Evidence suggests that some students have learned procedures with little or no underpinning understanding while others have a much more connected and conceptual levels of understanding. In this article, the work of four primary students is discussed in terms of their contextual understanding of multiplicative concepts. The difference between…

Specific acts of problem solving with rates and ratios were interpreted using a journey metaphor derived from Skemp's (1979) construct of director systems. To successfully undertake a problem solving journey a learner must recognise their starting place (present state), have a sense of destination (goal state), and co-ordinate sub-routes along the…

There are many potential ways to represent arithmetical tasks, but students' choices may be limited by beliefs that only certain standardised representations are "legitimate" in school mathematics. Furthermore, concern for the quantity and speed of "work done" can override opportunities for meaningful engagement with the…

The goal of this qualitative study is to provide theoretical knowledge and design principles for a creative educational environment characterized by simultaneous study and exploration of science or math, and the arts: Write a Science Opera (WASO). To do so, we used a theory of creativity in education which links collaborative co-creation in…

This study examined immigrant parents' involvement in early years mathematics learning, focusing on learning of multiplication in in- and out-of-school settings. Ethnographic interviews and workshops were conducted in an urban city in Japan, to examine out-of-school practices of immigrant families. Drawing from sociocultural theory of learning and…

The move from additive to multiplicative thinking requires significant change in children's comprehension and manipulation of numerical relationships, involves various conceptual components, and can be a slow, multistage process for some. Unit arrays are a key visuospatial representation for supporting learning, but most research focuses on 2D…

Multiplicative thinking has been widely accepted as a critically important "big idea" of mathematics and one which underpins much mathematical understanding beyond the primary years of schooling. It is therefore of importance to consider the capacity of children to think multiplicatively but also to consider the capacity of their…

This research documents the aims and the impact of a teaching experiment on how the absorption of light depends on the thickness of the absorbing medium. This teaching experiment is more specifically characterized as bringing to bear a "concept-driven interactive pathway". It is designed to make students analyse the absorption of light…

This research aims to analyze ways of understanding of students with mathematics learning disabilities when learning fraction. The research was conducted in an Inclusive Junior High School in the West Java Province, Indonesia. This study is qualitative, with the single-case (holistic) designs. The case will focus on three students who suspected of…

This paper reports on 13 Grade 3 students' approaches to partitive and quotitive division word problems. Of particular interest was the extent to which students drew on their knowledge of multiplication to solve division problems. The findings suggest that developing a relationship between multiplication and division is more significant than…

This paper presents findings as part of a wider study that investigated the professional knowledge of teachers when teaching mathematics for numeracy in the primary school classroom. This paper focuses on teachers in action as they taught two lessons on multiplication. It outlines the specific pedagogical categories the teachers used and the…

This paper shares research from a pilot study in which young children were introduced to multiplication and division problems in their first year of school. The focus was on building children's conceptual understanding of the idea of "repeated groups" as a fundamental aspect of multiplication and its relation to division. The particular…

We examine how rectangular array model can support students' spatial structuring in learning multiplication. To begin, we define what we mean by spatial structuring as the mental operation of constructing an organization or form for an object or set of objects. For that reason, the eggs problem was chosen as the starting point in which the…

This study investigated children's understanding of area measurement, including the concept of area and the area formula of a rectangle, as well as their strategic knowledge for solving area measurement problems. Twenty-two fourth-graders from three classes of a public elementary school in Taipei, Taiwan, participated in a one-on-one interview.…

This paper reports on an investigation into pre-service teachers' mathematical content knowledge and their ability to interpret students' responses to a multi-digit multiplication task and make subsequent appropriate teaching decisions. Using a combination of quantitative and qualitative methods, the researchers tested aspects of the mathematical…