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…

Fluency in mathematics is more than adeptly using basic facts or implementing algorithms. It is not about speed or recall. Real fluency is about choosing strategies that are efficient, flexible, lead to accurate solutions, and are appropriate for the given situation. Developing fluency is also a matter of equity and access for all learners. The…

Using the theoretical constructs of double instrumental genesis and instrumental distance, in this article, I examine case studies of four primary school teachers (K-5) in British Columbia, Canada, who implemented the multi-touch, iPad application TouchTimes (hereafter, TT) into their mathematics teaching. This novel digital technology provides…

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 Akita approach is the guideline in learning that responds to rapid changes in the 21st century. This research aimed to determine the effectiveness of learning management based on the Akita approach in accordance with the criteria efficiency of 75/75 and to explore the index of effectiveness of learning management pursuant to the Akita…

This paper substantiates the notion of inclusive materialism by illustrating the role of signs and micro-perception in the learning process of multiplication. Inclusive materialism, offered by de Freitas and Sinclair, considers mathematical learning a process of becoming a learning assemblage, which embodies various materials in a Deleuzian sense.…

Teacher knowledge is one of the main factors in the quality of mathematics learning. Many mathematics teachers have difficulty using proportional reasoning. Proportional reasoning is one of the essential aspects of the middle school mathematics curriculum to develop students' mathematical thinking. Teachers should realize that developing…

The purpose of this study was to investigate how teachers understand one specific aspect of proportional reasoning. We were interested in understanding the extent to which practicing teachers were able to make sense of reasoning that involved the fixed number of variable-sized parts perspective. We used two items, drawn from a larger dataset, that…

Numeracy is one of the essential competencies that the objectives of teaching math to primary students should be towards. However, many research findings show that the problem of "innumeracy" frequently exists at primary schools. That means children still do not feel at home in the world of numbers and operations. Therefore, the paper…

Multiplicative thinking underpins much of the mathematics learned beyond the middle primary years. As such, it needs to be understood conceptually to highlight the connections between its many aspects. This paper focuses on one such connection; that is how the array, place value partitioning and the distributive property of multiplication are…

Fluent recall of basic facts is essential to the development of more complex math skills. Therefore, failure to develop fluency with basic facts may impede the development of these skills. The present study used a between-group experimental design to investigate whether a basic facts fluency program, implemented within a self-regulated learner…

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…

Examining the use of conceptual frameworks such as the instructional hierarchy (IH) to drive academic interventions represents an important area of inquiry in order to understand why an intervention was effective. However, to date, the IH has only been examined retroactively to explain the effectiveness of math interventions and has not been used…

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…

This paper describes results from a case study about how a primary pre-service teacher (PST) used curriculum materials (CMs) when planning for a mathematics lesson during her final practicum. The data is drawn from a doctoral study (in progress) and results show how the PST initiated an active process of searching for and sifting through CMs on a…