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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 teaching for understanding and procedural teaching is highlighted. An analogy is drawn with Charles Dickens' character, Mr Thomas Gradgrind, who would have endorsed procedural teaching and abhorred teaching that encouraged understanding. The work of four primary students is discussed in terms of their contextual understanding of multiplicative concepts. The difference between teaching for understanding and procedural teaching is highlighted.

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 multiplicative thinking. They contend that many primary-aged children are taught algorithms for multiplication and division without an appropriate understanding of the mathematical structure and concepts that underpin those algorithms.

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 important ideas--grid multiplication and extended multiplication facts. The graphic combines a number of learning pathways that lead to those two ideas. The article goes on to demonstrate how each element builds part of the underpinning structure needed to understand algorithms and to use them efficiently.

A journey into multiplicative thinking by three teachers in a primary school is reported. A description of how the teachers learned to identify gaps in student knowledge is described along with how the teachers assisted students to connect multiplicative ideas in ways that make sense.

It is important for students to make informed decisions about computation. This article highlights this importance and develops a framework which may assist teachers to help students to make effective computational choices.

Multiplicative thinking is accepted as a "big idea" of mathematics that underpins important mathematical concepts such as fraction understanding, proportional reasoning, and algebraic thinking. It is characterised by understandings such as the multiplicative relationship between places in the number system, basic and extended number facts, and properties of operations and associated relationships. Using examples from a current Year 6 research project, this article highlights the importance of a combination of conceptual understanding and procedural fluency in developing multiplicative thinkers.

The relationships between three critical elements, and the associated mathematical language, to assist students to make the critical transition from additive to multiplicative thinking are examined in this article by Chris Hurst.

Most adults' attitudes to mathematics come from their experiences of mathematics in school when they were children. Children's mathematical worlds are complex places containing both cognitive and affective elements. One cannot ignore the affective domain if one wishes to understand children's mathematical learning. Teacher education students consistently use the following words in describing their memories of mathematics: repetition, monotony, boredom, not understanding, mystery, failure irrelevance, and humiliation. These memories group into feelings relating to the teaching of mathematics and their personal reaction to themselves as mathematics learners. In this article, the authors remind readers of the importance of helping students to develop positive attitudes to mathematics and provide practical suggestions as to how to engage students in a variety of stimulating activities. (Contains 4 figures.)

The plethora of data that confronts teachers on a daily basis requires that they know more than simply being able to calculate. It demands that they understand the context in which the mathematical ideas are embedded and what those ideas are telling them in relation to that context. Also, it necessitates that they are prepared to question how data, and the associated claims, are presented. For primary teachers, the implication is that they must be prepared to help their students to develop these skills. The "Finding the Maths" strategy described in this article has been shown to be successful in helping students achieve this outcome. The aim of using this strategy is to encourage students to develop the three types of thinking referred to in the numeracy framework (fluent operator, mathematical learner, and critical mathematician) and to assume the roles of the "mathematical learner" and "critical mathematician." This strategy was used in tandem with the "Debriefing the Numeracy" strategy that was designed to focus student thinking on the embedded mathematics in a particular context. (Contains 2 figures and 6 student work samples.)

The article discusses the efforts and studies of three primary school teachers within their professional learning community regarding developing multiplicative thinking in students. Topics discussed include the impact of mathematical concepts taught by the primary school, suggestions for teachers on how to develop an action plan, and the development of mental computation strategies.

The article discusses the significance of a combination of conceptual understanding and procedural fluency in developing multiplicative thinkers in the field of mathematics. Multiplicative thinking underpins important mathematical concepts and is characterized by an ability to recognise and solve a range of mathematical problems.