Learning theories such as behaviourism, Piagetian theories and cognitive psychology, have been dominant influences in education this century. This article discusses and supports the recent claim that Constructivism is an alternative paradigm, that has rich and significant consequences for mathematics education. In the United States there is a growing body of published research that claims to demonstrate the distinct nature of the implications of this view. There are, however, many critics who maintain that this is not the case, and that the research is within the current paradigm of cognitive psychology. The nature and tone of the dispute certainly at times appears to describe a paradigm shift in the Kuhnian model. In an attempt to analyse the meaning of Constructivism as a learning theory, and its implications for mathematics education, the use of the term by the intuitionist philosophers of mathematics is compared and contrasted. In particular, it is proposed that Constructivism in learning theory does not bring with it the same ontological commitment as the Intuitionists’ use of the term, and that it is in fact a relativist thesis. Some of the potential consequences for the teaching of mathematics of a relativist view of mathematical knowledge are discussed here.

Excerpt: Constructivism is certainly the dominant theory, but it is being subjected to much criticism. Not that this is new for constructivism; it gained in support during the 1980s despite strong attacks and even political manœuvrings in its early days. In this chapter I will attempt to create the written equivalent of a snapshot. What can be seen in the picture is a scene at one instant. By the time the snapshot has been developed, when the book is published, the scene will look different, people and places will have moved on. Yet the snapshot will have captured something, although I do not pretend that the snapshot has captured any ‘truth’, however temporary. It is my fiction as I write it, and the reader’s fiction as it is read. It is a photo-journalist’s creation: the angle, the light, the subjects, all chosen to convey the effect the photo-journalist wishes to be seen, to carry that particular story.

Radical constructivism is currently a major, if not the dominant, theoretical orientation in the mathematics education community, in relation to children’s learning. There are, however, aspects of children’s learning that are challenges to this perspective, and what appears to be “at least temporary states of intersubjectivity” (Cobb, Wood, & Yackel, 1991, p. 162) in the classroom is one such challenge. In this paper I discuss intersubjectivity and through it offer an examination of the limitations of the radical constructivist perspective. I suggest that the extension of radical constructivism toward a social constructivism, in an attempt to incorporate intersubjectivity, leads to an incoherent theory of learning. A comparison of Piaget’s positioning of the individual in relation to social life with that of Vygotsky and his followers is offered, in support of the claim that radical constructivism does not offer enough as an explanation of children’s learning of mathematics.

In their response to my (1996) article, Steffe and Thompson argued that I have taken an early position of Vygotsky’s and that his later work is subsumed in and developed by von Glasersfeld. I argue that the two theories, Vygotsky’s and radical constructivism, are, on the contrary, quite distinct and that this distinction, when seen as a dichotomy, is productive. I suggest that radical constructivists draw on a weak image of the role of social life. I argue that a thick notion of social leads to a complexity of sociocultural theories concerning the teaching and learning of mathematics, a perspective that is firmly located in the debates surrounding cultural theory of the last 2 decades.

At successive meetings of the British Society for Research in Learning Mathematics in 1988/89, we initiated discussion about the nature of ‘radical constructivism’ as a theory of knowledge and its possible implications first for the working mathematician, and then for mathematics education. We proposed that the radical statement of constructivism does not deny the existence of the real world, but makes it similar to an undecidable statement. Recognising that attitudes to the nature of mathematics affect working mathematicians, including teachers, we summarise the discussion here, quote from some interviews, and propose that this position potentially empowers one to engage mathematically with the world around.

This chapter addresses issues concerning epistemology, as they relate to mathematics and education. It commences with an examination of some of the main epistemological questions concerning truth, meaning and certainty, and the different ways they can be interpreted. It examines epistemologies of the ‘context of justification’ and of the ‘context of discovery’, foundationalist and non-foundationalist epistemologies of mathematics, historico-critical, genetic, socio-historical and cultural epistemologies, and epistemologies of meaning. \\In the second part of the chapter, after a brief look at epistemology in relation to the statements of mathematics education, epistemologies in mathematics education become the main focus of attention. Controversial issues within a number of areas are considered: the subjective-objective character of mathematical knowledge; the role in cognition of social and cultural context; and relations between language and knowledge. The major tenets of constructivism, socio-cultural views, interactionism, the French didactique, and epistemologies of meaning are compared. Relationships between epistemology and a theory of instruction, especially in regard to didactic principles, are also considered.