Tall, Dark, and Mysterious

3/15/2005

The texts, they are a-changin’.

File under: Sound And Fury, Those Who Can't, Queen of Sciences. Posted by Moebius Stripper at 10:22 pm.

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 could figure 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 plus width, 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.

Exactly.

25 Comments

  1. >Larson, Hostetler, and Edwards

    Ha! That was my text in high school. Yuck. Stewarts. . .so much better.

    I dunno. I kinda think every citizen should know calculus. And multivariable calculus and analytic geometry. But yeah, it would be nice if they first learned algebra at a reasonable age.

    - Saheli — 3/16/2005 @ 1:09 am

  2. Integration by tables = how to look numbers up in a big list? (My other image of this makes far less sense.)

    Also, you called it “yan” instead of tan.
    (Oops, corrected - MS)

    - wolfangel — 3/16/2005 @ 5:17 am

  3. In this age of textbook stickers, perhaps you need one to give your students for theirs:

    “This text sold by weight, not by usefulness.  Some adipose material may have accumulated since the last edition.”

    - Engineer-Poet — 3/16/2005 @ 5:59 am

  4. Ooh, integration by tables! That’s all my Calc II instructor ever did, really. Of course, he was responsible for selling the CRC Handbooks for our math society, so there you go. I still have my 29th edition with 596 integrals in it. And since the integrals he put on his tests were things like 1/(x*sqrt(2x^5-3)) (integral #277 in my book), he made a lot of sales.

    - tabstop — 3/16/2005 @ 8:32 am

  5. Saheli, may I ask why you think every student should know calculus? I’m so completely in disagreement with that that I’m interested in your reasons.

    And yes, integration by tables = looking up integrals in tables.

    I’ve taken to suggesting to math departments where I work that they find an out-of-print calc text from the 60’s and get permission from the publisher to photocopy it and sell it. Odds are, the book is of a higher quality than the ones in use today, doesn’t rely on fancy calculators, and would be a lot cheaper for the students.

    - Moebius Stripper — 3/16/2005 @ 8:52 am

  6. Not necessarily enough calculus that they can sit around doing classical mechanics. But when I moved from science/math to journalism/policy, I realized that besides the credit card stuff you were talking about before, people just don’t know how to think about a lot of crucial stuff. The ideas of integrating over a function, looking at how a function changes instantaneously, looking at inflection points, taking a function of multiple variables and analyzing eacn influence separately by holding the other still . ..all just missing from most people’s brains, to the great slowness of their policy processing. Financial math, global warming, looking at things like oil consumption, etc., etc.

    I realize I’m not buidling a very solid case. What I should do is keep a list of concrete examples. I’ve often noticed these deficiencies in the heat of the moment. Maybe I’ll start keeping a list now. ;-)

    Does even Stewart’s having graphing calculator exercises?

    - Saheli — 3/16/2005 @ 9:22 am

  7. Does even Stewart’s having graphing calculator exercises?

    Yes. And it’s huuuuge.

    >Larson, Hostetler, and Edwards

    Is this Edwards as in Edwards and Penney? That’s what we use here (because they have online homework. Crummy, hardly-ever-used online homework…) and it’s kinda miserable.

    - rosona — 3/16/2005 @ 11:08 am

  8. Ah, yes, books. I’m sure we all know that when they said “much has changed…” they meant “in this book…”, not “in calculus”. There probably aren’t more than a handful of new theorems since then (at least, not of interest to 1st year students).

    The books are bigger because the pages are bigger (hold on, we’re getting there) and have enormous amounts of white space and hundreds of illustrations, many of which have something to so with the material.

    And we all know that textbook publishers would be out of business in 4 years if they kept using the same ones over and over. (I’ve seen some math books from the early 1900s - they cover a lot of stuff, they don’t fool around, and you can learn a lot from them.)

    What do the current set of books your classes use cost? I read now and again of the astronomical cost of college texts - many over $100 each.

    - Mike — 3/16/2005 @ 11:52 am

  9. Forget the calc book from the 60’s.  Get one from the teens (which is going to be out of copyright), edit lightly to update the terminology, and go to town.

    Hell, I’ll volunteer to help edit scans if you’re willing to use OpenOffice format.

    - Engineer-Poet — 3/16/2005 @ 1:51 pm

  10. When I was in school, you could tell the difference between the different levels of Freshman Calc at my school by the textbook.

