The AHA! experience is a term that captures the essence of illumination. It is the experience of having an idea come to mind with brevity, suddenness, and a sense certainty. This book examines this extra-logical process within a number of different mathematical contexts in pursuit of three questions: What is the essence of the AHA! experience? What is the effect of an AHA! experience on a learner? Can the AHA! experience be controlled, and if so can it be invoked? The data for this pursuit comes from three related studies focused on three distinct groups of participants – undergraduate mathematics students, prominent mathematicians, and preservice teachers. The results from each of these studies combine to present some surprising conclusions regarding the very nature of the AHA! experience and the profundity of the effect that this experience has on learners and mathematicians alike. The methodology used as well as the finer details of the analysis will serve well anyone who is either interested in gaining a better understanding of the AHA! experience or curious about how to capitalize on the potential affordances it offers as a teaching and learning tool.

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