Thesis (Ph. D.)--University of Texas at Austin, 2002. / Vita. Includes bibliographical references. Available also from UMI Company.

Thesis (Ph.D)--University of Adelaide, Dept. of Pure Mathematics, 2001. / Includes bibliographies and index. Also available in an electronic version via ADT (Australian Digital Theses) Program.

Thesis (Ph. D.)--Illinois State University, 1997. / Title from title page screen, viewed June 8, 2006. Dissertation Committee: Graham A. Jones, John A. Dossey (co-chairs), Robert L. Fisher, Cynthia W. Langrall, Jane O. Swafford. Includes bibliographical references (leaves 251-257) and abstract. Also available in print.

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Music Learning and Mathematics Achievement: A Real-World Study in English Primary Schools Edel Marie Sanders Abstract This study examines the potential for music education to enhance children's mathematical achievement and understanding. Psychological and neuroscientific research on the relationship between music and mathematics has grown considerably in recent years. Much of this, however, has been laboratory-based, short-term or small-scale research. The present study contributes to the literature by focusing on specific musical and mathematical elements, working principally through the medium of singing and setting the study in five primary schools over a full school year. Nearly 200 children aged seven to eight years, in six school classes, experienced structured weekly music lessons, congruent with English National Curriculum objectives for music but with specific foci. The quasi-experimental design employed two independent variable categories: musical focus (form, pitch relationships or rhythm) and mathematical teaching emphasis (implicit or explicit). In all other respects, lesson content was kept as constant as possible. Pretests and posttests in standardised behavioural measures of musical, spatial and mathematical thinking were administered to all children. Statistical analyses (two-way mixed ANOVAs) of student scores in these tests reveal positive significant gains in most comparisons over normative progress in mathematics for all musical emphases and both pedagogical conditions with slightly greater effects in the mathematically explicit lessons. This investigation addresses concerns that UK and US governments' quests for higher standards in mathematics typically result in impoverished curricula with limited access to the arts. In showing that active musical engagement over time can improve mathematical achievement, as hypothesised, this work adds to a growing body of research suggesting that policy-makers and educationalists should reconsider curriculum balance.

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 Nez. 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 individuals 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.

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