Timeline for What should be offered in undergraduate mathematics that's currently not (or isn't usually)?
Current License: CC BY-SA 2.5
21 events
| when toggle format | what | by | license | comment | |
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| Apr 25, 2016 at 2:23 | comment | added | galois | Maybe a functional programming course using Haskell would suffice instead | |
| Mar 14, 2010 at 17:56 | comment | added | Tim Porter | (to continue) Although a related topic was only done in student projects, I did get students calculating in the (MAX,+) semiring i.e. tropical algebra! Again they had to grapple with rings and modules in this context and some categorical style definitions in order to understand how to construct quite a neat model for a simplified real life system. These are good examples of ALGEBRA at work and as Andrew seems to say you can treat the constructions categorically without making a fuss about it. We may tend too often to go for very safe options on traditional lines! | |
| Mar 14, 2010 at 17:50 | comment | added | Tim Porter | I tried an interesting experiment some years ago. I introduced discrete event systems and Petri nets into an OR course. The version of `linear algebra' that that needed was over the natural numbers and positive reals since negative quantities of actual things to be produced (I will have -5 icecreams thank you!) don't make much everyday sense. Students did seem to get the point that we had to think of vectors, linear transformations etc in a new way. I actually was feeding categorical methods into things in a light way to get definitions that worked at the PRACTICAL level. | |
| Dec 7, 2009 at 12:09 | comment | added | David Lehavi | It is downright dangerous to do abstract nonsense before you have enough examples. Unless you have a really amazing taste you'll figure out that worthy mathematics is only what's true for (almost) every category, simply because these are the only things you know how to do, and you have no examples for anything else. | |
| Nov 18, 2009 at 22:28 | comment | added | Ryan Budney | Courses where you touch lightly on many topics run the risk of students not learning anything. I want courses to be things where there are multiple layers of redundancy, so that if something doesn't stick, the students have several other passes at the same ideas from other points of view. I think that's much of the reason people choose a narrow, deep topic, rather than many "unifying" topics. | |
| Nov 18, 2009 at 22:27 | comment | added | Ryan Budney | IMO it's not so much an issue of being too abstract, it's an issue of wanting to have courses where there's a deep, sustained development. A major theme where you start from something basic and build up a theory with some depth. Groups, rings and modules give you an avenue for that kind of development -- start off talking about the notion of groups, you can build up a familiarity with mathematical formalism and eventually get to Galois theory, or the classification of modules over PIDs, Jordan Canonical Form, etc. | |
| Nov 18, 2009 at 7:48 | comment | added | Andrew Stacey | Ryan, I agree about not having a dedicated course on category theory. But then I'm not convinced about the value of a course on, say, groups or rings-and-modules. I think we compartmentalise our courses too much and leave it up to the students to see the connections. However, in the case of category theory (and also set theory) there's a tendency to not even mention it when it would be useful to do so because it's "too abstract" and so, by implication, too difficult for our poor students to cope with. | |
| Nov 18, 2009 at 4:36 | comment | added | Ryan Budney | Hi Andy. There's nothing wrong with liking category theory. But I don't see it as something that'd be very useful as an undergraduate course at many institutions. Here at U.Vic at least, there would be no natural place to put such a course into the undergraduate curriculum because you'd have so few, and largely really boring examples. IMO categorical notions are better to be tossed into the mix in algebra, topology, analysis and geometry courses. Most kids that get that far can start connecting the dots on their own. | |
| Nov 12, 2009 at 19:44 | comment | added | Steven Gubkin | The point is that category theory does enable you too see how much of what you are doing is trivial. It is a huge conceptual organizer: Free groups, tensor products, sheafification, ... These things used to be hard to understand (and compute with!) before people routinely used universal mapping properties. You don't have to teach a course on category theory, but emphasizing universal properties first, before giving the crazy constructions particular to each subject area, is a great unifying tool that helps to organize any subject. | |
