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 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.

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Filed under Abstract algebra, Calculus, Education, Math, Number theory, Sage

7 responses to “Keeping things in context

  1. Part of the question is, do the authors know who the readers will be? And the answer is clearly, no. Or yes, but they haven’t made the necessary pedagogical adaptations, which wouldn’t matter, except these are text books.

    You can do algebra all day without number theory, although there is meaning and reason that the latter adds to the study of the former. If you can do algebra, it it is not required that you know why you are doing it.

    But it’s exactly those who need to know why who the text book authors miss. They might have assumed that those students would never reach calculus. No matter, they have, and if the authors don’t adapt, the instructors will still be forced to.


  2. mvngu


    Just out of curiosity, how did you know about Sage?

    • I think I first learned about Sage through a slashdot article a year or so ago. Had been meaning to download it and mess around with it for some time and finally got my chance over Christmas break. I came to the conclusion that to really understand the full potential of the software, I needed to know a lot more about algebraic number theory.

  3. readers are essentially irrelevant; what matters are adoption committees.
    and like all committees, these tend to be dumber than their dumbest member.
    hell, *authors* are mostly irrelevant; this is why all the books fail badly
    *in the same ways* again and again.

    one should mention here that *james* stewart–textbook millionaire–
    and *ian* stewart– co-author of the most accessible text on its subject
    that, anyway, *i* was able to find fifteen years ago– are different people.
    love the “stuff named after gauss” link. wow.

    • Yes! Ian Stewart (who wrote the NT book I referenced and many other things besides) and James Stewart are different people! Although it makes me think that Ian Stewart would write an awesome calculus book.

      • mvngu

        Yep. Ian Stewart’s ANT book is awesome and I think accessible at the undergraduate level. Although it doesn’t cover as much and in such depth as Richard Mollin’s book on the same topic, Stewart’s book is small-ish, reasonably well-contained, and very light to carry around in my bag. As regards Sage, I think one of its strong features is number theory. Plus the support community is very open and friendly, well, from my experience anyway. It’s good to know that another person finds Sage useful.

  4. I love Number Theory. Fun stuff. Wish I had more time to fiddle around with math on the side, but being a pastor is quite busy enough.