Ackermann E. K. (2004) Constructing knowledge and transforming the world. In: Tokoro M. & Steels L. (eds.) A learning zone of one’s own: Sharing representations and flow in collaborative learning. IOS Press, Amsterdam: 15–37. https://cepa.info/3894

The first part of this paper examines the differences between Piaget’s constructivism, what Papert refers to as“constructionism,” and the socio-constructivist approach as portrayed by Vygotsky. All these views are developmental, and they share the notion that people actively contribute to the construction of their knowledge, by transforming their world. Yet the views also differ, each highlighting on some aspects of how children learn and grow, while leaving other questions unanswered. Attempts at integrating these views [learning through experience, through media, and through others] helps shed light on how people of different ages and venues come to make sense of their experience, and find their place – and voice – in the world. Tools, media, and cutural artifacts are the tangible forms, or mediational means, through which we make sense of our world and negociate meaning with others. In the second part of this paper, I speak to the articulations between make-believe activities and creative symbol-use as a guiding connection to rethink the aims of representations. Simulacrum and simulation, I show, play a key role besides language in helping children ground and mediate their experience in new ways. From computer-based microworlds for constructive learning (Papert’s turtle geometry, TERC’s body-syntonic graphing), to social virtual environments (MUDing). In each case, I discuss the roles of symbolic recreation, and imaginary projection (people’s abilities to build and dwell in their creations) as two powerful heuristic to keep in touch with situations, to bring what’s unknown to mind’s reach, and to explore risky ideas on safe grounds. I draw implications for education.

Bausch K. (2010) Body wisdom: Interplay of body and ego. Ongoing Emergence Press, Atlanta GA. https://cepa.info/370

Body Wisdom applies a non-dualist and phenomenological approach. It contends that our bodies are microcosms of the universe and bearers of its unspoken secrets. It describes how our egos arise from our bodies through language. When we focus our hearts on troubling questions, our bodies come through for us. Open questions posed to the unconscious act as the strange attractors of chaos theory. They enable the creative speech of discovery. Nietzsche, Merleau Ponty, holograms, chaos theory, Zen Buddhism, and fractal geometry bear witness.

Greenstein S. (2014) Making sense of qualitative geometry: The case of amanda. Journal of Mathematical Behavior 36: 73–94. https://cepa.info/1195

This article presents a case study of a seven-year-old girl named Amanda who participated in an eighteen-week teaching experiment I conducted in order to model the development of her intuitive and informal topological ideas. I designed a new dynamic geometry environment that I used in each of the episodes of the teaching experiment to elicit these conceptions and further support their development. As the study progressed, I found that Amanda developed significant and authentic forms of geometric reasoning. It is these newly identified forms of reasoning, which I refer to as “qualitative geometry,” that have implications for the teaching and learning of geometry and for research into students’ mathematical reasoning. Relevance: This article describes a study for which I used Steffe & Thompson’s teaching experiment methodology to produce a learning trajectory (Steffe 2003, 2004) resulting from the actual teaching of children. In order to perform the conceptual analysis, the theoretical framework draws on von Glasersfeld’s scheme theory, which is an interpretation of Piaget’s theory of cognitive development.

Inselberg A. & Foerster H. von (1970) A mathematical model of the basilar membrane. Mathematical Biosciences 7: 341–363.

A two-parameter basilar membrane model with uniform geometry, mass, and stiffness distribution is studied. The exact solution of the equation of motion is obtained. For certain model configurations the displacement patterns of the membrane consist of traveling waves and damped standing waves. A place principle is observed with the direction of the shift governed by the relative magnitudes of the model parameters. The qualitative effect of a stiffness gradient along the membrane on the place principle is discussed. Thresholds, with respect to frequency, are found that suggestthat the location of the low-frequency threshold depends only on the membrane length.

