A teacher of mathematics knows mathematics as a teacher and as a mathematician. Whilst the existing research on teacher knowledge contributes to our understanding of the ways of knowing mathematics as a teacher, little is known about ways of knowing mathematics as a mathematician. Guided by the conceptual framework of mathematical practices (MPs)…

Mathematical reasoning, which plays a critical role in students' capacity to make sense of mathematics, is now emphasised more strongly in various curricula internationally. However, reasoning is sometimes difficult for teachers to recognise, let alone teach. This case study considers video of one teacher's implementation of a problem-solving…

Duncan Symons and Derek Holton discuss the different types of mathematical reasoning and what each of these might look like in the classroom. By suggesting language that can be used to describe the different methods of reasoning, they hope to provide teachers with the tools they need to better recognise and assess student reasoning.

When young students engage with activities aimed at improving spatial reasoning, there is potential to also improve learning and performance in practical mathematics problem-solving tasks, scientific inquiries, and other STEM-based tasks. In this study, students worked in pairs to problem solve through 3D construction. The aim was to assess…

In this brief article, Peter, Larissa and Doug present a single-page scaffold that challenges us to think of the mathematical proficiencies in a whole new way. By thinking of the proficiencies as verbs, they can be seen as purposeful learner actions and offer clues as to what it might mean to be a mathematician.

Estimation skills are critical for students in the 21st century. The ability to estimate or approximate is widely accepted as fundamental to problem solving, and important for checking the reasonableness of results. (Dowker, 2003; Reys & Reys, 1989). Therefore, estimation should be an important aspect of problem solving in all year levels but…

The mathematics standards for many countries include teaching and learning ratios, typically with efficient algorithms and strategies (Australian Curriculum: Mathematics, Assessment and Reporting Authority, 2017; National Council of Teachers of Mathematics, 2014). In the 1990s, Cramer and Post (1993) claimed that school mathematics mostly involves…

This professional practice paper is underpinned philosophically by the indisputable centrality of proof to the discipline of mathematics. Proof offers students the opportunity to deepen their own understanding of mathematical ideas, to construct and defend logical arguments, and to think critically about the veracity of mathematical statements.…

Mathematical reasoning is a key proficiency in mathematics, however primary school teachers may have trouble noticing students' mathematical reasoning mid-lesson. In this study, students' mathematical reasoning has been analysed using video data collected as one grade of Year 5 and 6 students worked through the 'Painted Cube' task. Analysis…

What could two people stand to gain from sharing a taxi ride? We aimed to explore the extent to which this challenging yet accessible financial context might stimulate students' mathematical exploration of multiplicative thinking and proportional reasoning. Through teaching experiment methodology, data were collected from 37 Year 5 and 6 students…

In this paper, we report on the interim findings of a study that seeks to identify the characteristics of children's Critical Mathematical Thinking CMT. Characteristics of CMT were initially generated from a synthesis of relevant research literature and then validated using a case study methodology via trials in early childhood classrooms. This…

The importance of mathematical reasoning is unquestioned and providing opportunities for students to become involved in mathematical reasoning is paramount. The open-ended tasks presented incorporate mathematical content explored through the contexts of problem solving and reasoning. This article presents a number of simple tasks that may be…

This article presents "The Two Children Problem," published by Martin Gardner, who wrote a famous and widely-read math puzzle column in the magazine "Scientific American," and a problem presented by puzzler Gary Foshee. This paper explains the paradox of Problems 2 and 3 and many other variations of the theme. Then the authors…

Explaining appears to dominate primary teachers' understanding of mathematical reasoning when it is not confused with problem solving. Drawing on previous literature of mathematical reasoning, we generate a view of the critical aspects of reasoning that may assist primary teachers when designing and enacting tasks to elicit and develop…

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…