Much of elementary mathematics, including calculus, has traditionally been presented as a toolbox of formulas, to be learned largely by rote; the only skill involved other than memorization is the ability to determine which formula to use when. Those of us who use mathematics in our daily lives eventually develop a more coherent view of the subject, but this is not often visible to students in coursework up to and including the lower division.
This talk presents an alternative vision, in which coherence is an essential feature. Examples will be drawn from recent efforts at OSU to reform the teaching of both first-year calculus (the informal Calculus Working Group) and vector calculus (the NSF-funded Vector Calculus Bridge Project), but many of the underlying techniques are applicable at other levels. Notable among these are the use of conceptual ("clicker") questions (ConcepTests), and the attempt to develop a Calculus Concept Inventory to measure student gains in conceptual understanding, along the lines of the highly successful Force Concept Inventory developed by the physics community.