Starting with Woodworth and Thorndike’s classical experiment published in 1901, major periods in mathematics education throughout 20th century and on into the current century are reviewed in terms of competing epistemological and psychological paradigms that were operating within as well as across the major periods. The periods were marked by attempts to make changes in school mathematics by adherents of the dominant paradigm. Regardless of what paradigm was dominant, the attempts essentially led to major disappointments or failures. What has been common across these attempts is the practice of basing mathematics curricula for children on the first-order mathematical knowledge of adults. I argue that rather than repeat such attempts to make wholesale changes, what is needed is to construct mathematics curricula for children that is based on the mathematics of children. Toward that end, I present several crucial radical constructivist research programs.

Key words: learning trajectories, research methods, cognition, curriculum.

This short paper revisits Bertalanffy’s open systems theory and Luhmann’s theory of autopoietic social systems in order to highlight the linkage between systemic complexity and the carrying capacity of the environment. Being paradigmatically focused on biological complexity, Bertalanffy’s work shows this linkage to be relatively unproblematic. In contrast, Luhmann argued that autopoietic social systems are likely to develop excessive complexity which overstrains the environment’s carrying capacity. The paper synthesizes these conceptions into a framework of discretionary social decision-making aimed at preventing systemic complexity from becoming excessive. Rooted in the idea of balancing the system and the environment, this framework determines the optimal level of systemic complexity as that level which systems can maximally attain without incurring the risk of self-destruction. The system–environment balance is shown to be the general systems theory core of Ulrich’s critical systems heuristics and critical systems thinking more generally.

Key words: open systems, autopoietic systems, complexity, environment, critical systems heuristics.

Some of the problems faced by Cybertherapy along the last two decades are far from being restricted to technical issues. They entail new challenges of medical education, mainly related with the adequate insertion of new technologies in therapeutic processes without distorting the relation between medical professionals and clients. We contend that the acknowledgment of the effects of the systemic effects of therapeutic applications of virtual reality is not fully predictable and can only be achieved attending to the way the patient enacts certain tasks oriented by goals. Enaction means the patient is placed at the centre of the treatment processes, not only as an informed agent, but also as the agent of change through practice. Focusing on the requirements of Cybertherapy applied to Post-Traumatic Stress Disorder, we propose a theoretic reflection on the conditions of training and treatment in virtual settings. We underline the decisive role of Health Care professionals in applying and improving the potentialities of biometric sensors, graphic and aural engines in virtual (and hybrid) settings. This role can only be adequately understood within a framework of different levels of recursion of the therapeutic system. Two main levels are referred, the first encompassing the patients adaptation and learning to “move within” the interfaces, the second requiring a reflection on the architecture and design of the physical setting and the computerized rendering of sensory data. Further levels concern the larger framework of therapy, relating to its allocation of resources and the social ends that therapeutic technologies, particularly those concerning mental health, must accomplish.

This article compares the potential of classical and connectionist computational concepts for explanations of innate infant knowledge and of its development. It focuses on issues relating to: the perceptual process; the control and form(s) of perceptual-behavioural coordination; the role of environmental structure in the organization of action; and the construction of novel forms of activity. There is significant compatibility between connectionist and classical views of computation, though classical concepts are, at present, better able to provide a comprehensive computational view of the infant. However, Varela’s “enaction” perspective poses a significant challenge for both approaches.

Key words: action, connectionism, learning and development, enaction, human infancy, innate knowledge

Excerpt: Sincerity and ethics are two elusive and multifaceted constructs. As people and contexts change, ethical questions look similar, whereas answers do not. From my point of view, talking about ethics means asking oneself why lying or conflicting, for example, could be the preferable choices for someone. It means bringing both one’s own and other people’s assumptions into question, and to recognise that doubt is perhaps the path toward otherness. The goal of the present chapter is to explore the role of epistemological doubt in dealing with different people and political dissent.

Excerpt: During the brief history of Constructivism, many authors (Foerster & Glasersfeld, 2000; Glasersfeld, 1985, 1992, 2007; Hermans, 2003; Mahoney, 1988; Mancuso, 2000) have claimed the Neapolitan philosopher Giambattista Vico (1668–1744) as a forerunner of constructivist epistemology. ’Verum est ipsum factum’ (to be true is one and the same as to be made) has a high level of commonality with Kellian epistemology through two statements: (i) science is the knowledge of how things are construed; (ii) man knows only what is construed by himself (Glasersfeld, 1985). These assumptions subsume a circular constructive process (Foerster, 2003) and a metaphor of man as a system organizing himself and his world (Maturana & Varela, 1972; Piaget, 1937). In Vico’s words (Vico, 1744/1783) we are our poetics (etym. Greek’poiéin’: to make), we embody and stage this so-called world.

