Tag Archives: abstract algebra

April 22, 2009, 5:19 am

Free textbooks: It can be done

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 …

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March 1, 2009, 4:40 pm

Keeping things in context

Part of Article 131 in the first edition (1801...
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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 a…

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November 7, 2007, 4:13 pm

Five positive student outcomes from the textbook-free algebra class

We’ve got just 4-5 weeks left in the semester and until the textbook-free Modern Algebra course will draw to a close. It’s been a very interesting semester doing the course this way, with no textbook and a primarily student-driven class structure. In many ways it’s been your basic Moore Method math course, but with some minor alterations and usage of technology that Prof. Moore probably never envisioned.

As I mentioned in this lengthy post on the design of the course, students are doing a lot of the work in our class meetings. We have course notes, and students work to complete “course note tasks” outside of class and then present them in class for dissection and discussion. The tasks are either answering questions posed in the notes (2 points), working out exercises which can be either short proofs or illustrative computations (4 points), or proving theorems (8 points). We have a…

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November 1, 2007, 8:00 am

Retrospective: Escaping textbooks (3.28.2007)

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.

Escaping textbooks

Originally posted: March 28, 2007

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I’ve blogged before about my…

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October 16, 2007, 9:32 pm

Fear, courage, and place in problem solving

Sorry for the slowdown in posting. It’s been tremendously busy here lately with hosting our annual high school math competition this past weekend and then digging out from midterms.

Today in Modern Algebra, we continued working on proving a theorem that says that if \(a\) is a group element and the order of \(a\) is \(n\), then \(a^i = a^j\) if and only if \(i \equiv j \ \mathrm{mod} \ n\). In fact, this was the third day we’d spent on this theorem. So far, we had written down the hypothesis and several equivalent forms of the conclusion and I had asked the students what they should do next. Silence. More silence. Finally, I told them to pair off, and please exit the room. Find a quiet spot somewhere else in the building and tell me where you’ll be. Work on the proof for ten minutes and then come back.

As I wandered around from pair to pair I was very surprised to…

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October 3, 2007, 9:19 am

Textbook-free Modern Algebra update

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.) (more…)