Several common threads run through *Toward a Lean and Lively
Calculus*, *Calculus for a New Century*, discussions at many
professional meetings, the report of the NCTM/MAA Joint Task Force on
Curriculum for Grades 11-13, and the NCTM *Curriculum and
Evaluation Standards for School Mathematics*. First, there is a
consistent call for *leaner and more conceptual courses*, driven by
and focused on central ideas. Second, there is a realization that
courses should *reflect modern technology both in content and in
pedagogy*.

**A diagnosis.**
Many calculus courses, we believe, slight the conceptual foundations of the
subject and overemphasize routine techniques-formal differentiation,
antidifferentiation, convergence testing, etc. Analytic objects
(integral, derivative, convergence, etc.) are represented and
manipulated only algebraically (i.e., *via* symbolic manipulation
of explicit elementary functions). For example, textbooks
often treat limits, derivatives, and integrals-all *analytic*
objects-only as *algebraic* operations on *algebraic*
functions. We try to take a broader view.

Whether one views calculus as an introduction to pure mathematics or as a
foundation for applications (or both!), the conclusion is
the same-concepts, not
techniques, are truly fundamental to the course.
Whatever uses they make of the calculus, students need more than a
compendium of manipulative techniques. The *sine qua non* for a
useful command of the calculus is a conceptual understanding that is
deep and flexible enough to accommodate diverse applications.

**A prescription.**
Our key strategy for improving conceptual understanding is combining,
comparing, and moving among graphical, numerical, and algebraic
``representations'' of central concepts. This strategy
pervades and unifies our
exposition. Bringing graphical and numerical, as well as algebraic,
viewpoints to bear on calculus ideas is the philosophical foundation of
our text. By representing and manipulating calculus ideas and objects
graphically, numerically, and algebraically, we believe that students
gain a better, deeper, and more useful understanding.