Several working memory processes have been hypothesized to influence different arithmetic operations. Working memory has been compartmentalized into a number of different sub-processes, such as phonological memory and visuospatial memory that are believed to have unique contributions to the performance of two distinct arithmetic operations:…

The purposes of this study are to report current Preservice Teachers' (PTs) decimal abilities regarding the types of error patterns that they commit and to analyze characteristics of the errors in the problems posed for multiplication and division in decimals. We have determined common concept- and procedure-based errors made by the PTs. 37 PTs in…

This study compared two goal-setting approaches found in the Precision Teaching literature, namely the minimum celeration line and the beat your personal best during the mathematical practice of three male students diagnosed with autism, aged 8-9. An adapted alternating treatments design with a control condition was embedded in a concurrent…

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…

In combinatorics, combinatorial notation, e.g., C(n, r), is explicitly defined as a numerical value, a cardinality. Yet, we do not use another symbol to signify the set of outcomes--the collection of objects being referenced, whose cardinality is, for example, C(n, r). For an expert, this duality in notation, of signifying both cardinality and…

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…

For many years, mathematics research has focused on finding ways to help students improve their math performance. These efforts have led to the conclusion that building a strong mathematics foundation is associated with greater achievement across time. Students who develop number sense and learn to use efficient strategies at an early age are more…

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.

This article explores the role that gestures play in the development of mathematical understanding. Using Pirie Kieren's (1994) notion of image making and Lunney Borden's (2011) idea of verbing mathematics, we share two examples of how students respond to teacher requests to demonstrate what they know about arrays.

Early years mathematics textbooks are support material for teachers and learners as they contain visual representations which communicate and clarify mathematical concepts. In this paper, we report on the types and functions of visual representations of multiplication and division in South African early years mathematics textbooks. There seems to…

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…

The purpose of this mixed-methods study was to measure the impact of a sequential use of manipulatives and visual models on third-grade students' multiplication fluency, math attitudes, and fact retention. According to results of this study, there was significant evidence that the students who gained a conceptual understanding of multiplication…

We examine opportunities and challenges of applying a single, explicit definition of multiplication when modeling situations across an important swathe of school mathematics. In so doing, we review two interrelated conversations within multiplication research. The first has to do with identifying and classifying situations that can be modeled by…