Contemporary Environmental and Mathematics Education Modelling Using New Geometric Approaches

Geometries of Liberation

Susan Gerofsky

• Publisher: Palgrave Pivot
• Year: 2018
• Print ISBN:
  9783319725222
• eBook ISBN:
  9783319725239
• Topic: Education
• Subtopic: Education General

This book takes a fresh approach to using educational tools to solve profound problems in societies. The authors bring perspectives from curriculum studies, mathematics education, environmental education, and Indigenous epistemologies to a new consideration of "geometries to think with". These tools reveal the wealth of resources and interrelationships in our world that have the potential to reconfigure and revitalize education. The transdisciplinary nature of the chapters and authors emphasizes the need for thinking beyond boundaries, while respecting the wisdom inherent in intellectual disciplines and traditions.

© The Author(s) 2018

Susan Gerofsky (ed.)Contemporary Environmental and Mathematics Education Modelling Using New Geometric Approacheshttps://doi.org/10.1007/978-3-319-72523-9_1

Begin Abstract

1. Introduction: Geometries of Liberation

Susan Gerofsky¹

(1)

Department of Curriculum & Pedagogy, University of British Columbia, Vancouver, BC, Canada

Susan Gerofsky

Email: [email protected]

Abstract

This book takes a fresh and provocative approach to a deep problem of our times: the need for new tools, models, and ‘geometries to think and act with’ that will allow us to live in new ways so that humans and all our greater-than-human relations are able to survive and thrive. The key premise of Geometries of Liberation is that Cartesian/Enlightenment/Modernist era geometries of straight lines and rectangular grids, the geometries that dominate our thoughts and social organization of space and time, prevent us from perceiving innovative approaches to living and learning—approaches that might even save our lives.

Keywords

Geometriesenvironmental educationmathematics educationIndigenous epistemologiescurriculum studies, arts-based approachesgrid, rectilinear, transdisciplinary alternativesmodelling space/timegreater-than-human world

End Abstract

This book takes a fresh and provocative approach to a deep problem of our times: the need for new tools, models, and ‘geometries to think and act with’ that will allow us to live in new ways so that humans and all our greater-than-human relations are able to survive and thrive. The key premise of Geometries of Liberation is that Cartesian/Enlightenment/Modernist era geometries of straight lines and rectangular grids, the geometries that dominate our thoughts and social organization of space and time, prevent us from perceiving innovative approaches to living and learning—approaches that might even save our lives.

The authors bring perspectives from mathematics and mathematics education, environmental education, curriculum studies, and Indigenous epistemologies to a new consideration of ‘geometries to think with’. These new tools reveal the wealth of resources and interrelationships in our world that have the potential to reconfigure and revitalize education and solve profound problems in our societies. The transdisciplinary nature of the chapters and authors emphasizes the need for thinking beyond boundaries, while respecting the wisdom inherent in intellectual disciplines and traditions.

The book is a collaboration among six leading thinkers and authors from Canada and the United States, including both new and established scholars. The six chapters, originally conceived as papers in a collective interactive symposium at the 2013 American Educational Research Association (AERA) conference, are coherent as a brief volume, leading from philosophical and theoretical grounding, to specific examples and personalized reflections, to a brilliant commentary that brings together the first five chapters in the context of Indigenous ways of knowing and their intersection with formal mathematics and geometries.

We hope that readers will find this short book concise and thought-provoking, an exhilarating conversation among stimulating writers and a brief, timely, and exciting volume. It is both philosophically challenging and full of hope for the future.

What Is Meant by ‘Geometries’ in This Book?

Most people encounter geometry in their elementary and secondary schooling in the form of Euclidean geometry: figures such as triangles, circles, parallelograms, and straight lines lying in an imagined flat plane extending infinitely in all directions. Euclidean geometry is all about axioms and proofs, and the sense of shape and space is quickly set aside in many geometry classes as geometry is used as an example of logical argumentation.

Cartesian algebraic geometry is also familiar to many people from pre-calculus and calculus courses in secondary and tertiary education, and its linear and curvilinear forms (e.g. straight lines, sine curves, parabolas, cubic functions) translate algebraic functions into shapes on a gridded flat plane. Once again, the sense of shape and space is often treated as subordinate to the learning of algebra, so that algebraic notation takes up much more instructional time than the geometry of curves on a grid.

Mathematicians are familiar with other geometries that may be less well known in general. Most non-Euclidean, non-Cartesian geometries were developed in the late nineteenth and twentieth centuries, relatively recently by the standards of mathematics. This multiplicity of ‘geometries’, all consistent systems within their own rules of engagement, includes spherical, hyperbolic, projective, affine, finite, elliptic, and fractal geometries. What is more, most mathematical conceptualizations can be represented spatially through sketches, diagrams, models, and computer modelling, so that concepts that are not considered fully developed geometric systems may still suggest geometries that are quite startlingly different from Euclidean lines, squares, and triangles.

