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Permanent link (DOI): https://doi.org/10.7939/R3R63W

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Making connections: network theory, embodied mathematics, and mathematical understanding Open Access

Descriptions

Other title
Subject/Keyword
metaphor
scale-free
nature of mathematical knowledge
exponent
representations
embodied mathematics
exponentiation
network theory
mathematical understanding
mathematics education
complex systems
mathematics cognition
Type of item
Thesis
Degree grantor
University of Alberta
Author or creator
Mowat, Elizabeth M.
Supervisor and department
Simmt, Elaine (Secondary Education)
Davis, Brent (Secondary Education)
Examining committee member and department
McGarvey, Lynn (Elementary Education)
Mason, John (Education, University of Oxford)
Pimm, David (Secondary Education)
Department
Department of Secondary Education
Specialization

Date accepted
2009-12-10T22:03:00Z
Graduation date
2010-06
Degree
Doctor of Philosophy
Degree level
Doctoral
Abstract
In this dissertation, I propose that network theory offers a useful frame for informing mathematics education. Mathematical understanding, like the discipline of formal mathematics within which it is subsumed, possesses attributes characteristic of complex systems. As the techniques of network theorists are often used to explore such forms, a network model provides a novel and productive way to interpret individual comprehension of mathematics. A network structure for mathematical understanding can be found in cognitive mechanisms presented in the theory of embodied mathematics described by Lakoff and Núñez. Specifically, conceptual domains are taken as the nodes of a network and conceptual metaphors as the connections among them. Examination of this ‘metaphoric network of mathematics’ reveals the scale-free topology common to complex systems. Patterns of connectivity in a network determine its dynamic behavior. Scale-free systems like mathematical understanding are inherently vulnerable, for cascading failures, where misunderstanding one concept can lead to the failure of many other ideas, may occur. Adding more connections to the metaphoric network decreases the likelihood of such a collapse in comprehension. I suggest that an individual’s mathematical understanding may be made more robust by ensuring each concept is developed using metaphoric links that supply patterns of thought from a variety of domains. Ways of making this a focus of classroom instruction are put forth, as are implications for curriculum and professional development. A need for more knowledge of metaphoric connections in mathematics is highlighted. To exemplify how such research might be carried out, and with the intent of substantiating ideas presented in this dissertation, I explore a small part of the proposed metaphoric network around the concept of EXPONENTIATION. Using collaborative discussion, individual interviews and literature, a search for representations that provide varied ways of making sense of EXPONENTIATION is carried out. Examination of the physical and mathematical roots of these conceptualizations leads to the identification of domains that can be linked to EXPONENTIATION.
Language
English
DOI
doi:10.7939/R3R63W
Rights
License granted by Elizabeth Mowat (emowat@ualberta.ca) on 2009-12-02T16:46:44Z (GMT): Permission is hereby granted to the University of Alberta Libraries to reproduce single copies of this thesis and to lend or sell such copies for private, scholarly or scientific research purposes only. Where the thesis is converted to, or otherwise made available in digital form, the University of Alberta will advise potential users of the thesis of the above terms. The author reserves all other publication and other rights in association with the copyright in the thesis, and except as herein provided, neither the thesis nor any substantial portion thereof may be printed or otherwise reproduced in any material form whatsoever without the author's prior written permission.
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