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-
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-
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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