1. The widely used textbook by James Stewart has (of course?) an early chapter on limits and continuity. Once upon a time I was a graduate teaching assistant leading a discussion section in a calculus course that lasted through a ten-week term followed by final exams, in which one of the students was the son of a famous professor of something-or-other.  This young man was very bright and hard-working.  One of the problems on the final was: compute the following limit — and there followed a fraction in which the numerator and the denominator both approached zero. His answer: It is undefined because it is zero over zero.  That is not merely a wrong answer; it is an answer that showed he had missed what should be the main point of an early chapter on limits.  (BTW, I think this was the Salas & Hille textbook.)  The T.A.s were given zero discretion in assigning course grades (often with other lecturers they could decide quite a lot) and this student got an “A”.  Numerous other examples I can recall show the same thing in just as extreme a way, but with more noise; this time all was crystal-clear without looking at a more complex picture. In fact, this is typical. Stewart’s textbook and the many others that differ from it and each other only in the author’s names and the particular shade of blue used in Figure 3.47 go through every technique needed to compute vast numbers of different sorts of limits, some involving zero over zero, some involving infinity minus infinity, some involving infinity over infinity, some involving infinite limits (vertical asymptotes) and limits at infinity (horizontal asymptotes), some involving one-sided limits with different values on the left and right, some involving things like sin(1/x) as x approaches 0, etc., etc.  Students finish the course never suspecting that the main reason for studying limits is the ones involving zero over zero, and that there is a simple comprehensible reason for that.  They are overwhelmed by all the details.  Instructors probably feel that it would be terrible for students to miss some of this, simply because it might have been terrible for one whose life’s course resembles the instructor’s to miss some of this.  If students miss the main point, that cannot be good.  It would be a serious mistake to think the solution lies in strengthening their preparation.  The result of that is well known: people who don’t go into mathematical fields do not use that preparation and go through life thinking that learning mathematics consists of memorizing lots of meaningless algorithms.  And there is no reason why they ought to have any stronger preparation in technical aspects of math than what little they get, and they know that.  There is, however a reason why they ought to find out why calculus is a great scientific and aesthetic achievement. A different kind of course would tell students about zero-over-zero: the car that goes zero miles in zero hours is moving at a speed of …. what, at that instant? That is the reason for telling them about limits.  Preparing them to understand and deal with all sorts of limits is destructive in that it camouflages the main point of the chapter.  I understood all those details, including how to prove them with epsilon-delta methods, before finishing first-semester calculus, and perhaps most others who teach calculus did, but it is a fact that nearly all who now take calculus have interests and abilities incompatible with a requirement that they learn every sort of idea about limits that can be presented at the freshman level and also learn that only one of those kinds of limits is essential to the first semester’s topic.
2. The stagnant standard course includes the mean value theorem. It is trivially easy to make it clear even to the weakest students what the theorem says, and indeed to make it intuitively apparent why it must be true.  But can one make it clear why the theorem matters? This theorem is included because if the student’s career path is like those of the professors who write the textbooks and teach the courses, then they need to know it.  If a student is capable of understanding how to prove and use the mean value theorem at the age of seven, or any other age, then I’m all in favor of it, but the vast hordes of students who are encouraged to take calculus who have altogether different interests and talents will never face the question of how to write a rigorous proof of the intuitively obvious fact that if a function’s second derivative is everywhere negative, then all of the chords of its graph lie below the graph, or the derivation of error bounds in Taylor’s theorem, etc.  The time spent on this could instead be spent on something that would contribute to their education, rather than to their dumb false views about what mathematics is.
3. The standard textbook (Stewart or any of a zillion other authors) has a section on “related rates”, which leads the student to think that “related rates” are just one topic in differential calculus.  “Related rates” are in fact the whole of differential calculus.  The phrase “related rates”, used in the sense in which it is intended in that section of the standard text, is synonymous with “differential calculus”.  The student who learns nothing else in differential calculus should learn that the subject is about related rates.
4. Stewart’s textbook has a section purporting to be about applications of differential calculus in science.  It says something like “The following differential equation arises in the science of whateverology” and then asks the student to do some things with it.  Speaking with a professor who supervises numerous adjuncts, I said a more honest approach would tell students how a differential equation is derived from scientific laws about rates of change.  She said that might be a good thing but there’s not enough time.  I said “So scrap ninety percent of it and then there’s time.” She didn’t want to continue that discussion.  A list of topics that “must” be covered in a course, including coverage of technical issues concerning every sort of computation of every kind of limit at the expense of any account of what the subject is about, is considered as trumping all considerations of respect for students.  As an employment of the talents of those who earn a Ph.D. in mathematics, the work of maintaining compliance with such a list of obligatory topics at the expense of opportunities to contribute to the education of students is contemptible.  Designing a calculus course appropriate for the droves of students whose interests and talents are primarily in other subjects is, on the other hand, worthy of the talents of mathematicians.

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