1. Calculus: Reform or Renewal?

''The great obstacle to progress is not ignorance but the illusion of knowledge.'' {Daniel Boorstin, Sept. 9, 1987, in a television interview on the occasion of his retirement as director of the Library of Congress, quoted by Chickering [4], p. 57.}

The primary qualification for teaching mathematics in an American university or college is a Ph.D. in mathematics. We take for granted that anyone who has mastered the subject at this level is prepared to teach. After all, we had excellent role models. If we do what they did, we will be successful -- it worked for us. This is not ignorance but a dangerous illusion of knowledge: Good teaching engendered learning in us, so our job is good teaching -- learning will follow. {For a counterexample, see [15].} If it doesn't, the students must be at fault.

In the mid-1980's there was widespread recognition that something was wrong with this theory, at all levels of mathematics education. Calculus was chosen as the first target for ''reform'' because it was both the capstone course for secondary education and the entry course for collegiate mathematics. Thus was born the Calculus Reform Movement, whose history, philosophy, and practice are described in [17] and [19].

The first National Science Foundation calculus grants were awarded 10 years ago. Since then we have seen development and implementation of several new approaches to teaching calculus, with widespread acceptance on some campuses, and rejection and backlash on others. Our own approach is to treat calculus as a laboratory science course that emphasizes real-world problems, hands-on activities, discovery learning, writing, teamwork, intelligent use of tools, and high expectations of students.

At the time of development, we had little or no theoretical support for our choice of strategies. In place of theory, we relied on careful empirical work. The following sections develop the theoretical base that we lacked 10 years ago. The results from cognitive psychology were in the literature then but unknown to us and most of the other developers. The results from neurobiology have come to fruition just in this decade, and they confirm the cognitive theories that fit with our empirical observations. Thus, we are replacing the illusion of knowledge with real knowledge about learning and the teaching strategies that engender learning.

In hindsight, ''reform'' was not a good choice of name. The word has stuck, and most people recognize the course types to which it refers. However, it is an emotionally charged word -- in the area of religion, wars have been fought over it. One source of the current controversy is that people with deeply held beliefs feel they are under attack. ''Renewal'' would be a better descriptor -- perhaps we can discuss rationally whether the new aspects are also good, and whether renewal of pedagogical strategies from time to time is itself a good thing to do.


| Title page |
| Students | Problems |
| Cognitive Psychology | Brain Research |
| Technology and Learning | Technology and Curriculum |
| Renewal in Calculus Courses | References |