This paper stems from calls for methodological advancement in two areas of research: primary mathematics education and dialogical education. We respond to these calls from the commognitive standpoint by introducing the "Grove of Realizations" as a tool for capturing and visualizing individual students' learning through collaborative work…

This paper reports on exploratory research into visual imagery in the learning of early mathematics. It aims to present a theoretical position in relation to perceptual perspectives of visual imagery. Students' visual imagery is explored in one example of a teacher's use of a protocol to encourage students to replicate dot patterns. This example…

A robust understanding of place value is essential. Using a problem-based approach set within meaningful contexts, students' attention may be drawn to the multiplicative structure of place value. By using quotitive division problems through a concrete-representational-abstract lesson structure, this study showed a powerful strengthening of Year 3…

In this paper we report on two assessment tasks extracted from a larger study. The tasks involved number-line placements on two different number lines (0-to-10 and 0-to-20) and place-value understanding. Participants were 119 children from four different classes (Years 1-3). Children's placements were more accurate on the 0-to-20 than the 0-to-10…

This paper outlines a framework to explain the early development of place-value understanding based on an analysis of data from 84 five- to seven-year-old children from diverse cultural and linguistic backgrounds. The children were assessed individually on number knowledge tasks (recalled facts, subitizing, counting, place-value understanding) and…

This paper focuses on children's number fact knowledge from a study that explored the impact of using multiplication and division contexts for developing number understanding with 34 five- and six-year-old children from diverse cultural and linguistic backgrounds. After a series of focused lessons, children's knowledge of number facts, including…

This paper focuses on selected findings from a study that explored the use of multiplication and division with 34 five- and six-year-old children from diverse cultural and linguistic backgrounds. The focus of instructional tasks was on working with groups of ten to support the understanding of place value. Findings from relevant assessment tasks…

The importance of understanding the various uses of the literal symbol in algebra, and in particular the idea of generalised number, is well documented in the literature. Many research findings have also reported student difficulties with this vital and central concept. This research study examines the use of a combination of historical and…

This paper challenges the emphasis on counting in New Zealand's Numeracy Development Project (NDP), arguing that subitizing provides an alternative pathway to quantification. Longitudinal data is presented showing that children's subitizing skills at the age of five years were a strong predictor of their later success in mathematics at the age of…

This paper examines how a teacher-led cluster of rural schools used evidence of student achievement to improve the quality of learning opportunities for students in mathematics. The teachers began to collect evidence in order to promote constructive dialogue and stimulate action. An inquiry cycle provided feedback to schools through the cluster…

A considerable body of research exists on students' understanding of decimal fractions and the prevalence and persistence of common misconceptions related to this understanding. Results from major studies such as the National Assessment of Educational Progress (NAEP) in the United States and the Concepts in Secondary Mathematics and Science (CSMS)…

This paper reports on an ethnographically based study of entry level supermarket work. The study, carried out in a large suburban supermarket in Auckland, New Zealand, focused on the literacy and numeracy practices of supermarket assistants, all who had English as an additional language. It found that skills such as oral communication, personal…

The importance of student understanding of the concept of place value cannot be underestimated. Place value is a "gate keeper" in developing mathematical understanding. The purpose of this study was to examine and develop a teacher-made test of place value knowledge. The questions were developed using the progressions from the Number…

This publication contains the conference proceedings of the New Zealand Council for Educational Research's (NZCER) mathematics seminar in 1998. Practical insights are offered into topics such as Maori students' participation and performance in mathematics, a conceptual development for multiplication using thinking strategies, practical processes…

Results at the end of one year showed that the experimental group mastered an average of 32 objectives while the control group averaged 15.5, suggesting that hierarchically sequenced mathematics curricula may provide an effective approach to the teaching of number concepts to the severely and moderately retarded. (Author/DLS)