I think there is a serious disconnect between what we actually teach in a calculus class, and what students ought to be learning from (and taking away from) a calculus class.
In my syllabus for calculus, and regularly in class, I give students the following definition of calculus:
Calculus is the study of the mathematics of quantities that undergo change.
In addition to that overall slogan, I let them know that everything we do in calculus is targeted at answering one or more of the following basic questions:
- How can we describe, in a precise way, the manner in which one quantity depends upon another? [We answer this question using the idea of a function.]
- How can we determine the rate at which a quantity is changing at a given moment and describe the overall manner in which this change occurs? [We answer that question using the idea of a derivative, which needs the idea of a limit.]
- How can we determine how much change has accumulated in a quantity over a certain period, if the rate at which the quantity changes is not constant? [We answer this question with the idea of the integral and with the Fundamental Theorem of Calculus.]
I think these are all accurate ways to describe the subject which allow students to think about why calculus would be useful — enough so that a four-hour course in it is required — for what they are going on to study. A course in calculus ought to be focused on asking and answering variations on these questions in different kinds of contexts — some purely mathematical, some applied to real questions in applied areas. The focus, in other words, ought to be on the questions and the way in which we answer them.
But after teaching calculus for over a decade, it seems to me that the only goal that calculus textbooks and course desginers really have in mind is to have students get really, really good at algebraic calculations of a certain type, and the students who can do the most complicated and esoteric calculations are the “best” students. Just look at some of the grotesque limit, derivative, and integral calculations students are asked to do in exercises. Some of these are justifiably complicated, because they arise from actual scientific or economic formulas that are themselves complicated. But some are merely contrived in order to present the maximum amount of complication, and for no good reason.
Calculus becomes a course in applied algebra — when in fact, algebra is only one of a whole arsenal of quantitative techniques used to answer the main questions of calculus, and it’s not especially more common or important than the other methods such as numerical or graphical approximation. People who actually use calculus do not use algebra the majority of the time; but we spend a lot more than the majority of time in a calculus class teaching algebra and algebraic techniques.
Why is this the case? Should we really be focusing so much on algebraic techniques in calculus? Could we teach students just as well and have just as effective a course — in terms of actually being able to use the subject matter in a real setting — if we introduced only what was needed to actually answer an important question in front of us, rather than the umpteenth limit calculation involving rationalizing a denominator, or the derivative of ln(ln(ln(ln(x)))), and so on? Might the calculus course be more effective if we did so?