This article presents results of a research study. Sixty-one first graders' responses to interview questions about negative integer values and order and directed magnitudes were examined to characterize the students' mental models. The models reveal that initially, students overrelied on various combinations of whole-number principles as…

This study investigated the development of length measurement ideas in students from prekindergarten through 2nd grade. The main purpose was to evaluate and elaborate the developmental progression, or levels of thinking, of a hypothesized learning trajectory for length measurement to ensure that the sequence of levels of thinking is consistent…

Quantitative and qualitative analyses of 37 first-graders' solutions to addition problems were conducted to re-examine inconsistencies in children's progress from counting-all to counting-on. The authors applied the constructivist's participatory-anticipatory stage distinction as a tool for fine-grained assessment. Among solutions given by…

This study examines the impact of a nationally disseminated professional development program, Developing Mathematical Ideas (DMI), on teachers' specialized knowledge for teaching mathematics and illustrates how such research could be conducted. Participants completing 2 DMI modules were compared with similar colleagues who had not taken DMI.…

Recent research on mathematics reforms in the United States indicates that the reforms are not yet widely implemented. Generally, this claim results from looking at the extent to which teachers use curricular materials or engage in particular classroom practices. This article moves beyond disparate questions of use and practice to examine…

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…

The framework of Tharp and Gallimore (1988) was adapted to form a ZPD (Zone of Proximal Development) Model of Mathematical Proficiency that identifies two interacting kinds of learning activities: instructional conversations that assist understanding and practice that develops fluency. A Class Learning Path was conceptualized as a classroom path…

The study reported in this article examined relations between ethnicity, out-of-school activities, and arithmetical achievements in Latin American and Korean American children in first, second, and third grades. Three types of data were collected: parents' educational attitudes and beliefs, parents' reports of children's everyday activities with…