What evolutionary account explains our capacity to reason mathematically? Identifying the biological provenance of mathematical thinking would bear on education, because we could then design learning environments that simulate ecologically authentic conditions for leveraging this universal phylogenetic inclination. The ancient mechanism coopted for mathematical activity, I propose, is our fundamental organismic capacity to improve our sensorimotor engagement with the environment by detecting, generating, and maintaining goal-oriented perceptual structures regulating action, whether actual or imaginary. As such, the phenomenology of grasping a mathematical notion is literally that – gripping the environment in a new way that promotes interaction. To argue for the plausibility of my thesis, I first survey embodiment literature to implicate cognition as constituted in perceptuomotor engagement. Then, I summarize findings from a design-based research project investigating relations between learning to move in new ways and learning to reason mathematically about these conceptual choreographies. As such, the project proposes educational implications of enactivist evolutionary biology.

Key words: constructivism, design, education, enactivism, mathematics

In a previous paper, an interpretation of spirituality along constructivist lines was proposed (Gash and Shine Thompson, 2002). One of the lines of exploration discussed personal transformation as a possible consequence of an experience of an epiphany – a moment of grace. Epiphanies are first, grounded in constructivist psychology as moments when a person shifts levels to reach new understandings (Gregory Bateson, 1987). Epiphanies are also moments of insight that allow the possibility of personal transformation, and hence potentially desirable experiences of spiritual growth. In the present paper we outline a series of experiences of epiphanies in children’s learning in the context of a project on constructionist learning led by one of us – Deirdre Butler. The purpose of the paper is to make a case for the importance of such moments as providing opportunities for personal growth, encapsulated in the title of the project EmpoweringMinds. Relevance: The value of wonder in education; using digital technology in classrooms

Key words: Constructivism, spirituality, children, creative learning

An understanding of Piaget’s theoretical position on the knowing process is necessary in order to appreciate fully the impact of his research. His theory of knowledge has many significant points in common with that of the German philosopher Kant. This article sketches the epistemologies of the 17th and 18th century movements of rationalism, empiricism, and romanticism which preceded Kant, Kant’s revolutionary conclusions concerning reality and the knowing process, and some parallels and areas of divergence between the Kantian and Piagetian theories of knowledge

Two aspects of Vico’s constructivist epistemology are germane to contemporary cognitive developmental psychology. These aspects are Vico’s account of cognitive operations and of the limits to human knowledge of the world. Drawing on Vico’s epistemological treatise, and on contemporary commentary on Vico, it is argued that this eighteenth-century constructivist epistemology is useful in two ways. First, by being a consistent, and so radical, constructivism it may be helpful in clarifying the meaning of the environment in Piaget’s theory. Second, the description of mental operations may provide a way of overcoming objections to the overly formal quality of Piaget’s basic concrete-operational structures.

Key words: Cognitive development, Genetic epistemology, Piaget, Realism, Theory formulation, Vico

As traditional religious observances decline and Ireland’s consumer society competes in a global economy, we have evidence that many Irish young people find their heroes in American culture. It is timely therefore to reflect on our legacy of constructs of spirituality. Here we open a conversation grounded on Bateson’s analysis of epistemology, in order to begin an examination the Irish spiritual heritage within the discourse of constructivism. We explore the idea that it is in shifting between Bateson’s contextual levels that epiphanies are possible. This allows movement between understandings, so facilitating the construction of alternative realities. Psychotherapists call certain forms of this awareness insight, the sliding is central to being in love, and theologians call it grace.

The terms, period, phase, stage, and level – all frequent in the literature on development – are examined in order to specify conceptual differences. The authors suggest that while level has no inherent temporal component, the other three refer to stretches of time and should be differentiated according to whether or not they refer to (a) part of a progression; (b) a recurrent state or event; (c) quantitative change, and (d) qualitative change. The four terms can be distinguished as different combinations of the given criterial elements.

Key words: cognition, development

Empiricist and nativist approaches to understanding the origins and development of conceptions of logical necessity are proposed and criticized. A constructivist alternative is proposed, incorporating contemporary ideas regarding the role of structure, mechanisms of development, the metacognitive basis of stages, and the relation of development and learning. Three stages are postulated. The first involves the development of various classification and seriation behaviors. The second (concrete) stage is marked by a reflection on these behaviors, involving the construction of implicit concepts of necessity which, given the truth of certain premises, require certain conclusions. Finally, the third (formal) stage is marked by a further metacognitive reconstruction of logical necessity on a plane still further removed from empirical truth, thereby yielding the concept of inferential validity. Relevant research is briefly reviewed.

Piaget’s logical model of formal operations is problematic since it is unclear how Piaget’s logic should be understood. In a recent review, Braine and Rumain conclude that neither of the two available interpretations is adequate. Under one interpretation, formulae which should be compatible are incompatible in Piaget’s logic. Under the other interpretation, formulae which should be incompatible are compatible in Piaget’s logic. A constructivist interpretation is outlined so as to overcome these weaknesses. It is proposed that the attribution of formal operational thinking depends upon an individual’s generalization of classificatory abilities which are present during concrete operations. Each of the formal operations has a unique specification in terms of the 16 patterns inherent in such classifications. Access to formal operations does not require conscious awareness of propositional symbolism. The interpretation is shown to be minimally adequate in avoiding the objections which invalidate existing interpretations. The interpretation is taken to exemplify two features of Piaget’s constructivism, namely the differentiation and integration of understanding which has a modal character.