Tag Archives: stewart calculus

Calculus reform’s next wave

There’s a discussion going on right now in the Project NExT email list about calculus textbooks, the merits/demerits of the Stewart Calculus textbook, and where — if anywhere — the “next wave” of calculus reform is going to come from. I wrote the following post to the group, and I thought it would serve double-duty fairly well as a blog post. So… here it is:

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I’d like to add my $0.02 worth to this discussion just because (1) I’m a longtime Stewart Calculus user, having used the first edition (!) when I was an undergrad and having taught out of it for my entire career, and (2) I’m also a fairly consistent critic of Stewart’s calculus and of textbooks in general.

I try to see textbooks from the viewpoints of my students. From that vantage point, I unfortunately find very little to say in favor of Stewart’s franchise of  books, including the current edition, all of the previous five editions, the CCC version (which is almost exactly the same as the non-reform version of the book but with less clarity in its language), or the “Essential” calculus edition. Stewart has a relentlessly formalistic approach to calculus that, while admirable in its rigor, renders it all but impenetrable to students who are not used to such an approach, which is certainly nearly every student I teach and I would imagine a large portion of the entire population of beginning calculus students.

If you don’t believe me, go check out his introductory section on the definite integral (Section 5.2 in the sixth edition). Stewart hopelessly confuses the essentially very simple idea of the definite integral by hitting students with an avalanche of sigma-notation right out of the gate. Or, try the section on exponential functions (1.5), in which Stewart for some reason feels like it’s necessary to explain how it is we can define an exponential function at rational and irrational inputs. This is all well and good, but does the rank-and-file beginning calculus student need to know this stuff, right now?

As a result, I find myself having to tell students NOT to read certain portions of the book, and then remixing and rewriting large parts of the rest of it. But that leads to the ONE thing I can say in the positive sense about Stewart, which I can’t say about many “reform” books: Stewart is what you make it. The book does not force me to teach in a certain way, and if I want to totally ignore certain parts of it and write my own stuff, then this generally doesn’t cause problems down the road. For example, at my college we don’t cover trigonometry in the first semester. In most other books we’ve examined, trig and calculus are inextricable, and so the books are unusable for us. With Stewart, though, given a judicious choice of exercises to omit, you can actually pull off a no-trig Calculus I course with very little extra work on the prof’s part.

I can also say, regarding Stewart CCC, that the ancillary materials are excellent. The big binder of group exercises that comes with the instructor edition is much better than the book itself.

I don’t think that I have yet seen a calculus book that is really fundamentally different from the entire corpus of calculus textbooks, with the possible exception of Hughes & Hallett. They all cover the same topics in the same order, more or less, and in the same ways. If you’re looking for the next wave of calculus reform, therefore, you’ll have to find it outside the confines of a textbook, or at least the textbooks that are currently on the market. Textbooks almost by definition are antithetical to reform. Perhaps real reform will come with the rejection of textbooks as authoritative oracles on the subject in the first place. That could mean designing courses with no centralized information source, or using “inverted classroom” models utilizing online resources like the videos at Khan Academy (http://www.khanacademy.org) or iTunesU, or some combination of these.

Actually, more likely the next wave of reform will be in the form of reconsidering the place of calculus altogether, as the CUPM project did several years ago. Is it perhaps time to think about replacing calculus with a linear combination (pardon the pun) of statistics, discrete math, and linear algebra as the freshman introduction to college mathematics, or at least letting students choose between calculus and this stat/discrete/linear track? Is calculus really the best possible course for freshmen to take? I think that’s a discussion worth having, or reopening.

Enjoy,
Robert Talbert
Franklin College
Peach Dot 1997-1998

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A simple idea for publishers to help students (and themselves)

OXFORD, ENGLAND - OCTOBER 08:  A student reads...
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I’m doing some research, if you can call it that, right now that involves looking at past editions of popular and/or influential calculus books to track the evolution of how certain concepts are developed and presented. I’ll have a lot to say on this if I ever get anywhere with it. But in the course of reading, I have been struck with how little some books change over the course of several editions. For example, the classic Stewart text has retained the exact wording and presentation in its section on concavity in every edition since the first, which was released in the mid-80’s. There’s nothing wrong with sticking with a particular way of doing things, if it works; but you have to ask yourself, does it really work? And if so, why are we now on the sixth edition of the book? I know that books need refreshing from time to time, but five times in 15 years?

Anyhow, it occurred to me that there’s something really simple that textbook companies could do that would both help out students who have a hard time affording textbooks (which is a lot of students) and give themselves an incentive not to update book editions for merely superficial reasons. That simple thing is: When a textbook undergoes a change in edition, post the old edition to the web as a free download. That could be a plain PDF, or it could be a  Kindle or iBooks version. Whatever the format, make it free, and make it easy to get.