The connected chapters in this book consider a wide variety of geometries of these sorts in the context of environmental education, mathematics teaching and learning, arts-based approaches, and Indigenous ways of knowing. In taking up ‘geometries’ that go beside, beyond, beneath, betwixt, between, and alongside the very familiar Euclidean planes and right angles that predominate in contemporary mainstream culture, the authors are also suggesting new models for conceiving of the world, making sense of phenomena, and finding better ways to cope with difficulties and opportunities. Having new ‘geometries to think and act with’ brings with it the potential to be differently in the world, including possibilities for more regenerative, ecological, sustainable ways of life where human and greater-than-human life can thrive. The authors experiment with these models and possibilities.

What Is Meant by ‘Liberation’ in This Book?

Since Brazilian educator Paulo Freire’s hugely influential work in the 1960s and 1970s, and particularly after the publication of his Pedagogy of the oppressed in the late 1960s, the term ‘liberation’ has been linked in educational circles with the Critical Pedagogy movement. Liberation theologists, including many bishops within the Catholic Church in Latin America from the 1950s and 1960s onwards, have worked and struggled within the mainstream church for social justice and human rights. Freire’s work within the field of popular education drew inspiration from similar conflicts and sources of strength within Brazilian society, and his work advocated strongly for an educational system that would help create equity, dignity, self-determination, and strength of community, especially for those who currently suffered oppression and poverty. The word ‘liberation’ in our subtitle, Geometries of liberation, will likely lead readers to connect this work with a Freirian critical pedagogy approach—and in fact, the authors are deeply sympathetic to this approach.

However, this book cannot really be characterized as a work of Freirian critical pedagogy. Where critical pedagogical writing focuses on human rights, oppression, inequity, and societal injustices, this book is concerned with liberation at a different scale and location—the liberation of the very geometric models we think with. The authors of this volume are concerned that the binary, rectilinear, Modernist conventions of visualization and linguistic metaphor are already binding our thoughts and actions to oppositional, boxed-in, hyperrational, colonial ways of being and doing in the world. Our concern is to experiment with the liberation of geometric metaphors from the solely binary and linear, using geometric modelling that is being developed in mathematics, the arts, and ecology, and which has been highly developed through millennia of traditions in Indigenous cultures.

Non-‘grid-like’ ways of thinking, doing, and being is suggested here as a substitute and/or supplement to the grid. Linear models, Euclidean geometry, and binary logic are not entirely vilified, but alternate geometries are seen as tools for modelling and conceptualizing the world in ways that promote better relations with the greater-than-human world, and within human societies as well. ‘Geometries of liberation’ ought to entrain some aspects of the kinds of human liberation Critical Pedagogy aims for, even though their starting points are not identical.

Chapters, Contributions, Connections

There are seven chapters in this book, connected by the themes of the rectilinear ‘grid’ and the alternate geometries that might offer us other ways of thinking, doing, and being in the world. Here is an account of the six chapters that follow this Introduction, and their contributions to the whole.

Chapter 2: ‘Shaped by the Places We Reason? Contrasting the Rectilinearity of Western Educational Thought with Other Possibilities’ by Brent Davis

Historically, the logico-rational mode of argumentation co-evolved with particular mathematical systems and particular geometrically informed manners of interpreting experience and perception. This chapter examines some of the ways these geometries continue to shape the sensibilities, practices, and structures of much of educational discourse, in spite of the well-developed critiques of their associated logics. Davis compares manufactured living environments with those of other cultures—specifically, sub-Saharan villages and remnants of several Plains First Nations settlements—drawing on fractal geometry to highlight a complementary mode of organizing cultural spaces. He develops the suggestions that (1) the available logics are associated with the available geometries of one’s living spaces and (2) fractal geometry is a mathematical analogue to such discourse fields as postmodernism, post-structuralism, and ecological theory.

Davis argues that 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. Finally, this chapter examines briefly how fractal geometry can inform discussions of learning theory, curriculum development, and teaching approaches.

Chapter 3: ‘Ecofractal Poetics: Five Fractal Geometries for Creative, Sustainable, and Just Educational Design’ by Marna Hauk

Hauk’s research concerns the use of fractal patterns to spark group creative collaboration for sustainability education design. Fractals have been shown to model the process of divergent creativity. Fractal patterns in nature can serve as templates or archetypes for innovative social structures and processes, addressing the poverties of reductionism in education itself.

Research in the sustainability education field leverages natural pattern, chaos, and critiques of reductive grid patterning, and these critiques include fractal geometries of liberation. These geometries have the potential to ...

Table of contents

• Cover
• Front Matter
• 1. Introduction: Geometries of Liberation
• 2. Shaped by the Places We Reason? Contrasting the Rectilinearity of Western Educational Thought with Other Possibilities
• 3. Ecofractal Poetics: Five Fractal Geometries for Creative, Sustainable, and Just Educational Design
• 4. Always an Abundance: Interstitial/Liminal Space, Time, and Resources that Are Invisible to the Grid
• 5. The Curricular Geometries of *SAMBA* Schools: Fractal Dimensions, Surface, Depth, and Recursion
• 6. Of Grids and Gardens: School Gardening and the Unsettling Attachments of Teaching Beside the Grid
• 7. Off the Grid
• Back Matter