Marks-Tarlow T. (2004) Semiotic Seams: Fractal Dynamics of Re-entry. Cybernetics & Human Knowing 11(1): 49–62. https://cepa.info/3388

This essay concerns fractal geometry as a bridge between the imaginary and the real, mind and matter, conscious and unconscious. The logic rests upon Jungs theory of number as the most primitive archetype of order for linking observers with the observed. Whereas Jung focused upon natural numbers as the foundation for order that is already conscious, I offer fractal geometry, with its endlessly recursive iteration on the complex number plane, as the underpinning for a dynamic unconscious destined never to become fully conscious. Everywhere in nature, fractal separatrices articulate a paradoxical zone of bounded infinity that both separates and connects natures edges. By occupying the ‘space between’ dimensions and levels of existence, fractal boundaries exemplify reentry dynamics of Varela’s autonomous systems, as well as Hofstadters ever-elusive ‘tangled hierarchy’ where brain and mind are most entwined. At this second-order, cybernetic frontier, the horizon of observers observing the observation process remains infinitely complex and ever receding from view. I suggest that the property of self-similarity, by which the pattern of the whole permeates fractal parts at different scales, represents the semiotic sign of identity in nature.

Moran D. T., Rowley III J. C., Zill S. N. & Varela F. J. (1976) The mechanism of sensory transduction in a mechanoreceptor. Journal of Cell Biology 71(3): 832–847. https://cepa.info/2054

This paper describes the ultrastructural modifications that cockroach campaniform sensilla undergo at three major stages in the molting cycle and finds that the sensilla are physiological functional at all developmental stages leading to ecdysis. Late stage animals on the verge of ecdysis have two completely separate cuticles. The campaniform sensillum sends a 220-mum extension of the sensory process through a hole in its cap in the new (inner) cuticle across a fluid-filled molting space to its functional insertion in the cap in the old (outer) cuticle. Mechanical stimulation of the old cap excites the sensillum. The ultrastructural geometry of late stage sensilla, coupled with the observation they are physiolgically functional, supports the hypotheses (a) that sensory transduction occurs at the tip of the sensory process, and (b) that cap identation causes the cap cuticle to pinch the tip of the sensory process, thereby stimulating the sensillum.

Noë A. (2008) Précis of Action in Perception. Philosophy and Phenomenological Research 76(3): 660–665. https://cepa.info/6713

Excerpt: We perceive as much as we do because, in a way, we perceive so little. Traditional approaches to perception have found it difficult to accommodate the distinctive amodality of perceptual consciousness. Take vision, for example. If we think of what is visible in terms of projective geometry and artificial perspective, then, however paradoxical it may sound, vision is not confined to the visible. We visually experience much more than that. We experience what is hidden (occluded) and what is out of view. For example, we have a sense of the visual presence of the back of a tomato when we look at one sitting before us, even though the back of the tomato is out of view; and we experience the circularity of a plate, its actual shape, even when, seen from an angle, the circularity itself can’t be seen. Or consider your sense of the detail of the scene before your eyes now. You have a sense of the presence the detail; the scene is replete with detail. But it is not the case that you seem to yourself actually to see all the detail; you can no more see every bit of detail in sharp focus and high resolution than you can see the tomato from all sides at once. Just as the back of the tomato shows up in your experience although it is hidden from view, so the detailed scene before you shows up in your experience, although the detail outstrips by far what can be taken in at a glance. The world outstrips what we can take in at a glance; but we are not confined to what is available in a glance.

Panorkou N. & Maloney A. (2015) Elementary Students’ Construction of Geometric Transformation Reasoning in a Dynamic Animation Environment. Constructivist Foundations 10(3): 338–347. https://cepa.info/2146

