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This volume focuses on two related questions that are central to both the psychology of mathematical thinking and learning and to the improvement of mathematics education: what is the nature of arithmetic expertise? and how can instruction best promote it? Contributors from a variety of specialities, including cognitive, developmental, educational and neurological psychology; mathematics education; and special education, offer theoretical perspectives amd empirical evidence about these issues. As reported in this volume, both theory and research indicate that the nature of arithmetic expertise and how best to promote it are are far more complex than conventional wisdom and many scholars, past and present, have suggested. The results of psychological, educational and clinical studies using a wide range of arithmetic tasks and populations (including "normally" and atypically developing children, non-injured and brain-injured adults and savants) all point to the same conclusion: the heart of arithmetic fluency, in general, and the flexible and creative use of strategies, in particular, is what is termed "adaptive expertise" (meaningful or conceptually based knowldge). The construction of adaptive expertise is examined across various arithmetic topics and age groups. The book should be useful for researchers and graduate students interested in mathematical cognition and learning (including mathematics educators, developmental and educational psychologists and neuropsychologists), educators (including teachers, curriculum supervisors and school administrators) and others interested in improving arithmetic instruction (including officials in national and local education departments, the media and parents).