The number sequences describe a hierarchy of students' concepts of number. This research uses two defining cognitive structures of the number sequences--units coordination and the splitting operation--to model middle-grades students' abilities to write linear equations representing the multiplicative relationship between two unknowns. Results…

For many students, developing mathematical reasoning can prove to be challenging. Such difficulty may be explained by a deficit in the core understanding of many arithmetical concepts taught in early school years. Multiplicative reasoning is one such concept that produces an essential foundation upon which higher-level mathematical thinking skills…

In an effort to examine the impact of the changes being made at San Jose State University (California) in the calculus curriculum, multiple measures were collected and analyzed. This study focuses on the relationship between performance on a pretest and the class grade. Through written responses on the pretest, a belief and knowledge profile for…

Describes environments for concretely enacting multiplication and division. Discusses difficulties occurring when students use one of the concrete environments to model situations involving modified environments. (YP)

Studied were the misconceptions that preservice elementary teachers have about multiplication and division. Results indicated that they are influenced by the same primitive models as students; the most common errors made by both groups are quite similar. (MNS)

Describes the characteristics of progressive schematization with regard to column multiplication and column division. Contrasts this with column arithmetic based on progressive complexity. Presents a summary of research data concerning column arithmetic. (TW)