The last time I taught abstract algebra, I used no textbook but rather my own homemade notes. That went reasonably well, but in doing initial preps for teaching the course again this coming fall I realized my notes needed a serious overhaul; and since I’m playing stay-at-home dad to three kids under 6 this summer, this is looking more like a sabbatical project than something I can get done before August. So last month I set about auditioning textbooks.
I looked at the usual suspects — the excellent book by Joe Gallian which I’ve used before and really liked, Hungerford’s undergraduate text*, Rotman — but in the end, I went with Abstract Algebra: Theory and Applications by Tom Judson. I would say it’s comparable to Gallian, with a little more flexibility in the topic sequencing and a greater, more integrated treatment of applications to coding theory and cryptography. (This last was something I was really looking for.) There’s even a free companion to the book which incorporates Sage, which I am sorely tempted to use as well because learning Sage has been a pet project of mine.
But what’s really different about this book is that it’s free, licensed under the GNU Free Documentation License. I am having the bookstore prepare print copies for the students — I asked the students if they wanted a print version in addition to the free PDF’s online, and they said “yes” — which the bookstore will sell for a whopping $16.95, just enough to cover the costs of copying and 3-hole punching the 400+ pages of the book. I’m happy because I found a book that really fits my needs; the students are happy because they get a good book too, for a tremendous bang-to-buck ratio.
In the long and contentious comment thread for my post about James Stewart’s new $24M mansion, I suggested that Stewart should consider topping off his impressive (and apparently lucrative) teaching and writing career by making his Calculus book freely available online for anybody who wants it. That suggestion was met with shocked incredulity: “If you had any idea how much work it was to write and maintain a textbook, you’d never consider making it free.” Well, I’m happy to report that hard work and good writing need not necessarily be mutually exclusive with giving it away.
In fact, as more well-written textbooks appear for free online — and there were even more free abstract algebra e-books I did not end up selecting — the commercial market might find itself in trouble.
* Actually, I requested the Hungerford algebra book, complete with a crystal-clear note that I needed to have it in hand by April 10 in order to be able to adopt it in time for our bookstore. To this date I have not received it. Another problem with commercial textbooks: the distribution model for review copies is dreadful. I’m always receiving multiple copies of books I neither need nor am interested in, and not getting the books I do need and am interested in.
I’ve started reading through Stewart and Tall’s book on algebraic number theory, partly to give myself some fodder for learning Sage and partly because it’s an area of math I’d like to explore. I’m discovering a lot about algebra in the process that I should have known already. For example, I didn’t know until reading this book that the Gaussian integers were invented to study quadratic reciprocity. For me, the Gaussian integers were always just this abstract construction that Gauss invented evidently for his own amusement (which maybe isn’t too far off from the truth) and which exists primarily so that I would have something to do in abstract algebra class. Here are the Gaussian integers! Now, go and find which ones are units, whether this is a principal ideal domain, and so on. Isn’t this fun?
Well, yes, actually it is fun for me, but that’s because I like abstract nonsense and I like just constructing rings out of nowhere and seeing what works and what doesn’t. But this approach to algebra is a lot harder to convince others to adopt, particularly college math majors whom I teach, most of whom struggle with abstraction. For them, any connection, no matter how tenuous, to the real world is a comfort and a reason to study. Quadratic residues aren’t exactly in the same league as designing airplanes in terms of “real world” utility, but it’s at least something that’s easy enough to understand and explain. Even if you care nothing for real world utility, it’s important to know why something was invented when you are setting about studying it. Otherwise you learn a subject in abstraction and without connections to its roots.
In fact, it seems like a lot of what we take as being canonical in abstract algebra was invented to study number theory. And yet, I have never taken a number theory course, and the number theory that was included in my studies of algebra was added mainly to set up the study of abstract groups and rings, as if to say that number theory exists to make studying algebra easier instead of the other way around as appears to be the case. And it’s not because I had a bad algebra education; I studied under some of the best algebraists around, but I never got the memo that abstract algebra was for something. I learned algebra mainly in isolation for the sole purpose of calculating homotopy groups. Likewise, my entire grad school training was focused on topology, which is supposedly a branch of geometry, but the only course in geometry I have in my background was Mrs. Buttrey’s class at William James Junior High School in the eighth grade — and that didn’t exactly give me the disciplinary perspective I needed to put topology in its proper context. (Even though it was a really good geometry class — thanks Mrs. B!)
I’ve been thinking that my post about the, er, pedagogically challenged way that Stewart Calculus does its examples about instantaneous velocity is really about the idea that you need to make sure that a person learning a new idea has some reason to learn it, before you give it to them in full complexity. Or at least before they’ve finished a course in it. Perhaps this idea extends to all of mathematics and maybe even beyond.
Editorial: This is article #8 in this weeklong series of reposts of “classic” articles here at CO9s. The article I’m posting below probably has the most references to it of any article I’ve written. It’s the culmination of a bunch of prior posts about the nature of college textbooks, and it kicked off a pretty major experiment of my own that is currently underway — the design and execution of an abstract algebra course that does not use a textbook. The story of the textbook-free algebra course is still unfolding, and there’s a lot of good coming out of my little experiment.
We hear a lot about “innovation” in education, almost as if it were an end in itself. But I like to think about and write about ways of doing college differently that actually make students’ college education better.
Originally posted: March 28, 2007
I’ve blogged before about my ambivalence towards textbooks, at least in mathematics (here, here, here, here, and here). But a couple of recent events have really motivated me to think seriously about not using textbooks at all in my courses. And this fall I will be taking the plunge, requiring no textbooks except for my precalculus class (which has to have a book because the course has to be somewhat standard across five different sections). Continue reading
It’s been a while since I last said anything about the textbook-free Modern Algebra class experiment. This is mainly because the class itself is now underway, five weeks into the semester, and it’s only now that I’ve got enough perspective to give a reasonable first look at how it’s going. So, let me give an update. (Click to get the whole, somewhat lengthy article.) Continue reading