The development of conceptual multiplication knowledge will assist students in making progress within current mathematics standards. Previous research has shown the concrete-representational-abstract (CRA) sequence to be successful in teaching multiplication with regrouping with an emphasis on conceptual understanding while developing fluency and…

Students in upper secondary education encounter difficulties in applying mathematics in physics. To improve our understanding of these difficulties, we examined symbol sense behavior of six grade 10 physics students solving algebraic physic problems. Our data confirmed that students did indeed struggle to apply algebra to physics, mainly because…

The part-whole multiplicative relationship, as a topic that gives rise to the concept of fraction, is fundamental in education at the primary school level, and must therefore be included in training courses for prospective primary school teachers (PSTs). In this paper, we introduce a first study of a larger project, which aims to understand the…

Incremental rehearsal (IR) has consistently led to effective retention of newly learned material, including math facts. The number of new items taught during one intervention session, called the intervention set, could be used to individualize the intervention. The appropriate amount of information that a student can rehearse and later recall…

A deep understanding of fraction concepts and operations is necessary if pre-service teachers (PSTs) are to present the concepts in multiple forms to learners. Such an understanding needs to be grounded in rich conceptual knowledge. In the present study, we explore the development of this understanding by supporting a cohort of 103 PSTs, who had…

Multiplicative thinking is a critical component of mathematics which largely determines the extent to which people develop mathematical understanding beyond middle primary years. We contend that there are several major issues, one being that much teaching about multiplicative ideas is focussed on algorithms and procedures. An associated issue is…

There is a renewed debate about whether educated adults solve simple addition problems (e.g., 2 + 3) by direct fact retrieval or by fast, automatic counting-based procedures. Recent research testing adults' simple addition and multiplication showed that a 150-ms preview of the operator (+ or ×) facilitated addition, but not multiplication,…

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…

Students learning to separate variables in order to solve a differential equation have multiple ways of correctly doing so. The procedures involved in "separation" include "division" or "multiplication" after properly "grouping" terms in an equation, "moving" terms (again, at times grouped) from…

In this paper, we present, analyse and critique an episode from a secondary school lesson involving an introduction to the definition of the scalar product. Although the teacher attempted to be explicit about the difference between a definition and a theorem, emphasizing that a definition was just an arbitrary assumption, a student rejected the…

This article investigates the impact of research findings on the representation of proportional reasoning in two commonly used Swedish mathematics textbook series for grades 7-9. A research-based framework that identifies five learning goals for understanding of proportional reasoning was used to analyse the textbooks. The results brought to…

Lesson study cultivates teachers' capacity for formative assessment by placing student thinking front and center throughout. Lesson study is a form of professional development in which a team of teachers determines a mathematical focus, collaboratively studies student thinking about the topic, designs a lesson about this content, implements the…

Parity helps us determine whether an arithmetic equation is true or false. The current research examines the development of sensitivity to parity cues in multiplication in typically achieving (TA) children (grades 2, 3, 4 and 6) and in children with mathematics learning disabilities (MLD, grades 6 and 8), via a verification task. In TA children…

Experts think in patterns and structures using the specific "language" of their domains. For mathematicians, these patterns and structures are represented by numbers, symbols and their relationships (Stokes, 2014a). To determine whether elementary students in the United States could learn to think in mathematical patterns to solve…

Designing and developing games for learning is a difficult endeavor. Educational game designers must not only make an engaging and motivating game, but must also ensure that learning takes place as a result of gameplay. Educational researchers have sought to define design principles in order to lessen the difficulty involved with game design. In…