Counting is fundamental to early mathematics. Most studies of teaching counting focus on teachers observing children count. The present study compares mathematical ideas that 12 PK, transitional kindergarten (TK), and kindergarten teachers noticed from observing their own students count during a classroom session of Counting Collections with ideas…

The belief that mathematics ability is a fixed trait is particularly common and may be a key reason for many students' disinterest and underperformance in mathematics. This study investigates how mathematics teaching practices might contribute to students' beliefs about mathematics ability being a fixed or malleable trait (mindset). Through a…

To learn mathematical practices, students need opportunities to engage in them. But simply providing such opportunities may not be sufficient to support all students. Simultaneously, explicitly teaching mathematical practices could be problematic if instruction becomes prescriptive. This study investigated how teachers might make mathematical…

We identify and document 3 cognitive obstacles, 3 cognitive affordances, and 1 type of integer understanding that can function as either an obstacle or affordance for learners while they extend their numeric domains from whole numbers to include negative integers. In particular, we highlight 2 key subsets of integer reasoning: understanding or…

Research in experimental and developmental psychology, cognitive science, and neuroscience suggests that tool fluency depends on the merging of perceptual and motor aspects of its use, an achievement the authors call "perceptuomotor integration." Just as expertise in playing a piano relies on the interanimation of finger movements and…

How issues of power and identity play out in mathematical practices are described by Na'ilah Suad Nasir and Maxine McKinney de Royston. Their article offers a perspective on how to better understand the sociopolitical nature of teaching and learning mathematics. They present data from studies of mathematics teaching and learning in out-of-school…

Mathematics classrooms are conceptualized as heterogeneous spaces in which multiple figured worlds come into contact. The study explores how a group of high school students drew upon several figured worlds as they navigated mathematical discussions. (Contains 5 excerpts and 2 footnotes.)

A case study examines a 5th-grade teacher's orchestration of discourse and interaction to create opportunities for English language learners to participate in the repair of mathematical errors during a unit on finding the area of geometric shapes. The findings detail how the teacher used gestures in grounding her questioning, revoicing students'…

There is widespread agreement that effective teachers have unique knowledge of students' mathematical ideas and thinking. However, few scholars have focused on conceptualizing this domain, and even fewer have focused on measuring this knowledge. In this article, we describe an effort to conceptualize and develop measures of teachers' combined…

California is one of 4 states that have accelerated addition and subtraction basic-facts memorization. This article reports on teacher practices, first-grade achievement of the standard, and a broader conception of basic-facts competence. Even among students from the highest performing schools, fewer than 11% made progress toward the memorization…

A yearlong experimental study showed positive effects of a professional development project that involved 19 urban elementary schools, 180 teachers, and 3735 students from one of the lowest performing school districts in California. Algebraic reasoning as generalized arithmetic and the study of relations was used as the centerpiece for work with…

Assessment test data from 1976-79 on California children in grade six were analyzed. Loglinear models were used to evaluate the consistency of response of each sex. A significant interaction between sex and choice of distractor occurred for a large majority of the items. (MNS)

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…