A Sensible Calculus
Over the past few years there has been much discussion of the quality and quantity of calculus instruction that should be required as part of an undergraduate education in mathematics. We have heard plans to trim down the calculus instruction to a leaner form and to add more computer interaction, graphics, and applications to make it livelier. Others want the calculus course used as a pump to push more students on to advanced scientific studies, not as a filter to screen out students who don't match the profile for an elite few preparing for a career in mathematics or physics. Certainly it is unhelpful to blame inadequate student preparation for the bulk of the problems in calculus courses today.
It is time to recognize the obvious: The students taking calculus courses today have backgrounds, needs, and goals different from those of students of 20 years ago. We need to adjust in a meaningful way to the legitimate demands for change. If we look critically at what most textbooks and our own lectures are saying to students, we may be a little less likely to defend the current courses as presenting calculus as a triumph of coherent human thought.
Who are we trying to kid? The malaise in the calculus curriculum, which is manifest in the frustrations and distress of our students, has a rather simple source: What we teach does not make sense as a coherent body of knowledge. We give our students a little of this and that; here an application to graphing, there an application to physical science. There are very few common threads to hold together the fabric of the calculus we now present. And even when those threads are present, we don't pay enough attention to them for our students to notice their importance.
For a moment or two let's face some hard questions.
Does your text or your classroom presentation have a response to these questions? It's not an adequate response to say the reasons lie in some later course. For most of the students, there will not be any later course; and calculus is very likely the last chance for mathematics to make sense to them.
The key to solving the "calculus problem" will not be discovered by wandering through a maze of applications and computer graphics. Nor will it be found in a move toward "harder" problems and more rigor. We must try to make sense in our calculus instruction. The topics we discuss should make sense both internally and in context, to ourselves as instructors and to our students as learners. This criterion will provide the knife for cutting and the thread for reassembling the calculus curriculum of the next 40 to 50 years. If a topic is sensibly organized by itself and sensibly placed with regard to the other topics, then it should remain a part of the course. But if it fails to make sense locally or globally, it needs careful reassessment and revision.
In reviewing the calculus for sensibility, we will benefit from agreeing on some basic principles. Both instructor and student will find it easier to make sense of a topic if they can refer to two or three principal themes that are developed in calculus. I suggest here three themes that I believe are particularly significant for reviewing and revising the calculus curriculum, namely,
The first theme is that everything in a calculus course can be related to the study of differential equations. Consider how this theme can affect the understanding of the derivative form of the fundamental theorem of calculus. What was formerly treated as a very useful tool for evaluating definite integrals can be viewed also as a theorem about the existence (and uniqueness) of a solution to a differential equation. Or take the treatment of the differential for estimating the value of a function; this technique makes much more sense when followed up with a discussion of using Euler's method to estimate particular solutions to a differential equation with an initial condition.
The treatment of the differential also ties in well with the second theme, that estimation is valuable for both numerical and conceptual development. Another example of this theme assisting in the organization of the course is in the treatment of infinite series. Infinite series can be thought of as the continuation of efforts to estimate function values with polynomials (as in Taylor's Theorem) and thereby serve to estimate some difficult definite integrals as well as solve some differential equations.
The final theme I suggest is the importance of models as sources for concepts and interpretations as well as for applications.Although practically every treatment of the differential presents a graphical interpretation of this tool, none to my knowledge gives the simple and very convincing position-velocity interpretation: that is, if you know the velocity and position of a moving object at a certain time and you wish to estimate the change in position for a short change in time, you should multiply the velocity by the time change.
The modeling theme also could provide a more sensible approach for the study of the transcendental functions. Rather than treating them abstractly, with applications placed almost as afterthoughts, it might make more sense to discuss first some (differential equation) models that these functions solve. Indeed, the models often provide meaningful questions that lead to many of the most important properties of these functions.
I think those are enough examples to give you a sense
of these themes. Now, go forth, and try them on your courses. And may your
solutions be sensible to yourselves and your students.
Humboldt State University
Arcata, California 95521