A great discovery solves a great problem but there is a grain of discovery in the
solution of any problem. Your problem may be modest; but if it challenges your
curiosity and brings into play your inventive faculties, and if you solve it by your
own means, you may experience the tension and enjoy the triumph of discovery.
G E O R G E P O L Y A
The art of teaching, Mark Van Doren said, is the art of assisting discovery. I have tried to
write a book that assists students in discovering calculus—both for its practical power and
its surprising beauty. In this edition, as in the first six editions, I aim to convey to the student a sense of the utility of calculus and develop technical competence, but I also strive
to give some appreciation for the intrinsic beauty of the subject. Newton undoubtedly
experienced a sense of triumph when he made his great discoveries. I want students to
share some of that excitement.
The emphasis is on understanding concepts. I think that nearly everybody agrees that
this should be the primary goal of calculus instruction. In fact, the impetus for the current
calculus reform movement came from the Tulane Conference in 1986, which formulated
as their first recommendation:
Focus on conceptual understanding.
I have tried to implement this goal through the Rule of Three: “Topics should be presented
geometrically, numerically, and algebraically.” Visualization, numerical and graphical experimentation, and other approaches have changed how we teach conceptual reasoning in fundamental ways. The Rule of Three has been expanded to become the Rule of Four by
emphasizing the verbal, or descriptive, point of view as well.
In writing the seventh edition my premise has been that it is possible to achieve conceptual understanding and still retain the best traditions of traditional calculus. The book
contains elements of reform, but within the context of a traditional curriculum.
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