    Math 130s - calc for people with no calc/pre-calc experience, or pre-meds who wanted a (relatively) easy A - was purple, and had a picture of a tree on it.

    150s - for science majors and people with some experience - was white, and had some graphs, and a few geometric figures on it.

    Honors calc was an all-black 6x8″ clothbound textbook, with just the word ‘Calculus’ and the name of the authors printed in white letters on the cover. It was about 400 pages long, had just a few graphs to depict the different functions.

    The thing is, despite its resemblance to the Necronomicon, it was probably the easiest book to read. It wasn’t used for the other classes because it spent much more time on proofs and theory, but the explanations were much more lucid and concise. Plus, you can’t say this about a math textbook very often, but this book was… funny. I wish I could remember the name of the author - I was in 150s, and would borrow it from my genius friends all the time.

    - Independent George — 3/16/2005 @ 1:59 pm

  11. My Stewart cost about $130 used. If I didn’t buy from shady dealers on ebay, I’d never spend less than $400/semester.

    I work at my school’s bookstore, and if anything will supercede the bigger-better-pricier-edition scam, it’s the interactive technology scam.

    We have one college algebra book (~$100) that is required of all college algebra sections (~40 people per section in 50 sections 2x a year, roughly. Campus of 20,000 + 10,000 on satillite campuses) Each book comes with a *required* copy of some graphing software, unresellable at the end of the year. Every semester we stock a new pile of textbooks identical to last semesters…a pile that can easily block an aisle and reach my chest. And that’s just us, there are two other bookstores. Every semester we buy back about 200, because the only people who can use the used, non-besoftwared version are the satillite campus students.

    Then the huge lecture rooms were fitted with “remote control” technology which you must have seen by now. For $4 a student buys a toy remote control, and for $15 gets the registration code that lets the student actually use it (not that they work, but if they did, you’d need it properly registered). Or, registration is FREE with purchase of new textbook.

    - K. — 3/16/2005 @ 3:46 pm

  12. First, MS, I agree with Saheli, at least superficially, for totally different reasons. I believe that any educated person should have a serious exposure to most areas of human work. A lot of people have put a lot of blood, sweat and tears into the calculus as we know it. I find it quite unfortunate that mathematics is a secret teaching only for the initiate, to be handed out only to those who can demonstrate a need for it (and then only in a watered down form). I feel there is a deep problem if my psychologist friends (for instance) can proudly tell me that they have never studied calculus (or any other mathematics) but it is reasonable for them to take me for a total idiot if I don’t recognize Piaget.

    I recognize that I am begging the question here : I am only justifying teaching a solid, “technical” [=with proofs] mathematics course with an emphasis on the pure elements of the subject. This does lead to a second point : many of the problems studied in contemporary mathematics involve differential equations or are motivated by them. A lot of the historical work in the development of the theory of differentiable manifolds, of Lie groups, of functional analysis… come from partial differential equation work. Furthermore, most of the interesting applications of mathematics involve modelling some problem with a differential equation (Navier-Stokes, Black-Scholes, Schroedinger, etc.) While I don’t expect every student to have a grasp of anything approaching the finer points of these theories (just as I don’t know much about Pushkin’s poetry), I do feel it is imperative that a large number of students have some idea of what it might mean to do research in mathematics. I think it’s reprehensible that someone with a PhD in philosophy can think that I add up pages of numbers, whereas I have a pretty good handle on what type of issues he/she is addressing in his work (not at any level of detail or understanding, but I am able to respect the interest of the work and have some notion as to what might be going on).

    While I’m on this subject, I have to say that mathematics is not a distinguished subject in this regard. I get the sense that many “useless” academic disciplines are becoming more like mathematics in this respect. If there are any literature ppl out there, please correct me if I am wrong, but I get the sense that it is more and more OK for people to have no idea who various literary figures are (e.g. Wordsworth). This may relate to the two conflicting roles of the university : educate (in some ill-defined, “academic” sense) and train (for the job market).

    Indpt George : I suspect you are talking about Spivak’s book. It’s a strange size, is very dark, and is called “Calculus”. And it’s rather funny (including many silly index entries).

    K : I have no idea what you mean. can you please describe your remote control technology?

    - Sam — 3/16/2005 @ 7:10 pm

  13. Re the student question “why are we studying this stuff?” I had a math professor who answered such questions as follows:
    Student: But prof, how do you *use* all this?
    Prof: You use it to build the atomic bomb.
    Student: (pause) *How* do you use it to build the atomic bomb?
    Prof: That’s classified!