This would be a win-win-win for publishers, authors, and students:

  • By charging the regular full price for the “premium” (= most up-to-date) edition of the book, the publisher wouldn’t experience any big changes in its revenue stream, provided (and this is a big “if”) the premium edition provides significant additional value over the old edition. In other words, as long as the new edition is really new, it would cost the publisher nothing to give the old version away.
  • But if the premium edition is just a superficial update of the old one, it will cost the publisher big money. So publishers would have significant incentive not to update editions for no good reason, thereby costing consumers (students) money they didn’t really need to spend (and may not have had in the first place).
  • All the add-ons like CD-ROMs, websites, and other items that often get bundled with textbooks would only be bundled with the premium edition. That would provide additional incentive for those who can afford to pay for the premium edition to do so. (It would also provide a litmus test for exactly how much value those add-ons really add to the book.)
  • It’s a lot easier to download a PDF of a deprecated version of a book, free and legally, then to try your luck with the various torrent sites or what-have-you to get the newest edition. Therefore, pirated versions of the textbook would be less desirable, benefitting both publishers and authors.
  • Schools with limited budgets (including homeschooling families) could simply agree not to use the premium version and go with the free, deprecated version instead. This would always be the case if the cost of the new edition outweighs the benefits of adopting it — which again, puts pressure on the publishers not to update editions unless there are really good reasons to do so and the differences between editions are really significant.
  • The above point also holds in a big, big way for schools in developing countries or in poverty-stricken areas in this country.
  • Individual students could also choose to use the old edition, and presumably accept responsibility for the differences in edition, even if their schools use the premium edition. Those who teach college know that many students do this now already, except the old editions aren’t free (unless someone gives the book to them).
  • All this provides publishers and authors to take the moral high road while still preserving their means of making money and doing good business.

Some individual authors have already done this: the legendary Gil Strang and his calculus book, Thomas Judson and his abstract algebra book (which I used last semester and really liked), Fred Goodman and his algebra book. These books were all formerly published by major houses at considerable cost, but were either dropped or deprecated, and the authors made them free.

How about some of the major book publishers stepping up and doing the same?

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Where the money for your calculus book goes

stewart

You too can own a massive house if you sell enough calculus books.

There’s a new, five-story, 18000 square foot, $24 million house in Toronto that is built of curves and glass and boasts its own professional-quality concert hall. The owner? Not a billionaire financier, head of state, movie or sports star, or anything of the sort — it’s James Stewart, author of the Stewart Calculus franchise of books.

From the Wall Street Journal article:

As visitors descend into the house, the fins disappear and the views widen. On the first floor, push a button and a 24-foot wall of glass windows vanishes into the floor, opening the pool area to the outside. Curves are everywhere, down to the custom door handles and light fixtures. The architects are even working with Steinway to create a coordinating piano. […]

An hour before five friends arrived for dinner, Mr. Stewart ambled around his kitchen, marinating some pork tenderloin chunks and tossing chopped leeks, red peppers and corn into a deep soup pot to simmer. He laid some ready-made sushi on a large red platter and then leaned back against a green-hued quartz countertop to relax.

Mr. Stewart say he isn’t overwhelmed by his home. “I just enjoy wandering around it,” he says. “Even now I’m still discovering details, and I’ve lived here for more than a year.”

Go to the article and look at the slideshow for more. It’s indeed a beautiful home (in a way it reminds me of St. Procopius Abbey near Chicago, which I visited last year).  I’m certainly not going to be down on Prof. Stewart for building his dream home, for which he apparently saved up money for 60 years. But it certainly destroys the old idea that professors never make money off of textbooks they write. And it also makes you wonder, if you recently spent $150 on a Stewart Calculus book, what part of that house you have a legitimate claim to. If you’re a Stewart Calculus book owner, I’d say you have a right to stop in at his place for sushi unannounced at any time.

A proposition for Prof. Stewart: Now that you’ve built your dream home and established your legacy, take all of your calculus books and make them available as free PDF downloads under a Creative Commons license, so students who are spending down to their last dime on textbooks can have a shot at saving for their dream houses, too.

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Why do we overcomplicate calculus like this?

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In the Stewart calculus text, which we use here, the first chapter is essentially a precalculus review. The second chapter opens up with a treatment of tangent lines and velocities, with the idea of secant line slopes converging to tangent line slopes and average velocities converging to instantaneous velocities taking center stage.

Calculating average velocity is just a matter of identifying two time values and two position values and then performing two subtractions and a division. It is not complicated. Doing this several times for shorter and shorter time periods is also not complicated, and then using the results to guess the instantaneous velocity is a little complicated but not that bad once you understand the (essentially qualitative, not quantitative) idea behind shrinking the length of the interval to get an instantaneous value out of a sequence of averages.

