Preface
Particularly during the last thirty years, many criticisms have been directed
at the school mathematics curriculum. In response, a number of movements
have left their trace-New Mathematics, Real-world Applications, Problem
Solving and now Back to the Basics. Moreover, with so many students
encouraged to take mathematics for the sake of their careers, educators
have tried to respond in a practical way to the difficulties they find in the
subject.
The result is that mathematics in school is suffering from ecological over-
load. The attempt to respond in a piecemeal way to often conflicting advice
has threatened the enterprise with being swamped. Whatever the merits of
the criticisms of the traditional mathematics program and however com-
pelling the psychological and political consequences of high failure rates,
the attempt at a resolution seems often to have resulted in a denatured
curriculum, one from which any depth, sophistication or joy has been rig-
orously expunged. Students nibble at topics, abandoning them before they
discover any reason to master them. The mathematics taught is quickly
lost to memory and must be reviewed at a later stage (often in a remedial
class).
Rather than fragment mathematics, it may be more productive to take
an integrated approach, in which students are encouraged to dwell on a
mathematical topic long enough to sense how it is put together and what
its proper context is. Formerly, students might spend a whole year in a
single area of mathematics-Euclidean geometry, the analytic geometry of
conic sections, trigonometry and statics, theory of equations. They had the
chance to learn many techniques and experience through astute reasoning
and manipulation the power of mathematics. Better students would develop
a sensitivity to pattern and elegance, and find mathematics both substantial
and satisfying.
This book is not a textbook. Nor is its topic being particularly recom-
mended for inclusion, indiscriminately, into the school curriculum. How-
ever, it should convey some of the breadth and depth found close to the
traditional school and college curricula, and encourage the reader not only
to follow up on some of the historical and technical references, but to pull
out pen and paper to tackle some problems of special interest. Some of the
mathematics will be difficult, but I believe that it will all be accessible.
The intended audience consists of students at both high school and col-
lege who wish to go beyond the usual curriculum, as well as teachers who