Math Is Alive

152 Pages·2017·0.82 MB·English

by  Stacy K. Stockton

Math Is Alive: The Metaphor of Living School Disciplines
and Some of the Educational Implications
by
Stacy K. Stockton

A Thesis submitted to the Faculty of Graduate Studies of the University of Manitoba
in partial fulfillment of the requirements of the degree of

MASTER OF EDUCATION

Department of Curriculum, Teaching and Learning
Faculty of Education
University of Manitoba
Winnipeg

Copyright © 2017 by Stacy K. Stockton
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Abstract

This thesis is concerned with an unconventional understanding of the mathematics
taught in schools and, thus, the role that students’ engagement with mathematics plays
for mathematics as a discipline. The discipline of mathematics has a rich
philosophical history. In this thesis I explore mathematics through the lens of
metaphor theory first developed by George Lakoff and Mark Johnson. This area of
scholarship is used to develop the idea that all abstract concepts are conceived of
metaphorically. From this perspective, the metaphors at the foundation of several
major philosophies of mathematics are analyzed. This analysis concludes by asking if
there might be another metaphor that allows for a different, more productive,
understanding of school mathematics than commonly used mathematical metaphors.
The metaphor “Math is Alive” is offered as an alternative through a detailed analysis
of living systems in the tradition of Humberto Maturana and Francisco Varela’s
theory of autopoietic organization. The final chapter looks specifically at school
mathematics as a part of mathematics as a discipline, and how “Math is Alive” might
alter how educators view the role of children in mathematics.
MATH IS ALIVE ii

Acknowledgements

This thesis would not have been possible without the guidance of my academic
advisor, Thomas Falkenberg, and the support of my committee members, Barbara
McMillan and Paul Betts.
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Table of Contents Page

Introduction ………………………………………………………………………… 1
Chapter One: Metaphor …………………………………………………………….. 4
The Integrated Theory of Metaphor ……………………………………………... 5
Some Attributes of Metaphor ……………………………………………………. 6
Categories of Metaphor ………………………………………………………….. 8
Where Metaphors Come From ………………………………………….………. 10
New Metaphors …………………………………………………………………. 13
Conclusion ………………………………………………………………………. 15
Chapter Two: What Mathematics Is (metaphorically speaking) …………………… 17
Mathematics and Metaphor ……………………………………………………... 18
Philosophy of Mathematics …………………………………………………...… 20
Objectivist Philosophies ……………...……………………………………… 20
Platonism ………………………...……………………………………….. 20
Structuralism ……………………...………………………………………. 21
Deductive Philosophies …………...…………………….……………………. 23
Logicism …………………………………………………………………... 23
Formalism …………………………………………………………………. 25
Intuitionism …………………………...…………………………………... 26
Humanist Philosophies ……………………………………………………….. 29
Social Constructivism ……………...…………………...……………….... 29
Embodied Mathematics …………………………………………………… 32
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Conclusion ………………………………………………………………………. 34
Chapter Three: Being Alive ……………………………………………………….... 36
Characteristics of Living Systems ………………………………………………. 38
Pattern of Organization ………………………………………………………. 39
Maintenance ………………………………………………………………….. 44
Growth, Adaptation and Evolution …………………………………………... 50
Conclusion ………………………………………………………………………. 60
Chapter Four: Making a Metaphor …………………………………………………. 62
Potential Mappings ……………………………………………………..……….. 63
Pattern of Organization ………………………………………………………. 65
Maintenance ………………………………………………………………….. 72
Growth, Adaptation and Evolution …………………………………………... 78
Conclusion ………………………………………………………………………. 87
Chapter Five: School Mathematics and Metaphor …………………………………. 89
School Mathematics as a Localized Area of Mathematics ……………………… 90
The Pervasiveness of Platonism …………………………………………..….. 92
Constructivism as Hidden Formalism ……………………………..………..... 96
Beyond Platonism ……………………………………………………..……... 98
Metaphors for Mathematics in Schools …………………………………….. 102
Research into Metaphors for Mathematics …………………………………. 103
Math is Alive …………………………………………………………………... 106
Pattern of Organization ……………………………………………………... 110
Maintenance ………………………………………………………………… 116
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Growth, Adaptation and Evolution …………………………………………. 122
Conclusion ..……………………………………………………………………. 125
References .………………………………………………………………………... 134
MATH IS ALIVE 1

“It is inorganic, yet alive, and all the more alive for being inorganic”
(Deleuze & Guattari, 1987, p. 520).

