The New Zealand Ministry of Education has introduced a Numeracy Project for students aged 5-14 years in selected schools. The project encourages the adoption of flexible strategies for solving numerical problems, and discourages reliance on standard computational algorithms. One potential benefit of the project is that the methods students acquire…

In this paper, we discuss issues in planning and conducting research into mathematics learning. We emphasize two central themes: (a) the learners mathematics (especially the issues and ideas, in given problem situations, that learners' choose to think about and to present) and (b) the kinds of knowledge that learners may be building (including…

This paper contributes to knowledge about principled action which makes a difference to learners' attainment. We report on the Improving Attainment in Mathematics Project, a project focusing on low-attaining secondary students. The purpose of the project was to introduce innovations in practice through action research with 10 teachers over 2…

The world wide web is becoming wider at an increasing rate. It is virtually impossible to take any kind of accurate snapshot of the state of its development. Undergraduate mathematics courses presented on the internet are relative newcomers to the race but nevertheless seem to be increasing in numbers also at a phenomenal rate. Internet education…

Internet education in mathematics is developing as a new mode of teaching with its own characteristics and possibilities, different from the traditional way of teaching. In a study presented in two parts, we attempt to capture the world of Internet teaching of undergraduate mathematics. In the first part of the study (Engelbrecht, J. and Harding,…

Development in the ability of 11-year-olds to solve numerical problems of addition, multiplication, and proportion was analysed by means of three Rasch models of change. The students, who had participated in a New Zealand numeracy project in 2002, comprised two groups that differed in socio-economic status: 1,274 students came from low…

We start from the assumption that school mathematics knowledge could be better explained if social practices were considered to be generators of knowledge. This perspective changes the way we look at what school mathematics knowledge is and what it takes to teach and learn it. In this article, we will present a teaching situation about periodic…

This paper applies APOS Theory to suggest a new explanation of how people might think about the concept of infinity. We propose cognitive explanations, and in some cases resolutions, of various dichotomies, paradoxes, and mathematical problems involving the concept of infinity. These explanations are expressed in terms of the mental mechanisms of…

In research on teachers' beliefs, a distinction is often made between what teachers state ("professed beliefs") and what is reflected in teachers' practices ("attributed beliefs"). Researchers claim to have found both consistencies and inconsistencies between professed and attributed beliefs. In this paper, methods and research designs typically…

These seem to be very special times for mathematics education. The public interest in the topic has never been greater. Probably the most prominent among the occurrences that occasioned this recent leap in popularity are international comparative studies such as TIMSS and PISA. The fact that, in spite of the ongoing efforts toward reform in…

In this paper, we analyze an investigative situation proposed to a class of 5th graders in a primary school. The situation is based on the following task: In a sale with group rates on a sliding scale, the students must find the lowest possible purchase price for a given number of tickets. A study of students' arguments made it possible to…

This paper is based on a long-term didactic engineering about division problems (only in a numerical setting) at primary school. Situations and students' work are analyzed by means of a double theoretical framework: the theory of situations and the theory of conceptual fields (Vergnaud 1991). The analysis focuses mainly on classroom interactions…

The purpose of the present article is to give an account of the emergence of knowledge pertaining to the transition from arithmetic to algebra in the course of a debate in a grade 7 classroom. This debate follows two other instances of work: (1) the adidactic interaction between each student and a given problem, (2) the adidactic interaction of…

In this paper, we use the theory of didactic situations to characterize a mathematics teaching practice, currently used in secondary schools in France, which we have called "interactive synthesis discussion." We have studied this practice in ordinary classes, i.e. classes where the researcher intervenes neither in the preparation nor in the…

This paper outlines some theoretical categories (i.e. the "meso-", "topo-", "chrono-geneses," the "milieu," the didactical contract and the learning games), providing a "model" to study mathematics teacher's action. In order to show what this model brings to the didactical analysis, we present the action of two teachers, on the same content, and…