Q: How ``rigorous'' is the O/Z approach to calculus?
A: ``Rigor'' is commonly used in 2 different ways: (i) in reference to a traditional, Bourbaki-style mathematical sense; (ii) as a rough synonym for ``hard'' or ``challenging''. The O/Z approach is certainly not rigorous in the first sense. (Nor, for that matter, are most other beginning texts. For instance, a rigorous approach to trigonometric functions should be preceded with a proof of existence of arclength for circular segments.) Rigor in this sense is, of course, essential to knowing and doing mathematics---major students should certainly encounter it seriously in their courses. But we think that a beginning calculus course is too ``early'' for this encounter to occur in a concerted way. A pedagogically sounder goal for beginning calculus students (including prospective mathematics majors) is to become familiar and proficient with the objects and methods of calculus. This approach seems more practical for non-major students; it can also help prospective mathematics majors better understand and use the very objects they'll encounter later in more theory-based courses.
On the other hand, we certainly regard our approach as rigorous in the sense of being challenging. One source of challenge is the fact that we expect students to grapple seriously with the concepts as well as the manipulations of calculus.
Q: Why are derivatives defined before formal discussion of limits?
A: In a fully formal development, a full treatment of limits would of course come first. We prefer, though, to first introduce limits (and other concepts, for that matter) by showing them in action where the need for limits arises naturally---in finding derivatives. So we first introduce limits informally, illustrating their meaning and uses in context. We study limits in their own right after a section or two of using limits in context. We think that students benefit from having some sandbox time with limits before treating them formally.
Q: Why do DEs appear so early?
A: Since they're so important for solving real problems---much more important than other things. Since they express laws. Since they explain why log and exp fns and trig fns are so important.
Q: Do students now need to learn DEs as well as calculus?
A: No; we do not pretend to teach a comprehensive course in DEs. What we do want mathematically is for students to learn is the idea and language of DEs and how to check whether something is a solution.
Q: (From a first-year student) What is calculus?
A: Good questions. Like all big subjects, calculus is difficult to encapsulate in a few words, but here's a try. I'd say that the main theme of calculus is the study of continuously changing quantities. (I don't mean ``continuous'' in any terribly technical sense---it's just that we're talking about things like physical position and velocity, not discretely changing quantities, like the readout on a digital watch. The most basic problem in calculus has to do with the relation between a varying quantity (say f(x) ) and the rate of change of that quantity (say f'(x) ). Finding f' from f is the derivative problem; going the other way is the antiderivative problem. There are tons of applications of these ideas, ranging from physical ones (e.g., relating position, velocity, acceleration, etc.) to geometric ones (e.g., finding curves with certain shapes, perhaps for computer graphic applications).
Q: How do you ensure that your students emerge from the course with good skills in the basic procedures --- e.g. symbolic differentiation, symbolic integration, and finding limits?
A: Our texts treat all of the usual differentiation rules (see Chapters 2 and 3) and all of the standard antidifferentiation methods (substitution, integration by parts, partial fractions, trigonometric integrals and substitutions --- see Chapters 5 and 8 in the second edition). On the other hand, ``covering'' a set of techniques doesn't in itself ``ensure that students emerge from the course with good [basic] skills''---perhaps especially in a course that, like ours, tries to go beyond building basic skills to concentrate explicitly on conceptual understanding. The ``gateway test'' or ``proficiency test'' is one practical strategy we and others have used, we think successfully, to convince ourselves (and our students) that students can really carry out symbolic algorithms correctly and efficiently. I usually give one of these in differentiation and another in antidifferentiation. My own typical gateway tests include, say, 10 straightforward but not completely trivial symbolic derivatives or antiderivatives; students need to get 9 out of 10 completely correct (no partial credit for sign errors, etc.) to pass the gateway test. Passing is worth enough points to make the effort worthwhile. I retest (more than once, if necessary) those who fail on a given attempt. In my experience, only a minority pass on the first try, but very few take more than 3 attempts. Now and then a student never passes the test, but not very often---in this event the student loses a substantial but not catastrophic number of points.
Q: Are there any such skills that you deliberately downplay in the course, and if so, why?
A: I wouldn't say that we ``downplay'' specific skills. In fact, we've been accused of being on the right wing of the reform movement in this sense---we think students DO need a reasonable facility with symbols, partly as a practical tool, partly because the standard elementary functions of calculus ARE important in their own right, and and partly because manipulating objects symbolically can be a useful pedagogical vehicle for learning what these objects really are and what makes them tick. What I would say is that we try to present a view of calculus that balances symbolic skills with other desirable outcomes, such as a good conceptual and graphical understanding of the derivative and the integral in Calculus I, and some numerical intuition for such things as integrals and infinite series in Calculus II. We think that traditional approaches have often been, although perhaps quite successful at teaching students to perform symbolic algorithms, not very good at helping students ``unpack'' the symbols they use to see the mathematical ideas beneath. For instance, it's quite possible to ``know'' that (x^2)'=2x , or even that (x^2 * sin(x))'=2x*sin(x) + x^2*cos(x) , without really having any idea what derivatives are or what they tell about functions.
Q: Could you briefly describe the motivation for the calculus reform movement, and for texts like yours?
A: Some of what we are aiming for in our text is implicit in what's above, especially the bit about conveying a sense of what calculus is really about---objects and processes that aren't just pure symbols on which to perform mysterious symbolic operations, but that have a geometric and numerical reality as well. (This doesn't deny that the symbolic view is important, too---but it's not the only game in town.) Quite a bit more (but not TOO much, I hope) on these subjects is in the Preface for Instructors in the beginning of the book. What's above is really about our book, not necessarily about the ``calculus reform movement'' as a whole. My sense of it is that the phrase ``calculus reform'' has come to mean so many things to so many people---ranging from real ideas to pure advertising hype---that it's hard to know what it really means, if anything. So I personally avoid the phrase. Still, there are probably some general common tenets of ``reform'' approaches, including such things as emphasis on graphs, a willingness to use computing tools, some focus on conceptual understanding (whatever that means---it's not the same for different people), and some RELATIVE de-emphasis on symbolic manipulation as the principal topic of the course.
Q: There are those that claim that understanding of mathematics comes through doing many routine problems. How do you respond?
A: I agree that this is one way of learning; indeed, symbolic drill (I think that's what the question refers to) is essential for learning some things, at some times. But it's certainly not the only way to learn, and it's a poor way to learn some things.
Q: Why do the sections on theory of continuity and differentiability (Sections 4.8 and 4.9 in the second edition) appear so late? Doesn't one need these properties much earlier?
A: These topics indeed out of place in the sense of mathematical development, but that's not the same thing as pedagogical development. There are many places in our text (in all calculus texts, for that matter) where ideas are used before being formally nailed down. (For example, one could argue that exponential functions can't be rigorously defined until after the log, which is in turn only rigorously defined by the integral.) The idea behind the present placement of theoretical discussion of continuity and differentiability is that these sections represent brief excursions into the theoretical side of the subject. Students (so the thinking goes) are better primed and therefore more ``ready'' for such discussion after having spent some informal time with ideas like continuity and differentiability.
Click here to return to the O/Z calculus home page.