This study used units coordination as a theoretical lens to investigate how whole number and fraction reasoning may be related for preservice teachers at the conclusion of a math methods class. The study contributes quantitative evidence that units coordination provides a common foundation for both mathematical knowledge for teaching whole number…

Learning the concept of fractions is among the most challenging topics in school mathematics. One of the main sources of difficulties in learning fractions is related to "natural number bias" (Van Hoof, Verschaffel & Van Dooren, 2015). Applying properties of the natural numbers incorrectly in situations involving rational numbers can…

To explain children's difficulties learning fraction arithmetic, Braithwaite et al. (2017) proposed FARRA, a theory of fraction arithmetic implemented as a computational model. The present study tested predictions of the theory in a new domain, decimal arithmetic, and investigated children's use of conceptual knowledge in that domain. Sixth and…

To understand misconceptions with rational numbers (i.e., fractions, decimals, and percentages), we administered an assessment of rational numbers to 331 undergraduate students from a 4-year university. The assessment included 41 items categorized as measuring foundational understanding, calculations, or word problems. We coded each student's…

Introduction: In this paper, we present how seventh-grade students become aware of rational numbers during mathematics instruction through the social process of objectification when comparing two magnitudes. Methods: Theoretical elements of the Theory of Objectification and the Onto-Semiotic Approach to Mathematical Instruction and Cognition guide…

This scoping review describes mathematics intervention studies that include initial skill moderation analyses. Sixteen studies were included that focused on students with or at risk for mathematics difficulties (MD). The results revealed that mathematics intervention studies with initial skill moderation analyses focused on students in…

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…

Analogy is a powerful learning mechanism for children to learn novel, abstract concepts from only limited input, yet also requires cognitive supports. My dissertation sought to propose and examine number lines as a mathematical schema of the number system to facilitate both the development of rational number understanding and analogical reasoning.…

This study used fraction number line estimation tasks to evaluate students' developmental patterns, and the relationship of such tasks with whole number bias was explored. In total, 189 fourth-grade students in a northern Taiwan elementary school were followed over 2 years. The results demonstrated that the students' fraction learning development…

Given its pedagogical appeal, the number line is a commonly used representation in the teaching and learning of fractions. However, behind its apparent simplicity, this mathematical object may involve layers of complexity when looked at from the perspective of affordances as is the case in this study. In particular, this in situ exploration…

In this article, I argue that the common practice across many school mathematics curricula of using a variety of different representations of number may diminish the coherence of mathematics for students. Instead, I advocate prioritising a single representation of number (the number line) and applying this repeatedly across diverse content areas.…

This study aimed to reveal the common errors in fractions, the associated thinking strategies among 5th graders, and the persistence of these errors. A quantitative method was applied in this study through calculating the percentages of every type of error the students made in the diagnostic test, The qualitative part was performed through…

This paper shares how fourth grade students solved a unique type of number line task. Instead of giving students the unit and asking them to locate a fraction on the number line, students were given two points on the number line and asked to either find 1, or another number on the number line. This type of task is called "reconstructing the…

The current study investigated the effects of the concrete-representational-abstract integrated sequence (CRA-I) for teaching fifth grade students receiving tier two interventions within a multi-tiered system of support (MTSS). The research questions addressed the following: the extent to which students' rational number skills changed as a result…

Building proficiency with fraction arithmetic poses a consistent challenge for students with learning difficulties or disabilities in mathematics. This article illustrates how teachers can use the number line model to support struggling learners in making sense of fraction arithmetic. Number lines are a powerful tool that can be used to help…