| Nov 11, 2009 at 7:54 | comment | added | Andrew Stacey | Oh, and same to you, Andy! | |
| Nov 11, 2009 at 7:53 | comment | added | Andrew Stacey | And (how annoying is this character limit!) I think a course like "semi-simple Lie algebras" could, if done right, be a great way to bring topics together (if done badly it could be down right awful!). There's room for both strategies, indeed for lots of different strategies. Just as not everyone likes topology (hard to believe, I know), not everyone likes learning from case studies. One of our roles as educators is to make ourselves redundant (in any particular student's life, that is) in that we equip students with the tools to go out an learn for themselves. | |
| Nov 11, 2009 at 7:50 | comment | added | Andrew Stacey | If I were feeling belligerent, then I would say that the sentence "Using categories achieves this unification at only a superficial level." shows that you've not grokked category theory. However, I'm not, so I shan't. As I said, I wouldn't teach a course called "Category Theory", but then I'm equally suspicious about "Rings and Modules". My feeling is more that we deliberately go out of our way to avoid talking about categories and so miss out on a wonderful opportunity to show students that mathematics is really one subject and not a load of disconnected, disjointed, random topics. | |
| Nov 11, 2009 at 6:52 | comment | added | Andy Putman | By the way, I spent a few minutes confused as to why Ryan was complaining that I liked category theory too much! You have a great name, Andrew... | |
| Nov 11, 2009 at 6:51 | comment | added | Andy Putman | Using categories achieves this unification at only a superficial level. A better strategy is to have "capstone" courses on subjects like, say, semisimple Lie algebras that are pretty accessible but use real ideas and results from several different undergraduate areas. | |
| Nov 10, 2009 at 7:51 | comment | added | Andrew Stacey | Gosh! Two words can really spark a debate! My serious point is that there is a tendency to view mathematics courses as separate entities. This is particularly the case in universities that run on modular systems. It can be hard for students to see mathematics as a single thing within the morass that we throw at them. If we allow the ideas of category theory to infect our courses, there's a chance that we give the students the opportunity to see how things relate to each other. Exactly how to do this is more complicated (and I'm running out of charac | |
| Nov 9, 2009 at 23:59 | comment | added | Ryan Budney | Hi Andrew, I understand where you're coming from but what real advantage would you get by having categories around at that level? I tend to avoid terminology unless it's necessary for a major theorem. In that regard, most everything categorical at the undergraduate level can be de-categorified without any real loss. Without the language of category theory, students see the trend building up. My experience is that most students that are inclined that way teach themselves basics of categories without any formal instruction, nor do they even need to look at a book. | |
| Nov 9, 2009 at 8:54 | comment | added | Andrew Stacey | Okay, so make it "category theory done right"! But I wouldn't teach a class that was called "Category Theory". That would be like offering a class in, oh I don't know, "Rings and modules". What's the point of that? But category theory could easily be introduced into a more general course well before they reached the "Algebraic Topology" stage where we can finally use the word "functor" without shame. And I disagree completely with your statement "A good class needs some deep theorems". No! A good class needs a story. | |
| Nov 9, 2009 at 8:04 | comment | added | Andy Putman | I think there is a big danger in making undergraduate classes (especially classes with a strong algebraic flavor) consist entirely of definitions and trivial lemmas. A good class needs some deep theorems! Dwelling on things like category theory often ends up making this problem worse. | |
| Nov 7, 2009 at 21:32 | comment | added | Andrew Stacey | I disagree completely. I'm teaching a course this semester that introduces, in quick succession, metric spaces, normed vector spaces, hilbert spaces, and abstract vector spaces. So often I've found myself essentially teaching about categories without actually saying the words. And these are mainly non-mathematics students. | |
| Nov 6, 2009 at 22:40 | comment | added | Ryan Budney | IMO category theory is best kept until the student has a context for it. So picking up bits and pieces as they learn algebraic topology is a natural thing to do, making it graduate level material for most students. | |
| Nov 3, 2009 at 19:47 | history | answered | Andrew Stacey | CC BY-SA 2.5 |