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.
The ICTCM is coming up fast, and I’ll be there, mostly to give a talk on using wikis in upper-level math courses (like this one from my topics course in Cryptology) and take a minicourse on Camtasia. But I’ll also be checking out the latest and greatest (?) ideas and products in educational technology. One general category I am quite interested in is making all this technology that we use — especially computer algebra systems — portable and accessible from all different locations, in particular so that commuter students aren’t left out of the loop.
The fact that commuter students are left out is a growing concern for me, at least. We have Derive and Maple installed on my campus, but it’s a network install — and you have to be on campus to use it. Some campuses have a network installation that works from off campus, but we (and other places like us) also have a network that cannot be accessed unless you are physically on campus. (I suppose that theoretically, if you’re in wi-fi range of campus you could get on.) So, we give all this training and emphasis on computer software, and then what happens if you live in Indianapolis and have to drive an hour to get here?
Having all this fancy technology doesn’t do any good if a growing population of students (commuters, especially those who are older students with kids who can’t just drop everything and drive to the campus library at any moment) can’t even get to the software when they have the time to work. (Which if they have kids, is usually after the kids are in bed.)
There are some promising and free web-based applications, like xFunctions and the Integrator, that do the sorts of things that previously were restricted to locally-installed CAS’s and high-end graphing calculators. But I’d like to see more. Sage looks good too, but it’s a little too raw for the average student at this point.
If you’ve got thoughts or examples of commuter-friendly technology like this, leave them in the comments.
I just had a visit from one of our IT people to help me upgrade from Maple 10 to Maple 11. (Hoping that this would clear up the Maple/Leopard incompatibility issue.) We have a multiple-seat license for Maple that involves having some of the licenses on our campus network. Only a certain number of copies of Maple can be open at any given time.
But the install disc never made it out of the paper sleeve. The IT guy told me that, with Maple 11, you must be connected to the campus server in order to use it, due to the nature of the network license. And he said this was a new “feature” of Maple 11. With Maple 10, I installed the software along with a license file, and then I could use it wherever I wanted, network connection or no. But apparently Maple 11 can only be used on campus and when I am connected to the campus network. (I can’t connect from off-campus.)
I handed the disc back to the IT guy and apologized for wasting his time, as I have no use for software that I cannot use off campus.
Is this really true about Maple 11? If so, I’m very disappointed and that much more eager to learn Sage.
Lots of activity on the software front lately.
OmniFocus, the GTD app which I wrote about here, was released in version 1.0 today. I’ve been very satisfied with OmniFocus since settling on it for my GTD needs, especially since I managed to combine discounts to get it for under $20. I don’t know how many of those discounts are still available, but definitely the educational pricing is still there (though you have to look around for it at the Omni web site).
Bento, called the “missing database from iWork”, was released out of beta today as well. I’ve been demoing Bento for the last few days as a tracking system for students, and it’s very nice and visual. But I found the $49 price tag to be a little pricey, especially when the entire iWork ’08 suite is $79.
Sage, an open-source computer algebra system comparable to Matlab, has been gathering lots of buzz. With all my issues with Maple 10 not working under OS X Leopard, I’ve made learning Sage to be one of my January projects. I’ve got it downloaded and installed — which was no small feat, since there is no DMG package for OS X and it has to be built from source — but I haven’t had a chance to test drive it much. More later if I do.
Jott is not exactly software but rather a voice-to-text service that is really quite amazing. You call up a central phone number, address your voice message using voice commands, and then speak your message — and Jott converts it to text and sends it to the addressee as an email, SMS message, or both. You can also set Jott up to post to Google Calendar, Twitter, even blogging services (which unfortunately excludes WordPress.com). I used to want a digital voice recorder for capturing thoughts for my GTD inbox while not able to write things down or get to my laptop, but now I just call up Jott and have it send me an email. Brilliant — and free! (This has been around for a while, but I realized I hadn’t blogged about how enthused I was about it.)