Strange attractors and chaos as a paradigm for understanding student attitudes toward mathematics.

摘要

This study provides a new theoretical framework for understanding the dynamics and development of student attitudes toward mathematics (SATM). This framework is based on the mathematical ideas of strange attractors and chaos and on numerous examples of their application to the hard and social sciences. In particular, the dissertation explains why strange attractors and chaos are appropriate models for educational research, and in particular, for the study of attitudes toward mathematics. The theoretical portion of the dissertation is supplemented with an empirical study of the dynamics of college student attitudes toward mathematics. The empirical study is based on a survey administered in one of Indiana University's T104 classes "Mathematics for Elementary Teachers via Problem Solving" each day of one semester.; One of the major theoretical findings of the dissertation is that the dynamics of student attitudes toward mathematics can be explained as a complex nonlinear irregular process, which might not be a result of measurement error, but the very nature of the phenomenon. The dissertation also provides additional analytical instruments and metaphors for interpreting and explaining the development and evolution of attitudes toward mathematics. The introduction of the strange attractor and chaos paradigm provided in the dissertation gives insights into understanding features of SATM, including sensitive dependence on initial conditions (the "butterfly effect"), unpredictability, irreversibility, and self-organization through sequences of structural changes in patterns and forms (bifurcations).; The empirical component of the dissertation shows how attitudes toward T104 form irregular nonlinear patterns similar to patterns provided by strange attractors. It also demonstrates that chaos in attitudes toward mathematics might develop through bifurcations caused by variation in some well-known educational parameters, such as SAT or ACT scores. Finally, the analysis of the T104 mathematics curriculum revealed some relation among the order and character of specific themes and activities and the dynamics of SATM. At the same time, it is unlikely that all the fluctuations of SATM could be explained in terms of the T104 curriculum without applying the strange attractor and chaos paradigm.