Competence with rational numbers is essential for mathematics proficiency in secondary mathematics. However, many students struggle with rational number concepts, and students with mathematics difficulties struggle even more. The purpose of this study was to examine the effects of an intervention that incorporated the use of explicit instruction…

Multiplication can be presented to students through different models, each one with its pros and cons. In this contribution we focus on the repeated sum and the array model to investigate the relations between the two models and those between them and multiplication properties. Formal counterparts are presented. Taking both a mathematical and…

In this article, we report the findings of a study conducted with 6 Grade 8 students in the United States. The students posed stories for open number sentences involving addition and subtraction of integers. We analysed the stories posed by the students to build models that describe the conceptual structures behind these posed stories -- the…

The number line and the neutralization model have been used very extensively in teaching integer additions and subtractions for decades. Despite their advantages, issues concerning subtractions on these models are still debatable. Therefore, the neutralization on an empty number line (NNL) model is proposed in this research to help students better…

Thomas Kuhn (1962/2012) introduced the term "paradigm shift" to the scientific literature to describe how knowledge in science develops. The aims of this article are to identify paradigm shifts, or revolutions, that have occurred in mathematics, and to discuss their relevance to teaching mathematics in schools. The authors argue that…

Practicing teachers as well as researchers, mathematicians, and teacher educators have offered opinions and theoretical critiques of the multiple models used to teach integer arithmetic. Few studies, however, have investigated what students learn with models or empirically compared affordances and constraints of integer models. This led me to…

Over the past few decades there has been increased interest in how students and teachers think and learn about negative numbers from a variety of perspectives. In particular, there has been debate about when integers should be taught and how to teach them to best support students' learning. This book brings together recent work from researchers to…

The author wants her students to see any new mathematics--fractions, negative numbers, algebra--as logical extensions of what they already know. This article describes two students' efforts to make sense of their conflicting interpretations of 1/2 × -6, both of which were compelling and logical to them. It describes how discussion, constructing…

Although the general development of mathematical abilities in primary school has been the focus of many researchers, the development of place value understanding has rarely been investigated to date. This is possibly due to the lack of conceptual approaches and empirical studies related to this topic. To fill this gap, a theory-driven and…

The present study revalidated a measurement model describing the nature of early number sense. Number sense was shown to be composed of elementary number sense, conventional arithmetic and algebraic arithmetic. Algebraic arithmetic was conceptualized as synthesis of number patterns, restrictions and functions. Two hundred and four 1st grade…

The concept of exponents has been shown to be problematic for students, especially when expanding it from the domain of positive whole numbers to that of exponents that are negative and later rational. This paper presents a theoretical analysis of the concept of exponentiation as a continuous operation and examines the deficiencies of existing…

This study examined 361 Chinese and 345 Singaporean sixth-grade students' performance and problem-solving strategies for solving 14 problems about speed. By focusing on students from two distinct high-performing countries in East Asia, we provide a useful perspective on the differences that exist in the preparation and problem-solving strategies…

The authors asked fifth-grade and eighth-grade students to pose stories for number sentences involving the addition and subtraction of integers. In this article, the authors look at eight stories from students. Which of these stories works for the given number sentence? What do they reveal about student thinking? When the authors examined these…

In 2007, a group of mathematics educators and researchers met to examine rational numbers and why children have such an issue with them. An extensive review of the literature on fractional understanding was conducted. The ideas in that literature were then consolidated into a theoretical framework for examining fractions. Once that theoretical…

The intention of the present study is to know how the pupils can learn to make a group of ten to understand the idea of unitizing. The pupils were given a contextual problem "Counting the Beads" in order to promote their understanding about the idea of unitizing. The process of designing the problem was based on the 5 tenets of…