xvi Preface we are a dying breed, that the very reproduction of mathematics as a social institution and a professional community is under threat. I approach the problems of mathematical education from this view- point which should not be easily set aside: what kind of mathemat- ics teaching allows for the production of future professional math- ematicians? What is it that makes a mathematician? What are the specific traits which need to be encouraged in a student if we want him or her to be capable of a rewarding career in mathematics? I hope that my observations and questions might be interesting to all practitioners and theorists of general mathematical education. But I refrain from any critique of, or recommendations for, school mathematics teaching. Alexandre Borovik, aged 11 The unity of mathematics means that there are no boundaries between “recreational”, “elementary”, “under- graduate”, and “research” mathematics in my book, I freely move throughout the whole range. Nevertheless, I try to keep the book as non-technical as possible. I hope that the book will find readers among school teachers as well as stu- dents. In a few instances, the mathematics used appears to be more technical. This usually happens when I have to re- sort to metamathematics, a mathematical description of the structure and role of mathematical theories. But even in such cases, mathematical concepts are no more than a pre- sentation tool for a very informal description of my observa- tions. Occasionally I could not resist the temptation to include some comments on matters of my own professional interest however, such comments are indicated in the text by smaller print. Photographs in this book I come from childhood as from a homeland. Antoine de Saint-Exupery, ´ Pilot de guerre I tried to place in the margins of the book a photograph of every living mathematician/computer scientist/historian of math- ematics/philosopher of mathematics/scholar of mathematics men- tioned or quoted in the book. The catch is, I am using childhood photographs. In my book, I write a lot about children and early mathematical education, and I wish my book to bear a powerful re- minder that we all were children once. I hope that the reader agrees that the photographs make a fascinating gallery—and my warmest thanks go to everyone who contributed his or her photograph.

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