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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Casting Out Nines 
If calculus is regarded as more of a rite of passage and a means to an end rather than as an end unto itself (and this is the case, I’m afraid, as it reflects the attitudes of business, parents, and most of the engineers I have worked with over the years, especially when the high opportunity costs of education is taken into consideration) then why does the pedagogy really matter as long as the students are exposed to enough rigor and head-banging – what my engineering father called “walking the hot sands?” Professional teachers and mathematicians are always seeking to make their subject more appreciated, but in the end no one really wants to render their prized discipline by way of some amazing new trick or reformed learning technique “so easy a cave man could do it,” now do they? The craft, the sociology of knowledge, demands a certain achievement as an honorarium for admittance (passing the course satisfactorily).
In 1975 my calculus professor dissed all contemporary calculus texts as watered down and incoherently useless, a bit strong in my view, but it does illustrate that this problem has long been an issue and probably has no end in sight. Calculus books are exceeding $200 now and there are hundreds to choose from over the last 60 years. Will there never be an agreed upon body of knowledge and approach by which to communicate it and work it through? Silly me!