Excerpt: I draw from recent developments in philosophy, biology, ecological thought, phenomenology, and curriculum theory in an effort to re-formulate a response to the question, Why teach mathematics?

Excerpt: We are not much concerned with who has it right and who has it wrong, nor with the sorts of contortions necessary to embrace two or more variations. Rather, we find it much more instructive to read across versions and interpretations and to highlight common elements as well as points of disjuncture. This is our project in the first section of this essay, and we use it to explore a possible shared intuition, one that actually points to the impossibility of bringing together different theoretical offerings. This examination is followed by a discussion of the manners in which constructivist vocabularies have been assimilated into the discourses of professional practice, curriculum development, and related projects. The main intention here is not to critique but to develop the assertion that, at the level of practical action, most constructivist discourses were not originally conceived as educational discourses. Oriented by this point, in the third section of the essay we speculate on the sorts of revisions and accommodations that might help to frame relevant constructivist insights in terms that are useful and appropriate to discussions of teaching and schooling. We look to several cross-disciplinary discourses in this section, both as means to read across different versions of constructivism and as sources of advice on a new vocabulary.

Through several collaborative inquiries with teachers in elementary and middle schools, we have noticed a troublesome trend: teachers have become familiar with many of the key terms and catchphrases of various constructivist discourses, yet they tend to be relatively unfamiliar with the developments in epistemology that have driven the rapid emergence of these vocabularies. In consequence, our efforts to invite teachers into current discussions of cognition have often been frustrated and frustrating. We argue that this situation is in large part due to two circumstances. First, the vocabularies chosen by constructivists are often too readily aligned with commonsense understandings of personal knowing and collective knowledge. Second, and closely related, educational theorists and researchers have not always been sufficiently attentive to the contexts of their work. As such, rather than prompting a break from deeply entrenched habits of thinking, constructivist discourses have often been co-opted to support renewed and regressive embraces of Platonic and Cartesian assumptions. Somewhat ironically, then, the work of many educational theorists and researchers appears to be carried out in ignorance of the tentative and participatory dynamics that are argued to be at the root of cognitive processes.

This book draws from current constructivist and enactivist theory to critically examine prevalent norms and taken for granted assumptions about knowing, learning, and teaching that continue to frame formal education. The authors argue from the transphenomenality of complex learning systems where teaching might be enacted as a kind of practicable consciousness for triggering and otherwise occasioning learning possibilities better fitted to the contingencies of time and place in an ever-emergent world.

As we witness a push toward studying spatial reasoning as a principal component of mathematical competency and instruction in the twenty first century, we argue that enactivism, with its strong and explicit foci on the coupling of organism and environment, action as cognition, and sensory motor coordination provides an inclusive, expansive, apt, and fit framework. We illustrate the fit of enactivism as a theory of learning with data from an ongoing research project involving teachers and elementary-aged children’s engagement in the design and assembly of motorized robots. We offer that spatial reasoning with its considerations of physical context, the dynamics of a body moving through space, sensorimotor coordination, and cognition, appears different from other conceptual competencies in mathematics. Specifically, we argue that learner engagements with diverse types of informationally ‘dense’ visuo-spatial interfaces (e.g., blueprints, programming icons, blocks, maps), as in the research study, afford some of the necessary experiences with/in a vast number of cases described by Varela et al. (1991) that enable the development of other mathematical competencies.

Key words: enactivism, sensorimotor coordination, spatial reasoning, robotics

This article seeks to interrogate conceptions of cognition and knowledge, explicit and implicit, that underpin conventional projects of educational research. Developed around our own efforts to make sense of the contingent and complex nature of a recent action research project, the discussion develops an enactivist account of cognition that is offered as an alternative both to subject-centred orientations (e.g. representationism and constructivism) and culture-privileging accounts (e.g. critical and sociocultural theories). The relevance of enactivism for educational action research – conceived as a site for learning, and hence tranformative of both individual and collective – is examined in terms of the practical and moral dimensions of the activity.