key information in the problem. We describe the model method and a three-phase theoretical framework supporting its use. We conducted 2 studies to…

The goal of this article is to examine students' mathematical development that occurs as a teacher works within each of 2 zones of learning: students' zones of proximal development (ZPD) and students' zones of potential construction (ZPC). ZPD, proposed by Vygotsky, is grounded in a social constructivist perspective on learning, whereas ZPC,…

This article describes one prevalent practice that a group of 6th- and 7th-grade students engaged in when they used fractions in the context of area and perimeter, decimal operations, similarity, and ratios and proportions. The study's results revealed that students did not simply take the concepts and skills learned in formal fractions units and…

The overreliance on linear methods in students' reasoning and problem solving has been documented and discussed by several scholars in the field. So far, however, there have been no attempts to assemble the evidence and to analyze it is a systematic way. This article provides an overview and a conceptual analysis of students' tendency to use…

Research on the impact of "Standards"-based mathematics and reform calculus curricula has largely focused on changes in achievement and attitudes, generally ignoring how students experience these new programs. This study was designed to address that deficit. As part of a larger effort to characterize students' transitions into and out of reform…

This "Brief Report" summarizes results from a study that investigated joint effects of two innovations adopted at a high school in an affluent suburban community in the northeast United States: 4 x 4 block scheduling and the "Standards"-based curriculum, the Interactive Mathematics Program (IMP).

This article describes one student teacher's interactions with mathematics curriculum materials during her internship in a kindergarten classroom. Anne used curriculum materials from two distinct programs and taught lessons multiple times to different groups of children. Although she used each curriculum in distinct ways, her curriculum use was…

There are three articles in this month's "Research Commentary" section. Analyses of disparities in students' mathematics experiences and outcomes are an essential part of efforts to promote equity. Scholars concerned about equity should not write off such analyses as mere "gap gazing." Research on gaps between underserved groups and their more…

A substantial amount of research in mathematics education seeks to document disparities in achievement between middle-class White students and students who are Black, Latina/Latino, First Nations, English language learners, or working class. I outline the dangers in maintaining an achievement-gap focus. These dangers include offering little more…

In this rejoinder, the authors further detail their positions on the role of gaps analyses in mathematics education research as outlined in the previous 2 articles. They clarify areas of agreement and probe areas of disagreement, focusing on the benefits and dangers they see in either emphasizing educational disparities between groups or shifting…