Diagrams and in general the use of visualization and manipulative material play an important role in mathematics teaching and learning processes. Although several authors warn that mathematics objects should be distinguished from their possible material representations, the relations between these objects are still conflictive both from an…

In this study, we present a mathematical analysis distinguishing two conceptions of equivalence: "proportional equivalence" and "unit equivalence." These two conceptions have distinct meanings in relation to equivalent fractions: one is grounded in proportionality, while the other is grounded in equal wholes. We argue that (a)…

Fractions are an imperative skill for students to master to achieve success in future mathematics concepts and classes. Yet many students, especially those experiencing math difficulties and/or characteristics of emotional and behavioral disorders, continue to exit elementary school without a concrete foundation of fractions skills required to…

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 paper conceptualizes children's mathematical thinking from a materialist perspective on language and mathematics. This perspective considers human and non-human bodies as ontologically equivalent; that is, as both being agentive, vibrant, and animated, thus resisting static representations. This conceptualization is an alternative to the…

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…

In the target article, Xing and colleagues (2021) claimed that 6- to 8-year-olds who spontaneously referenced the midpoint of 0-100 number lines made more accurate magnitude estimates and scored higher on a standardized math achievement test than other children. Unlike previous studies, however, the authors found no relation between accuracy on…

Nat Banting, argues that the "invert and multiply" algorithm for division of fractions is, in fact, not an algorithm. He suggests that Liping Ma's (1999) algorithm, which involves writing the fractions with a common denominator and then dividing the numerator of the first fraction by the numerator of the second fraction, is better as it…

Developing an understanding of fractions is critical and is an educational focus, as reflected in national standards and principles. This study conducted a quantitative synthesis of 22 experimental studies to investigate the overall efficacy of previously conducted fraction interventions compared to standard instruction. Instruction type and…

In this paper, we explore theoretical approaches to Japanese teachers' lesson designs involving the adaptation of mathematics textbooks for instructional change. In Japan, although most teachers use textbooks as the main resource for lesson planning, called "kyouzai kenkyuu," the ways in which they use textbooks can differ. In a…

The intent of this commentary is to identify elements of fraction sense and note how the research studies provided in this special issue, in related but somewhat different ways, validate the importance of such understandings. Proficiency with fractions serves as a prerequisite for student success in higher level mathematics, as well as serving as…

"Fraction Face-Off!" is a supplemental math program designed to support fourth-grade students who need assistance solving fraction problems. Teachers use program materials with individual students or small groups to promote understanding of the magnitude of fractions, to compare two fractions, to put three fractions in order, and to…

"Fraction Face-Off!" is a supplemental math program designed to support fourth-grade students who need assistance solving fraction problems. Teachers use program materials with individual students or small groups to promote understanding of the magnitude of fractions, to compare two fractions, to put three fractions in order, and to…

"Fraction Face-Off!" is a supplemental math program designed to support fourth-grade students who need assistance solving fraction problems. Teachers use program materials with individual students or small groups to promote understanding of the magnitude of fractions, to compare two fractions, to put three fractions in order, and to…

Felix Klein's Erlangen program classifies geometries based on the kinds of geometric transformations that preserve key properties of their figures, rather than focusing on the geometric figures themselves. This shift in perspective, from figurative to operative, fits Piaget's characterization of mathematical development. This paper considers how…