In my article, I propose to discuss constructivism and realism in terms of actions instead of doing that in a usual way, in terms of theories, philosophers or general positions. To enable this, I offer two conceptual tools. First, I use modified model of four types of knowledge introduced by Andrzej Zybertowicz. It approaches any knowledge-building process as a cultural game, and recognizes reproduction, discovery, redefinition, and design of a new game. Second, I use Stanislaw Lem’s model of three types of geniuses. I illustrate my approach briefly using examples from Plato, Spinoza and Berkeley.

Key words: Constructivism, realism, cultural theory, actions, knowledge.

Upshot: Emergence and Embodiment is a highly worthwhile and well-crafted collection of essays on second-order cybernetics that draws together ideas related to self-organization, autopoiesis, organizational closure, self-reference, and neurophenomenology. Chapters include articles by Heinz von Foerster, Francesco Varela, Niklas Luhmann, George Spencer-Brown, and Evan Thompson and external commentaries on them that analyze the relevance of their ideas in the context of social and cultural theory. Despite some projective distortions to cybernetics that arise from the internal imperatives of culture criticism, the book contains many valuable insights and analyses of core ideas of cybernetics that significantly advance our understanding of them.

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.

Purpose: To show the connections and differences between Mitterer’s concept, cultural theory, and sociology of knowledge in order to reproduce the development of non-dualizing philosophy. Problem: Mitterer’s non-dualizing philosophy explicitly places emphasis on the continuation and coherence of discourses. Consequently, it grants an epistemological option that does not focus on the object as the end of cognition and description, but rather as the beginning. This perspective not only helps to overcome fundamental philosophical problems; it also concedes that the whole concept of non-dualizing philosophy refers theoretical descriptions, on the one hand, to the status of “so far” and, on the other hand, can be described as “from now on.” Solution: It seems necessary to exemplify obvious and hidden connections to cultural theories, especially those of the early 20th century (i.e., Karl Mannheim, Heinrich Rickert, Max Weber, William James), which predominantly concentrate on the relations between language and object, experience and world. The illustration of those relations should bring out Mitterer’s arguments, as well as how his argumentation can be applied to itself. Benefits: To explain and avoid the epistemological problems of realism, as well as of constructivism, which emerge within a dualistic perspective.

Key words: subject/object, cultural theory, epistemology

The caption that I have chosen for my discussion of the papers by Peter Renshaw and Paul Cobb is particularly relevant because it encapsulates much of what I want to say. Both authors emphasize that socio-cultural theory, whatever its form, has come to the fore in early childhood mathematics education. Renshaw provides an excellent overview of Vygotsky’s socio-cultural theory and how it has influenced mathematics education at the Academy of Pedagogical Sciences in Moscow, and Cobb provides an equally insightful comparison and contrast of Soviet Activity Theory and social constructivism. Rather than attempt to carry on with comparing and contrasting the two theories, my goal is to bring Piaget’s genetic epistemology squarely into sociocultural theory and to explore the consequences of doing so. Piaget based his genetic epistemology on interaction as a hard core principle, so in my view it is unnecessary to keep genetic epistemology and sociocultural theory separate as we create our visions of what early childhood mathematics might be like. In fact, I believe that including Piaget’s genetic epistemology in sociocultural theory is especially important in the context of early childhood mathematics education.