The process of learning mathematics, more precisely of learning to be a mathematician, is a long and exacting one. It requires special discipline, and peculiar determination. It also requires a certain level of intelligence, but we will see in the discussions of the present book that intelligence is not the primary or determining factor.

As with any serious task, one should not embark on it unless one fully realizes what one is getting into. The purpose of this book is to explore what the task entails, who should engage in it, and what the rewards are.

The centerpiece of the mathematical education of any student is the intellectual development of that student. In grade school the child learns arithmetic and other basic mathematical operations. In middle school and high school there begins an exposure to algebra and other more abstract mathematical ideas. Geometry, trigonometry, and the theory of functions (that is, what is a function, and what does it do, and how do we manipulate functions?) follow in good order.

In today's world, however, the American K–12 student passing through this standard curriculum gets little or no exposure to rigor or to the concept of proof. Sophisticated problem-solving and analytical skills are not developed. As a result, the students that we have in our freshman calculus classes do not know what a proof is or what serious problem-solving is.