The aim of this study was to determine learning difficulties and misunderstanding in multipliers and factors of middle school students in Kayseri, Turkey. One hundred and seven students from 6th grade and 48 students from 8th grade were selected randomly from three middle schools for the study. A questionnaire, which was developed by the first…

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…

After being introduced to the distributive law in meaningful contexts, students need to extend its scope of application to unfamiliar expressions. In this article, a process model for the development of structure sense is developed. Building on this model, this article reports on a design research project in which exercise tasks support students…

The reduced interest of students in learning mathematics and consider mathematics to be a difficult subject to become a serious problem in Indonesia, especially in one of the provinces in Indonesia, namely in Lampung. The purpose of this study was to determine the effectiveness of the demonstration method assisted by Multiplication Board props to…

The number sequences describe a hierarchy of students' concepts of number. This research uses two defining cognitive structures of the number sequences--units coordination and the splitting operation--to model middle-grades students' abilities to write linear equations representing the multiplicative relationship between two unknowns. Results…

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…

The term numeracy is used widely in schools today and brings with it the expectation that students will be taught both how to do the mathematics, alongside an understanding of the concepts associated with the procedural application. One issue, which has arisen with the terminology 'numeracy classroom', is how to best support teachers to enhance…

The rural area's student difficulties in learning the concept of number operation had been documented by several studies, especially for the case of multiplication. The teacher typically introduces the multiplication concepts using the formula without involving the concept itself. Furthermore, this study aims to design learning trajectory on…

This is a position paper in which we argue for "Teaching for Mathematising" as a pedagogy that supports teaching mathematics for understanding in primary school. Mathematics education in Australia currently emphasises teaching for mathematical understanding, a shift that redirects children's learning from merely memorising computation…

Introducing new concepts to learners in an order of increasing complexity appears to be beneficial for learning, but typically introduction of concepts does not always adhere to this principle. We examined whether introducing new algebra concepts in a contrasted manner or in an order of increasing complexity instead of a different more typical…

This article examines the importance of developing the notion of place value as a rate of ten. In exploring how to nurture this concept, the author looks at the role of the language of value, the problem types of multistep multiplication and addition along with measurement division, each with ten as an organizing unit, as well as strategically…

This small study sought to determine students' knowledge of multiplication and division and whether they are able to use sets of bundling sticks to demonstrate their knowledge. Manipulatives are widely used in primary and some middle school classrooms, and can assist children to connect multiplicative concepts to physical representations.…

How do students cope with and make sense of polysemy in mathematics? Zazkis (1998) tackled these questions in the case of 'divisor' and 'quotient'. When requested to determine the quotient in the division of 12 by 5, some of her pre-service teachers operated in the domain of integers and argued for 2, while others adhered to rational numbers and…

In our exploration of the order of operations we focus on the following claim: "In the conventional order of operations, division should be performed before multiplication." This initially surprising claim is based on the acronym BEDMAS, a popular mnemonic used in Canada to assist students in remembering the order of operations. The…

Three-in-a-Row Lucky Numbers is an engaging, enjoyable, mathematically meaningful, game-based activity involving dice and a hundred chart, which can be used to introduce students to multiplication. The game provides a mechanism for students to explore the structure of multiplication, experiment with the distributive property, and begin to…