Content

Changes in Course Content

Almost everyone agrees that students must constantly experience calculus concepts in a rich interplay of symbolic, numerical, and graphical forms -- what the Harvard Consortium project and others have popularized as "The Rule of Three." In the last year or so, Deborah Hughes Hallett (Harvard) has been saying publicly, "We really should have called it The Rule of Four," the fourth form of representation being writing.

Every calculus reform program focuses on developing thinking skills and conceptual understanding -- on eliminating the possibility of students being certified as having learned calculus when all they have demonstrated is modest proficiency at memorizing formulas and manipulating symbols. Some projects (e.g., Duke University's Project CALC) have had a strong writing component from the outset; others are now coming to the conclusion that writing is an important tool both for student learning of concepts and for faculty observation and measurement of that learning.

Several conference participants suggested a "Rule of Five," with oral communication considered on a par with the other four. To some extent, the subject of multiple representations is at least as much an issue of pedagogy as of content; we will see in the next section that these issues cannot be completely separated.

Most content developers see a need for a steady interplay between the discrete and the continuous -- in the phenomena being studied, in the models of those phenomena, and in the methods for dealing with those models. Here's an example: Growth of populations and spread of epidemics are powerful motivators for understanding exponential and other forms of growth. Biological populations are inherently discrete, but sufficiently large populations with no minimal inter-reproduction time can be conveniently and accurately modeled by continuous models: simple, variable-separable differential equations. With sufficiently simple assumptions about the growth model (e.g., growth rate proportional to the population), solutions may be obtained by undoing differentiation formulas. In other cases (e.g., logistic growth studied before integration techniques), symbolic solutions may not be accessible, but students can generate numerical and graphical solutions via Euler's method, a discrete approximation to the model. On the one hand, Euler's method is nothing more than "rise equals slope times run," which students grasp with little difficulty. On the other hand, Euler's method is what traditional texts call "the differential approximation." When every student has a calculator in hand, it makes no sense to use differentials to approximate square roots, but it makes a lot of sense to use Euler's method to generate interesting, new, and useful functions.

A few years ago, there was a popular misconception (it may still exist in some places) that calculus reform meant using technology (calculators and/or computers) in calculus. In fact, rapid advances in affordable technologies have provided a powerful stimulus for rethinking mathematics curricula. But some reform projects (e.g., those at New Mexico State University and Ithaca College) started out with no emphasis on technology, and others have found that sophisticated technology alone does not assure any lasting or substantial reform. The emerging consensus recognizes the importance of using "appropriate technology," which means different things on different campuses and for different groups of students, depending on available resources, the particular focus of the course, and many other factors.

The use of technology is not just a trendy thing to do, and its appearance in reformed courses is not an artificial implant -- as in, "We have to use it because it's there." In fact, computers and calculators are the tools that enable us to break out of the mindless symbol-pushing mode and allow students to experience the interplays of graphical, numerical, and symbolic representations and of discrete and continuous models and methods. Furthermore, computer-based word processing, especially with symbolic and graphic capabilities, provides an environment in which students can write and rewrite reports in which they synthesize and explain their conceptual understanding -- without the heavy overhead imposed by handwritten or typewritten documents.

Finally, while many are looking at the content of calculus as what we had always hoped the traditional course would become, we are seeing some very real changes occurring in the syllabus. First, there is now a consensus that there need not be just one syllabus -- indeed, that the course will be healthier if there is more than one. Second, there is recognition that the course must not be measured out in 50-minute sound bites and 4-minute exercises.

Beyond those consensus aspects, we see a lot of different things happening, but with a number of common themes. For example, many are discovering the value of real-world problems, not as afterthoughts, but as up-front motivators, as "hooks" to capture the interest of students who have (or think they have) no interest in mathematics for its own sake. (That describes practically all students enrolled in calculus courses.) After all, one doesn't have to know much mathematics to state a substantial and interesting problem, and the problem itself can keep students focused on the task of developing mathematical tools. Here is an example from Keith Stroyan's project at the University of Iowa: Why is it that we can eradicate diseases such as polio and smallpox but not diseases such as measles and rubella?

One of the issues on which there is no consensus at all is the role (if any) of Advanced Placement (AP) calculus in a changing curricular environment. John Kenelly (Clemson University) argues elsewhere in this volume that AP as an institution can be a positive force for change. Many of the conference participants disputed that and saw AP more as a part of the problem than as a part of the solution. No one disputes that a very large number of communities all across the country have an enormous stake in AP -- a stake with political, social, economic, and emotional dimensions that sometimes outweigh educational considerations.

Bill Medigovich (Redwood High School, CA) suggests that universities -- the folks who created AP in their own image -- take responsibility for high schools (and, by implication, community colleges) in their own areas by supporting reformed high school (and transfer) calculus courses for which they will grant credit and placement, whether they are called "AP" or something else. An example: Jim Hurley (University of Connecticut) has an NSF-supported project to disseminate his technology-based calculus course to over 100 high schools across Connecticut, with credit granted by the University.