The Pythagorean Theorem, arguably one of the best-known results in mathematics, states that a triangle is a right triangle if and only if the sum of the squares of the lengths of two of its sides equals the square of the length of its third side. Closely associated with the Pythagorean Theorem is the concept of Pythagorean triples. A "Pythagorean…

Effects of guided (GI) and direct instruction (DI) in solving subtraction problems for mathematically low performers in regular schools were compared. In the GI condition, self-development of solution procedures was encouraged whereas in the DI condition one prescribed strategy was to be used. Forty children (M[subscript age] = 9.3 years) were…

In this technical report, we describe the development of cognitive diagnostic test items that form the basis of the diagnostic system for Project DIVIDE (Dynamic Instruction Via Individually Designed Environments). The construct underlying the diagnostic test is division of fractions. We include a description of the process we used to identify the…

This study determined the effect of Cuisenaire Rods' approach on some Nigeria primary pupils' achievement in decimal fractions. Three hypotheses guided the study. A total of 200 Primary six pupils (that is, 6th grade) from randomly selected schools in Makurdi metropolis of Benue State of Nigeria served as the sample for the study. A Mathematics…

General Certificate of Secondary Education (GCSE) mathematics assessment previously included a piece of coursework that required learners to develop a general equation to describe the relationship between two groups of numbers on a number grid. In this author's experience, it became clear that Year 10 students often struggled to form a link…

Background: Developmental dyscalculia is a heterogeneous disorder with largely dissociable performance profiles. Though our current understanding of the neurofunctional foundations of (adult) numerical cognition has increased considerably during the past two decades, there are still many unanswered questions regarding the developmental pathways of…

This article uses one of the authors' classroom experiences to explore how teachers can create technology-rich learning environments that support upper elementary students' mathematical understanding of algebra and number and operations. They describe a unit that presents a common financial problem (the use of credit cards) to engage sixth graders…

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…

Dividing by zero can be confusing for students at all levels. If teachers are to provide clear and understandable explanations of this topic, they must possess a strong conceptual understanding of it themselves. In this article, the authors present qualitative data on fourth- through eighth-grade teachers' perceptions of division by zero. They…

A widely recognized concern in elementary school mathematics education is that teachers' understanding of the mathematical curricular content generally appears quite fragmented, sparsely connected, and procedurally oriented. This pilot study applies methods of educational neuroscience to investigate and improve preservice teachers' learning and…

As a group, children from disadvantaged, low-income families perform substantially worse in mathematics than their counterparts from higher-income families. Minority children are disproportionately represented in low-income populations, resulting in significant racial and social-class disparities in mathematics learning linked to diminished…

The current study tested the development of working memory involvement in children's arithmetic strategy selection and strategy efficiency. To this end, an experiment in which the dual-task method and the choice/no-choice method were combined was administered to 10- to 12-year-olds. Working memory was needed in retrieval, transformation, and…

In the 1970s, the movement to the metric system (which has still not completely occurred in the United States) and the advent of hand-held calculators led some to speculate that decimal representation of numbers would render fractions obsolete. This provocative proposition stimulated Zalman Usiskin to write "The Future of Fractions" in 1979. He…

It is one thing to be able to count and share items proficiently, but it is another thing to know how counting and sharing establish and identify quantity. The aim of the study was to identify which measures of numerical knowledge predict children's success on simple number problems, where counting and set equivalence are at issue. Seventy-two…

This study examines the development of children's ability to reason about proportions that involve either discrete entities or continuous amounts. Six-, 8- and 10-year olds were presented with a proportional reasoning task in the context of a game involving probability. Although all age groups failed when proportions involved discrete quantities,…