    - David Foster — 3/16/2005 @ 7:32 pm

  14. Why teach Calculus? Mostly Calculus is a matter of culture.

    The concepts of derivatives and integrals are not very practical ideas to be frank. It is far more practical to learn how to drive a car.

    By Calculus is part of what I would call the “scientific culture”. You don’t know Calculus? You are missing a piece of culture…

    - Daniel Lemire — 3/17/2005 @ 6:40 am

  15. SPIVAK!!! That’s the one! Thank you, Sam, this was driving me nuts. Yes, I loved the Spivak book. Actually, now that I know who wrote it, I think I’m going to order a copy from Amazon. No geek should be without it.

    As to whether everyone should take Calculus or not… I would make it a qualified no. As far as the math is concerned, I don’t see much practical application beyond algebra and geometry. But, the more important thing you learn from math is logic and mathematical reasoning - for me, Calculus was just the vector. Let’s put it this way: nobody’s asked me to prove the mean value theorem in ten years, but it taught me logical methods which I’ve used every day since. People look at me funny when I say this, but Calculus was the best writing class I ever took.

    That said, it is a difficult subject with very little direct utility. It’s impossible to learn economics, statistics, or any of the sciences without it, but only a small portion of the population will ever need advanced study in those subjects. I think there are alternatives to calculus which would offer many of the same benefits, without the added stress.

    - Independent George — 3/17/2005 @ 7:10 am

  16. Wowsers… I checked Amazon for the Spivak textbook, and I found this.

    My curiosity has been piqued. It sounds like a calc guide for people who don’t plan on formal study, but still wanted to learnt he basics. [burns]Excellent[/burns].

    Hey, MS - does wordpress not allow previews? I’m worried my shoddy html is going to ruin your comments.

    - Independent George — 3/17/2005 @ 7:41 am

  17. >>Larson, Hostetler, and Edwards
    >Is this Edwards as in Edwards and Penney?

    No - Edwards from Edwards and Penny is C. Henry Edwards while LHE is Bruce Edwards (my undergraduate advisor from University of Florida).

    Also I used another Spivak book - Calculus on Manifolds - for my first grad school calculus course. It’s all about higher dimensional derivatives and is _very_ dense (137 pages through Stokes Theorem). That adjective may be the fundamental reason behind all of the bloat in textbooks - to reduce density! A dense text requires lots of work on the part of the student to make sure you understand each and every single word in a sentence before you move on. For example, Alhfors’ _Complex_Analysis_ (321 pages) or H.L. Royden’s _Real_Analysis_ (434 pages) are both very tough to get through - they require substantial annotation and commentary by a student until the concepts become clear. I spent three semesters working through Carleson and Gamelin’s _Complex_Dynamics_ and it’s only 160 pages! By putting in 1500 pages of examples, laws, pretty four-color boxes around theorems, Maple figures, margin sidebars, and projects the authors are making the material less dense and easier to get through. Imagine the student thinking, “Wow, I must be learning something, I’ve gotten through 500 pages of this book!” At least, that’s what some of them are thinking…

    - Joshua Sasmor — 3/17/2005 @ 10:04 am

  18. Sam: The remote control “technology” my school uses comes from these guys, but I’m sure there’s others. Basically, they fit a room up with some recievers and sell students plastic remotes. These things are the cheapest looking toys you’ve ever seen. They have 8 buttons on them,you can see the batteries right in them…just cheap. They’re marketed as ways to take attendence or easily give multiple-choice quizzes to large numbers of students. The only thing is, they’re pieces of retro-fitted crap that don’t work. My roommate dropped a class rather than spend a semester sitting in a lecture hall, impotently pressing buttons and shaking the remote in the general direction of the nearest reciever. Several profs used them on the reccommendation of other profs and came away swearing…it was easier to to send a grad student off to grade scantrons than to deal with the supercool “technology.”

    The remotes have a registration numbers assigned to them. As a student, if you want your attendence noted or your quiz recorded, you have to have one registered in your name. The remote control unit is $4, but that doesn’t include the registration. The registration is bundled with new textbooks. You can use the same remote for all your classes, but license only lasts one term…so next semester, you get to spend another $15 if you take another class that uses them.

    There are other, better ones. I know that our business school just got a mock-stock exchange that has unique, much higher quality remotes. But only one or two really small classes use them. The blue ones are used (in my school) by the intro to journalism classes (400 people/semester) some intro to business classes (couldn’t even begin to guess) and some science classes.

