The teaching of fractions is often identified as the point in primary school instruction when the "wheels tend to fall off". While there could be many reasons for this, the overuse of one 'preferred' resource at the expense of a range of representations will often create gaps in understanding at best, and misconceptions at its worst. In…

Getting the right answers in maths is only half the problem. Understanding why what you're doing works is the part that often stumps students and teachers alike. The essential guide for mathematics teachers and those training to teach, "Yes, but Why?" answers all your questions, and sheds light on the hidden connections between…

Procedural flexibility is an important skill for algebra. Although prior work has focused on measuring students' procedural flexibility using arithmetic problems, word problems may also capture students' flexibility because of their open-ended nature. To date, no published study has examined the use of word problems as another measure of…

In the present study, we illuminate students' multiplicative reasoning in the context of their units-coordinating activity. Of particular interest is to investigate students' use of three levels of units as given material for problem-solving activity, which we regard as supporting a more advanced level of multiplicative reasoning. Among 13 middle…

Mathematical coherence is a goal within the Common Core State Standards for Mathematics. One aspect of this coherence is how student mathematical thinking is developed across concepts. Unfortunately, mathematics is often taught as isolated ideas across grades. The multiplicative field is an area of study that needs to be examined as a space to…

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…

This paper reports on a key representation, a triple number line, designed as part of the first author's doctoral study. The study sought ways to represent multiple constructs of fractions in the context of merging music and mathematics to support learners' understanding of fractions. A problem scenario was designed guided by Realistic Mathematics…

The concept of fractions given in learning is the concept of part of whole and part of unit. The development of student's concepts of fractions can be built through fraction schemes. A partitive fraction scheme is a scheme that estimates the size of the fraction in the form of non-units to the whole that is not partitioned. The concept of…

This study focuses on the knowledge revealed and developed by Elementary Mathematics teachers, in a teacher education course related to the representation of fraction division and the flexibility of the reference unit. The teachers solved a task aimed at mobilizing (and accessing) their knowledge related to their approaches to the sense of…

The purpose of this article is to validate the relevance of a concept inventory on fractions by measuring the presence and evolution of misconceptions among prospective primary and pre-school teachers, including the overcoming of their misconceptions during and at the end of the instructional intervention. Seven text statements were defined and…

The algorithm for fraction multiplication is relatively easy to memorize and implement with accuracy. The simplicity of the algorithm masks the conceptual complexity involved in making sense of what fraction multiplication means in quantitative situations. One important interpretation of fraction multiplication involves fraction composition,…

The purpose of this study was to understand children's operations on fragmenting schemes while they are engaged with three different visual models, including circle, rectangle, and length model. This study was a sequential explanatory mixed method, which included two phases that happened sequentially with the dominant use of the quantitative…

There is emerging evidence that professional noticing is embodied. Yet, there is still a need to better under embodied noticing at a fundamental level, especially from the preservice teachers. This study used traditional and holographic video, along with eye-tracking technology, to examine how preservice teachers' physical act of looking interacts…

In this article, the authors introduce and illustrate a new activity that involves "follow-up equations," which are equations that follow up on ideas children have already expressed when solving fraction story problems. Specifically, teachers pose story problems and purposely attend to fraction ideas that arise in children's strategies…

Children display an early sensitivity to implicit proportions (e.g., 1 of 5 apples vs. 3 of 4 apples), but have considerable difficulty in learning the explicit, symbolic proportions denoted by fractions (e.g., "1/5" vs. "3/4"). Theoretically, reducing the gap between representations of implicit versus explicit proportions…