A theoretical model is proposed that explicates the generation of conceptual structures from unitary sensory objects to abstract constructs that satisfy the criteria generally stipulated for concepts of “number”: independence from sensory properties, unity of composites consisting of units, and potential numerosity. The model is based on the assumption that attention operates not as a steady state but as a pulselike phenomenon that can, but need not, be focused on sensory signals in the central nervous system. Such a view of attention is compatible with recent findings in the neurophysiology of perception and provides, in conjunction with Piaget’s postulate of empirical and reflective abstraction, a novel approach to the analysis of concepts that seem indispensable for the development of numerical operations.

Key words: philosophy of mathematics, cognition, development

Excerpt: In the current social and critical moment, perhaps no project is more overdue than the articulation of a posthumanist theoretical framework for a politics and ethics not grounded in the Enlightenment ideal of “Man.” Within postmodern theory, that humanist ideal is critiqued as forcefully as anywhere in the early and middle phase of Michel Foucault’s career, whose “genealogical” aim is to “account for the constitution of knowledges, discourses, domains of objects, etc., without having to make reference to a subject which is either transcendental in relation to the field of events or runs in empty sameness throughout the course of history” by virtue of his – and it must be “his” – privileged relation to either the presence or the absence of the phallus, language, the symbolic, property, productive capacity, toolmaking, reason, or a soul. In Foucault, however, this call for posthumanist critique is more often than not accompanied, as many critics have noted, by a dystopianism that imagines that the end of the humanist subject is the beginning of the total saturation of the social field by power, domination, and oppression. And the later Foucault, as if compensating for his early dystopianism, evinces a kind of nostalgia for the Enlightenment humanism powerfully critiqued in his early and middle work but approached much more sympathetically in the History of Sexuality project.

This chapter explores the theoretical intersection between Niklas Luhmann’s systems theory and Gilles Deleuze’s philosophy as an avenue for post-structuralist political thought. Against the predominant reception of Luhmann’s thought as analytically positivist and politically conservative, it highlights the potential of Luhmann’s work as a critical theory. Linking Deleuze’s and Luhmann’s theories of sense, the chapter first develops an immanent perspective on onto-genetic creativity. For both theorists, sense-expression is ungrounded but self-grounding. In sense, material and epistemic singularities are always already interlinked so that neither can be assumed to hold constitutive primacy. Second, the chapter uses Luhmann’s political sociology in conjunction with Deleuze and Guattari’s work to sketch out a post-structuralist theory of how democratic politics functions in contemporary capitalist societies. Luhmann shows how, under the conditions of general functional differentiation, which mirror Deleuze and Guattari’s account of capitalism, politics functions autopoietically and reproduces itself through providing orientation in sense.

Excerpt: The difficulties of embracing a constructivist perspective towards teaching mathematics are evident when we analyze future teachers’ difficulties in building models of children’s understanding. Our understanding of these difficulties can be further shaped by considering characteristics of future teachers as to their own approaches to learning mathematics and their perceptions of the nature of mathematical knowledge and mathematical learning.

Constructivism in science education has been the subject of considerable debate in the science education literature. The purpose of this study was to facilitate chemistry teachers’ understanding that the tentative nature of scientific knowledge leads to the coexistence and rivalries among different forms of constructivism in science education. The study is based on 17 in-service teachers who had registered for a 11-week course on ‘Epistemology of Science Teaching’ as part of their Master’s degree program. The course is based on 17 readings drawing on nature of science and a critical evaluation of constructivism. Course activities included written reports, classroom discussions based on participants’ presentations and written exams. Based on the results obtained, it is plausible to suggest that participant teachers experienced the following transitions leading to greater understanding, as they acquired experience with respect to constructivism: (a) Active participation of students as a pre-requisite for change; (b) Different forms of constructivism represent competing and conflicting interpretations of progress in science; (c) Acceptance of the present state of constructivism as a Kuhnian paradigm; (d) Social constructivism as the preferred form of constructivism; (e) Critical appraisal of social constructivism; (f) Despite its popularity, social constructivism does not constitute a Kuhnian paradigm (due to controversies, there is no consensus in the science education community); (g) Contradictions faced by constructivism in science education provide the base for its advance and evolution towards more progressive forms, and hence the need to consider, whither constructivism?

Key words: Constructivism, Science education, Nature of science, Tentative nature of scientific knowledge.

Excerpt: In the educational technology literature the term instructional design has so many meanings that its use has little purpose without further elaboration. In some publications, the term roughly refers to the field called educational technology in this article (e.g., Reigeluth, 1983). In others, the term is used to describe the practice of educational technology from a particular theoretical perspective such as behaviorism (Gropper, 1987). Instructional design may also be the umbrella term indicating the components that should be included in an instructional package (Merrill, 1988). The phrase has also been used to indicates the process of designing instruction (McAlpine, 1992), and still others have used it to mean a particular model or theory that can guide the design of instruction (Wright & Conroy, 1988). In this article, instructional design (ID) refers to the process of designing instructional materials, and the term instructional design model (ID model) refers to a model or theory that can guide the instructional design process.

In recent years a new movement in the philosophy of education has arisen. It flies the banner of “constructivism, ” taking its name, Ernst von Glasersfeld tells us, from comments of the eighteenth century Neapolitan philosopher Giambattista Vico. Vico was an admirer of the work of René Descartes, the seventeenth century philosopher, mathematician, and scientist, who offered rational explanations of a wide range of natural phenomena, for which he is famous to this day. Unfortunately, Descartes’ writings had been banned by the Church, suffering the same fate as those of Copernicus and Galileo. And that region of the Italian peninsula was then ruled by a Spanish viceroy, who, like the king he served, had little tolerance for ideas that might be seen as challenging religious authorities. For this reason, Vico faced a difficult problem in publishing his philosophy of history, which sought rational and historical explanations of the origin of human cultures from early primitive beginnings to the complexities of European states of that day. Such explanations might seem to conflict with the conventional wisdom that human societies were the work of Divine providence, being properly ruled by kings responsible only to God. His solution was to tell us that, as God had created the world, only He could truly know it. The best that mere humans could do was to construct their own ideas about it, without pretending that these could ever reach the stature of His knowledge. By putting matters in this way, Vico was successful in getting the publication of his work permitted by the Church, and his philosophy of history stands today as a historical landmark on its own.