Dr. David Ssevviiri was in Maseno the last couple days; he’s a Ugandan mathematician currently in Kampala, educated in South Africa. David’s an incredibly bright guy, very interested in a kind of narrow research area of prime ideals (and prime modules) in ring theory. But he has a voracious appetite for learning new things, and seemed very happy to soak up as much algebraic combinatorics as I was willing to throw at him.
One of the things that was really clear in our conversations was the need for easier connections between African mathematicians and the rest of the world mathematics community. South Africa has been doing quite a lot (with plenty of international support) to ensure chances for the best mathematical brains in Africa to get a quality graduate education. Unfortunately, when the resulting mathematicians go back home, they are often cut off almost completely from the broader research world. This is because the research communities in most African countries are very small, and most of these countries don’t have a science budget to support research activities. Since much of the research funding in the developed world comes from national agencies like the NSF (in the US), this means there are very few opportunities for Africans to receive funding. Additionally, for international grants they are in direct competition with international researchers, and the University systems in Africa largely aren’t preparing people (yet) to be able to compete on level footing for these kinds of grants.
I’m spending the afternoon grading student papers from the Foundations course, and realizing that it might have helped to separate out the process of mathematics a bit more. We gave them a take-home assignment to write up a proof that we discussed in class, in detail. The issue is, though, that our classroom discussion included a lot of exploration and kind of side-conversations, which have worked themselves into the submitted proofs in interesting (in the not-great sense) ways.
There’s a great course in the Budapest Semesters in Mathematics program called ‘Conjecture and Proof,’ which combines a proof-writing class with a problem-solving class. (But it’s really focused on the problem solving.) In Foundations we’ve been striving to get across the importance of rigor and proof, while teaching basics of proof-writing and techniques of proof. Inevitably, though, such a project has to be mixed with some problem solving alongside: students need to write proofs that they haven’t seen before, and that involves solving problems. So we’ve ended up a bit reversed from the C&P structure, which places a lot more emphasis on the problem-solving than the proof writing. (And I feel a bit like we’re falling on the wrong side of history in this sense…)
At the main maths camp this year, we had five different topic areas that we structured the camp around. All of them used computers in a big way, except one: Programming. The justification for this was that we had really nice, user-friendly programs for illustrating ideas in statistics, geometry, and so on, but actually throwing the students into a programming environment would almost certainly be too overwhelming. A significant number of the students had never touched a computer before, and really taking them into a code environment seemed a bit of a stretch for people still figuring out the idea of a right-click.
That’s not to say that good computer tools don’t exist; just that we haven’t managed to review them yet. (MIT’s Scratch, for example, looks well worth checking out.) Furthermore, given the time-scale we were working on, I think there was a lot of value in separating the programming concepts from the physical object of the computer. This makes the concepts available in a larger context than the computer, which, as a maths camp, we were eager to do. The idea of setting some basic rules from which we can extrapolate is a basic idea of mathematics. Getting across the idea of the need for precision was also of key importance.
To celebrate the fact that I’ve finally got latex working on the blog, I’m going to bore you with a post about combinatorics and categories…
After some nice student questions in the Foundations course the other day, David and I were playing around with the number of surjections that exist between finite sets. And somehow this led down the road of writing the number of maps between two finite sets as a sum over the number of pre-image sets. There’s a very nice formula for this, actually. The number of surjections from set of things to a set of things is given by , where is the Sterling number of the second kind, which counts the ways of partioning an n-element set into k subsets. (And it’s pretty easy to backwards reason why the number of surjections is this number!) The number of injections is the falling factorial, .
Yesterday I spent five hours in front of a board! Getting some good honest work in… Two hours of normal class,doing the Foundations of Pure Math course, two hours of seminar, and an hour making a video lecture for the algebraic structures class.
Here are the videos for the algebraic structures class; it seems like a nice deliverable! It’s for the first section of the notes, which gives an introduction to the definition of a group, along with a bunch of examples. Group Definition Integers modulo n Permutation Group Dihedral Group General Linear Group
The videos were shot in David’s house; the clip-on microphone that I picked up last year all but eliminated the terrible echo in the room. On the other hand, there’s a good bit of noise in the audio, which it would be great to figure out how to eliminate.
Lots of teaching things going on right now! David and I were tasked (with extremely short notice) with developing an online abstract algebra course (entitled ‘algebraic structures’) for second-year undergraduates, and have been hashing out a good direction for the course to take.
One of the main things we realized (in a few hours of back and forth) is that the usual first course in abstract algebra probably isn’t really locally appropriate. This is largely because we can’t be assured that the later maths classes — which hook abstract algebra into a wide variety of different contexts — will ever be available to the students. The other big idea we came up with is that it would be great if we got students to the point where they could look at a mathematical problem and find the influence of algebra, and (in a perfect world) craft solutions to the problem appropriately.
As the strike drags on, I’ve had some time to actually do some maths. In particular, I’m preparing to run a weekly seminar on using representation theory for certain statistical problems. The plan is to work from some old lecture notes by Persi Diaconis entitled ‘Group Representations in Probability and Statistics.’ These deal with, for example, how long one should apply a random shuffling process before the thing which is being shuffled is well-mixed. Particular examples include the question of how many times one should shuffle a deck of cards, and how long one should let a random walk on Z_n run before we can be reasonably sure that every point has been reached. There are numerous real-world applications of the results, and it uses a lot of first-rate representation theory along the way!
The over-riding event of the last week and a half has been the teacher’s and lecturer’s strikes going on across the country. As an outsider who’s just shown up in-country, I can’t really speak to the nuances of what’s going on. Essentially the teachers struck to try for a pay increase, and recently the lecturers followed suit. On Monday, a meeting was held by the Maseno Vice-Chancellor to try and end the strike: the attendees decided to go back to work the next day. But the actual lecturer’s union was having a separate concurrent meeting on the other side of campus and thus wasn’t considered in the vote; on Tuesday morning they went department to department shutting things down and locking up offices. In fact, before the action on Tuesday, someone told me that the ‘Vice-Chancellor had called off the strike.’ I heard that and thought, hmmm, that’s not usually how these things work…
As it stands, we don’t really know how long the strike will last. It’s given us plenty of time to work on non-class related things, like processing work permits and thinking about what to do for our research seminar(s). (Which I’ll post more on later, of course!)
Yesterday was the first day of class! I’m teaching a class entitled Foundations of Pure Mathematics for first year students. David was along for the first lecture, and will likely be along for more; we seem to be pretty good at improvising a good show.
For the first class, we wanted to hone in on the idea that good definitions are important, since definitions are usually where you start on any mathematical problem. Thus, we asked the students to define addition, and then worked with their definitions until we managed to get something really precise. Continue reading →