Brent Davis

Understanding Learning Systems: Mathematics Education and Complexity Science.

Complexity science may be described as the science of learning systems, where learning is understood in terms of the adaptive behaviors of phenomena that arise in the interactions of multiple agents. Through two examples of complex learning systems, we explore some of the possible contributions of complexity science to discussions of the teaching of mathematics. We focus on two matters in particular: the use of the vocabulary of complexity in the redescription of mathematical communities and the application of principles of complexity to the teaching of mathematics. Through the course of this writing, we attempt to highlight compatible and complementary discussions that are already represented in the mathematics education literature.

Challenging Images of Knowing: Complexity Science and Educational Research.

This article represents an attempt to reconcile discussions of aspects of educational research with recent developments in complexity science. It is argued that current characterizations of and distinctions among research methodologies in education are potentially counterproductive, in large part because they tend to be defined against or in terms of principles and methods that have been rendered problematic within the sciences. To develop this point, the authors draw on several contemporary discourses: poststructuralist methods are used to foreground the Euclidean (plane) geometric roots of much of the vocabulary of educational research; fractal geometry is taken as a source of images and analogies to support alternative conceptions of knowledge, learning and teaching; informed by poststructuralist and fractal geometric notions, the authors turn to complexity science and argue that it is fitted to and offers important elaborations of current discussions of educational research methodologies. In the process, they suggest that it may be time to abandon some of the prominent distinctions used to describe educational research, including ‘quantitative versus qualitative’ and ‘sciences versus humanities’.

Why Aren't They Getting This? Working through the regressive myths of constructivist pedagogy

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.

Curriculum forms : on the assumed shapes of knowing and knowledge

Historically, the logico-rational mode of argumentation co-evolved with particular mathematical systems and particular geometrically-informed manners of interpreting experience and perception. We examine some of the ways these geometries continue to shape the sensibilities, practices and structures of much of curriculum discourse, in spite of the well-developed critiques of their associated logics. We draw on fractal geometry, a new field of mathematical study, to illustrate the pervasiveness and the constraining tendencies of classical geometries. We develop the suggestion that fractal geometry is a mathematical analogue to such discourse fields as post-modernism, post-structuralism and ecological theory. We attempt to show how, through the visual metaphor of a fractal image, conventional theories of knowing and knowledge might be seen as not only compatible, but as nested in and suggestive of one another. We briefly examine how fractal geometry can complement and inform other emergent sensibilities in curriculum, in particular, those discussions critical of the linear structures associated with classical logic.

Complexity science and educational action research: toward a pragmatics of transformation

Abstract In this article, the authors extend Phelps & Hases (2002) explorations of the theoretical and methodological connections of complexity theory and action research by emphasizing complexity science as the study of learning systems. By emphasizing the importance of ‘complexity thinking’, an argument is made for conceptualizing action research as a ‘pragmatics of transformation’ that explicitly aims to bring together the self-interests of autonomous agents into grander collective possibilities. In addition, the authors offer pragmatic advice that is relevant for the ‘educational’ and ‘learning’ aspects of educational action research by describing a number of conditions that need to be in place in order for complex learning systems to develop and thrive.

Canadian Identity and Curriculum Theory: An Ecological, Postmodern Perspective

In this article, we develop the thesis that curriculum studies work in Canada might be characterized in terms of some persistent and consistent theoretical commitments, ones that we suggest might have been prompted in part by the nations history and by popular commentaries on national identity. We draw on ecological and postmodern discourses in efforts to conceptualize and to describe a relationship between Canadian culture(s) and the development of theories of curriculum within the Canadian context.