    So lets see…they sell the recieving unit (start around $3,000) then remotes at $3-4 a piece, plus an equal number of registrations (at $15, or FREE with a new textbook, but that means you have to use a textbook that is bundled with the software (or vice versa?). No idea how that arrangement works). Coincidentally, in the journalism class case, the difference in price between the new and used textbook was…*gasp* $15.

    But if you’re a teacher who convices the university to spend $3000+ on a system, and agrees to get his 50-400 students to buy remotes, you get a free t-shirt!

    The funniest part was after two semesters of selling these things, my electromagnetic theory prof brought them up one day in class. Now, he’s politically active and is always giving lectures on things like teaching methods and being an effective science teacher and things like that. He’s also hard to impress. For example, if you brag about using technology in a classroom, and then show him a computer, he’s going to laugh at you unless you have something really new and cool loaded on that machine. So the eInstruction people came in and presented thier stuff, and he was unimpressed. And they made thier claims, and he was unimpressed, and then they offered him a free t-shirt. And of course he bought six units right there. No, actually, he came back and told us (all 8 of us in his only class), and we laughed, and still make fun of the remotes in class whenever possible.

    - K. — 3/17/2005 @ 5:08 pm

  19. When I first read this post I thought: “Hmmm, the names Larson, Hostetler, and Edwards sound vaguely familiar.” I then went to retrieve from my shelf the treasured text from which I learned Calculus at age nineteen and which I have kept for the last decade and found that, sure enough, they were its authors! Mine is the fifth edition, circa 1994. I’m totally with you on the issue of it being useless to keep issuing edition after edition of these things. When I leaf through my 5th edition, it seems (and always has seemed) frighteningly comprehensive, and it’s hard to imagine that they could have improved it enough to warrant two more editions. Money makes the world go ’round, I guess.

    Side note: I kind of love my Calculus text, and still look up something in it from time to time, when I encounter something Calculus-related in the graduate studies I’m doing.

    - wes — 3/17/2005 @ 7:06 pm

  20. Everyone Hates Sitemeter
    And Technorati. And The System That Shall Be Neither Named nor Linked Here. Yes, yes, but sometimes it pays to check one or the other of them anyway, just to see who’s linking you. I have no idea why this…

    - Ilyka Damen — 3/18/2005 @ 4:18 pm

  21. I used to make my living calculating the length of a piece of steel required to bend a chain link. The steel would stretch, and a correction factor was required. After much research and analysis, I determined that for the purpose of my calculations, Pi=3.

    - triticale — 3/20/2005 @ 2:06 pm

  22. George: You went to U of C, no? I took the class with Spivak’s book.

    - Paul — 3/22/2005 @ 4:10 pm

  23. Spivak’s Calculus is a classic book that my department in the 80s refused to use because it was too rigorous. Of course, the department eventually made it possible for a student to complete his math major requirements without ever seeing a ε-δ proof. Spivak also wrote The Joy of TeX: A Gourmet Guide to Typesetting with the AMS-TeX macro Package.

    Still, the book I would recommend were it in print is the one I used in my childhood, Lipman Bers’ Calculus. I haven’t seen a book since that better describes how to graph a function.

    I also have a secret admiration for Landau’s old calculus textbook. One chapter has in quick succession the definition of the derivative, Theorem 100: “If a function be differentiable at a point, it is continuous there,” and Theorem 101: “There exists a function that is everywhere continuous but nowhere differentiable.” This would probably put today’s student into shock.

    - Eric jablow — 3/23/2005 @ 6:56 pm

  24. Spivak is the man. For an intro calculus class for pure math majors, his text is the one to use, no question. I wouldn’t use it with any other group, partly because other groups wouldn’t appreciate it, and partly because it doesn’t teach the stuff that’s useful to non-majors.

    Functions that are continuous everywhere but differentiable nowhere probably wouldn’t put my students into shock - simply because they wouldn’t appreciate what’s so unusual about that. On a related note - I told my precalc students that a continuous function is “one that you can draw without lifting your pencil.” I trust, though, that the purveyors of mathematical rigour will forgive me this one.

    - Moebius Stripper — 3/23/2005 @ 7:09 pm

  25. That’s the definition we’ve all used, M.S. No one will introduce the Cantor function to precalculus students, I hope.

    - Eric jablow — 3/23/2005 @ 8:32 pm

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