Despite substantial research exploring multiplicative thinking and students' difficulty in the domain, the topic of multi-digit multiplication is under-researched. In this paper, I share a learning trajectory for multi-digit multiplication that combined social and cognitive perspectives of learning. Using Design Research methods and involving 45…

Research has shown that many primary students experience transition barriers between additive and multiplicative thinking. This paper analysed responses from 253 Years 3 to 6 students to a diagnostic assessment which consists of whole number multiplication and division problems involving equal groups, arrays, multiplicative comparison and…

Children's multiplicative thinking as the recognition of equal group structures and the enumeration of the composite units was the subject of this research. In this paper, we provide an overview of the Multiplication and Division Investigations project. The results were obtained from a small sample of Australian children (n = 21) in their first…

This paper reports on 73 Grades 3 to 6 students' written responses to equal groups, arrays, multiplicative comparison and Cartesian product word problems. It is part of a larger study relating to students' development of multiplicative thinking (MT). The potential of using multiplicative word problems as a diagnostic assessment to reveal students'…

The Times Tables Drill, in which students recite multiplication facts at high speed, is based on a 200 year-old American military education practice. Emphasising "commands," "obedience," "rules," "silence," "separation," "surveillance," "speed," "competition,"…

The array is a powerful tool that builds students' understanding in multiplication. Students' interactions with the array changes through the course of an instructional sequence, which can be viewed as a process of "reification." In this paper, I report the findings of a research study conducted with 45 Year 5 students in Sydney. The…

Problem solving and reasoning are two key components of becoming numerate. Reports obtained from international assessments show that Australian students' problem solving ability is in a long-term decline. There is little evidence that teachers are embracing problem solving as part of the classroom routine. In this study, we analyse 598 Year 7 to…

Progressing from additive to multiplicative thinking is a key outcome of school mathematics, making ratios an essential topic of study in junior secondary. In this study, 15 Australian Year 8 students were administered a ratio test followed by semi-structured interviews to explore their conceptions of ratio prior to formal instruction. In this…

The shift from additive to multiplicative thinking is challenging for students. A professional learning program was developed that focused on an identified area of need by teachers, namely multiplicative thinking. Program content focused on concepts underpinning multiplicative thinking, pedagogical approaches, challenging tasks, and application to…

The authors define two mathematical cognitive verbs which are fundamental to the development of mathematical thinking and reasoning. They distinguish between 'describing' and 'explaining' in relation to doing mathematics, rather than using them interchangeably.

The purpose of this work is to explore alternative geometric pedagogical perspectives concerning justifications to 'fast' multiplication algorithms in a way that fosters opportunities for skill and understanding within younger, or less algebraically inclined, learners. Drawing on a visual strategy to justify these algorithms creates pedagogical…

In the context of a multiplicative problem, our study investigated young children's ability to visualise and draw equal groups. This paper reports the results obtained from 18 Australian children in their first year of school (age 5-6 years). The task "12 Little Ducks," taught by their classroom teacher, provoked children to visualise…

Making connections within and between different aspects of mathematics is recognised as fundamental to learning mathematics with understanding. However, exactly what these connections are and how they serve the goal of learning mathematics is rarely made explicit in curriculum documents with the result that mathematics tends to be presented as a…

Specialised Content Knowledge (SCK) is defined by Ball, Hoover-Thames, and Phelps (2008) as mathematical knowledge essential for effective teaching. It is knowledge of mathematics that is beyond knowledge which would be required outside of teaching; for instance, the capacity to determine what misconception(s) may lie behind an error in…

Multiplication and division are complex constructs for young children to understand. In this article, the authors explore young children's ability to visualise and make groups of objects before they had formally been introduced to multiplication and division. The results provide some valuable insights into students' early understandings of…