The Importance of Teachers’ Mathematical Awareness for In-the-Moment Pedagogy

We propose that what teachers can say or write about their views of mathematics and about a range of pedagogic strategies and didactic tactics that they use is of minor importance compared to what comes to mind moment by moment when they are planning or leading a lesson. Beginning with a quote from a teacher that provoked a focus on the role and nature of in-the-moment choices made in the midst of teaching, we offer some comments provoked in us. Following a concise history of attempts to articulate ‘mathematics needed for teaching’, three further brief accounts of incidents led us to propose that what really matters, in the words of Heidegger and others, is one’s ‘mathematical being’, because it orients awareness and is the source and basis for choices, whether consciously or unconsciously made. The development of mathematical being is described using the discourse of awareness in the sense of Gattegno, making use of the discipline of noticing to educate awareness. These are then used to extend the analysis of prepositional suffixes for the gerund knowing begun by Ryle in order to focus on the development of ‘connective tissue’ between mathematical awareness and in-the-moment pedagogical choices of actions.RésuméÀ notre avis, ce que les enseignants peuvent dire ou écrire sur ce qu’ils pensent de certaines questions mathématiques et des différentes stratégies pédagogiques et didactiques qu’ils utilisent, a peu d’importance comparativement à ce qui leur vient à l’esprit sur le moment lorsqu’ils planifient ou lorsqu’ils donnent un cours. Partant du commentaire d’un enseignant qui nous a conduits à centrer notre attention sur le rôle et la nature des choix faits sur le moment pendant les cours, nous présentons nos propres commentaires. Après un bref compte rendu des tentatives précédentes d’expliciter « les connaissances mathématiques nécessaires pour enseigner », trois autres brefs comptes rendus d’incidents nous ont conduits à l’hypothèse suivante: ce qui compte réellement, comme le dit entre autres Heidegger, c’est « l’être mathématique », car il oriente la prise de conscience et est à la base des choix qui sont faits, qu’il s’agisse de choix conscients ou inconscients. Le développement de « l’être mathématique » est décrit en termes de prise de conscience au sens ou` l’utilise Gattegno, c’est-a`-dire qu’il consiste à remarquer pour former la conscience. Suivant Ryle, nous procédons ensuite à une analyse des suffixes prépositionnels associés au gérondif « connaissant » pour tenter de mieux définir le développement des « tissus conjonctifs » qui lient la conscience mathématique et les choix pédagogiques spontanés.

Obstacles to the Dissemination of Mathematics Education Research

Mathematics education research has often been undertaken with an expectation that the results would have some influence on teachers’ practice and on educational policies including curriculum, but numerous obstacles seem to limit that influence. This investigation of the topic begins by exploring the meanings of the words obstacles, dissemination, mathematics, education, and research. In section 2, the development process and the relationship between research, policy, and practice are examined. Next, some variables that influence change are considered. In the fourth section, some of the obstacles are discussed under the headings ‘research and researchers’, ‘teachers and schools’, and ‘systems’. Finally, three strategies for the future are suggested; these relate to communication and complexity, changing the development model, and extending the boundaries of research.

Dynamic Responsive Pedagogy: Implications of Micro-Level Scaffolding

In mathematics education, scaffolding is often viewed as a mechanism to provide temporary aid to learners to enhance mathematical understanding. Micro-level scaffolding is process by which the teacher returns the student(s) to a conceptual point where scaffolding is not needed. Then the teacher creates a series of incrementally more complex tasks leading to the original task. This process is dynamic, as it often requires multiple steps, and it is responsive because involves moment-by-moment assessment, which shapes each increment. In this paper, we present data on how experienced teachers in the Math Minds Initiative employ micro-level scaffolding. Implications of micro-level scaffolding are discussed.

Consciousness and the Literary Engagement: Toward a Bio-Cultural Theory of Reading and Learning

In this paper, we develop the understanding that in context of expanding notions of the literary engagement, consciousness is understood as a process that both participates in the acts of reading and response, and at the same time, is potentially transformed by those acts. Yet, uninterrogated understandings of consciousness – what it is, what it does, what it feels like - continue to shape the way we structure our experiences with literature in the classroom in predominantly implicit ways. In the context of an enactivist understanding of cognition, consciousness emerges as an orienting feature that brings together the cultural and biological aspects of the literary experience.