In this second of a series of four articles, the author deconstructs the primary national strategy's (PNS) approach to written subtraction. The approach to subtraction is divided into three stages: using the empty number line; partitioning; and expanded layout leading to column methods (called "standard methods" in the consultation document). The…

Four hundred and three 3rd- and 5th-grade Chinese students took the Multiplication Estimation Test or participated in the interview on it, designed to assess their computational estimation performance on whole-number multiplication. Students perform better when tasks are presented visually than orally. Third graders tend to use rounding based…

The goal of this study is to uncover the successes and challenges that preservice teachers are likely to experience as they unpack lesson-level mathematical learning goals (i.e., identify the subconcepts and subskills that feed into target learning goals). Unpacking learning goals is a form of specialized mathematical knowledge for teaching, an…

Children in third and fourth grades suffering from developmental dyscalculia (DD) and typically developing children were asked to compare numbers to a standard. In two separate blocks, they were asked to compare a number between 1 and 9 to 5, or a two-digit number between 10 and 99 to 55. In the single-digit comparisons, DD children were…

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…

I describe how data pooling and data visualization can be employed in the first-semester general chemistry laboratory to introduce core statistical concepts such as central tendency and dispersion of a data set. The pooled data are plotted as a 1-D scatterplot, a purpose-designed number line through which statistical features of the data are…

This article focuses on spontaneous and progressive knowledge building in ''the arithmetic of the child.'' The aim is to investigate variations in the behavior patterns of eight pupils attending a school for the intellectually disabled. The study is based on the epistemology of radical constructivism and the methodology of multiple clinical…

Preschoolers made numerical comparisons between sets with varying degrees of shared surface similarity. When surface similarity was pitted against numerical equivalence (i.e., crossmapping), children made fewer number matches than when surface similarity was neutral (i.e, all sets contained the same objects). Only children who understood the…

This qualitative study explored the integration of religion and spirituality in counseling among six African American Christian college students ranging from age 26-48. Specifically, attention was given to the circumstances that either facilitated or discouraged the student's willingness to include their faith system in counseling. Thematic…

Sets of numbers where not only their sums are equal but the sums of other powers are also equal have been called multigrades. This article presents several mathematical equations that portray how multigrades are generated. By further extension of the process outlined in this article, students can generate higher-order multigrades. (Contains 1…

Despite widespread agreement that the activity of "reasoning-and-proving" should be central to all students' mathematical experiences, many students face serious difficulties with this activity. Mathematics textbooks can play an important role in students' opportunities to engage in reasoning-and-proving: research suggests that many decisions that…

Two intervention studies are described. Both were designed to study the effects of teaching children about the inverse relation between addition and subtraction. The interventions were successful with 8-year-old children in Study 1 and to a limited extent with 5-year-old children in Study 2. In Study 1 teaching children about inversion increased…

There are two different conceptions of the innate basis for numerical abilities. On the one hand, it is claimed that infants possess a "number module" that enables them to construct concepts of the exact numerosities of sets upon which arithmetic develops (e.g. Butterworth, 1999 ; Gelman & Gallistel, 1978). On the other hand, it has been proposed…

There is evidence for both semantic and asemantic routes for naming Arabic digits, but neuropsychological dissociations suggest that number-fact retrieval (2x3=6) can inhibit the semantic route for digit naming. Here, we tested the hypothesis that such inhibition should slow digit naming, based on the principle that reduced access to multiple…

This study investigated the effect of preschool experience (two types of preschool: Madrasa and non-Madrasa) on the cognitive development of children in East Africa. In the three countries studied (Kenya, Uganda, and Tanzania/Zanzibar) preschool education is burgeoning and government standards are being set. This quasi experimental evaluation used…