This paper proposed the application of the constructivist approach in virtual university where learners can learn based on their learning style, information and skills to succeed in life and also in their job. Constructivist learning and the strategies in constructivist learning can foster in-depth learning and practical application. Integration of communication and information technologies into curricula offers significant potentials for designing new learning environments, and advancing research and development in learning theories. Based on the main aspects of the constructivist approach, traditional universities and classroom cannot provide the conditions for learners to construct the knowledge for themselves, for this reason virtual university with the communication and information technologies (ICT) can implement constructivist strategies in the process of teaching and learning. In virtual university, constructivism promotes the learner’s skills to solve real-life problems and practical problems.

Key words: virtual university, constructivist learning, constructivist approach in virtual university.

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

Inspired by Enactivist philosophy yet in dialog with it, we ask what theory of embodied cognition might best serve in articulating implications of Enactivism for mathematics education. We offer a blend of Dynamical Systems Theory and Sociocultural Theory as an analytic lens on micro-processes of action-to-concept evolution. We also illustrate the methodological utility of design-research as an approach to such theory development. Building on constructs from ecological psychology, cultural anthropology, studies of motor-skill acquisition, and somatic awareness practices, we develop the notion of an “instrumented field of promoted action”. Children operating in this field first develop environmentally coupled motor-action coordinations. Next, we introduce into the field new artifacts. The children adopt the artifacts as frames of action and reference, yet in so doing they shift into disciplinary semiotic systems. We exemplify our thesis with two selected excerpts from our videography of Grade 4–6 volunteers participating in task-based clinical interviews centered on the Mathematical Imagery Trainer for Proportion. In particular, we present and analyze cases of either smooth or abrupt transformation in learners’ operatory schemes. We situate our design framework vis-à-vis seminal contributions to mathematics education research.

Key words: mathematics, education, motor learning, conceptual learning, motor problem, conceptual metaphor

Enactivism theorizes thinking as situated doing. Mathematical thinking, specifically, is handling imaginary objects, and learning is coming to perceive objects and reflecting on this activity. Putting theory to practice, Abrahamson’s embodied-design collaborative interdisciplinary research program has been designing and evaluating interactive tablet applications centered on motor-control tasks whose perceptual solutions then form the basis for understanding mathematical ideas (e.g., proportion). Analysis of multimodal data of students’ hand- and eye- movement as well as their linguistic and gestural expressions has pointed to the key role of emergent perceptual structures that form the developmental interface between motor coordination and conceptual articulation. Through timely tutorial intervention or peer interaction, these perceptual structures rise to the students’ discursive consciousness as “things” they can describe, measure, analyze, model, and symbolize with culturally accepted words, diagrams, and signs – they become mathematical entities with enactive meanings. We explain the theoretical background of enactivist mathematics pedagogy, demonstrate its technological implementation, list its principles, and then present a case study of a mathematics teacher who applied her graduate-school experiences in enactivist inquiry to create spontaneous classroom activities promoting student insight into challenging concepts. Students’ enactment of coordinated movement forms gave rise to new perceptual structures modeled as mathematical content.

A rising epistemological paradigm in the cognitive sciences – embodied cognition – has been stimulating innovative approaches, among educational researchers, to the design and analysis of STEM teaching and learning. The paradigm promotes theorizations of cognitive activity as grounded, or even constituted, in goal-oriented multimodal sensorimotor phenomenology. Conceptual learning, per these theories, could emanate from, or be triggered by, experiences of enacting or witnessing particular movement forms, even before these movements are explicitly signified as illustrating target content. Putting these theories to practice, new types of learning environments are being explored that utilize interactive technologies to initially foster student enactment of conceptually oriented movement forms and only then formalize these gestures and actions in disciplinary formats and language. In turn, new research instruments, such as multimodal learning analytics, now enable researchers to aggregate, integrate, model, and represent students’ physical movements, eye-gaze paths, and verbal–gestural utterance so as to track and evaluate emerging conceptual capacity. We – a cohort of cognitive scientists and design-based researchers of embodied mathematics – survey a set of empirically validated frameworks and principles for enhancing mathematics teaching and learning as dialogic multimodal activity, and we synthetize a set of principles for educational practice.

