Excerpt: It is widely recognized that a variety of different forms of constructivism exist, both radical and otherwise (Ernest, 1991b). However it is the radical version which most strongly prioritizes the individual aspects of learning. It thus regards other aspects, such as the social, to be merely a part of, or reducible to, the individual. A number of authors have criticized this approach for its neglect of the social (Ernest 1991b, 1993d; Goldin 1991; Lerman, 1992, 1994). Thus in claiming to solve one of the problems of the psychology of mathematics education, radical constructivism has raised another: how to account for the social aspects of learning mathematics? This is not a trivial problem, because the social domain includes linguistic factors, cultural factors, interpersonal interactions such as peer interaction, and teaching and the role of the teacher. Thus another of the fundamental problems faced by the psychology of mathematics education is: how to reconcile the private mathematical knowledge, |69| skills, learning, and conceptual development of the individual with the social nature of school mathematics and its context, influences and teaching? In other words: how to reconcile the private and the public, the individual and the collective or social, the psychological and the sociological aspects of the learning (and teaching) of mathematics?

Key words: philosophy, radical constructivism, mathematics

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.

Excerpt: Complementarity of subject and object makes up the starting point of epistemology. Epistemology is not independent of meta-physics, because if we insist on identifying the object with the definition theory gives of it, we also, perhaps unwittingly, tend to define the human subject. This is in opposition to the idea that the essence of man is existential freedom. Mathematics may in part construct its own reality but always in face of the continuum of yet undefined real possibility. Otherwise such a construction loses its subject becoming instead a quasi-mechanized process, as in the case of radical constructivism.

In recent years, the social interaction involved in the construction of the mathematics of children has been brought into the foreground in order to sptcify its constructive aspects (Bauersfeld, 1988; Yackel, etc., 1990). One of the basic goals in our current teaching experiment is to analyse such social interaction in the context of children working on fractions in computer microworlds. In our analyses, however, we have found that social interaction does not provide a full account of children’s mathematical interaction. Children’s mathematical interaction also includes enactment or potential enactment of their operative mathematical schc:mes. Therefore, we conduct our analyses of children’s social interaction in the context of their mathematical interaction in our computer microworlds. We interpret and contrast the children’s mathematical interaction from the points of view of radical constructivism and of Soviet activity theory. We challenge what we believe is a common interpretation of learning in radical constructivism by those who approach learning from a social-cultural point of view. Renshaw (1992), for example, states that ‘In promulgating an active. constructive and creative view of learning,… the constructivists painted the learner in close-up as a solo-pia yer, a lone scientist, a solitary observer, a me:ming maker in a vacuum.’ In Renshaw’s interpretation, learning is viewed as being synonymous with construction in the absence of social interaction with other human beings. To those in mathematics education who use the teaching experiment methodology, this view of learning has always seemed strange because we emphasize social interaction as a primary means of engendering learning and of building models of children’s mathematical knowledge (Cobb and Steffe, 1983; Steffe, 1993).

Excerpt: Being grateful to von Glasersfeld for having pointed out to me what I now accept, I have no desire to attack radical constructivism. There are, however, three ways in which I find it seriously, even radically, deficient, and I want in this chapter to set them out in the hope that my doing so will be some use where I, and apparently von Glasersfeld, care most about usefulness, in education. These deficiencies are a lack of due emphasis on the construction of the self, whether over against the world or as a part of it, the denial of the possibility of knowledge of the world, and von Glasersfeld’s ignoring of the massive social assistance in one’s construction of one’s notion of the world. The latter two of these deficiencies have considerable importance for education, and that is my reason for airing my criticisms in this place.