Purpose: The paper discusses possible implications of Heinz von Foerster’s notion of second‐order cybernetics for management thinking. The purpose of this paper is to outline challenges of as well as prospective further developments for management theory that emanate from second‐order cybernetics. Design/methodology/approach – As a conceptual paper, the paper tries to develop its findings through theoretically applying von Foerster’s insights to management thinking’s conventional assumptions. When looking for applications of von Foerster’s approach within the social sciences, at least in german‐speaking countries one sooner or later comes across Niklas Luhmann’s system sociology. Hence, Luhmann’s version of the theory of the observer is introduced and its take on organization and management is briefly outlined. Drawing upon von Foerster’s and Luhmann’s reflections, possible implications for management thinking are presented – ideas that might be disagreeable for “classical” management science but might set out a path for further developments of management thinking. Findings: What difference might second‐order cybernetics (and autopoietic systems sociology) make for management thinking? As a conclusion, deliberately poignant statements are formulated, calling for a higher degree of self‐reflection, for critical readings of conventional texts, for more complex descriptions of organizations and for a more modest, low‐key take on management theory’s endeavours. Originality/value – Whereas first‐order cybernetics has been fairly well‐received in management theory, second‐order cybernetics, which poses troubling questions to conventional epistemologies, remained relatively unpopular. Acts of “observing observers” reclaim these questions, possibly leading to valuable insights for researchers and reflected practitioners alike.

Key words: Cybernetics, systems theory, management science

The author’s purpose in this article was to respond to two questions raised by Roth and Lawson in the September, 1993, issue of the Journal of Research in Science Teaching. Question 1: Would a radical constructivist step out of the path of an approaching vehicle? Question 2: In the conduct of inquiry, would a radical constructivist employ a controlled experiment, test a hypothesis, and quantitatively analyze the data? The author answers each question affirmatively, using selected work of Heinz von Foerster, Ernst von Glasersfeld, and others in developing the answers. Issues central to the development include the nature of truth and knowledge, the concept of fit versus match, and the notion that inquiry is driven by questions, with methods as subordinate to questions.

Law and Environment enter a connection of disrupted continuum. The recent ‘turns’ in law towards materiality, spatiality, corporeality, disconnect the usual distance between law and its environment and enhance the visibility of materiality continuum between the two. Law is no longer abstract but spatially emplaced, corporeally felt, materially present. The environment, be it in the form of human/non-human bodies, technology, weather phenomena, and the wider, open ecology of material presence, destabilises the system, rendering it more precarious and more distant from its usual self-description. Building on Luhmann’s theory of autopoiesis, I present my reading of what I call Critical Autopoiesis, namely the autopoietics of materiality, spatiality and corporeality as they emerge from contemporary legal theory. I am employing Deleuze and Guattari in connection to Luhmann in order to multiply and indeed fractally explode the Luhmannian boundaries between law and its environment.

Key words: critical autopoiesis, space, body, law, environment, materiality, deleuze, luhmann.

Excerpt: To be vibrant is to be “pulsating with life, vigor, or activity” (Mish 2003). Science education, like science, is a vibrant discipline. It pulsates due to competition among individuals and groups holding disparate views, as portrayed above (Hull 1988). One source of pulsation is the question: Can we justify that anything we know represents some aspect of reality? My purpose herein is to review an on-going dialectical discussion between communities of scholars that hold different views about whether or not knowledge represents reality, the nature of knowledge, and the process of coming to know. The adversaries, realism and constructivism, constitute different paradigms (Kuhn 1970) or models for characterizing knowledge and the || process of coming to know, for conducting research, and for recommending best practices in teaching and learning science. To achieve my purpose, I will take five steps: (1) define and describe knowledge; (2) describe realism, constructivism, and truth; (3) cite points of divergence, convergence, and peaceful coexistence; (4) review the key issue over which realism and constructivism collide from a constructivist perspective; and (5) offer a closing thought.