In this chapter, I focus on one of the aspects of constructivist theory that Glasersfeld (Ch. 1) identifies as in need of further development. This aspect of the theory involves locating students’ mathematical development in social and cultural context while simultaneously treating learning as a process of adaptive reorganization. In addressing this issue, I illustrate the approach that I and my colleagues currently take when accounting for the process of students’ mathematical learning as it occurs in the social context of the classroom. In the opening section of the chapter, I clarify why this is a significant issue for us as mathematics educators. I then outline my general theoretical orientation by discussing Glasersfeld’s constructivism and Bauersfeld’s interactionism. Against this background, I develop criteria for classroom analyses that are relevant to our interests as researchers who develop learning environments for students in collaboration with teachers. Next, I illustrate the interpretive framework that I and my colleagues currently use by presenting a sample classroom analysis. Finally, in the concluding sections of the chapter, I reflect on the sample analysis to address four more general issues. These concern the contributions of analyses of the type outlined in the illustrative example, the relationship between instructional design and classroom-based research, the role of symbols and other tools in mathematical learning, and the relation between individual students’ mathematical activity and communal classroom processes.

Lewin’s chapter [Lewin 2000] focuses on the need for a constructivist ethics. His position calls for an analysis of ways in which knowledge and relationships are associated. This response is concerned with the following themes: first, connections between ways of knowing and ways of relating in constructivist writings; second, some constructivist issues in Lewin’s chapter; and third, I describe a series of constructivist intervention studies designed to apply constructivist educational strategies for ethical reasons.

Excerpt: There are at present two aspects that need to be developed further. One of them is to find new ways of expressing the fundamental instrumentalist idea in order to make it less prone to metaphysical misinterpretation. The other is to achieve a far more detailed analysis of the complex area covered by the generic term ‘social interaction’.

Key words: radical constructivism, metaphysics, social

Excerpt: I want to discuss the implications of Glasersfeld’s epistemology as it applies to education. One might say, without stretching the truth, that I – the kind of person I am, the kind of thinking I do – am at least in part one of those implications. I will be critical in some of my comments, but please understand that mine is criticism made from within a frame of respect and gratitude and continuing affection. The power of constructivism lies in how it underwrites viability, in how it extends permission for being. Ernst granted me that permission at a key moment in my education and for that I will always be grateful.

From the conclusion: a comprehensive philosophy of language and communication would be well advised to observe and to analyze its subjects under a fourfold perspective, viz. a biological, a psychological, a sociological, and a culture-theoretical one, integrating genetic and structural points of view, respectively. The man who has paved the way to this important insight is Ernst Glasersfeld. He has been the trailblazer out of the behaviorist desert into cognitive constructivist pastures with inspiring new perspectives, and an abundance of new interesting problems. I think Ernst Glasersfeld has proved that radical constructivism is much more than one approach among others: it is a philosophy which opens up the way to a better, more human way of living.

Lewin raises four major issues for constructivism in the opening chapter of this section: how we think of ‘self’ within a constructivist framework; how constructivism addresses the notion of paideia – ‘the complete process of education through which one becomes a competent participant within a culture’; how science and mathematics differ from the humanities because of different constraints and forms of viability in them; and whether constructivism should produce an ethics of the field. Lewin’s concern with ‘self’ is that without some understanding of how our being arises from our situatedness in time, we are cut off from understanding our conditions as biological organisms and historical persons. Although asking questions about what constitutes self may seem unrelated to mathematics and science education, the way in which students construct their relationship to knowledge (mathematical or scientific) and to their learning and doing of the knowledge in question is one of the major problems confronting constructivist educators in these fields.

Excerpt: I would like to bring the book full circle, returning to two points raised by Ernst von Glasersfeld in his opening chapter. These are the fact that radical constructivism is misinterpreted so persistently by its critics, and the need for radical constructivism to provide a clear model of social interaction. I return to these points not only to give the book a particular rhetorical structure, but because they penetrate many of the controversies both internal to mathematics and science education and at the boundaries of radical constructivism. At the same time, I will point out the importance of conceptual analysis in Glasersfeld’s method and urge more people to use it in mathematics and science education.

Excerpt: It is not possible within the scope of this chapter to mount a full-scale philosophical defence of radical constructivism. What is possible is an attack from the radical constructivist point of view using a weapon readily adaptable to a wide range of opponents.