## Looking Backwards

The term has come to a close, finishing the first half of my Fulbright year, which provides a bit of time to look back over what I’ve done, what’s worked well, and what’s worked less well.  A big part of the plan for the first time was to try out the existing structures, get to know what’s going on in the university, and figure out interesting ways forward that might work in the local context.  There were a lot of failures this term, places where things didn’t work as expected, where it’s clear that things need to happen differently next time around.  So if this post sounds bleak in some ways, rest assured that I’m already working hard on projects for next term that will try to get around these difficulties in one way or another.

• #### Teaching Face-to-Face

Despite my mandate to work on electronic education, I felt it was very important to teach a face-to-face course in order to better understand the undergraduate students and their context.  To that end, I co-taught Foundations of Mathematics with David Stern.

The course went reasonably well, but has definitely made me consider the degree of work necessary to really address the problems in the education system. We were working with first-year students, which is ideal in many ways. It’s easier to do something revolutionary with first-years, simply because they haven’t lowered their expectations too far yet. (This was true even when I was teaching at the University of California; the first-years are a lot more open to non-traditional techniques, simply because they expect University to be different from secondary.) Continue reading

## Explore, Conjecture, Test, Prove.

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…)

## A Small Exercise in Categorification

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 ${rm Sur}^n_k$ from set of $n$ things to a set of $k$ things is given by $k!S(n,k)$, where $S(n,k)$ 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 ${rm Inj}^n_k$ is the falling factorial, $(k)_n=k(k-1)cdots (k-n)$.

## Teaching, teaching, teaching….

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.

## First Day Teaching – Defining Addition

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