I first discovered the science of complexity in a course on arts education. My exploration of
its application to the arts in schools inspired my particular interest in autopoietic systems, and led
me to a study of biological living systems. Maturana and Varela’s Tree of Knowledge (1987),
Capra’s Web of Life (1996), and Deleuze and Guattari’s Thousand Plateaus (1987) altered my
understanding of being alive, and invited the possibility of living ideas.
As a teacher, I am constantly confronted with uncertainties and differences in paradigm when
exploring conceptions of disciplines through discussions with others and through professional
development and readings. Paradigmatic changes in disciplines over time and competing current
conceptions became more apparent to me. I also began to notice curious resemblances between
living systems and disciplines. This included similarities in their patterns of organization,
similarities in the way they behaved, and similarities in their processes of evolution.
As I continued to read further, seeking more recent explorations of living concepts from a
complexity perspective, I was unable to answer my questions about disciplines in particular, and
what it would mean for education if they were conceived of as being alive. Just as Deleuze relies
on his metaphor of rhizome to allow us to see our thinking about concepts differently, I have
begun this thesis to ask if the metaphor of a living discipline might allow us to think differently
about disciplines. Further, if we conceive of disciplines in this way, I wonder what this implies
for the way that we currently engage with them in classrooms, and what this could become.
This inquiry will be philosophical, relying on philosophical inquiry as described by Burbules
and Warnick (2006), Floden and Buchmann (1990), and to some extent, John Wilson (1963). It
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is not about finding answers, but about asking fundamental questions about what it means to be
alive, and what we perceive our role to be in the production of disciplined knowledge. It is about
the potential impact of these beliefs on disciplines themselves, which it would be impossible to
empirically measure. This is the realm of philosophical inquiry. The methods I have selected
represent several of Burbules and Warnick’s (2006) ten methods of philosophical inquiry.
These will include the use of concept analysis (method 1, Burbules & Warnick, 2006, p. 491)
through an extensive literature review on metaphor, the philosophy of mathematics, complex
living systems, and school mathematics methods and metaphors. My thesis deals with the
meanings that we assign to the terms we use in the contexts in which we use them, and how our
conceptions guide our actions.

Philosophical inquiry often relies on analyses of language or concepts. Inquiry will
focus on terms that seem to play an important role in an argument or practice, trying
to answer the question, what do these words or phrases mean or entail? (Floden &
Buchmann, 1990, p. 3)

Chapter one focuses on metaphor and its importance of our understanding of abstract
concepts. Chapters two and three explore the target and source domains in a living mathematics
metaphor, current conceptions of mathematics and the biology of living systems.
Using information gathered regarding mathematics and living systems, a metaphor is then
constructed in chapter four. This making of a metaphor will be shown to be a conceptual
mapping, “an ambitious review of conceptual mapping may include a review of related concepts
that share certain features with the primary object” (Burbules & Warnick, 2006, p. 492). The
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creation of a metaphor can identify correspondences in structure, maintenance, and reproduction
in complex systems and mathematics, following method 10, “synthesizing disparate research
from philosophy itself or other fields to find meaning and implications for educational theory and
practice” (Burbules & Warnick, 2006, p. 491). Here I ask: what does living have to do with
mathematics? Through this, I will be required to show that the vitality of a discipline is one of
the “ends or purposes education should achieve,” reflecting method 6 (Burbules & Warnick,
2006, p. 491).
Finally, in chapter five I turn to method 7, “speculating about alternative systems or practices
of education, whether utopian or programmatic, that contrast with and challenge conventional
educational understandings and practices” (Burbules & Warnick, 2006, p. 491), in order to
conceive of how this alternate view might impact the way mathematics is taught in schools. I
will consider the implications of a living mathematics in educational practice, and how this
might resolve difficulties with current perspectives, and create healthier relationships between
children and mathematics.
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CHAPTER ONE
Metaphor

“Language is a map, not a tracing”
(Deleuze & Guattari, 1987, p. 98).

Metaphor is a term most often used to describe a literary device. It is a type of figurative
language used to compare objects with like properties. At least this is what I tell my grade six
students. In 1980, Lakoff and Johnson challenged this assumption, by noticing the pervasiveness
of everyday metaphors and their potential to shape our thinking (Kovecses, 2002, viii). These
authors, as well as Gibbs (2008) and others continue to challenge our assumptions about
metaphor comprehension, as knowledge about the inner workings of our brains increases.
Interestingly, George Lakoff will reappear again later in this thesis when discussing philosophies
of mathematics.
In cognitive linguistics, metaphors have come to be regarded as any conceptual domain
defined in terms of another (Kovecses, 2002, p. 4). It is surprising to realize how often we use
metaphor in our everyday discourse, and how difficult it is to describe many things without it.
Some metaphors become embedded in a concept, and become a part of that concept to the point
where we are barely able to understand them as metaphors at all, and the concepts themselves
mean little without them. “Just living an everyday life gives you the experience and suitable
brain activations to give rise to a huge system of the same primary metaphorical mappings that
are learned around the world without any awareness” (Lakoff as referenced in Gibbs, 2008, p.
26).