There seem to be as many definitions of constructivism as there are minds to construct them. At least this is how it felt after reading the chapters by Duit (chap. 14), Saxe (chap. 15), and Spivey (chap. 16). Or perhaps there are as many questionings of constructivist ideas as practices in the field of education. For example, a teacher experiences a different set of constraints than a researcher or a designer, and these constraints in turn shape theories of learning in different ways.

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.

This paper examines the shared roots and crossed paths between Jean Piaget’s constructivism, what Seymour Paper refers to as “constructionism,” and socio-cultural theories as epitomized by Lev Vygotsky. We do so in the light of more situated, pragmatic, and ecological approaches to human cognition. All these views are developmental (stressing the genesis children’s interests and abilities over time), experiential (in the sense that knowledge is rooted in sensori-motor activity) and interactionist (people are seen as constructing their knowledge by transforming the world). Yet, the views also differ, each highlighting some aspects of how children grow and learn, while leaving other questions unanswered. Piaget’s main contribution was to flesh out what is common in children’s ways of thinking at different stages of their cognitive development and, more important, how consistent, robust, and generally “adapted” their views are. The theory stresses the progressive de-contextualization of knowledge (from here-and-now to then-and-there) and identifies some of the hidden mechanisms (internal reorganizations) that drive human cognitive development. Papert, in contrast, stresses how individuals learn in context and how they use their own – and other people’s – externalizations as objects to think with, especially as their convictions break down. His approach is more situated. Papert is particularly interested the role of new media in human learning. Both Papert and Vygotsky shed light on the articulations between direct and mediated experience (from action and tool-use to enactments, language, and symbol-use). Yet Vygotsky and the Russian school have paid much closer attention to the role of caring adults and peers in a child’s initiation to her culture. They remind us that it takes a whole village to raise a child. Integrating the views helps rethink how children come to make sense of their experiences, and how they find their own places – and voices – in the world. At once world-makers, world-readers, and dwellers in the world, human infants are granted from birth with the abilities to optimize exchanges with people and things by moving in and out of contexts, by shifting perspectives, and by switching roles or standpoint. They are extraordinary learners, and much can be learned from them. Lastly, while mostly inner-driven and curious, children need caring adults, secure grounds, and engaging peers and props to thrive and grow. Tools, media, and cultural artifacts are the tangible forms through which they explore their surrounds, express their thoughts, and share the fun with others – and the traces left by those who came before (cultural heritage) become a terrain for newcomers to create their paths.

Key words: Constructivism, Piaget, Papert, Vygotsky, situated learning, embodied cognition, ecology of mind

In the drive to improve standards, the collection and dissemination of numerical data still directs much contemporary educational policy. However, recent publications and debates seemingly attempt to reorient discussion from performance to learning. In support, constructivism is often referenced as a contributor in this endeavour. However, constructivism is not a single unified theory either of knowledge or pedagogy. This article identifies one version of constructivist thinking, social constructivism, both in terms of its underlying epistemology (theory of knowledge) and related pedagogy. Contemporary educational theories are then outlined to demonstrate that many practical solutions and theoretical ideas now presented as ‘good learning and teaching’ have much in common with social constructivist thinking. Finally, the article concludes by identifying two issues that require further discussion and debate if pedagogy of a social constructivist nature is to be considered.

Education has institutionalized a process that reifies cultures, ecological communities, and ultimately evolution itself. This enclosure has lessened our sensitivity to the pedagogical (eteragogical) nature of our lived relations with other people and with other living beings. By acknowledging that learning and teaching go on between species, humans can regain an eteragogical sense of the interspecies curricula within which they exist. This article explores interspecies lived curricula through a selection of ideas from ecopragmatist Anthony Weston, and cybernetician Gregory Bateson, and through lived experiences with shorebirds of Lake Ontario. Some gulls and a tern teach the author to enrich and diversify, rather than constrict, the potentiality of life. In so doing, being ecological and being educative become unified concepts. Relevance: The publication is concerned with the relational implications between humans and other species of Bateson’s cybernetic theory of learning.