Visualising positive and negative numbers on a number line is helpful for exploring problems involving operations with positive and negative numbers. This is because number lines lend themselves to exploring problems involving continuous linear contexts such as travelling distances and temperature. Teachers in a professional development program…

The main goal of this paper is to show how Vasily Davydov's powerful ideas about the nature of mathematical thinking and learning can transform the teaching and learning of additive word problem solving. The name Vasily Davydov is well known in the field of mathematics education in Russia. However, the transformative value of Davydov's theoretical…

In this commentary, I examine some of the distinctive, foundational difficulties in learning fractions and other types of rational numbers encountered by students with a mathematical learning disability and how these differ from the struggles experienced by students classified as low achieving in math. I discuss evidence indicating that students…

This paper analyses the strategies used by pre-service primary school teachers for solving simple addition problems involving negative numbers. The findings reveal six different strategies that depend on the difficulty of the problem and, in particular, on the unknown quantity. We note that students use negative numbers in those problems they find…

This article sets out to evaluate the English Early Years Foundation Stage Goal for Numbers, in relation to research evidence. The Goal, which sets out to provide "a good foundation in mathematics", has greater breadth of content and higher levels of difficulty than previous versions. Research suggests that the additional expectations…

"What do you do with the remainder when you divide?" Mrs. Thompson asked her third-grade students. They replied with such comments as, "You can't share that, because they won't be equal!" and "It's not going to come out even because you can't do that!" These answers were consistent with third- and fourth-grade student performance in a pretest and…

As an alternative to looking solely at linear functions, a three-lesson learning progression developed for Year 6 students that incorporates triangular numbers to develop children's algebraic thinking is described and evaluated.

This paper focuses on children's number fact knowledge from a study that explored the impact of using multiplication and division contexts for developing number understanding with 34 five- and six-year-old children from diverse cultural and linguistic backgrounds. After a series of focused lessons, children's knowledge of number facts, including…

The purpose of the present review was to evaluate the quality of the research and evidence base for representation of problems as a strategy to enhance the mathematical performance of students with learning disabilities and those at risk for mathematics difficulties. The authors evaluated 25 experimental and quasiexperimental studies according to…

Preschoolers' conceptual understanding and procedural skills were examined so as to explore the role of number-words and concept-procedure interactions in their additional knowledge. Eighteen three- to four-year-olds and 24 four- to five-year-olds judged commutativity and associativity principles and solved two-term problems involving number words…

The residue number system (RNS) has been an important research field in computer arithmetic for many decades, mainly because of its carry-free nature, which can provide high-performance computing architectures with superior delay specifications. Recently, research on RNS has found new directions that have resulted in the introduction of efficient…

Children's symbolic number sense was examined at the beginning of first grade with a short screen of competencies related to counting, number knowledge, and arithmetic operations. Conventional mathematics achievement was then assessed at the end of both first and third grades. Controlling for age and cognitive abilities (i.e., language, spatial,…

This study examined how four domain-specific skills (arithmetic procedural skills, number fact retrieval, place value concept, and number sense) and two domain-general processing skills (working memory and processing speed) may account for Chinese children's mathematics learning difficulties. Children with mathematics difficulties (MD) of two age…

We begin by answering the question, "Which natural numbers are sums of consecutive integers?" We then go on to explore the set of lengths (numbers of summands) in the decompositions of an integer as such sums.

Memorizing the basic number combinations, such as 9 + 7 = 16 and 16 - 9 = 7, is a punishing and insurmountable task for children with difficulties learning mathematics. Two perspectives on such learning lead to different conclusions about the primary source of this key learning difficulty. According to the conventional wisdom (the Passive Storage…