Teachers report that engaging students in solving contextual problems is an important part of supporting student understanding of algorithms for fraction division. Meaning for whole-number operations is a crucial part of making sense of contextual problems involving rational numbers. The authors present a developed instructional sequence to…

Today, the Common Core State Standards for Mathematics (CCSSI 2010) expect students in as early as eighth grade to be knowledgeable about irrational numbers. Yet a common tendency in classrooms and on standardized tests is to avoid rational and irrational solutions to problems in favor of integer solutions, which are easier for students to…

In this paper, the use of guided hyperlearning, unguided hyperlearning, and conventional learning methods in mathematics are compared. The design of the research involved a quasi-experiment with a modified single-factor multiple treatment design comparing the three learning methods, guided hyperlearning, unguided hyperlearning, and conventional…

This paper analyses the strategies used by pre-service primary school teachers for solving simple addition problems involving negative numbers. The findings reveal six different strategies that depend on the difficulty of the problem and, in particular, on the unknown quantity. We note that students use negative numbers in those problems they find…

This paper focuses on selected findings from a study that explored the use of multiplication and division with 34 five- and six-year-old children from diverse cultural and linguistic backgrounds. The focus of instructional tasks was on working with groups of ten to support the understanding of place value. Findings from relevant assessment tasks…

The purpose of this article is to examine one possible extension of greatest common divisor (or highest common factor) from elementary number properties. The article may be of interest to teachers and students of the "Australian Curriculum: Mathematics," beginning with Years 7 and 8, as described in the content descriptions for Number…

Understanding fractions remains problematic for many students. The use of the number line aids in this understanding, but requires students to recognise that a fraction represents the distance from zero to a dot or arrow marked on a number line which is a linear scale. This article continues the discussion from "Identifying Fractions on a…

In this article, we obtain a genetic decomposition of students' progress in developing an understanding of the decimal 0.9 and its relation to 1. The genetic decomposition appears to be valid for a high percentage of the study participants and suggests the possibility of a new stage in APOS Theory that would be the first substantial change in…

The development of numerical concepts is described from infancy to preschool age. Infants a few days old exhibit an early sensitivity for numerosities. In the course of development, nonverbal mental models allow for the exact representation of small quantities as well as changes in these quantities. Subitising, as the accurate recognition of small…

Recent research on fractions has broadened and deepened theories of numerical development. Learning about fractions requires children to recognize that many properties of whole numbers are not true of numbers in general and also to recognize that the one property that unites all real numbers is that they possess magnitudes that can be ordered on…

In this article, we present some mathematical tasks designed to support pre-service teachers (PSTs) in obtaining a broader, more connected understanding of decimal numbers and quantities. We describe classroom scenarios from a content course in which PSTs experienced these cognitively demanding decimal tasks, enacted in ways that maintain high…

Recent work by researchers has focused on synthesizing and elaborating knowledge of students' thinking on particular concepts as core progressions called learning trajectories. Although useful at the level of curriculum development, assessment design, and the articulation of standards, evidence is only beginning to emerge to suggest how learning…

Analysis of the patterns of signs of infinitely differentiable real functions shows that only four patterns are possible if the function is required to exhibit the pattern at all points in its domain and that domain is the set of all real numbers. On the other hand all patterns are possible if the domain is a bounded open interval.

Before they enter preschool, children vary greatly in their numerical and mathematical knowledge, and this knowledge predicts their achievement throughout elementary school (e.g. Duncan et al., 2007; Ginsburg & Russell, 1981). Therefore, it is critical that we look to the home environment for parental inputs that may lead to these early…

Individual differences in inhibitory control have been shown to relate to general mathematics achievement, but whether this relation varies for specific areas within mathematics is a question that remains open. Here, we evaluate if inhibitory processes play a specific role in the particular case of fraction comparison, where learners must ignore…