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.

The concept of self is explored from the perspective of systems theory, particularly 2nd-order cybernetics. The author believes that the self, like the family, is a legitimate autonomous system (i.e., a unity with organization and structure that functions in a manner that is always self-referential). The concepts of organizational closure and structural coupling, as developed by H. Maturana and F. Varela (1980), are employed to describe how the self and the family represent nonintersecting systems that have mutual influence.

It is argued in the paper that social systems are self-referential. As such, they can be best viewed as autopoietic systems. Various implications of this view are discussed in the paper.

Key words: Society, social system, self-reference, evolution, autopoietic system, complexity.

Reverse micelles hosting the internal production of the surfactant are proposed as experimentally feasible models of simple (or ‘minimal’) autopoietic systems. We describe the conditions under which these may be formed and their possible biological implications. The micellar systems considered here turn out also to exhibit a capacity for self-reproduction through fragmentation under plausible conditions, thus constituting also a minimal experimental model for prebiotic self-reproduction.

We monitored in normal adult BALB/c mice the serum concentrations of four natural IgM antibodies, two of which show idiotypic complementarity in in vitro assays. In each individual, serum concentration of all four idiotypes were found to fluctuate in complex dynamical patterns with low correlation. The spectral power of some such patterns was found to be compatible with the existence of a chaotic regime. Groups of normal adult mice were injected intravenously with low (10 ng) or moderate (10 micrograms) doses of either of the two complementary idiotypes in saline. This treatment resulted in a pronounced inhibition of the fluctuation in the serum concentration of both complementary idiotypes for periods up to 3 months. Such compensations were not detected for the two unrelated natural idiotypes and were specifically induced, for they did not occur following the injection of unrelated antibodies. These results indicate the functional operation of an idiotypic network among natural antibodies.

Comments on the articles on constructivism by N. M. Agnew and J. L. Brown and expands on three issues raised. These relate to definitions and boundaries of modern constructivism and constructivist meta theory.