My calculus students, I should mention, are a dream. They come to class. They do their homework. They can add fractions. They don’t whine that sometimes I make them think. Some of them are failing, but none to the point that I’m led to wonder whether they have ever done any math in their lives, ever.
I can’t extend such praise to the textbook (Larson, Hostetler, and Edwards), though. Before last week, I’d been more or less happy with it, with few relatively minor exceptions, but that was before I decided to see what I’d be teaching next week, and found that Chapter Seven is basically an amalgamation of everything that the authors think should be in a calculus textbook but couldn’t figure out where to put. Well, that, along with the material they couldfigure out where to put, but decided was worth repeating. Sure, we did all of the stuff in Section 7.1 back in 5.9, but who’s keeping track?
Not only have my calculus students, apparently, forgotten everything they learned a whole month ago in the class, they’ve also forgotten basic algebra. And since we’re about to teach them how to integrate by parts, decompose algebraic expressions into partial fractions, and slaughter any concoction of trigonometric functions we can fit under the integral sign, there’s never been a better time for them to brush up on of the stuff they did back in grade seven. Witness, for instance, the aside at the bottom of page 485:
NOTE Remember that you can separate numerators but not denominators. Watch out for this common error when fitting integrands to basic rules.
Do not separate denominators.
This is the calculus book. To be honest, part of me is positively delighted that for the first time since I’ve started teaching here, I have a textbook that actually underestimates my students, to the point of cautioning them – in red ink – against a mistake that not one of them is making. But still, it’s hard to keep up with the cutting-edge field of integral calculus, so I’m willing to cut the textbook authors some slack. From the preface of the text:
Welcome to Calculus of a Single Variable, Seventh Edition. Much has changed since we wrote the first edition – nearly 25 years ago.
Yes, if there’s one field that has evolved beyond recognition in the past quarter-century, it’s introductory calculus. Why, back when I was a tot, the area of a rectangle was length pluswidth, the derivative of sin x was 5, and we only had whole numbers, so whenever we needed to compute the area of a circle we had to use pi=3, AND YOU NEVER HEARD US COMPLAIN.
No, math hasn’t changed, at least not nearly as much as would justify the proliferation of calculus textbooks and their many editions; but the dissemination of the first-year mathematics canon sure has. In particular, a quarter century ago, it wasn’t assumed that everyone and their dog needed a university education, let alone one that required calculus credits, and so calculus classes weren’t populated with students who couldn’t add fractions.
A quarter century ago, graphing calculators didn’t exist, so high schools had to find some other useless skill to teach students instead of math. A quarter century ago, calculus textbooks were shorter, contained more geometry, left out the tables containing four thousand trig integrals, and didn’t mediate every concept through a Maple program or a TI83+. In other words – “much has changed” since the first edition because the textbook industry has made it so.
From the department of plus ça change, the best article on the topic, written by Underwood Dudley in 1988. It’s worth reading in full (especially if you come here for snarky prose about teaching math; I got nothin’ on Dudley), but some parts in particular are worth highlighting:
CALCULUS BOOKS ARE TOO LONG…If you plot the books’ numbers of pages against their year of publication, you have a chart in which an ominous increasing trend is clear. The 1000-page barrier, first pierced in 1960, has been broken more and more often as time goes on. New highs on the calculus-page index are made almost yearly. Where will it all end? We can get an indication. The magic of modern statistics packages produces the least-squares line: Pages = 2.94 (Year) – 5180, showing that in the middle of the next millennium, the average calculus book will have 2,270,pages and the longest one, just published, will have 3,783 pages exclusive of index.
My book far exceeds the 712 pages predicted by the line of best fit; but are we really surprised that the function increases more quickly than linearly? (ETA: actually, if you don’t count the appendices, LHE is 713 pages. Damn.)
Dudley mentions that he used a 416-page calculus textbook for four semesters. It was probably a lot cheaper than the seventh edition of LHE, too.
Why do we need 1000 pages to do what L’Hospital did in 234, Loomis in 309, Thompson in 301, and the text I learned calculus from, used exclusively for four whole semesters, 14 semester-hours in all, in 416? There are several reasons. One, of course, is the large number of reviewers of prospective texts. No more can an editor make up his mind about the merits of a text, it has to go out to fifteen different people for opinions. And if one of them writes that the author has left out the tan(x/2) substitution in the section on techniques of integration, how can he or she do that, we won’t be able to integrate 3/(4 + 5 sin 6x), how can anyone claim to know calculus who can’t do that; isn’t the easiest response to include the tan(x/2) substitution? Of course it is, in it goes, and in goes everything else that is in every other 1000-page text.
I think that that very same integral is in my textbook, along with plenty of others that are in that spectacularly useful section “Integration by Tables”. I’m not looking forward to fielding the question, “Why are we studying this stuff?” which my calculus students, though generally receptive to my lessons, will ask when the material gets particularly tedious. I have one week to think of a more suitable answer than, “Karma’s a bitch, isn’t it?”
Dudley also addresses the burgeoning market for applications, which further add to the lengths of these beasts. Calculus texts are long because they involve lots of applications because you’re terminally unemployable if you haven’t taken a calculus course, no matter your field:
The existence of all those calculus books with “Applications” in their titles implies a market for them, There must be students out there who are being forced to undergo a semester of calculus before they can complete their major in botany and take over the family flower shop. I cannot believe that any more than a tiny fraction of them will ever see a derivative again, or need one…I don’t know about you, but I long ago concluded…NOT EVERYONE NEEDS TO LEARN CALCULUS.
Amen to that. Now, as I said, my calc students are pretty solid, and even the ones who are doing poorly don’t seem irreparably out of their league. But that other course I teach? It’s called precalculus. Pre-CALCULUS. PRE. CALCULUS.