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,"…

In this article, the authors argue that teachers need to possess appropriate understanding and diverse pedagogies to successfully move learners from additive to multiplicative thinking. Herein, they offer five pedagogical strategies that teachers can use to help students make this transition. Some of the challenges of learning multiplicative…

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.

This article describes some of the essential mathematics that underpins the use of algorithms through a series of learning pathways. To begin, a graphic depicting the mathematical ideas and concepts that underpin the learning of algorithms for multiplication and division is provided. The understanding and use of algorithms is informed by two…

Children learn to find answers when multiplying two whole numbers (e.g., 3 × 7 = 21). To this end, they may repeatedly add one number (e.g., 7 + 7 + 7 = 21). But what meanings do they have for multiplication? The authors address this issue while sharing an innovative, playful task called Please Go and Bring for Me (PGBM). Drawing on the…

This is the first of two articles on the use of a written multiplication algorithm and the mathematics that underpins it. In this first article, the authors present a brief overview of research by mathematics educators and then provide a small selection of some of the many student work samples they have collected during their research into…

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…

Productive struggle leads to productive classrooms where students work on complex problems, are encouraged to take risks, can struggle and fail yet still feel good about working on hard problems (Boaler, 2016). Teachers can foster a classroom culture that values and promotes productive struggle by providing students with challenging tasks. These…

In order for students to move from using concrete materials to using mental strategies and from additive to multiplicative thinking, the use of arrays and visualisation is pivotal. This article describes a lesson in which students are taken through a Concrete-Representational-Abstract (CRA) approach that involves noticing structure, using…

The movie "Frozen" took the world by storm and this global popularity of the movie and its music can be harnessed by teachers of mathematics. This article builds on the "frozen fractal" lyric from "Let It Go" to incorporate fractal geometry into primary mathematics classrooms.

Lorraine Day and Derek Hurrell provide a convincing argument for using arrays to promote students' understandings of mental computation strategies for multiplication. They also provide a range of different examples that illustrate the benefits of arrays in the primary classroom.

In this article, Lorraine Jacob and Joanne Mulligan discuss how arrays can be used to promote students' early learning in relation to multiplication and division. They provide examples of activities that can be used from Foundation to Year 5.

Action research (AR) is defined by Macintyre to be: "an investigation, where, as a result of rigorous self-appraisal of current practice, the researcher focuses on a problem,or a topic or an issue which needs to be explained, and on the basis of information, plans, implements, then evaluates an action then draws conclusions on the basis of…

Describes a technique from Japan, the lesson study, where teachers study ways of teaching one lesson for several hours while focussing on the concepts of multiplication and division. (ASK)

Drawing upon research, theory, classroom and personal experiences, this paper focuses on the development of primary-aged children's computational fluency. It emphasises the critical links between number sense and a child's ability to perform mental and written computation. The case of multi-digit multiplication is used to illustrate these…