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…

Drawing on the work of Gattegno, it is suggested that a powerful way of teaching mathematics is to introduce symbols as relationships between visible or tangible resources. The symbols are abstract (formal) from the beginning and yet there are concrete resources to support their use. Drawing on data from a research project in primary schools in…

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…

Before starting school, many children reason logically about concepts that are basic to their later mathematical learning. We describe a measure of quantitative reasoning that was administered to children at school entry (mean age 5.8 years) and accounted for more variance in a mathematical attainment test than general cognitive ability 16 months…

In this third of a series of four articles, the author deconstructs the primary national strategy's approach to written multiplication. The approach to multiplication, as set out on pages 12 to 15 of the primary national strategy's "Guidance paper" "Calculation" (DfES, 2007), is divided into six stages: (1) mental…

In this study, the authors wish to explore the use of visual representations in facilitating the understanding of multiplication. In doing so, they examine the different aspects of multiplication that they can access through different representations. In addition, they draw on a study that they have been carrying out looking at pupils' actual use…

Describes how five-year-old children responded to the investigation of looking for patterns in a multiplication square. Provides a 10 x 10 multiplication square for the investigation. (YP)

Reanalyzes data obtained in 1987 on mental calculation strategies used by 11-year-olds in England, Wales, and Northern Ireland. Classifies mental strategies developed in the past decade in international research. Compares frequency and effectiveness of the strategies used by pupils of different levels of attainment. Discusses basic arithmetic…

Interviewed 59 2nd- and 44 3rd-year elementary school students concerning their solutions to arithmetic problems commensurate with their age and ability. The results indicate that as children progress through school, they continue to use counting as an important part of their problem-solving repertoire, combining counting skills in idiosyncratic…

Mathematics is a core subject in the UK curriculum, with high standards achievement a policy of the UK government. Innovative approaches and strategies have been advocated to achieve these high standards. One approach that has been advocated is Precision Teaching, supported by its successful application in schools in the United States. For this…

The authors use digital photographs to help children recall "key events" in a practical activity--those moments that provide rich learning opportunities. The captured images of particular events helped the children to recall what they were doing at a particular time and prevented confusion over which event was being discussed. They also…