This paper addresses issues linking research into the classroom teaching and learning of mathematics with the growth of knowledge in mathematics teaching, developments in the practice of teaching and the enhanced learning of mathematics by students in classrooms. A basic premise is that research promotes development. The paper considers both…

Acknowledging the complex relationships which the field of didactics of mathematics has with other research fields (e.g. mathematics, educational sciences, epistemology, history, psychology, semiotics, sociology, cognitive science), the authors analyze in this paper some cases of fruitful and some of failed dialogue between experts of the…

Previous research has shown that--due to the extensive attention spent to proportional reasoning in mathematics education--many students have a strong tendency to apply linear or proportional models anywhere, even in situations where they are not applicable. This phenomenon is sometimes referred to as the "illusion of linearity". For example, in…

This article aims to contribute to the theory of mathematics instruction by highlighting and analysing Leen Streefland's work as a teacher in a primary school. Core ideas of Streefland's are discussed as part of a recent wave of educational innovations using the idea of learning communities. I present a case study of a lesson co-taught by…

The purpose of this article is to describe how, within the Dutch approach to mathematics education, called "Realistic Mathematics Education" (RME), models are used to elicit students' growth in understanding of mathematics. First some background information is given about the characteristics of RME related to the role of models in this approach.…

This reaction to the papers in this PME Special Issue of "Educational Studies in Mathematics" draws a wider perspective on the issues addressed and some of the constructs used in research in Realistic Mathematics Education (RME). In particular, it tries to show that while the problems addressed existed within the world-wide arena of mathematics…

In this paper we present three cases of instructional design that illustrates both horizontal didactising, the activity of using already established principles to design instruction, and vertical didactising the activity of developing new tools and principles for instructional design. The first case illustrates horizontal didactising by…

Reflecting on the learning process of topologist Stephen Smale, certain properties and characteristics of that process are selected and analysed: the question and anticipation of future mathematics, the generalising potential thereof, and the shift from after-image or "model of..." to pre-image or "model for...". These features are explored for…

In early algebra students often struggle with equation solving. Modeled on Streefland's studies of students' own productions a prototype pre-algebra learning strand was designed which takes students' informal (arithmetical) strategies as a starting point for solving equations. In order to make available the skills and tools needed for manipulating…

This paper addresses one approach to reconciling theory, research, and practice, namely, a multitiered teaching experiment involving a models and modelling approach to learning. The four-tiered teaching experiment explored in this paper involves participants at different levels of development who work interdependently towards the common goal of…

In this paper we explore the issue of interdependency of theory and research findings in the context of research on the practice of mathematics teaching and learning at school. We exemplify how analyses of a lesson by using two different theoretical perspectives lead to different interpretations and understandings of the same lesson, and discuss…

This article explores some of the intellectual bases of the apparent cultural divide between the fields of mathematics and mathematics education research. The chasm is in part attributable to epistemologies or theoretical "paradigms", fashionable in education, that dismiss or deny the integrity of fundamental aspects of mathematical and scientific…

A broad view of mathematics education takes it as the study of how people learn and do mathematics. Starting with this view, the actual and potential relationships of mathematics education as a research discipline to mathematics as a field of knowledge and activity and to the mathematicians carrying out that activity are analyzed. This leads to…

Examines the use of mathematics in a computer-aided design and manufacturing setting, whether this mathematics is related to school mathematics, how technicians understand this mathematics, and the role of technology in the technicians' mathematics-related problem solving activities. Focuses on technician's calculations of the interval volume of…

Examines the relationship between mathematical knowledge and social practices of the Renaissance. Suggests that all efforts to understand the conceptual reality and the production of knowledge cannot restrict themselves to language and the discursive activity, but that they also need to include the social practices that underlie them. (Author/KHR)