Context: Technology has not only changed the way we teach mathematical concepts but also the nature of knowledge, and thus what is possible to learn. While geometric transformations are recognized to be foundational to the formation of students’ geometric conceptions, little research has focused on how these notions can be introduced in elementary schooling. Problem: This project addressed the need for development of students’ reasoning about and with geometric transformations in elementary school. We investigated the nature of students’ understandings of translations, rotations, scaling, and stretching in two dimensions in the context of use of the software application Graphs ’n Glyphs. More specifically, we explored the question “What is the nature of elementary students’ reasoning of geometric transformations when instruction exploits the technological tool Graphs ’n Glyphs?” Method: Using a design research perspective, we present the conduct of a teaching experiment with one pair of fourth-graders, studying translation and rotation. The project investigated how and to what extent activity using Graphs ’n Glyphs can elicit students’ reasoning about geometric transformations, and explored the constraints and affordances of Graphs ’n Glyphs for thinking-in-change about geometric transformations. Results: The students proved adept using the software with carefully designed tasks to explore the behavior of two-dimensional shapes, distinguish among transformations, and develop predictions. In relation to varied conditions of transformations, they formed generalizations about the way a shape behaves, including the use of referent points in predicting outcomes of translations, and recognizing the role of the center of rotation. Implications: The generalizations that the students developed are foundational for developing an understanding of the properties of transformations in the later years of schooling. Dynamic technological approaches to geometry may prove as valuable to elementary students’ understanding of geometry as it is for older students. Constructivist content: This study contributes to ongoing constructivism/constructionism conversations by concentrating on the transformation of ideas when engaging learners in activity through particular contexts and tools. Key Words: Geometry, transformations, constructionist technologies.

Quale A. (2012) On the Role of Constructivism in Mathematical Epistemology. Constructivist Foundations 7(2): 104–111. https://cepa.info/252

Context: the position of pure and applied mathematics in the epistemic conflict between realism and relativism. Problem: To investigate the change in the status of mathematical knowledge over historical time: specifically, the shift from a realist epistemology to a relativist epistemology. Method: Two examples are discussed: geometry and number theory. It is demonstrated how the initially realist epistemic framework – with mathematics situated in a platonic ideal reality from where it governs our physical world – became untenable, with the advent of non-Euclidean geometry and the increasing abstraction of the number concept. Results: Radical constructivism offers an alternative relativist epistemology, where mathematical knowledge is constructed by the individual knower in a context of an axiomatic base and subject items chosen at her discretion, for the purpose of modelling some part of her personal experiential world. Thus it can be expedient to view the practice of mathematics as a game, played by mathematicians according to agreed-upon rules. Constructivist content: The role played by constructivism in the formulation of mathematics is discussed. This is illustrated by the historical transition from a classical (platonic) view of mathematics, as having an objective existence of its own in the “realm of ideal forms,” to the now widely accepted modern view where one has a wide freedom to construct mathematical theories to model various parts of one’s experiential world.

Purpose: The purpose of this paper is to examine shared principles of “irreducibility” or “undecidability” in second-order cybernetics, architectural design processes and Leibniz’s geometric philosophy. It argues that each discipline constructs relationships, particularly spatio-temporal relationships, according to these terms. Design/methodology/approach – The paper is organized into two parts and uses architectural criticism and philosophical analysis. The first part examines how second-order cybernetics and post-structuralist architectural design processes share these principles. Drawing from von Foerster’s theory of the “observing observer” it analyses the self-reflexive and self-referential modes of production that construct a collaborative architectural design project. Part two examines the terms in relation to Leibniz’s account of the “Monad”. Briefly, developing the discussion through Kant’s theory of aesthetics, it shows that Leibniz provides a “prototype” of undecidable spatial relations that are also present in architectural design and second-order cybernetics. Findings: The paper demonstrates that second-order cybernetics, architectural design and metaphysical philosophy enable interdisciplinary understandings of “undecidability”. Practical implications: The paper seeks to improve understanding of the geometric processes that construct architectural design. Originality/value – The paper explores interdisciplinary connections between the disciplines, opening up potential routes for further examination Its analysis of the aesthetic and geometric value of the Monad (rather than its perspectival value) provides a particularly relevant link for discussing the aesthetic production and experience of spatial relations in second-order cybernetics and contemporary architectural design.