So I nearly hit the roof when a student came in this morning needing help understanding the Student Solutions Manual for the Stewart text on a problem where you had to find the average velocity of a moving object from 2 seconds to 2.5 seconds. A formula for position is given, y = s(t). The simple way to do this — the way that works, does not dumb the process down, and yet makes it understandable to the broadest possible audience and therefore sets  up general understanding of the more complicated idea of derivative calculations later — is to calculate s(2.5), calculate s(2), and then calculate \frac{s(2.5)-s(2)}{2.5 - 2}. Fifth-graders do this.

Instead, the Student Solution Manual does it like this:

  • Let h represent some positive number.
  • Calculate and fully simply the expression \frac{s(2+h)-s(2)}{h}.
  • Plug in h = 0.5.

This is crazy, absurd, and downright dangerous. It’s as if Stewart, and the person who wrote the manual, really believe that calculus is made up of algebra, and students who are in calculus are uniformly comfortable and skilled with algebra to the point that their way is just as transparent and simple as calculating distance divided by time — as if the algebraic work that ensues when you perform step (2) above were as natural as the concept of velocity itself and students spoke algebra like a first or second language.

Yes, the book’s approach works — and it closely mirrors what’s going to happen later when we want to get an exact value of the instantaneous velocity by letting h \rightarrow 0. But that’s not what students are doing right now. What students are doing is trying to understand the concept of average velocity. It’s not complicated. The complications should come, if at all, on the back end of the subject — where we are trying to make the concept of instantaneous velocity precise through limit calculations — but not on the front end when students are just trying to figure out what’s going on.

In the middle of typing this post out, another student came in, equally confused about the exact same problem. I told him to close his solutions manual. I asked him: What’s the definition of average velocity? He thought about it, and then gave me the right definition. “OK, then,” I said, “How would you get the average velocity from t=2 to t=2.5 here?” And he gave me an exactly right description of the process. The relief on his face was palpable. He understood this concept but the student solutions manual made it appear that he didn’t! How bad is it when you need a manual for the student manual?

Calculus is a really simple subject when you get to its core. I wish the book treated it that way.

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Deconstructing dx

Asking the following question may make me less of a mathematician in some people’s eyes, and I’m fine with that, but: How do you explain the meaning of the differential dx inside an integral? And more importantly, how do you treat the dx in an integral so that, when you get to u-substitutions, all the substituting with du and dx and so on means more than just a mindless crunching of symbols? 

Here’s how Stewart’s Calculus does it: 

  • In the section introducing the definite integral and its notation, it says: “The symbol dx has no official meaning by itself; \int_a^b f(x) \, dx is all one symbol.” (What kind of statement is that? If dx has “no official meaning”, then why is it there at all?) 
  • In the section on Indefinite Integrals and the Net Change Theorem, there is a note — almost an afterthought — on units at the very end, where there is an implied connection between \Delta t in the Riemann sum and dt in the integral, in the context of determining the units of an integral. But no explicit connection, such as “dx is the limit of \Delta x as n increases without bound” or something like that. 
  • Then we get to the section on u-substitution, which opens with considering the calculation of \int 2x \sqrt{x^2+1} \, dx (labelled as (1) in the book). We get this, er, explanation: 

Suppose that we let u be the quantity under the root sign in (1),  u = 1 + x^2. Then the differential of u is du = 2x dx. Notice that if the dx in the notation for an integral were to be interpreted as a differential, then the differential 2x dx would occur in  (1), and, so, formally, without justifying our calculation, we could write \int 2x \sqrt{1+x^2} \, dx = \int \sqrt{u} \, du

So, according to Stewart, dx has “no official meaning”. But if we were to interpret dx as a differential — he makes it sound like we have a choice! — then using purely formal calculations which we will not stoop to justify, we could write the du in terms of dx. That is, integrals contain these meaningless symbols which, although they have no meaning, we must give them some meaning — and in one particular way — or else we can’t solve the integral using these purely formal and highly subjunctive symbolic manipulations that end up getting the right answer. 

Er, right. 

To be fair, my usual way of handling things isn’t much better. I start by reminding students of the Leibniz notation for differentiation, i.e. the derivative of y with respect to x is dy/dx. Then I say that, although that notation is not really a fraction, it comes from a fraction — and that much is true, since dy/dx is the limit of \Delta y / \Delta x as the interval length goes to 0 — and so we can treat it like a fraction in the sense that, say, if u = x^2 + 1 then du/dx = 2x and so, “multiplying by dx”, we get du = 2x dx. But that’s not much less hand-wavy than Stewart. 

Can somebody offer up an explanation of the manipulation of dx that makes sense to a freshman, works, and has the added benefit of actually being true? 

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