To understand the ways that manipulatives might support changes in students' reasoning about algebraic generalizations, a constructivist teaching experiment was conducted with two sixth-grade students. The students interpreted numerical situations with units of one and could construct units of units in mental activity. Initially, the students'…

In this paper, we report how 5-year-olds' arithmetic skills developed through participation in an 8-month-long intervention. The intervention program aimed to enhance the children's ways of experiencing numbers' part-part-whole relations as a basis for arithmetic skills and was built on principles from the variation theory of learning. The report…

In order to give insights into cross-national differences in schooling, this study analyzed the development of multiplication and division of fractions in two curricula: "Everyday Mathematics" (EM) from the USA and the 7th Korean mathematics curriculum (KM). Analyses of both the content and problems in the textbooks indicate that…

The introduction of the concept of the variable represents a critical point in the arithmetic-algebraic transition. This concept is complex because it is used with different meanings in different situations. Its management depends on the particular way of using it in problem-solving. The aim of this paper was to analyse whether the notion of…

This article reviews two sets of research studies from outside of mathematics education to consider how they may be relevant to the study of bilingual mathematics learners using two languages. The first set of studies is psycholinguistics experiments comparing monolinguals and bilinguals using two languages during arithmetic computation (language…

A range of estimation techniques are described, and a series of mental computation procedures based on concepts of measurement and real numbers are presented in detail. (MP)

Two strands of research motivate this study. One is the interest in school students' development of understanding of the concept of average, historically part of the mathematics curriculum and prominent in the statistics curriculum introduced in the early 1990s. The other is the belief of some educators that students learn meaningfully when…

Uses interviews with 72 pupils in grades 2-6 to investigate awareness of the relation between situation and computation in simple quotitive and partitive division problems as informally and formally experienced. Concludes that formal division, understood as related to everyday situations, only depends in interplay with informal knowledge.…

This article considers skills and abilities of young children with number in the areas of cardinality and ordinality. Responses to number tasks by a sample of 3- and 4-year-old children are described to outline their differing approaches to determining the numerosity of a set, and to establish their application of ordinality language to describe…

This study provides an in-depth analysis of children's use of a taught mental calculation strategy. Three children (aged eight to nine years) who demonstrated contrasting spontaneous calculation approaches, were interviewed following the direct instruction of a calculation strategy. Their responses are explored in relation to constructivist and…

Suggests that strategy choice is a misleading characterization of efficient mental calculation and that teaching mental calculation methods as a whole is not conducive to flexibility. Proposes an alternative in which calculation is thought of as an interaction between noticing and knowledge. Presents an associated teaching approach to promote…

Interviews 53 sixth grade students individually in an attempt to discover whether wrong interpretations of the algebraic structure found in an algebraic context occurs in a purely numerical context. Concludes that students' difficulties with the algebraic structure were also found in purely numerical contexts. (Contains 27 references.) (Author/ASK)

Examines the computational strategies of 10 young street vendors in Beirut by describing, comparing, and analyzing computational strategies used in solving three types of problems: (1) transactions in the workplace; (2) word problems; and (3) computation exercises in a school-like setting. Indicates that vendors' use of semantically-based mental…

Describes the success, the methods (mental, informal written, standard algorithm) and the strategies of informal written arithmetic to be observed when 300 elementary students worked on six addition and six subtraction problems with three-digit numbers. (Author/MM)

Fourth graders' (n=117) solutions to addition problems were analyzed in terms of standard or idiosyncratic written algorithms. Students had not previously been taught pencil-and-paper algorithms. Preference for horizontal layout, working from left to right, and a wide variety of written algorithms were found. (Contains 48 references.) (Author/MKR)